Properties

Label 2016.2.a.r.1.1
Level $2016$
Weight $2$
Character 2016.1
Self dual yes
Analytic conductor $16.098$
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] = [2016,2,Mod(1,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, 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("2016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2016.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: \(16.0978410475\)
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 224)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.61803\) of defining polynomial
Character \(\chi\) \(=\) 2016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.23607 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-3.23607 q^{5} +1.00000 q^{7} +6.47214 q^{11} +0.763932 q^{13} -4.47214 q^{17} -1.23607 q^{19} -4.00000 q^{23} +5.47214 q^{25} +4.47214 q^{29} -2.47214 q^{31} -3.23607 q^{35} -4.47214 q^{37} +8.47214 q^{41} +6.47214 q^{43} -10.4721 q^{47} +1.00000 q^{49} +10.0000 q^{53} -20.9443 q^{55} +9.23607 q^{59} +11.2361 q^{61} -2.47214 q^{65} +4.00000 q^{67} +4.94427 q^{71} -2.94427 q^{73} +6.47214 q^{77} +12.9443 q^{79} +9.23607 q^{83} +14.4721 q^{85} +6.00000 q^{89} +0.763932 q^{91} +4.00000 q^{95} +12.4721 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 2 q^{7} + 4 q^{11} + 6 q^{13} + 2 q^{19} - 8 q^{23} + 2 q^{25} + 4 q^{31} - 2 q^{35} + 8 q^{41} + 4 q^{43} - 12 q^{47} + 2 q^{49} + 20 q^{53} - 24 q^{55} + 14 q^{59} + 18 q^{61} + 4 q^{65} + 8 q^{67} - 8 q^{71} + 12 q^{73} + 4 q^{77} + 8 q^{79} + 14 q^{83} + 20 q^{85} + 12 q^{89} + 6 q^{91} + 8 q^{95} + 16 q^{97}+O(q^{100}) \) 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
\(11\) 6.47214 1.95142 0.975711 0.219061i \(-0.0702993\pi\)
0.975711 + 0.219061i \(0.0702993\pi\)
\(12\) 0 0
\(13\) 0.763932 0.211877 0.105938 0.994373i \(-0.466215\pi\)
0.105938 + 0.994373i \(0.466215\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.47214 −1.08465 −0.542326 0.840168i \(-0.682456\pi\)
−0.542326 + 0.840168i \(0.682456\pi\)
\(18\) 0 0
\(19\) −1.23607 −0.283573 −0.141787 0.989897i \(-0.545285\pi\)
−0.141787 + 0.989897i \(0.545285\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 5.47214 1.09443
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) 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 \(-0.619823\pi\)
−0.367607 + 0.929981i \(0.619823\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 8.47214 1.32313 0.661563 0.749890i \(-0.269894\pi\)
0.661563 + 0.749890i \(0.269894\pi\)
\(42\) 0 0
\(43\) 6.47214 0.986991 0.493496 0.869748i \(-0.335719\pi\)
0.493496 + 0.869748i \(0.335719\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.4721 −1.52752 −0.763759 0.645501i \(-0.776648\pi\)
−0.763759 + 0.645501i \(0.776648\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 10.0000 1.37361 0.686803 0.726844i \(-0.259014\pi\)
0.686803 + 0.726844i \(0.259014\pi\)
\(54\) 0 0
\(55\) −20.9443 −2.82413
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.23607 1.20243 0.601217 0.799086i \(-0.294683\pi\)
0.601217 + 0.799086i \(0.294683\pi\)
\(60\) 0 0
\(61\) 11.2361 1.43863 0.719316 0.694683i \(-0.244456\pi\)
0.719316 + 0.694683i \(0.244456\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.47214 −0.306631
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.94427 0.586777 0.293389 0.955993i \(-0.405217\pi\)
0.293389 + 0.955993i \(0.405217\pi\)
\(72\) 0 0
\(73\) −2.94427 −0.344601 −0.172300 0.985044i \(-0.555120\pi\)
−0.172300 + 0.985044i \(0.555120\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.47214 0.737568
\(78\) 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
