Properties

Label 4032.2.a.bp.1.1
Level $4032$
Weight $2$
Character 4032.1
Self dual yes
Analytic conductor $32.196$
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] = [4032,2,Mod(1,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, 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("4032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2016)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 4032.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.23607 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-3.23607 q^{5} +1.00000 q^{7} -3.23607 q^{11} -4.47214 q^{13} +0.763932 q^{17} -2.47214 q^{19} -5.70820 q^{23} +5.47214 q^{25} -1.52786 q^{29} -2.47214 q^{31} -3.23607 q^{35} +4.47214 q^{37} +7.23607 q^{41} +12.9443 q^{43} -1.52786 q^{47} +1.00000 q^{49} -8.00000 q^{53} +10.4721 q^{55} -6.47214 q^{59} +4.47214 q^{61} +14.4721 q^{65} +8.00000 q^{67} -7.23607 q^{71} -14.9443 q^{73} -3.23607 q^{77} +12.9443 q^{79} +12.9443 q^{83} -2.47214 q^{85} +2.29180 q^{89} -4.47214 q^{91} +8.00000 q^{95} -6.94427 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 2 q^{7} - 2 q^{11} + 6 q^{17} + 4 q^{19} + 2 q^{23} + 2 q^{25} - 12 q^{29} + 4 q^{31} - 2 q^{35} + 10 q^{41} + 8 q^{43} - 12 q^{47} + 2 q^{49} - 16 q^{53} + 12 q^{55} - 4 q^{59} + 20 q^{65} + 16 q^{67} - 10 q^{71} - 12 q^{73} - 2 q^{77} + 8 q^{79} + 8 q^{83} + 4 q^{85} + 18 q^{89} + 16 q^{95} + 4 q^{97}+O(q^{100}) \) 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
\(5\) −3.23607 −1.44721 −0.723607 0.690212i \(-0.757517\pi\)
−0.723607 + 0.690212i \(0.757517\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.23607 −0.975711 −0.487856 0.872924i \(-0.662221\pi\)
−0.487856 + 0.872924i \(0.662221\pi\)
\(12\) 0 0
\(13\) −4.47214 −1.24035 −0.620174 0.784465i \(-0.712938\pi\)
−0.620174 + 0.784465i \(0.712938\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.763932 0.185281 0.0926404 0.995700i \(-0.470469\pi\)
0.0926404 + 0.995700i \(0.470469\pi\)
\(18\) 0 0
\(19\) −2.47214 −0.567147 −0.283573 0.958951i \(-0.591520\pi\)
−0.283573 + 0.958951i \(0.591520\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.70820 −1.19024 −0.595121 0.803636i \(-0.702896\pi\)
−0.595121 + 0.803636i \(0.702896\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.52786 −0.283717 −0.141859 0.989887i \(-0.545308\pi\)
−0.141859 + 0.989887i \(0.545308\pi\)
\(30\) 0 0
\(31\) −2.47214 −0.444009 −0.222004 0.975046i \(-0.571260\pi\)
−0.222004 + 0.975046i \(0.571260\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.23607 −0.546995
\(36\) 0 0
\(37\) 4.47214 0.735215 0.367607 0.929981i \(-0.380177\pi\)
0.367607 + 0.929981i \(0.380177\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.23607 1.13008 0.565042 0.825062i \(-0.308860\pi\)
0.565042 + 0.825062i \(0.308860\pi\)
\(42\) 0 0
\(43\) 12.9443 1.97398 0.986991 0.160773i \(-0.0513986\pi\)
0.986991 + 0.160773i \(0.0513986\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.52786 −0.222862 −0.111431 0.993772i \(-0.535543\pi\)
−0.111431 + 0.993772i \(0.535543\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −8.00000 −1.09888 −0.549442 0.835532i \(-0.685160\pi\)
−0.549442 + 0.835532i \(0.685160\pi\)
\(54\) 0 0
\(55\) 10.4721 1.41206
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −6.47214 −0.842600 −0.421300 0.906921i \(-0.638426\pi\)
−0.421300 + 0.906921i \(0.638426\pi\)
\(60\) 0 0
\(61\) 4.47214 0.572598 0.286299 0.958140i \(-0.407575\pi\)
0.286299 + 0.958140i \(0.407575\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 14.4721 1.79505
\(66\) 0 0
\(67\) 8.00000 0.977356 0.488678 0.872464i \(-0.337479\pi\)
0.488678 + 0.872464i \(0.337479\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7.23607 −0.858763 −0.429382 0.903123i \(-0.641268\pi\)
−0.429382 + 0.903123i \(0.641268\pi\)
\(72\) 0 0
\(73\) −14.9443 −1.74909 −0.874547 0.484940i \(-0.838841\pi\)
