Properties

Label 9216.2.a.bk.1.3
Level $9216$
Weight $2$
Character 9216.1
Self dual yes
Analytic conductor $73.590$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9216,2,Mod(1,9216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9216, 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("9216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.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: \(73.5901305028\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 256)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.517638\) of defining polynomial
Character \(\chi\) \(=\) 9216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.44949 q^{5} +1.03528 q^{7} +O(q^{10})\) \(q+2.44949 q^{5} +1.03528 q^{7} +1.26795 q^{11} +5.27792 q^{13} -3.46410 q^{17} -1.26795 q^{19} +6.69213 q^{23} +1.00000 q^{25} -2.44949 q^{29} +5.65685 q^{31} +2.53590 q^{35} -0.378937 q^{37} +6.92820 q^{41} -8.19615 q^{43} +9.79796 q^{47} -5.92820 q^{49} -6.03579 q^{53} +3.10583 q^{55} +10.7321 q^{59} +0.378937 q^{61} +12.9282 q^{65} +4.19615 q^{67} -6.69213 q^{71} -9.46410 q^{73} +1.31268 q^{77} +15.4548 q^{79} +8.19615 q^{83} -8.48528 q^{85} +9.46410 q^{89} +5.46410 q^{91} -3.10583 q^{95} -3.46410 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 12 q^{11} - 12 q^{19} + 4 q^{25} + 24 q^{35} - 12 q^{43} + 4 q^{49} + 36 q^{59} + 24 q^{65} - 4 q^{67} - 24 q^{73} + 12 q^{83} + 24 q^{89} + 8 q^{91}+O(q^{100}) \) 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
\(5\) 2.44949 1.09545 0.547723 0.836660i \(-0.315495\pi\)
0.547723 + 0.836660i \(0.315495\pi\)
\(6\) 0 0
\(7\) 1.03528 0.391298 0.195649 0.980674i \(-0.437319\pi\)
0.195649 + 0.980674i \(0.437319\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.26795 0.382301 0.191151 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0 0
\(13\) 5.27792 1.46383 0.731915 0.681396i \(-0.238627\pi\)
0.731915 + 0.681396i \(0.238627\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) −1.26795 −0.290887 −0.145444 0.989367i \(-0.546461\pi\)
−0.145444 + 0.989367i \(0.546461\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.69213 1.39541 0.697703 0.716387i \(-0.254206\pi\)
0.697703 + 0.716387i \(0.254206\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.44949 −0.454859 −0.227429 0.973795i \(-0.573032\pi\)
−0.227429 + 0.973795i \(0.573032\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.53590 0.428645
\(36\) 0 0
\(37\) −0.378937 −0.0622969 −0.0311485 0.999515i \(-0.509916\pi\)
−0.0311485 + 0.999515i \(0.509916\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) 0 0
\(43\) −8.19615 −1.24990 −0.624951 0.780664i \(-0.714881\pi\)
−0.624951 + 0.780664i \(0.714881\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 9.79796 1.42918 0.714590 0.699544i \(-0.246613\pi\)
0.714590 + 0.699544i \(0.246613\pi\)
\(48\) 0 0
\(49\) −5.92820 −0.846886
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.03579 −0.829080 −0.414540 0.910031i \(-0.636057\pi\)
−0.414540 + 0.910031i \(0.636057\pi\)
\(54\) 0 0
\(55\) 3.10583 0.418790
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 10.7321 1.39719 0.698597 0.715515i \(-0.253808\pi\)
0.698597 + 0.715515i \(0.253808\pi\)
\(60\) 0 0
\(61\) 0.378937 0.0485180 0.0242590 0.999706i \(-0.492277\pi\)
0.0242590 + 0.999706i \(0.492277\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.9282 1.60355
\(66\) 0 0
\(67\) 4.19615 0.512642 0.256321 0.966592i \(-0.417490\pi\)
0.256321 + 0.966592i \(0.417490\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.69213 −0.794210 −0.397105 0.917773i \(-0.629985\pi\)
−0.397105 + 0.917773i \(0.629985\pi\)
\(72\) 0 0
\(73\) −9.46410 −1.10769 −0.553845 0.832620i \(-0.686840\pi\)
−0.553845 + 0.832620i \(0.686840\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.31268 0.149593
\(78\) 0 0
\(79\) 15.4548 1.73880 0.869401 0.494107i \(-0.164505\pi\)
0.869401 + 0.494107i \(0.164505\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.19615 0.899645 0.449822 0.893118i \(-0.351487\pi\)
0.449822 + 0.893118i \(0.351487\pi\)
\(84\) 0 0
\(85\) −8.48528 −0.920358