\(83\) 9.23607 1.01379 0.506895 0.862008i \(-0.330793\pi\)
0.506895 + 0.862008i \(0.330793\pi\)
\(84\) 0 0
\(85\) 14.4721 1.56972
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0.763932 0.0800818
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 12.4721 1.26635 0.633177 0.774007i \(-0.281751\pi\)
0.633177 + 0.774007i \(0.281751\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.70820 0.169973 0.0849863 0.996382i \(-0.472915\pi\)
0.0849863 + 0.996382i \(0.472915\pi\)
\(102\) 0 0
\(103\) −5.52786 −0.544677 −0.272338 0.962202i \(-0.587797\pi\)
−0.272338 + 0.962202i \(0.587797\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.94427 −0.864675 −0.432338 0.901712i \(-0.642311\pi\)
−0.432338 + 0.901712i \(0.642311\pi\)
\(108\) 0 0
\(109\) 8.47214 0.811483 0.405742 0.913988i \(-0.367013\pi\)
0.405742 + 0.913988i \(0.367013\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 12.4721 1.17328 0.586640 0.809848i \(-0.300450\pi\)
0.586640 + 0.809848i \(0.300450\pi\)
\(114\) 0 0
\(115\) 12.9443 1.20706
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.47214 −0.409960
\(120\) 0 0
\(121\) 30.8885 2.80805
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.52786 −0.136656
\(126\) 0 0
\(127\) 8.94427 0.793676 0.396838 0.917889i \(-0.370108\pi\)
0.396838 + 0.917889i \(0.370108\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −11.7082 −1.02295 −0.511475 0.859298i \(-0.670901\pi\)
−0.511475 + 0.859298i \(0.670901\pi\)
\(132\) 0 0
\(133\) −1.23607 −0.107181
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −14.9443 −1.27678 −0.638388 0.769715i \(-0.720398\pi\)
−0.638388 + 0.769715i \(0.720398\pi\)
\(138\) 0 0
\(139\) 1.23607 0.104842 0.0524210 0.998625i \(-0.483306\pi\)
0.0524210 + 0.998625i \(0.483306\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.94427 0.413461
\(144\) 0 0
\(145\) −14.4721 −1.20185
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.94427 −0.241204 −0.120602 0.992701i \(-0.538483\pi\)
−0.120602 + 0.992701i \(0.538483\pi\)
\(150\) 0 0
\(151\) −8.94427 −0.727875 −0.363937 0.931423i \(-0.618568\pi\)
−0.363937 + 0.931423i \(0.618568\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000 0.642575
\(156\) 0 0
\(157\) 0.763932 0.0609684 0.0304842 0.999535i \(-0.490295\pi\)
0.0304842 + 0.999535i \(0.490295\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.00000 −0.315244
\(162\) 0 0
\(163\) −3.41641 −0.267594 −0.133797 0.991009i \(-0.542717\pi\)
−0.133797 + 0.991009i \(0.542717\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −23.4164 −1.81202 −0.906008 0.423261i \(-0.860886\pi\)
−0.906008 + 0.423261i \(0.860886\pi\)
\(168\) 0 0
\(169\) −12.4164 −0.955108
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −5.70820 −0.433987 −0.216993 0.976173i \(-0.569625\pi\)
−0.216993 + 0.976173i \(0.569625\pi\)
\(174\) 0 0
\(175\) 5.47214 0.413655
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.05573 0.527370 0.263685 0.964609i \(-0.415062\pi\)
0.263685 + 0.964609i \(0.415062\pi\)
\(180\) 0 0
\(181\) −12.1803 −0.905358 −0.452679 0.891674i \(-0.649532\pi\)
−0.452679 + 0.891674i \(0.649532\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 14.4721 1.06401
\(186\) 0 0
\(187\) −28.9443 −2.11661
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.94427 −0.357755 −0.178877 0.983871i \(-0.557247\pi\)
−0.178877 + 0.983871i \(0.557247\pi\)
\(192\) 0 0
\(193\) 0.472136 0.0339851 0.0169925 0.999856i \(-0.494591\pi\)
0.0169925 + 0.999856i \(0.494591\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.9443 −0.779747 −0.389874 0.920868i \(-0.627481\pi\)
−0.389874 + 0.920868i \(0.627481\pi\)
\(198\) 0 0