−0.874547 + 0.484940i \(0.838841\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.23607 −0.368784
\(78\) 0 0
\(79\) 12.9443 1.45634 0.728172 0.685394i \(-0.240370\pi\)
0.728172 + 0.685394i \(0.240370\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.9443 1.42082 0.710409 0.703789i \(-0.248510\pi\)
0.710409 + 0.703789i \(0.248510\pi\)
\(84\) 0 0
\(85\) −2.47214 −0.268141
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.29180 0.242930 0.121465 0.992596i \(-0.461241\pi\)
0.121465 + 0.992596i \(0.461241\pi\)
\(90\) 0 0
\(91\) −4.47214 −0.468807
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) −6.94427 −0.705084 −0.352542 0.935796i \(-0.614683\pi\)
−0.352542 + 0.935796i \(0.614683\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −17.7082 −1.76203 −0.881016 0.473086i \(-0.843140\pi\)
−0.881016 + 0.473086i \(0.843140\pi\)
\(102\) 0 0
\(103\) 18.4721 1.82011 0.910057 0.414484i \(-0.136038\pi\)
0.910057 + 0.414484i \(0.136038\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.2361 −1.08623 −0.543116 0.839658i \(-0.682756\pi\)
−0.543116 + 0.839658i \(0.682756\pi\)
\(108\) 0 0
\(109\) 10.9443 1.04827 0.524136 0.851635i \(-0.324389\pi\)
0.524136 + 0.851635i \(0.324389\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.9443 1.21769 0.608847 0.793287i \(-0.291632\pi\)
0.608847 + 0.793287i \(0.291632\pi\)
\(114\) 0 0
\(115\) 18.4721 1.72254
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.763932 0.0700295
\(120\) 0 0
\(121\) −0.527864 −0.0479876
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.52786 −0.136656
\(126\) 0 0
\(127\) −17.8885 −1.58735 −0.793676 0.608341i \(-0.791835\pi\)
−0.793676 + 0.608341i \(0.791835\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) −2.47214 −0.214361
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.52786 0.130534 0.0652671 0.997868i \(-0.479210\pi\)
0.0652671 + 0.997868i \(0.479210\pi\)
\(138\) 0 0
\(139\) −4.94427 −0.419368 −0.209684 0.977769i \(-0.567243\pi\)
−0.209684 + 0.977769i \(0.567243\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 14.4721 1.21022
\(144\) 0 0
\(145\) 4.94427 0.410599
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 17.8885 1.46549 0.732743 0.680505i \(-0.238240\pi\)
0.732743 + 0.680505i \(0.238240\pi\)
\(150\) 0 0
\(151\) 3.05573 0.248672 0.124336 0.992240i \(-0.460320\pi\)
0.124336 + 0.992240i \(0.460320\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) −16.4721 −1.31462 −0.657310 0.753620i \(-0.728306\pi\)
−0.657310 + 0.753620i \(0.728306\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.70820 −0.449869
\(162\) 0 0
\(163\) 8.00000 0.626608 0.313304 0.949653i \(-0.398564\pi\)
0.313304 + 0.949653i \(0.398564\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 11.4164 0.883428 0.441714 0.897156i \(-0.354371\pi\)
0.441714 + 0.897156i \(0.354371\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −17.7082 −1.34633 −0.673165 0.739492i \(-0.735066\pi\)
−0.673165 + 0.739492i \(0.735066\pi\)
\(174\) 0 0
\(175\) 5.47214 0.413655
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.180340 0.0134792 0.00673962 0.999977i \(-0.497855\pi\)
0.00673962 + 0.999977i \(0.497855\pi\)
\(180\) 0 0
\(181\) 8.47214 0.629729 0.314864 0.949137i \(-0.398041\pi\)
0.314864 + 0.949137i \(0.398041\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −14.4721 −1.06401
\(186\) 0 0
\(187\) −2.47214 −0.180780
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 26.6525 1.92851 0.964253 0.264984i \(-0.0853668\pi\)
0.964253 + 0.264984i \(0.0853668\pi\)
\(192\) 0 0
\(193\) 12.4721 0.897764 0.448882 0.893591i \(-0.351822\pi\)
0.448882 + 0.893591i \(0.351822\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.94427 −0.352265 −0.176132 0.984366i \(-0.556359\pi\)
−0.176132 + 0.984366i \(0.556359\pi\)
\(198\) 0 0
\(199\) 8.00000 0.567105 0.283552 0.958957i \(-0.408487\pi\)