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.46410 1.00319 0.501596 0.865102i \(-0.332746\pi\)
0.501596 + 0.865102i \(0.332746\pi\)
\(90\) 0 0
\(91\) 5.46410 0.572793
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.10583 −0.318651
\(96\) 0 0
\(97\) −3.46410 −0.351726 −0.175863 0.984415i \(-0.556272\pi\)
−0.175863 + 0.984415i \(0.556272\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 15.8338 1.57552 0.787759 0.615984i \(-0.211241\pi\)
0.787759 + 0.615984i \(0.211241\pi\)
\(102\) 0 0
\(103\) −4.62158 −0.455378 −0.227689 0.973734i \(-0.573117\pi\)
−0.227689 + 0.973734i \(0.573117\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −12.5885 −1.21697 −0.608486 0.793565i \(-0.708223\pi\)
−0.608486 + 0.793565i \(0.708223\pi\)
\(108\) 0 0
\(109\) −3.96524 −0.379801 −0.189901 0.981803i \(-0.560817\pi\)
−0.189901 + 0.981803i \(0.560817\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.928203 −0.0873180 −0.0436590 0.999046i \(-0.513902\pi\)
−0.0436590 + 0.999046i \(0.513902\pi\)
\(114\) 0 0
\(115\) 16.3923 1.52859
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.58630 −0.328756
\(120\) 0 0
\(121\) −9.39230 −0.853846
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.79796 −0.876356
\(126\) 0 0
\(127\) −5.65685 −0.501965 −0.250982 0.967992i \(-0.580754\pi\)
−0.250982 + 0.967992i \(0.580754\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 17.6603 1.54298 0.771492 0.636239i \(-0.219511\pi\)
0.771492 + 0.636239i \(0.219511\pi\)
\(132\) 0 0
\(133\) −1.31268 −0.113824
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −0.196152 −0.0166374 −0.00831872 0.999965i \(-0.502648\pi\)
−0.00831872 + 0.999965i \(0.502648\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.69213 0.559624
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.5211 1.18961 0.594806 0.803869i \(-0.297229\pi\)
0.594806 + 0.803869i \(0.297229\pi\)
\(150\) 0 0
\(151\) 4.62158 0.376099 0.188049 0.982160i \(-0.439784\pi\)
0.188049 + 0.982160i \(0.439784\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 13.8564 1.11297
\(156\) 0 0
\(157\) 4.52004 0.360739 0.180369 0.983599i \(-0.442271\pi\)
0.180369 + 0.983599i \(0.442271\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.92820 0.546019
\(162\) 0 0
\(163\) 13.2679 1.03923 0.519613 0.854402i \(-0.326076\pi\)
0.519613 + 0.854402i \(0.326076\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −23.6627 −1.83107 −0.915537 0.402234i \(-0.868234\pi\)
−0.915537 + 0.402234i \(0.868234\pi\)
\(168\) 0 0
\(169\) 14.8564 1.14280
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.03579 0.458893 0.229446 0.973321i \(-0.426308\pi\)
0.229446 + 0.973321i \(0.426308\pi\)
\(174\) 0 0
\(175\) 1.03528 0.0782595
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.19615 0.612609 0.306305 0.951934i \(-0.400907\pi\)
0.306305 + 0.951934i \(0.400907\pi\)
\(180\) 0 0
\(181\) −11.4896 −0.854013 −0.427007 0.904248i \(-0.640432\pi\)
−0.427007 + 0.904248i \(0.640432\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.928203 −0.0682429
\(186\) 0 0
\(187\) −4.39230 −0.321197
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.9706 1.22795 0.613973 0.789327i \(-0.289570\pi\)
0.613973 + 0.789327i \(0.289570\pi\)
\(192\) 0 0
\(193\) 2.39230 0.172202 0.0861009 0.996286i \(-0.472559\pi\)
0.0861009 + 0.996286i \(0.472559\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.44949 0.174519 0.0872595 0.996186i \(-0.472189\pi\)
0.0872595 + 0.996186i \(0.472189\pi\)
\(198\) 0 0
\(199\) −18.5606 −1.31573 −0.657865 0.753136i \(-0.728540\pi\)
−0.657865 + 0.753136i \(0.728540\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.53590 −0.177985
\(204\) 0 0
\(205\) 16.9706 1.18528
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.60770 −0.111207
\(210\) 0 0
\(211\) −0.196152 −0.0135037 −0.00675184 0.999977i \(-0.502149\pi\)
−0.00675184 + 0.999977i \(0.502149\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −20.0764 −1.36920
\(216\) 0 0
\(217\) 5.85641 0.397559
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −18.2832 −1.22986
\(222\) 0 0