\(199\) 15.4164 1.09284 0.546420 0.837511i \(-0.315990\pi\)
0.546420 + 0.837511i \(0.315990\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.47214 0.313882
\(204\) 0 0
\(205\) −27.4164 −1.91484
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −8.00000 −0.553372
\(210\) 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
\(215\) −20.9443 −1.42839
\(216\) 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
\(227\) 17.2361 1.14400 0.571999 0.820254i \(-0.306168\pi\)
0.571999 + 0.820254i \(0.306168\pi\)
\(228\) 0 0
\(229\) 23.5967 1.55932 0.779658 0.626205i \(-0.215393\pi\)
0.779658 + 0.626205i \(0.215393\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.8885 1.04089 0.520447 0.853894i \(-0.325765\pi\)
0.520447 + 0.853894i \(0.325765\pi\)
\(234\) 0 0
\(235\) 33.8885 2.21064
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 13.8885 0.898375 0.449188 0.893437i \(-0.351713\pi\)
0.449188 + 0.893437i \(0.351713\pi\)
\(240\) 0 0
\(241\) 12.4721 0.803401 0.401700 0.915771i \(-0.368419\pi\)
0.401700 + 0.915771i \(0.368419\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.23607 −0.206745
\(246\) 0 0
\(247\) −0.944272 −0.0600826
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −4.29180 −0.270896 −0.135448 0.990784i \(-0.543247\pi\)
−0.135448 + 0.990784i \(0.543247\pi\)
\(252\) 0 0
\(253\) −25.8885 −1.62760
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) −4.47214 −0.277885
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −11.0557 −0.681725 −0.340863 0.940113i \(-0.610719\pi\)
−0.340863 + 0.940113i \(0.610719\pi\)
\(264\) 0 0
\(265\) −32.3607 −1.98790
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.18034 0.254880 0.127440 0.991846i \(-0.459324\pi\)
0.127440 + 0.991846i \(0.459324\pi\)
\(270\) 0 0
\(271\) −24.0000 −1.45790 −0.728948 0.684569i \(-0.759990\pi\)
−0.728948 + 0.684569i \(0.759990\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 35.4164 2.13569
\(276\) 0 0
\(277\) 7.88854 0.473977 0.236988 0.971512i \(-0.423840\pi\)
0.236988 + 0.971512i \(0.423840\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −26.0000 −1.55103 −0.775515 0.631329i \(-0.782510\pi\)
−0.775515 + 0.631329i \(0.782510\pi\)
\(282\) 0 0
\(283\) −6.18034 −0.367383 −0.183692 0.982984i \(-0.558805\pi\)
−0.183692 + 0.982984i \(0.558805\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.47214 0.500094
\(288\) 0 0
\(289\) 3.00000 0.176471
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 12.7639 0.745677 0.372838 0.927896i \(-0.378385\pi\)
0.372838 + 0.927896i \(0.378385\pi\)
\(294\) 0 0
\(295\) −29.8885 −1.74018
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.05573 −0.176717
\(300\) 0 0
\(301\) 6.47214 0.373048
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −36.3607 −2.08201
\(306\) 0 0
\(307\) 1.81966 0.103853 0.0519267 0.998651i \(-0.483464\pi\)
0.0519267 + 0.998651i \(0.483464\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −8.00000 −0.453638 −0.226819 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −8.47214 −0.478873 −0.239437 0.970912i \(-0.576963\pi\)
−0.239437 + 0.970912i \(0.576963\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −9.05573 −0.508620 −0.254310 0.967123i \(-0.581848\pi\)
−0.254310 + 0.967123i \(0.581848\pi\)
\(318\) 0 0
\(319\) 28.9443 1.62057
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.52786 0.307579
\(324\) 0 0
\(325\) 4.18034 0.231884
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −10.4721 −0.577348
\(330\) 0 0
\(331\) 22.4721 1.23518 0.617590 0.786500i \(-0.288109\pi\)
0.617590 + 0.786500i \(0.288109\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −12.9443 −0.707221