0.283552 + 0.958957i \(0.408487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.52786 −0.107235
\(204\) 0 0
\(205\) −23.4164 −1.63547
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.00000 0.553372
\(210\) 0 0
\(211\) 24.0000 1.65223 0.826114 0.563503i \(-0.190547\pi\)
0.826114 + 0.563503i \(0.190547\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −41.8885 −2.85677
\(216\) 0 0
\(217\) −2.47214 −0.167820
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −3.41641 −0.229812
\(222\) 0 0
\(223\) 12.9443 0.866813 0.433406 0.901199i \(-0.357312\pi\)
0.433406 + 0.901199i \(0.357312\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 16.3607 1.08590 0.542948 0.839766i \(-0.317308\pi\)
0.542948 + 0.839766i \(0.317308\pi\)
\(228\) 0 0
\(229\) −12.4721 −0.824182 −0.412091 0.911143i \(-0.635201\pi\)
−0.412091 + 0.911143i \(0.635201\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 24.3607 1.59592 0.797961 0.602710i \(-0.205912\pi\)
0.797961 + 0.602710i \(0.205912\pi\)
\(234\) 0 0
\(235\) 4.94427 0.322529
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.76393 −0.566892 −0.283446 0.958988i \(-0.591478\pi\)
−0.283446 + 0.958988i \(0.591478\pi\)
\(240\) 0 0
\(241\) 7.88854 0.508146 0.254073 0.967185i \(-0.418230\pi\)
0.254073 + 0.967185i \(0.418230\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.23607 −0.206745
\(246\) 0 0
\(247\) 11.0557 0.703459
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.41641 −0.215642 −0.107821 0.994170i \(-0.534387\pi\)
−0.107821 + 0.994170i \(0.534387\pi\)
\(252\) 0 0
\(253\) 18.4721 1.16133
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −29.7082 −1.85315 −0.926573 0.376114i \(-0.877260\pi\)
−0.926573 + 0.376114i \(0.877260\pi\)
\(258\) 0 0
\(259\) 4.47214 0.277885
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 28.1803 1.73767 0.868837 0.495098i \(-0.164868\pi\)
0.868837 + 0.495098i \(0.164868\pi\)
\(264\) 0 0
\(265\) 25.8885 1.59032
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.23607 −0.197307 −0.0986533 0.995122i \(-0.531453\pi\)
−0.0986533 + 0.995122i \(0.531453\pi\)
\(270\) 0 0
\(271\) 7.41641 0.450515 0.225257 0.974299i \(-0.427678\pi\)
0.225257 + 0.974299i \(0.427678\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −17.7082 −1.06784
\(276\) 0 0
\(277\) 11.5279 0.692642 0.346321 0.938116i \(-0.387431\pi\)
0.346321 + 0.938116i \(0.387431\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −11.4164 −0.681046 −0.340523 0.940236i \(-0.610604\pi\)
−0.340523 + 0.940236i \(0.610604\pi\)
\(282\) 0 0
\(283\) 2.47214 0.146953 0.0734766 0.997297i \(-0.476591\pi\)
0.0734766 + 0.997297i \(0.476591\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 7.23607 0.427132
\(288\) 0 0
\(289\) −16.4164 −0.965671
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 32.1803 1.88000 0.939998 0.341181i \(-0.110827\pi\)
0.939998 + 0.341181i \(0.110827\pi\)
\(294\) 0 0
\(295\) 20.9443 1.21942
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 25.5279 1.47631
\(300\) 0 0
\(301\) 12.9443 0.746095
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −14.4721 −0.828672
\(306\) 0 0
\(307\) 18.4721 1.05426 0.527130 0.849785i \(-0.323268\pi\)
0.527130 + 0.849785i \(0.323268\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −11.4164 −0.647365 −0.323683 0.946166i \(-0.604921\pi\)
−0.323683 + 0.946166i \(0.604921\pi\)
\(312\) 0 0
\(313\) −15.8885 −0.898074 −0.449037 0.893513i \(-0.648233\pi\)
−0.449037 + 0.893513i \(0.648233\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.9443 1.17635 0.588174 0.808735i \(-0.299847\pi\)
0.588174 + 0.808735i \(0.299847\pi\)
\(318\) 0 0
\(319\) 4.94427 0.276826
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.88854 −0.105081
\(324\) 0 0
\(325\) −24.4721 −1.35747
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.52786 −0.0842339
\(330\) 0 0