\(223\) 5.65685 0.378811 0.189405 0.981899i \(-0.439344\pi\)
0.189405 + 0.981899i \(0.439344\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.12436 −0.207371 −0.103685 0.994610i \(-0.533064\pi\)
−0.103685 + 0.994610i \(0.533064\pi\)
\(228\) 0 0
\(229\) 1.69161 0.111785 0.0558925 0.998437i \(-0.482200\pi\)
0.0558925 + 0.998437i \(0.482200\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.46410 0.620014 0.310007 0.950734i \(-0.399669\pi\)
0.310007 + 0.950734i \(0.399669\pi\)
\(234\) 0 0
\(235\) 24.0000 1.56559
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.17260 0.463957 0.231979 0.972721i \(-0.425480\pi\)
0.231979 + 0.972721i \(0.425480\pi\)
\(240\) 0 0
\(241\) −18.3923 −1.18475 −0.592376 0.805661i \(-0.701810\pi\)
−0.592376 + 0.805661i \(0.701810\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −14.5211 −0.927717
\(246\) 0 0
\(247\) −6.69213 −0.425810
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.12436 −0.197208 −0.0986038 0.995127i \(-0.531438\pi\)
−0.0986038 + 0.995127i \(0.531438\pi\)
\(252\) 0 0
\(253\) 8.48528 0.533465
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.928203 0.0578997 0.0289499 0.999581i \(-0.490784\pi\)
0.0289499 + 0.999581i \(0.490784\pi\)
\(258\) 0 0
\(259\) −0.392305 −0.0243766
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 12.9038 0.795682 0.397841 0.917454i \(-0.369760\pi\)
0.397841 + 0.917454i \(0.369760\pi\)
\(264\) 0 0
\(265\) −14.7846 −0.908211
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.34847 −0.448044 −0.224022 0.974584i \(-0.571919\pi\)
−0.224022 + 0.974584i \(0.571919\pi\)
\(270\) 0 0
\(271\) −21.1117 −1.28244 −0.641221 0.767356i \(-0.721572\pi\)
−0.641221 + 0.767356i \(0.721572\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.26795 0.0764602
\(276\) 0 0
\(277\) 10.1769 0.611470 0.305735 0.952117i \(-0.401098\pi\)
0.305735 + 0.952117i \(0.401098\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.39230 0.262023 0.131011 0.991381i \(-0.458178\pi\)
0.131011 + 0.991381i \(0.458178\pi\)
\(282\) 0 0
\(283\) 12.1962 0.724986 0.362493 0.931986i \(-0.381926\pi\)
0.362493 + 0.931986i \(0.381926\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.17260 0.423385
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −17.1464 −1.00171 −0.500853 0.865533i \(-0.666980\pi\)
−0.500853 + 0.865533i \(0.666980\pi\)
\(294\) 0 0
\(295\) 26.2880 1.53055
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 35.3205 2.04264
\(300\) 0 0
\(301\) −8.48528 −0.489083
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0.928203 0.0531488
\(306\) 0 0
\(307\) −20.9808 −1.19744 −0.598718 0.800960i \(-0.704323\pi\)
−0.598718 + 0.800960i \(0.704323\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.2784 −0.582836 −0.291418 0.956596i \(-0.594127\pi\)
−0.291418 + 0.956596i \(0.594127\pi\)
\(312\) 0 0
\(313\) −28.7846 −1.62700 −0.813501 0.581563i \(-0.802441\pi\)
−0.813501 + 0.581563i \(0.802441\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.3190 1.36589 0.682946 0.730468i \(-0.260698\pi\)
0.682946 + 0.730468i \(0.260698\pi\)
\(318\) 0 0
\(319\) −3.10583 −0.173893
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.39230 0.244394
\(324\) 0 0
\(325\) 5.27792 0.292766
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10.1436 0.559234
\(330\) 0 0
\(331\) 4.58846 0.252204 0.126102 0.992017i \(-0.459753\pi\)
0.126102 + 0.992017i \(0.459753\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 10.2784 0.561571
\(336\) 0 0
\(337\) 33.7128 1.83645 0.918227 0.396055i \(-0.129621\pi\)
0.918227 + 0.396055i \(0.129621\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 7.17260 0.388418
\(342\) 0 0
\(343\) −13.3843 −0.722682
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 10.7321 0.576127 0.288063 0.957611i \(-0.406989\pi\)
0.288063 + 0.957611i \(0.406989\pi\)
\(348\) 0 0
\(349\) −28.4601 −1.52344 −0.761718 0.647909i \(-0.775644\pi\)
−0.761718 + 0.647909i \(0.775644\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.9282 −0.688099 −0.344049 0.938952i \(-0.611799\pi\)
−0.344049 + 0.938952i \(0.611799\pi\)