\(336\) 0 0
\(337\) 10.3607 0.564382 0.282191 0.959358i \(-0.408939\pi\)
0.282191 + 0.959358i \(0.408939\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −16.0000 −0.866449
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −6.47214 −0.347442 −0.173721 0.984795i \(-0.555579\pi\)
−0.173721 + 0.984795i \(0.555579\pi\)
\(348\) 0 0
\(349\) 26.6525 1.42667 0.713337 0.700821i \(-0.247183\pi\)
0.713337 + 0.700821i \(0.247183\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.8885 0.845662 0.422831 0.906209i \(-0.361036\pi\)
0.422831 + 0.906209i \(0.361036\pi\)
\(354\) 0 0
\(355\) −16.0000 −0.849192
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.9443 0.894284 0.447142 0.894463i \(-0.352442\pi\)
0.447142 + 0.894463i \(0.352442\pi\)
\(360\) 0 0
\(361\) −17.4721 −0.919586
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.52786 0.498711
\(366\) 0 0
\(367\) −22.8328 −1.19186 −0.595932 0.803035i \(-0.703217\pi\)
−0.595932 + 0.803035i \(0.703217\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.0000 0.519174
\(372\) 0 0
\(373\) 2.94427 0.152449 0.0762243 0.997091i \(-0.475713\pi\)
0.0762243 + 0.997091i \(0.475713\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.41641 0.175954
\(378\) 0 0
\(379\) −4.58359 −0.235443 −0.117722 0.993047i \(-0.537559\pi\)
−0.117722 + 0.993047i \(0.537559\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.4164 0.787742 0.393871 0.919166i \(-0.371136\pi\)
0.393871 + 0.919166i \(0.371136\pi\)
\(384\) 0 0
\(385\) −20.9443 −1.06742
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 4.47214 0.226746 0.113373 0.993552i \(-0.463834\pi\)
0.113373 + 0.993552i \(0.463834\pi\)
\(390\) 0 0
\(391\) 17.8885 0.904663
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −41.8885 −2.10764
\(396\) 0 0
\(397\) −15.2361 −0.764676 −0.382338 0.924022i \(-0.624881\pi\)
−0.382338 + 0.924022i \(0.624881\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 23.5279 1.17493 0.587463 0.809251i \(-0.300127\pi\)
0.587463 + 0.809251i \(0.300127\pi\)
\(402\) 0 0
\(403\) −1.88854 −0.0940751
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −28.9443 −1.43471
\(408\) 0 0
\(409\) −21.4164 −1.05897 −0.529487 0.848318i \(-0.677615\pi\)
−0.529487 + 0.848318i \(0.677615\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.23607 0.454477
\(414\) 0 0
\(415\) −29.8885 −1.46717
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −22.1803 −1.08358 −0.541790 0.840514i \(-0.682253\pi\)
−0.541790 + 0.840514i \(0.682253\pi\)
\(420\) 0 0
\(421\) −2.00000 −0.0974740 −0.0487370 0.998812i \(-0.515520\pi\)
−0.0487370 + 0.998812i \(0.515520\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −24.4721 −1.18707
\(426\) 0 0
\(427\) 11.2361 0.543751
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 28.0000 1.34871 0.674356 0.738406i \(-0.264421\pi\)
0.674356 + 0.738406i \(0.264421\pi\)
\(432\) 0 0
\(433\) 9.41641 0.452524 0.226262 0.974067i \(-0.427349\pi\)
0.226262 + 0.974067i \(0.427349\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.94427 0.236517
\(438\) 0 0
\(439\) −32.0000 −1.52728 −0.763638 0.645644i \(-0.776589\pi\)
−0.763638 + 0.645644i \(0.776589\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 13.8885 0.659865 0.329932 0.944005i \(-0.392974\pi\)
0.329932 + 0.944005i \(0.392974\pi\)
\(444\) 0 0
\(445\) −19.4164 −0.920426
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7.88854 0.372283 0.186142 0.982523i \(-0.440402\pi\)
0.186142 + 0.982523i \(0.440402\pi\)
\(450\) 0 0
\(451\) 54.8328 2.58198
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.47214 −0.115896
\(456\) 0 0