\(331\) −17.8885 −0.983243 −0.491622 0.870809i \(-0.663596\pi\)
−0.491622 + 0.870809i \(0.663596\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −25.8885 −1.41444
\(336\) 0 0
\(337\) −11.8885 −0.647610 −0.323805 0.946124i \(-0.604962\pi\)
−0.323805 + 0.946124i \(0.604962\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −20.7639 −1.11467 −0.557333 0.830289i \(-0.688175\pi\)
−0.557333 + 0.830289i \(0.688175\pi\)
\(348\) 0 0
\(349\) 20.4721 1.09585 0.547924 0.836528i \(-0.315418\pi\)
0.547924 + 0.836528i \(0.315418\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −12.1803 −0.648294 −0.324147 0.946007i \(-0.605077\pi\)
−0.324147 + 0.946007i \(0.605077\pi\)
\(354\) 0 0
\(355\) 23.4164 1.24281
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.7639 0.884766 0.442383 0.896826i \(-0.354133\pi\)
0.442383 + 0.896826i \(0.354133\pi\)
\(360\) 0 0
\(361\) −12.8885 −0.678344
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 48.3607 2.53131
\(366\) 0 0
\(367\) −8.00000 −0.417597 −0.208798 0.977959i \(-0.566955\pi\)
−0.208798 + 0.977959i \(0.566955\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −8.00000 −0.415339
\(372\) 0 0
\(373\) 11.8885 0.615565 0.307783 0.951457i \(-0.400413\pi\)
0.307783 + 0.951457i \(0.400413\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.83282 0.351908
\(378\) 0 0
\(379\) −24.0000 −1.23280 −0.616399 0.787434i \(-0.711409\pi\)
−0.616399 + 0.787434i \(0.711409\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 0 0
\(385\) 10.4721 0.533709
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.52786 −0.0774658 −0.0387329 0.999250i \(-0.512332\pi\)
−0.0387329 + 0.999250i \(0.512332\pi\)
\(390\) 0 0
\(391\) −4.36068 −0.220529
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −41.8885 −2.10764
\(396\) 0 0
\(397\) −0.472136 −0.0236958 −0.0118479 0.999930i \(-0.503771\pi\)
−0.0118479 + 0.999930i \(0.503771\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −17.5279 −0.875300 −0.437650 0.899145i \(-0.644189\pi\)
−0.437650 + 0.899145i \(0.644189\pi\)
\(402\) 0 0
\(403\) 11.0557 0.550725
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −14.4721 −0.717357
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −6.47214 −0.318473
\(414\) 0 0
\(415\) −41.8885 −2.05623
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.47214 −0.316185 −0.158092 0.987424i \(-0.550534\pi\)
−0.158092 + 0.987424i \(0.550534\pi\)
\(420\) 0 0
\(421\) 21.4164 1.04377 0.521886 0.853015i \(-0.325229\pi\)
0.521886 + 0.853015i \(0.325229\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.18034 0.202776
\(426\) 0 0
\(427\) 4.47214 0.216422
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −9.12461 −0.439517 −0.219759 0.975554i \(-0.570527\pi\)
−0.219759 + 0.975554i \(0.570527\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 14.1115 0.675042
\(438\) 0 0
\(439\) −22.8328 −1.08975 −0.544875 0.838517i \(-0.683423\pi\)
−0.544875 + 0.838517i \(0.683423\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.81966 −0.371523 −0.185762 0.982595i \(-0.559475\pi\)
−0.185762 + 0.982595i \(0.559475\pi\)
\(444\) 0 0
\(445\) −7.41641 −0.351571
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12.9443 0.610878 0.305439 0.952212i \(-0.401197\pi\)
0.305439 + 0.952212i \(0.401197\pi\)
\(450\) 0 0
\(451\) −23.4164 −1.10264
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 14.4721 0.678464
\(456\) 0 0
\(457\) −0.111456 −0.00521370 −0.00260685 0.999997i \(-0.500830\pi\)
−0.00260685 + 0.999997i \(0.500830\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −9.70820 −0.452156 −0.226078 0.974109i \(-0.572590\pi\)
−0.226078 + 0.974109i \(0.572590\pi\)
\(462\) 0 0
\(463\) 17.8885 0.831351 0.415676 0.909513i \(-0.363545\pi\)
0.415676 + 0.909513i \(0.363545\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.5279 1.18129 0.590644 0.806932i \(-0.298874\pi\)