\(354\) 0 0
\(355\) −16.3923 −0.870013
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −26.2880 −1.38743 −0.693715 0.720250i \(-0.744027\pi\)
−0.693715 + 0.720250i \(0.744027\pi\)
\(360\) 0 0
\(361\) −17.3923 −0.915384
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −23.1822 −1.21341
\(366\) 0 0
\(367\) 12.8295 0.669692 0.334846 0.942273i \(-0.391316\pi\)
0.334846 + 0.942273i \(0.391316\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.24871 −0.324417
\(372\) 0 0
\(373\) 14.3180 0.741358 0.370679 0.928761i \(-0.379125\pi\)
0.370679 + 0.928761i \(0.379125\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.9282 −0.665836
\(378\) 0 0
\(379\) 27.1244 1.39328 0.696642 0.717419i \(-0.254676\pi\)
0.696642 + 0.717419i \(0.254676\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 19.5959 1.00130 0.500652 0.865648i \(-0.333094\pi\)
0.500652 + 0.865648i \(0.333094\pi\)
\(384\) 0 0
\(385\) 3.21539 0.163871
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.9348 −0.554415 −0.277207 0.960810i \(-0.589409\pi\)
−0.277207 + 0.960810i \(0.589409\pi\)
\(390\) 0 0
\(391\) −23.1822 −1.17238
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 37.8564 1.90476
\(396\) 0 0
\(397\) −16.3886 −0.822518 −0.411259 0.911519i \(-0.634911\pi\)
−0.411259 + 0.911519i \(0.634911\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 34.3923 1.71747 0.858735 0.512420i \(-0.171251\pi\)
0.858735 + 0.512420i \(0.171251\pi\)
\(402\) 0 0
\(403\) 29.8564 1.48725
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.480473 −0.0238162
\(408\) 0 0
\(409\) 30.9282 1.52930 0.764651 0.644445i \(-0.222912\pi\)
0.764651 + 0.644445i \(0.222912\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 11.1106 0.546719
\(414\) 0 0
\(415\) 20.0764 0.985511
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3.80385 0.185830 0.0929151 0.995674i \(-0.470381\pi\)
0.0929151 + 0.995674i \(0.470381\pi\)
\(420\) 0 0
\(421\) −37.5002 −1.82765 −0.913824 0.406109i \(-0.866885\pi\)
−0.913824 + 0.406109i \(0.866885\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.46410 −0.168034
\(426\) 0 0
\(427\) 0.392305 0.0189850
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.17260 0.345492 0.172746 0.984966i \(-0.444736\pi\)
0.172746 + 0.984966i \(0.444736\pi\)
\(432\) 0 0
\(433\) 13.6077 0.653944 0.326972 0.945034i \(-0.393972\pi\)
0.326972 + 0.945034i \(0.393972\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.48528 −0.405906
\(438\) 0 0
\(439\) −31.9449 −1.52465 −0.762324 0.647196i \(-0.775941\pi\)
−0.762324 + 0.647196i \(0.775941\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.80385 0.180726 0.0903631 0.995909i \(-0.471197\pi\)
0.0903631 + 0.995909i \(0.471197\pi\)
\(444\) 0 0
\(445\) 23.1822 1.09894
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −24.2487 −1.14437 −0.572184 0.820125i \(-0.693904\pi\)
−0.572184 + 0.820125i \(0.693904\pi\)
\(450\) 0 0
\(451\) 8.78461 0.413651
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 13.3843 0.627464
\(456\) 0 0
\(457\) 20.7846 0.972263 0.486132 0.873886i \(-0.338408\pi\)
0.486132 + 0.873886i \(0.338408\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.175865 −0.00819087 −0.00409544 0.999992i \(-0.501304\pi\)
−0.00409544 + 0.999992i \(0.501304\pi\)
\(462\) 0 0
\(463\) 18.4863 0.859132 0.429566 0.903036i \(-0.358667\pi\)
0.429566 + 0.903036i \(0.358667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.2679 1.16926 0.584631 0.811300i \(-0.301239\pi\)
0.584631 + 0.811300i \(0.301239\pi\)
\(468\) 0 0
\(469\) 4.34418 0.200595
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −10.3923 −0.477839
\(474\) 0 0
\(475\) −1.26795 −0.0581775
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.62536 0.119956 0.0599778 0.998200i \(-0.480897\pi\)
0.0599778 + 0.998200i \(0.480897\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.48528 −0.385297
\(486\) 0 0
\(487\) −7.24693 −0.328390 −0.164195 0.986428i \(-0.552503\pi\)
−0.164195 + 0.986428i \(0.552503\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.1962 1.45299 0.726496 0.687171i \(-0.241148\pi\)