\(457\) −7.52786 −0.352139 −0.176069 0.984378i \(-0.556338\pi\)
−0.176069 + 0.984378i \(0.556338\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −21.7082 −1.01105 −0.505526 0.862811i \(-0.668702\pi\)
−0.505526 + 0.862811i \(0.668702\pi\)
\(462\) 0 0
\(463\) −35.7771 −1.66270 −0.831351 0.555748i \(-0.812432\pi\)
−0.831351 + 0.555748i \(0.812432\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 32.0689 1.48397 0.741985 0.670416i \(-0.233884\pi\)
0.741985 + 0.670416i \(0.233884\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 41.8885 1.92604
\(474\) 0 0
\(475\) −6.76393 −0.310350
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 8.58359 0.392194 0.196097 0.980584i \(-0.437173\pi\)
0.196097 + 0.980584i \(0.437173\pi\)
\(480\) 0 0
\(481\) −3.41641 −0.155775
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −40.3607 −1.83268
\(486\) 0 0
\(487\) 20.0000 0.906287 0.453143 0.891438i \(-0.350303\pi\)
0.453143 + 0.891438i \(0.350303\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −37.8885 −1.70989 −0.854943 0.518722i \(-0.826408\pi\)
−0.854943 + 0.518722i \(0.826408\pi\)
\(492\) 0 0
\(493\) −20.0000 −0.900755
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.94427 0.221781
\(498\) 0 0
\(499\) 21.8885 0.979866 0.489933 0.871760i \(-0.337021\pi\)
0.489933 + 0.871760i \(0.337021\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.94427 0.220454 0.110227 0.993906i \(-0.464842\pi\)
0.110227 + 0.993906i \(0.464842\pi\)
\(504\) 0 0
\(505\) −5.52786 −0.245987
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 41.1246 1.82282 0.911408 0.411503i \(-0.134996\pi\)
0.911408 + 0.411503i \(0.134996\pi\)
\(510\) 0 0
\(511\) −2.94427 −0.130247
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 17.8885 0.788263
\(516\) 0 0
\(517\) −67.7771 −2.98083
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 6.58359 0.288432 0.144216 0.989546i \(-0.453934\pi\)
0.144216 + 0.989546i \(0.453934\pi\)
\(522\) 0 0
\(523\) −4.29180 −0.187667 −0.0938336 0.995588i \(-0.529912\pi\)
−0.0938336 + 0.995588i \(0.529912\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.0557 0.481595
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.47214 0.280339
\(534\) 0 0
\(535\) 28.9443 1.25137
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.47214 0.278775
\(540\) 0 0
\(541\) −5.05573 −0.217363 −0.108681 0.994077i \(-0.534663\pi\)
−0.108681 + 0.994077i \(0.534663\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −27.4164 −1.17439
\(546\) 0 0
\(547\) −4.58359 −0.195980 −0.0979901 0.995187i \(-0.531241\pi\)
−0.0979901 + 0.995187i \(0.531241\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.52786 −0.235495
\(552\) 0 0
\(553\) 12.9443 0.550446
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.05573 −0.383704 −0.191852 0.981424i \(-0.561449\pi\)
−0.191852 + 0.981424i \(0.561449\pi\)
\(558\) 0 0
\(559\) 4.94427 0.209120
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 17.8197 0.751009 0.375505 0.926821i \(-0.377469\pi\)
0.375505 + 0.926821i \(0.377469\pi\)
\(564\) 0 0
\(565\) −40.3607 −1.69799
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.3607 −1.10510 −0.552549 0.833481i \(-0.686345\pi\)
−0.552549 + 0.833481i \(0.686345\pi\)
\(570\) 0 0
\(571\) −14.4721 −0.605640 −0.302820 0.953048i \(-0.597928\pi\)
−0.302820 + 0.953048i \(0.597928\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −21.8885 −0.912815
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 9.23607 0.383177
\(582\) 0 0
\(583\) 64.7214 2.68048
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.70820 0.153054 0.0765270 0.997068i \(-0.475617\pi\)
0.0765270 + 0.997068i \(0.475617\pi\)