0.590644 + 0.806932i \(0.298874\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −41.8885 −1.92604
\(474\) 0 0
\(475\) −13.5279 −0.620701
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 27.4164 1.25269 0.626344 0.779547i \(-0.284551\pi\)
0.626344 + 0.779547i \(0.284551\pi\)
\(480\) 0 0
\(481\) −20.0000 −0.911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 22.4721 1.02041
\(486\) 0 0
\(487\) 22.8328 1.03465 0.517327 0.855788i \(-0.326927\pi\)
0.517327 + 0.855788i \(0.326927\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 42.0689 1.89854 0.949271 0.314459i \(-0.101823\pi\)
0.949271 + 0.314459i \(0.101823\pi\)
\(492\) 0 0
\(493\) −1.16718 −0.0525673
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −7.23607 −0.324582
\(498\) 0 0
\(499\) 28.9443 1.29572 0.647862 0.761758i \(-0.275663\pi\)
0.647862 + 0.761758i \(0.275663\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −28.9443 −1.29056 −0.645281 0.763946i \(-0.723260\pi\)
−0.645281 + 0.763946i \(0.723260\pi\)
\(504\) 0 0
\(505\) 57.3050 2.55004
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −5.12461 −0.227144 −0.113572 0.993530i \(-0.536229\pi\)
−0.113572 + 0.993530i \(0.536229\pi\)
\(510\) 0 0
\(511\) −14.9443 −0.661096
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −59.7771 −2.63409
\(516\) 0 0
\(517\) 4.94427 0.217449
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −25.1246 −1.10073 −0.550365 0.834924i \(-0.685511\pi\)
−0.550365 + 0.834924i \(0.685511\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.88854 −0.0822663
\(528\) 0 0
\(529\) 9.58359 0.416678
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −32.3607 −1.40170
\(534\) 0 0
\(535\) 36.3607 1.57201
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.23607 −0.139387
\(540\) 0 0
\(541\) −38.3607 −1.64925 −0.824627 0.565677i \(-0.808615\pi\)
−0.824627 + 0.565677i \(0.808615\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −35.4164 −1.51707
\(546\) 0 0
\(547\) −38.8328 −1.66037 −0.830186 0.557487i \(-0.811766\pi\)
−0.830186 + 0.557487i \(0.811766\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.77709 0.160909
\(552\) 0 0
\(553\) 12.9443 0.550446
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.9443 0.887437 0.443719 0.896166i \(-0.353659\pi\)
0.443719 + 0.896166i \(0.353659\pi\)
\(558\) 0 0
\(559\) −57.8885 −2.44842
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −29.3050 −1.23506 −0.617528 0.786549i \(-0.711866\pi\)
−0.617528 + 0.786549i \(0.711866\pi\)
\(564\) 0 0
\(565\) −41.8885 −1.76226
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 17.5279 0.734806 0.367403 0.930062i \(-0.380247\pi\)
0.367403 + 0.930062i \(0.380247\pi\)
\(570\) 0 0
\(571\) −28.9443 −1.21128 −0.605640 0.795739i \(-0.707083\pi\)
−0.605640 + 0.795739i \(0.707083\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −31.2361 −1.30263
\(576\) 0 0
\(577\) −30.0000 −1.24892 −0.624458 0.781058i \(-0.714680\pi\)
−0.624458 + 0.781058i \(0.714680\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 12.9443 0.537019
\(582\) 0 0
\(583\) 25.8885 1.07219
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 19.4164 0.801401 0.400700 0.916209i \(-0.368767\pi\)
0.400700 + 0.916209i \(0.368767\pi\)
\(588\) 0 0
\(589\) 6.11146 0.251818
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −17.1246 −0.703224 −0.351612 0.936146i \(-0.614366\pi\)
−0.351612 + 0.936146i \(0.614366\pi\)
\(594\) 0 0
\(595\) −2.47214 −0.101348
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.7639 1.01183 0.505913 0.862584i \(-0.331156\pi\)
0.505913 + 0.862584i \(0.331156\pi\)
\(600\) 0 0
\(601\) 0.111456 0.00454639 0.00227320 0.999997i \(-0.499276\pi\)
0.00227320 + 0.999997i \(0.499276\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.70820 0.0694484
\(606\) 0 0
\(607\) −28.9443 −1.17481 −0.587406 0.809292i \(-0.699851\pi\)