0.726496 + 0.687171i \(0.241148\pi\)
\(492\) 0 0
\(493\) 8.48528 0.382158
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.92820 −0.310772
\(498\) 0 0
\(499\) −16.1962 −0.725039 −0.362520 0.931976i \(-0.618083\pi\)
−0.362520 + 0.931976i \(0.618083\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −32.4997 −1.44909 −0.724545 0.689227i \(-0.757950\pi\)
−0.724545 + 0.689227i \(0.757950\pi\)
\(504\) 0 0
\(505\) 38.7846 1.72589
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 30.5307 1.35325 0.676625 0.736328i \(-0.263442\pi\)
0.676625 + 0.736328i \(0.263442\pi\)
\(510\) 0 0
\(511\) −9.79796 −0.433436
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −11.3205 −0.498841
\(516\) 0 0
\(517\) 12.4233 0.546377
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −12.0000 −0.525730 −0.262865 0.964833i \(-0.584667\pi\)
−0.262865 + 0.964833i \(0.584667\pi\)
\(522\) 0 0
\(523\) 22.7321 0.994003 0.497002 0.867750i \(-0.334434\pi\)
0.497002 + 0.867750i \(0.334434\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −19.5959 −0.853612
\(528\) 0 0
\(529\) 21.7846 0.947157
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 36.5665 1.58387
\(534\) 0 0
\(535\) −30.8353 −1.33313
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −7.51666 −0.323765
\(540\) 0 0
\(541\) 19.9749 0.858786 0.429393 0.903118i \(-0.358728\pi\)
0.429393 + 0.903118i \(0.358728\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −9.71281 −0.416051
\(546\) 0 0
\(547\) −17.6603 −0.755098 −0.377549 0.925990i \(-0.623233\pi\)
−0.377549 + 0.925990i \(0.623233\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.10583 0.132313
\(552\) 0 0
\(553\) 16.0000 0.680389
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.13681 −0.0481683 −0.0240841 0.999710i \(-0.507667\pi\)
−0.0240841 + 0.999710i \(0.507667\pi\)
\(558\) 0 0
\(559\) −43.2586 −1.82964
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 39.1244 1.64889 0.824447 0.565938i \(-0.191486\pi\)
0.824447 + 0.565938i \(0.191486\pi\)
\(564\) 0 0
\(565\) −2.27362 −0.0956521
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −12.0000 −0.503066 −0.251533 0.967849i \(-0.580935\pi\)
−0.251533 + 0.967849i \(0.580935\pi\)
\(570\) 0 0
\(571\) −19.8038 −0.828765 −0.414383 0.910103i \(-0.636002\pi\)
−0.414383 + 0.910103i \(0.636002\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.69213 0.279081
\(576\) 0 0
\(577\) −16.1436 −0.672067 −0.336033 0.941850i \(-0.609085\pi\)
−0.336033 + 0.941850i \(0.609085\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.48528 0.352029
\(582\) 0 0
\(583\) −7.65308 −0.316958
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 15.1244 0.624249 0.312124 0.950041i \(-0.398959\pi\)
0.312124 + 0.950041i \(0.398959\pi\)
\(588\) 0 0
\(589\) −7.17260 −0.295542
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 7.85641 0.322624 0.161312 0.986903i \(-0.448427\pi\)
0.161312 + 0.986903i \(0.448427\pi\)
\(594\) 0 0
\(595\) −8.78461 −0.360134
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −29.8744 −1.22063 −0.610316 0.792158i \(-0.708958\pi\)
−0.610316 + 0.792158i \(0.708958\pi\)
\(600\) 0 0
\(601\) −19.6077 −0.799815 −0.399907 0.916556i \(-0.630958\pi\)
−0.399907 + 0.916556i \(0.630958\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −23.0064 −0.935341
\(606\) 0 0
\(607\) 30.9096 1.25458 0.627292 0.778785i \(-0.284163\pi\)
0.627292 + 0.778785i \(0.284163\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 51.7128 2.09208
\(612\) 0 0
\(613\) 32.6012 1.31675 0.658376 0.752689i \(-0.271244\pi\)
0.658376 + 0.752689i \(0.271244\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −23.3205 −0.938848 −0.469424 0.882973i \(-0.655538\pi\)
−0.469424 + 0.882973i \(0.655538\pi\)
\(618\) 0 0
\(619\) 4.58846 0.184426 0.0922128 0.995739i \(-0.470606\pi\)
0.0922128 + 0.995739i \(0.470606\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.79796 0.392547
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.31268 0.0523399
\(630\) 0 0
\(631\) 35.5312 1.41447 0.707237 0.706976i \(-0.249941\pi\)