\(588\) 0 0
\(589\) 3.05573 0.125909
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −32.8328 −1.34828 −0.674141 0.738603i \(-0.735486\pi\)
−0.674141 + 0.738603i \(0.735486\pi\)
\(594\) 0 0
\(595\) 14.4721 0.593300
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −17.8885 −0.730906 −0.365453 0.930830i \(-0.619086\pi\)
−0.365453 + 0.930830i \(0.619086\pi\)
\(600\) 0 0
\(601\) 29.7771 1.21463 0.607316 0.794460i \(-0.292246\pi\)
0.607316 + 0.794460i \(0.292246\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −99.9574 −4.06385
\(606\) 0 0
\(607\) 9.88854 0.401364 0.200682 0.979656i \(-0.435684\pi\)
0.200682 + 0.979656i \(0.435684\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.00000 −0.323645
\(612\) 0 0
\(613\) 29.4164 1.18812 0.594059 0.804422i \(-0.297525\pi\)
0.594059 + 0.804422i \(0.297525\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −34.3607 −1.38331 −0.691654 0.722229i \(-0.743118\pi\)
−0.691654 + 0.722229i \(0.743118\pi\)
\(618\) 0 0
\(619\) −48.0689 −1.93205 −0.966026 0.258446i \(-0.916790\pi\)
−0.966026 + 0.258446i \(0.916790\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 6.00000 0.240385
\(624\) 0 0
\(625\) −22.4164 −0.896656
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20.0000 0.797452
\(630\) 0 0
\(631\) −44.9443 −1.78920 −0.894602 0.446865i \(-0.852541\pi\)
−0.894602 + 0.446865i \(0.852541\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −28.9443 −1.14862
\(636\) 0 0
\(637\) 0.763932 0.0302681
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −14.5836 −0.576017 −0.288009 0.957628i \(-0.592993\pi\)
−0.288009 + 0.957628i \(0.592993\pi\)
\(642\) 0 0
\(643\) −43.7082 −1.72368 −0.861842 0.507177i \(-0.830689\pi\)
−0.861842 + 0.507177i \(0.830689\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 20.3607 0.800461 0.400230 0.916415i \(-0.368930\pi\)
0.400230 + 0.916415i \(0.368930\pi\)
\(648\) 0 0
\(649\) 59.7771 2.34646
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.41641 0.368493 0.184246 0.982880i \(-0.441016\pi\)
0.184246 + 0.982880i \(0.441016\pi\)
\(654\) 0 0
\(655\) 37.8885 1.48043
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −37.3050 −1.45319 −0.726597 0.687064i \(-0.758899\pi\)
−0.726597 + 0.687064i \(0.758899\pi\)
\(660\) 0 0
\(661\) −22.6525 −0.881079 −0.440540 0.897733i \(-0.645213\pi\)
−0.440540 + 0.897733i \(0.645213\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) −17.8885 −0.692647
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 72.7214 2.80738
\(672\) 0 0
\(673\) −2.94427 −0.113493 −0.0567467 0.998389i \(-0.518073\pi\)
−0.0567467 + 0.998389i \(0.518073\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −19.8197 −0.761731 −0.380866 0.924630i \(-0.624374\pi\)
−0.380866 + 0.924630i \(0.624374\pi\)
\(678\) 0 0
\(679\) 12.4721 0.478637
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −23.7771 −0.909805 −0.454902 0.890541i \(-0.650326\pi\)
−0.454902 + 0.890541i \(0.650326\pi\)
\(684\) 0 0
\(685\) 48.3607 1.84777
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.63932 0.291035
\(690\) 0 0
\(691\) −14.1803 −0.539446 −0.269723 0.962938i \(-0.586932\pi\)
−0.269723 + 0.962938i \(0.586932\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −37.8885 −1.43513
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 41.4164 1.56428 0.782138 0.623105i \(-0.214129\pi\)
0.782138 + 0.623105i \(0.214129\pi\)
\(702\) 0 0
\(703\) 5.52786 0.208487
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.70820 0.0642436
\(708\) 0 0
\(709\) 1.63932 0.0615660 0.0307830 0.999526i \(-0.490200\pi\)
0.0307830 + 0.999526i \(0.490200\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 9.88854 0.370329