−0.587406 + 0.809292i \(0.699851\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6.83282 0.276426
\(612\) 0 0
\(613\) −12.8328 −0.518313 −0.259156 0.965835i \(-0.583444\pi\)
−0.259156 + 0.965835i \(0.583444\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −46.4721 −1.87090 −0.935449 0.353462i \(-0.885004\pi\)
−0.935449 + 0.353462i \(0.885004\pi\)
\(618\) 0 0
\(619\) −11.0557 −0.444367 −0.222184 0.975005i \(-0.571318\pi\)
−0.222184 + 0.975005i \(0.571318\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.29180 0.0918189
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.41641 0.136221
\(630\) 0 0
\(631\) 17.8885 0.712132 0.356066 0.934461i \(-0.384118\pi\)
0.356066 + 0.934461i \(0.384118\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 57.8885 2.29724
\(636\) 0 0
\(637\) −4.47214 −0.177192
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 50.2492 1.98473 0.992363 0.123356i \(-0.0393657\pi\)
0.992363 + 0.123356i \(0.0393657\pi\)
\(642\) 0 0
\(643\) 23.4164 0.923453 0.461726 0.887022i \(-0.347230\pi\)
0.461726 + 0.887022i \(0.347230\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.4721 1.19798 0.598992 0.800755i \(-0.295568\pi\)
0.598992 + 0.800755i \(0.295568\pi\)
\(648\) 0 0
\(649\) 20.9443 0.822135
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −11.4164 −0.446759 −0.223379 0.974732i \(-0.571709\pi\)
−0.223379 + 0.974732i \(0.571709\pi\)
\(654\) 0 0
\(655\) −51.7771 −2.02310
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 30.6525 1.19405 0.597025 0.802222i \(-0.296349\pi\)
0.597025 + 0.802222i \(0.296349\pi\)
\(660\) 0 0
\(661\) −24.4721 −0.951856 −0.475928 0.879484i \(-0.657888\pi\)
−0.475928 + 0.879484i \(0.657888\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.00000 0.310227
\(666\) 0 0
\(667\) 8.72136 0.337692
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −14.4721 −0.558691
\(672\) 0 0
\(673\) 43.3050 1.66928 0.834642 0.550793i \(-0.185675\pi\)
0.834642 + 0.550793i \(0.185675\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 31.0132 1.19193 0.595966 0.803010i \(-0.296769\pi\)
0.595966 + 0.803010i \(0.296769\pi\)
\(678\) 0 0
\(679\) −6.94427 −0.266497
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 8.18034 0.313012 0.156506 0.987677i \(-0.449977\pi\)
0.156506 + 0.987677i \(0.449977\pi\)
\(684\) 0 0
\(685\) −4.94427 −0.188911
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 35.7771 1.36300
\(690\) 0 0
\(691\) 41.8885 1.59352 0.796758 0.604299i \(-0.206547\pi\)
0.796758 + 0.604299i \(0.206547\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.0000 0.606915
\(696\) 0 0
\(697\) 5.52786 0.209383
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 11.4164 0.431192 0.215596 0.976483i \(-0.430831\pi\)
0.215596 + 0.976483i \(0.430831\pi\)
\(702\) 0 0
\(703\) −11.0557 −0.416975
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −17.7082 −0.665986
\(708\) 0 0
\(709\) 2.94427 0.110574 0.0552872 0.998470i \(-0.482393\pi\)
0.0552872 + 0.998470i \(0.482393\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 14.1115 0.528478
\(714\) 0 0
\(715\) −46.8328 −1.75145
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.0557 −0.710659 −0.355329 0.934741i \(-0.615631\pi\)
−0.355329 + 0.934741i \(0.615631\pi\)
\(720\) 0 0
\(721\) 18.4721 0.687938
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −8.36068 −0.310508
\(726\) 0 0
\(727\) 12.3607 0.458432 0.229216 0.973376i \(-0.426384\pi\)
0.229216 + 0.973376i \(0.426384\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9.88854 0.365741
\(732\) 0 0
\(733\) 47.3050 1.74725 0.873624 0.486601i \(-0.161764\pi\)
0.873624 + 0.486601i \(0.161764\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −25.8885 −0.953617
\(738\) 0 0
\(739\) −40.0000 −1.47142 −0.735712 0.677295i \(-0.763152\pi\)