0.707237 + 0.706976i \(0.249941\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −13.8564 −0.549875
\(636\) 0 0
\(637\) −31.2886 −1.23970
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −34.3923 −1.35841 −0.679207 0.733947i \(-0.737676\pi\)
−0.679207 + 0.733947i \(0.737676\pi\)
\(642\) 0 0
\(643\) 20.1962 0.796459 0.398229 0.917286i \(-0.369625\pi\)
0.398229 + 0.917286i \(0.369625\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.69213 −0.263095 −0.131547 0.991310i \(-0.541995\pi\)
−0.131547 + 0.991310i \(0.541995\pi\)
\(648\) 0 0
\(649\) 13.6077 0.534149
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 46.1886 1.80750 0.903749 0.428062i \(-0.140804\pi\)
0.903749 + 0.428062i \(0.140804\pi\)
\(654\) 0 0
\(655\) 43.2586 1.69025
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −19.5167 −0.760261 −0.380131 0.924933i \(-0.624121\pi\)
−0.380131 + 0.924933i \(0.624121\pi\)
\(660\) 0 0
\(661\) 30.7338 1.19540 0.597702 0.801718i \(-0.296080\pi\)
0.597702 + 0.801718i \(0.296080\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −3.21539 −0.124687
\(666\) 0 0
\(667\) −16.3923 −0.634713
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0.480473 0.0185485
\(672\) 0 0
\(673\) 6.67949 0.257475 0.128738 0.991679i \(-0.458907\pi\)
0.128738 + 0.991679i \(0.458907\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −44.8759 −1.72472 −0.862360 0.506295i \(-0.831015\pi\)
−0.862360 + 0.506295i \(0.831015\pi\)
\(678\) 0 0
\(679\) −3.58630 −0.137630
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −29.6603 −1.13492 −0.567459 0.823402i \(-0.692073\pi\)
−0.567459 + 0.823402i \(0.692073\pi\)
\(684\) 0 0
\(685\) −29.3939 −1.12308
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −31.8564 −1.21363
\(690\) 0 0
\(691\) 33.3731 1.26957 0.634786 0.772688i \(-0.281088\pi\)
0.634786 + 0.772688i \(0.281088\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.480473 −0.0182254
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 37.7033 1.42403 0.712017 0.702162i \(-0.247782\pi\)
0.712017 + 0.702162i \(0.247782\pi\)
\(702\) 0 0
\(703\) 0.480473 0.0181214
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16.3923 0.616496
\(708\) 0 0
\(709\) −12.8023 −0.480799 −0.240399 0.970674i \(-0.577278\pi\)
−0.240399 + 0.970674i \(0.577278\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 37.8564 1.41773
\(714\) 0 0
\(715\) 16.3923 0.613037
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −46.3644 −1.72910 −0.864551 0.502545i \(-0.832397\pi\)
−0.864551 + 0.502545i \(0.832397\pi\)
\(720\) 0 0
\(721\) −4.78461 −0.178188
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.44949 −0.0909718
\(726\) 0 0
\(727\) −20.6312 −0.765169 −0.382584 0.923921i \(-0.624966\pi\)
−0.382584 + 0.923921i \(0.624966\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 28.3923 1.05013
\(732\) 0 0
\(733\) −35.6327 −1.31613 −0.658063 0.752963i \(-0.728624\pi\)
−0.658063 + 0.752963i \(0.728624\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.32051 0.195983
\(738\) 0 0
\(739\) 48.9808 1.80179 0.900893 0.434041i \(-0.142913\pi\)
0.900893 + 0.434041i \(0.142913\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.2784 0.377079 0.188540 0.982066i \(-0.439625\pi\)
0.188540 + 0.982066i \(0.439625\pi\)
\(744\) 0 0
\(745\) 35.5692 1.30316
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −13.0325 −0.476198
\(750\) 0 0
\(751\) −15.4548 −0.563954 −0.281977 0.959421i \(-0.590990\pi\)
−0.281977 + 0.959421i \(0.590990\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 11.3205 0.411995
\(756\) 0 0
\(757\) −25.8348 −0.938981 −0.469491 0.882937i \(-0.655562\pi\)
−0.469491 + 0.882937i \(0.655562\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 34.6410 1.25574 0.627868 0.778320i \(-0.283928\pi\)
0.627868 + 0.778320i \(0.283928\pi\)
\(762\) 0 0
\(763\) −4.10512 −0.148615
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 56.6429 2.04526
\(768\) 0 0
\(769\) 18.3923 0.663243 0.331622 0.943412i \(-0.392404\pi\)
0.331622 + 0.943412i \(0.392404\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −3.76217 −0.135316 −0.0676579 0.997709i \(-0.521553\pi\)