\(714\) 0 0
\(715\) −16.0000 −0.598366
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −51.1935 −1.90920 −0.954598 0.297898i \(-0.903714\pi\)
−0.954598 + 0.297898i \(0.903714\pi\)
\(720\) 0 0
\(721\) −5.52786 −0.205868
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 24.4721 0.908872
\(726\) 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
\(731\) −28.9443 −1.07054
\(732\) 0 0
\(733\) −4.76393 −0.175960 −0.0879799 0.996122i \(-0.528041\pi\)
−0.0879799 + 0.996122i \(0.528041\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 25.8885 0.953617
\(738\) 0 0
\(739\) −3.41641 −0.125675 −0.0628373 0.998024i \(-0.520015\pi\)
−0.0628373 + 0.998024i \(0.520015\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −24.9443 −0.915117 −0.457558 0.889180i \(-0.651276\pi\)
−0.457558 + 0.889180i \(0.651276\pi\)
\(744\) 0 0
\(745\) 9.52786 0.349074
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −8.94427 −0.326817
\(750\) 0 0
\(751\) 36.0000 1.31366 0.656829 0.754039i \(-0.271897\pi\)
0.656829 + 0.754039i \(0.271897\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 28.9443 1.05339
\(756\) 0 0
\(757\) 39.3050 1.42856 0.714281 0.699859i \(-0.246754\pi\)
0.714281 + 0.699859i \(0.246754\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.52786 0.127885 0.0639425 0.997954i \(-0.479633\pi\)
0.0639425 + 0.997954i \(0.479633\pi\)
\(762\) 0 0
\(763\) 8.47214 0.306712
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.05573 0.254768
\(768\) 0 0
\(769\) −18.3607 −0.662103 −0.331052 0.943613i \(-0.607403\pi\)
−0.331052 + 0.943613i \(0.607403\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −40.1803 −1.44519 −0.722593 0.691274i \(-0.757050\pi\)
−0.722593 + 0.691274i \(0.757050\pi\)
\(774\) 0 0
\(775\) −13.5279 −0.485935
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −10.4721 −0.375203
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.47214 −0.0882343
\(786\) 0 0
\(787\) −12.2918 −0.438155 −0.219078 0.975707i \(-0.570305\pi\)
−0.219078 + 0.975707i \(0.570305\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 12.4721 0.443458
\(792\) 0 0
\(793\) 8.58359 0.304812
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 52.1803 1.84832 0.924161 0.382003i \(-0.124765\pi\)
0.924161 + 0.382003i \(0.124765\pi\)
\(798\) 0 0
\(799\) 46.8328 1.65683
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −19.0557 −0.672462
\(804\) 0 0
\(805\) 12.9443 0.456226
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 17.4164 0.612328 0.306164 0.951979i \(-0.400954\pi\)
0.306164 + 0.951979i \(0.400954\pi\)
\(810\) 0 0
\(811\) −53.0132 −1.86154 −0.930772 0.365601i \(-0.880864\pi\)
−0.930772 + 0.365601i \(0.880864\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 11.0557 0.387265
\(816\) 0 0
\(817\) −8.00000 −0.279885
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −4.11146 −0.143491 −0.0717454 0.997423i \(-0.522857\pi\)
−0.0717454 + 0.997423i \(0.522857\pi\)
\(822\) 0 0
\(823\) −19.7771 −0.689386 −0.344693 0.938715i \(-0.612017\pi\)
−0.344693 + 0.938715i \(0.612017\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −15.0557 −0.523539 −0.261769 0.965130i \(-0.584306\pi\)
−0.261769 + 0.965130i \(0.584306\pi\)
\(828\) 0 0
\(829\) 11.2361 0.390245 0.195122 0.980779i \(-0.437490\pi\)
0.195122 + 0.980779i \(0.437490\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4.47214 −0.154950
\(834\) 0 0
\(835\) 75.7771 2.62237
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 21.5279 0.743224 0.371612 0.928388i \(-0.378805\pi\)
0.371612 + 0.928388i \(0.378805\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 40.1803 1.38225