−0.735712 + 0.677295i \(0.763152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 49.4853 1.81544 0.907720 0.419577i \(-0.137822\pi\)
0.907720 + 0.419577i \(0.137822\pi\)
\(744\) 0 0
\(745\) −57.8885 −2.12087
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −11.2361 −0.410557
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −9.88854 −0.359881
\(756\) 0 0
\(757\) 18.9443 0.688541 0.344271 0.938870i \(-0.388126\pi\)
0.344271 + 0.938870i \(0.388126\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 12.1803 0.441537 0.220768 0.975326i \(-0.429143\pi\)
0.220768 + 0.975326i \(0.429143\pi\)
\(762\) 0 0
\(763\) 10.9443 0.396209
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 28.9443 1.04512
\(768\) 0 0
\(769\) 27.8885 1.00569 0.502843 0.864378i \(-0.332287\pi\)
0.502843 + 0.864378i \(0.332287\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.7639 −0.746827 −0.373413 0.927665i \(-0.621813\pi\)
−0.373413 + 0.927665i \(0.621813\pi\)
\(774\) 0 0
\(775\) −13.5279 −0.485935
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17.8885 −0.640924
\(780\) 0 0
\(781\) 23.4164 0.837905
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 53.3050 1.90254
\(786\) 0 0
\(787\) 1.16718 0.0416056 0.0208028 0.999784i \(-0.493378\pi\)
0.0208028 + 0.999784i \(0.493378\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.9443 0.460245
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.34752 0.0477318 0.0238659 0.999715i \(-0.492403\pi\)
0.0238659 + 0.999715i \(0.492403\pi\)
\(798\) 0 0
\(799\) −1.16718 −0.0412920
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 48.3607 1.70661
\(804\) 0 0
\(805\) 18.4721 0.651057
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 32.0000 1.12506 0.562530 0.826777i \(-0.309828\pi\)
0.562530 + 0.826777i \(0.309828\pi\)
\(810\) 0 0
\(811\) −6.11146 −0.214602 −0.107301 0.994227i \(-0.534221\pi\)
−0.107301 + 0.994227i \(0.534221\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −25.8885 −0.906836
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −36.9443 −1.28936 −0.644682 0.764451i \(-0.723010\pi\)
−0.644682 + 0.764451i \(0.723010\pi\)
\(822\) 0 0
\(823\) 19.0557 0.664241 0.332120 0.943237i \(-0.392236\pi\)
0.332120 + 0.943237i \(0.392236\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 15.8197 0.550103 0.275052 0.961429i \(-0.411305\pi\)
0.275052 + 0.961429i \(0.411305\pi\)
\(828\) 0 0
\(829\) −31.5279 −1.09501 −0.547504 0.836803i \(-0.684422\pi\)
−0.547504 + 0.836803i \(0.684422\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.763932 0.0264687
\(834\) 0 0
\(835\) −36.9443 −1.27851
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 14.4721 0.499634 0.249817 0.968293i \(-0.419630\pi\)
0.249817 + 0.968293i \(0.419630\pi\)
\(840\) 0 0
\(841\) −26.6656 −0.919505
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −22.6525 −0.779269
\(846\) 0 0
\(847\) −0.527864 −0.0181376
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −25.5279 −0.875084
\(852\) 0 0
\(853\) 18.5836 0.636290 0.318145 0.948042i \(-0.396940\pi\)
0.318145 + 0.948042i \(0.396940\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 12.1803 0.416072 0.208036 0.978121i \(-0.433293\pi\)
0.208036 + 0.978121i \(0.433293\pi\)
\(858\) 0 0
\(859\) −34.4721 −1.17617 −0.588087 0.808798i \(-0.700119\pi\)
−0.588087 + 0.808798i \(0.700119\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 44.1803 1.50392 0.751958 0.659211i \(-0.229110\pi\)
0.751958 + 0.659211i \(0.229110\pi\)
\(864\) 0 0
\(865\) 57.3050 1.94843
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −41.8885 −1.42097
\(870\) 0 0
\(871\) −35.7771 −1.21226
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.52786 −0.0516512
\(876\) 0 0
\(877\) 8.83282 0.298263 0.149131 0.988817i \(-0.452352\pi\)
0.149131 + 0.988817i \(0.452352\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.12461 −0.0378891 −0.0189446 0.999821i \(-0.506031\pi\)