−0.0676579 + 0.997709i \(0.521553\pi\)
\(774\) 0 0
\(775\) 5.65685 0.203200
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −8.78461 −0.314741
\(780\) 0 0
\(781\) −8.48528 −0.303627
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11.0718 0.395169
\(786\) 0 0
\(787\) −18.3397 −0.653741 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −0.960947 −0.0341673
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.41044 −0.120804 −0.0604019 0.998174i \(-0.519238\pi\)
−0.0604019 + 0.998174i \(0.519238\pi\)
\(798\) 0 0
\(799\) −33.9411 −1.20075
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −12.0000 −0.423471
\(804\) 0 0
\(805\) 16.9706 0.598134
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.9282 −1.08738 −0.543689 0.839287i \(-0.682973\pi\)
−0.543689 + 0.839287i \(0.682973\pi\)
\(810\) 0 0
\(811\) −4.98076 −0.174898 −0.0874491 0.996169i \(-0.527872\pi\)
−0.0874491 + 0.996169i \(0.527872\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 32.4997 1.13842
\(816\) 0 0
\(817\) 10.3923 0.363581
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −45.2276 −1.57846 −0.789228 0.614101i \(-0.789519\pi\)
−0.789228 + 0.614101i \(0.789519\pi\)
\(822\) 0 0
\(823\) 18.5606 0.646983 0.323492 0.946231i \(-0.395143\pi\)
0.323492 + 0.946231i \(0.395143\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −6.33975 −0.220455 −0.110227 0.993906i \(-0.535158\pi\)
−0.110227 + 0.993906i \(0.535158\pi\)
\(828\) 0 0
\(829\) 31.2886 1.08670 0.543348 0.839507i \(-0.317156\pi\)
0.543348 + 0.839507i \(0.317156\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 20.5359 0.711527
\(834\) 0 0
\(835\) −57.9615 −2.00584
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −4.06678 −0.140401 −0.0702003 0.997533i \(-0.522364\pi\)
−0.0702003 + 0.997533i \(0.522364\pi\)
\(840\) 0 0
\(841\) −23.0000 −0.793103
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 36.3906 1.25188
\(846\) 0 0
\(847\) −9.72363 −0.334108
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.53590 −0.0869295
\(852\) 0 0
\(853\) 48.2591 1.65236 0.826181 0.563406i \(-0.190509\pi\)
0.826181 + 0.563406i \(0.190509\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.85641 −0.0634136 −0.0317068 0.999497i \(-0.510094\pi\)
−0.0317068 + 0.999497i \(0.510094\pi\)
\(858\) 0 0
\(859\) −38.4449 −1.31172 −0.655861 0.754882i \(-0.727694\pi\)
−0.655861 + 0.754882i \(0.727694\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −36.5665 −1.24474 −0.622369 0.782724i \(-0.713830\pi\)
−0.622369 + 0.782724i \(0.713830\pi\)
\(864\) 0 0
\(865\) 14.7846 0.502692
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 19.5959 0.664746
\(870\) 0 0
\(871\) 22.1469 0.750421
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −10.1436 −0.342916
\(876\) 0 0
\(877\) 11.4896 0.387975 0.193988 0.981004i \(-0.437858\pi\)
0.193988 + 0.981004i \(0.437858\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 4.14359 0.139601 0.0698006 0.997561i \(-0.477764\pi\)
0.0698006 + 0.997561i \(0.477764\pi\)
\(882\) 0 0
\(883\) 32.8756 1.10635 0.553177 0.833064i \(-0.313415\pi\)
0.553177 + 0.833064i \(0.313415\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10.2784 −0.345116 −0.172558 0.984999i \(-0.555203\pi\)
−0.172558 + 0.984999i \(0.555203\pi\)
\(888\) 0 0
\(889\) −5.85641 −0.196418
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −12.4233 −0.415730
\(894\) 0 0
\(895\) 20.0764 0.671080
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −13.8564 −0.462137
\(900\) 0 0
\(901\) 20.9086 0.696566
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −28.1436 −0.935525
\(906\) 0 0
\(907\) −35.9090 −1.19234 −0.596169 0.802859i \(-0.703311\pi\)
−0.596169 + 0.802859i \(0.703311\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 43.7391 1.44914 0.724570 0.689201i \(-0.242038\pi\)
0.724570 + 0.689201i \(0.242038\pi\)
\(912\) 0 0
\(913\) 10.3923 0.343935
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 18.2832 0.603766
\(918\) 0 0
\(919\) 55.1271 1.81848 0.909238 0.416277i \(-0.136665\pi\)
0.909238 + 0.416277i \(0.136665\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −35.3205 −1.16259
\(924\) 0 0
\(925\) −0.378937 −0.0124594
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.46410 0.113653 0.0568267 0.998384i \(-0.481902\pi\)
0.0568267 + 0.998384i \(0.481902\pi\)
\(930\) 0 0
\(931\) 7.51666 0.246349
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −10.7589 −0.351854
\(936\) 0 0
\(937\) −53.1769 −1.73721 −0.868607 0.495502i \(-0.834984\pi\)
−0.868607 + 0.495502i \(0.834984\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −41.2896 −1.34600 −0.673001 0.739641i \(-0.734995\pi\)
−0.673001 + 0.739641i \(0.734995\pi\)
\(942\) 0 0
\(943\) 46.3644 1.50983
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −22.7321 −0.738692 −0.369346 0.929292i \(-0.620418\pi\)
−0.369346 + 0.929292i \(0.620418\pi\)
\(948\) 0 0
\(949\) −49.9507 −1.62147
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −29.5692 −0.957841 −0.478920 0.877858i \(-0.658972\pi\)
−0.478920 + 0.877858i \(0.658972\pi\)
\(954\) 0 0
\(955\) 41.5692 1.34515
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −12.4233 −0.401170
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.85993 0.188638
\(966\) 0 0
\(967\) −7.24693 −0.233046 −0.116523 0.993188i \(-0.537175\pi\)
−0.116523 + 0.993188i \(0.537175\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −33.3731 −1.07099 −0.535496 0.844538i \(-0.679875\pi\)
−0.535496 + 0.844538i \(0.679875\pi\)
\(972\) 0 0
\(973\) −0.203072 −0.00651019
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −51.4641 −1.64648 −0.823241 0.567692i \(-0.807837\pi\)
−0.823241 + 0.567692i \(0.807837\pi\)
\(978\) 0 0
\(979\) 12.0000 0.383522
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 57.6038 1.83728 0.918638 0.395100i \(-0.129290\pi\)
0.918638 + 0.395100i \(0.129290\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −54.8497 −1.74412
\(990\) 0 0
\(991\) 16.5644 0.526186 0.263093 0.964770i \(-0.415257\pi\)
0.263093 + 0.964770i \(0.415257\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −45.4641 −1.44131
\(996\) 0 0
\(997\) 6.79367 0.215158 0.107579 0.994197i \(-0.465690\pi\)
0.107579 + 0.994197i \(0.465690\pi\)
\(998\) 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 9216.2.a.bk.1.3 4
3.2 odd 2 1024.2.a.g.1.1 4
4.3 odd 2 9216.2.a.bb.1.4 4
8.3 odd 2 inner 9216.2.a.bk.1.2 4
8.5 even 2 9216.2.a.bb.1.1 4
12.11 even 2 1024.2.a.j.1.3 4
24.5 odd 2 1024.2.a.j.1.4 4
24.11 even 2 1024.2.a.g.1.2 4
32.3 odd 8 2304.2.k.f.577.2 8
32.5 even 8 2304.2.k.k.1729.4 8
32.11 odd 8 2304.2.k.f.1729.1 8
32.13 even 8 2304.2.k.k.577.3 8
32.19 odd 8 2304.2.k.k.577.4 8
32.21 even 8 2304.2.k.f.1729.2 8
32.27 odd 8 2304.2.k.k.1729.3 8
32.29 even 8 2304.2.k.f.577.1 8
48.5 odd 4 1024.2.b.h.513.7 8
48.11 even 4 1024.2.b.h.513.1 8
48.29 odd 4 1024.2.b.h.513.2 8
48.35 even 4 1024.2.b.h.513.8 8
96.5 odd 8 256.2.e.b.193.4 yes 8
96.11 even 8 256.2.e.a.193.4 yes 8
96.29 odd 8 256.2.e.a.65.1 8
96.35 even 8 256.2.e.a.65.4 yes 8
96.53 odd 8 256.2.e.a.193.1 yes 8
96.59 even 8 256.2.e.b.193.1 yes 8
96.77 odd 8 256.2.e.b.65.4 yes 8
96.83 even 8 256.2.e.b.65.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
256.2.e.a.65.1 8 96.29 odd 8
256.2.e.a.65.4 yes 8 96.35 even 8
256.2.e.a.193.1 yes 8 96.53 odd 8
256.2.e.a.193.4 yes 8 96.11 even 8
256.2.e.b.65.1 yes 8 96.83 even 8
256.2.e.b.65.4 yes 8 96.77 odd 8
256.2.e.b.193.1 yes 8 96.59 even 8
256.2.e.b.193.4 yes 8 96.5 odd 8
1024.2.a.g.1.1 4 3.2 odd 2
1024.2.a.g.1.2 4 24.11 even 2
1024.2.a.j.1.3 4 12.11 even 2
1024.2.a.j.1.4 4 24.5 odd 2
1024.2.b.h.513.1 8 48.11 even 4
1024.2.b.h.513.2 8 48.29 odd 4
1024.2.b.h.513.7 8 48.5 odd 4
1024.2.b.h.513.8 8 48.35 even 4
2304.2.k.f.577.1 8 32.29 even 8
2304.2.k.f.577.2 8 32.3 odd 8
2304.2.k.f.1729.1 8 32.11 odd 8
2304.2.k.f.1729.2 8 32.21 even 8
2304.2.k.k.577.3 8 32.13 even 8
2304.2.k.k.577.4 8 32.19 odd 8
2304.2.k.k.1729.3 8 32.27 odd 8
2304.2.k.k.1729.4 8 32.5 even 8
9216.2.a.bb.1.1 4 8.5 even 2
9216.2.a.bb.1.4 4 4.3 odd 2
9216.2.a.bk.1.2 4 8.3 odd 2 inner
9216.2.a.bk.1.3 4 1.1 even 1 trivial