−0.0189446 + 0.999821i \(0.506031\pi\)
\(882\) 0 0
\(883\) 12.9443 0.435609 0.217805 0.975992i \(-0.430110\pi\)
0.217805 + 0.975992i \(0.430110\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −30.4721 −1.02315 −0.511577 0.859237i \(-0.670939\pi\)
−0.511577 + 0.859237i \(0.670939\pi\)
\(888\) 0 0
\(889\) −17.8885 −0.599963
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 3.77709 0.126395
\(894\) 0 0
\(895\) −0.583592 −0.0195073
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.77709 0.125973
\(900\) 0 0
\(901\) −6.11146 −0.203602
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −27.4164 −0.911352
\(906\) 0 0
\(907\) −12.9443 −0.429807 −0.214904 0.976635i \(-0.568944\pi\)
−0.214904 + 0.976635i \(0.568944\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −25.1246 −0.832416 −0.416208 0.909270i \(-0.636641\pi\)
−0.416208 + 0.909270i \(0.636641\pi\)
\(912\) 0 0
\(913\) −41.8885 −1.38631
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 16.0000 0.528367
\(918\) 0 0
\(919\) −44.9443 −1.48257 −0.741287 0.671188i \(-0.765784\pi\)
−0.741287 + 0.671188i \(0.765784\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 32.3607 1.06516
\(924\) 0 0
\(925\) 24.4721 0.804639
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 55.2361 1.81224 0.906118 0.423024i \(-0.139032\pi\)
0.906118 + 0.423024i \(0.139032\pi\)
\(930\) 0 0
\(931\) −2.47214 −0.0810210
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) 14.9443 0.488208 0.244104 0.969749i \(-0.421506\pi\)
0.244104 + 0.969749i \(0.421506\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 50.4296 1.64396 0.821978 0.569519i \(-0.192870\pi\)
0.821978 + 0.569519i \(0.192870\pi\)
\(942\) 0 0
\(943\) −41.3050 −1.34507
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −29.1246 −0.946423 −0.473211 0.880949i \(-0.656905\pi\)
−0.473211 + 0.880949i \(0.656905\pi\)
\(948\) 0 0
\(949\) 66.8328 2.16949
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −19.7771 −0.640643 −0.320321 0.947309i \(-0.603791\pi\)
−0.320321 + 0.947309i \(0.603791\pi\)
\(954\) 0 0
\(955\) −86.2492 −2.79096
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.52786 0.0493373
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −40.3607 −1.29926
\(966\) 0 0
\(967\) −11.7771 −0.378726 −0.189363 0.981907i \(-0.560642\pi\)
−0.189363 + 0.981907i \(0.560642\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −9.88854 −0.317338 −0.158669 0.987332i \(-0.550720\pi\)
−0.158669 + 0.987332i \(0.550720\pi\)
\(972\) 0 0
\(973\) −4.94427 −0.158506
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −50.2492 −1.60762 −0.803808 0.594889i \(-0.797196\pi\)
−0.803808 + 0.594889i \(0.797196\pi\)
\(978\) 0 0
\(979\) −7.41641 −0.237029
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −12.9443 −0.412858 −0.206429 0.978462i \(-0.566184\pi\)
−0.206429 + 0.978462i \(0.566184\pi\)
\(984\) 0 0
\(985\) 16.0000 0.509802
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −73.8885 −2.34952
\(990\) 0 0
\(991\) −9.16718 −0.291205 −0.145603 0.989343i \(-0.546512\pi\)
−0.145603 + 0.989343i \(0.546512\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −25.8885 −0.820722
\(996\) 0 0
\(997\) −50.3607 −1.59494 −0.797469 0.603359i \(-0.793828\pi\)
−0.797469 + 0.603359i \(0.793828\pi\)
\(998\) 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 4032.2.a.bp.1.1 2
3.2 odd 2 4032.2.a.bx.1.2 2
4.3 odd 2 4032.2.a.bo.1.1 2
8.3 odd 2 2016.2.a.u.1.2 yes 2
8.5 even 2 2016.2.a.v.1.2 yes 2
12.11 even 2 4032.2.a.bu.1.2 2
24.5 odd 2 2016.2.a.q.1.1 yes 2
24.11 even 2 2016.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2016.2.a.p.1.1 2 24.11 even 2
2016.2.a.q.1.1 yes 2 24.5 odd 2
2016.2.a.u.1.2 yes 2 8.3 odd 2
2016.2.a.v.1.2 yes 2 8.5 even 2
4032.2.a.bo.1.1 2 4.3 odd 2
4032.2.a.bp.1.1 2 1.1 even 1 trivial
4032.2.a.bu.1.2 2 12.11 even 2
4032.2.a.bx.1.2 2 3.2 odd 2