\(846\) 0 0
\(847\) 30.8885 1.06134
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 17.8885 0.613211
\(852\) 0 0
\(853\) 13.7082 0.469360 0.234680 0.972073i \(-0.424596\pi\)
0.234680 + 0.972073i \(0.424596\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 27.5279 0.940334 0.470167 0.882577i \(-0.344194\pi\)
0.470167 + 0.882577i \(0.344194\pi\)
\(858\) 0 0
\(859\) −33.8197 −1.15391 −0.576956 0.816775i \(-0.695760\pi\)
−0.576956 + 0.816775i \(0.695760\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 20.9443 0.712951 0.356476 0.934305i \(-0.383978\pi\)
0.356476 + 0.934305i \(0.383978\pi\)
\(864\) 0 0
\(865\) 18.4721 0.628071
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 83.7771 2.84194
\(870\) 0 0
\(871\) 3.05573 0.103539
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.52786 −0.0516512
\(876\) 0 0
\(877\) 13.4164 0.453040 0.226520 0.974007i \(-0.427265\pi\)
0.226520 + 0.974007i \(0.427265\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −24.8328 −0.836639 −0.418319 0.908300i \(-0.637381\pi\)
−0.418319 + 0.908300i \(0.637381\pi\)
\(882\) 0 0
\(883\) 23.0557 0.775887 0.387944 0.921683i \(-0.373186\pi\)
0.387944 + 0.921683i \(0.373186\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 33.3050 1.11827 0.559135 0.829076i \(-0.311133\pi\)
0.559135 + 0.829076i \(0.311133\pi\)
\(888\) 0 0
\(889\) 8.94427 0.299981
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12.9443 0.433164
\(894\) 0 0
\(895\) −22.8328 −0.763217
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −11.0557 −0.368729
\(900\) 0 0
\(901\) −44.7214 −1.48988
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 39.4164 1.31025
\(906\) 0 0
\(907\) 0.944272 0.0313540 0.0156770 0.999877i \(-0.495010\pi\)
0.0156770 + 0.999877i \(0.495010\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 34.8328 1.15406 0.577031 0.816722i \(-0.304211\pi\)
0.577031 + 0.816722i \(0.304211\pi\)
\(912\) 0 0
\(913\) 59.7771 1.97833
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −11.7082 −0.386639
\(918\) 0 0
\(919\) −35.7771 −1.18018 −0.590089 0.807338i \(-0.700907\pi\)
−0.590089 + 0.807338i \(0.700907\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 3.77709 0.124324
\(924\) 0 0
\(925\) −24.4721 −0.804639
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 47.3050 1.55203 0.776013 0.630717i \(-0.217239\pi\)
0.776013 + 0.630717i \(0.217239\pi\)
\(930\) 0 0
\(931\) −1.23607 −0.0405105
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 93.6656 3.06319
\(936\) 0 0
\(937\) −9.05573 −0.295838 −0.147919 0.988999i \(-0.547257\pi\)
−0.147919 + 0.988999i \(0.547257\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 35.5967 1.16042 0.580210 0.814467i \(-0.302970\pi\)
0.580210 + 0.814467i \(0.302970\pi\)
\(942\) 0 0
\(943\) −33.8885 −1.10356
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 4.58359 0.148947 0.0744734 0.997223i \(-0.476272\pi\)
0.0744734 + 0.997223i \(0.476272\pi\)
\(948\) 0 0
\(949\) −2.24922 −0.0730129
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −51.8885 −1.68083 −0.840417 0.541940i \(-0.817690\pi\)
−0.840417 + 0.541940i \(0.817690\pi\)
\(954\) 0 0
\(955\) 16.0000 0.517748
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −14.9443 −0.482576
\(960\) 0 0
\(961\) −24.8885 −0.802856
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.52786 −0.0491837
\(966\) 0 0
\(967\) 29.8885 0.961151 0.480575 0.876953i \(-0.340428\pi\)
0.480575 + 0.876953i \(0.340428\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −22.7639 −0.730529 −0.365265 0.930904i \(-0.619022\pi\)
−0.365265 + 0.930904i \(0.619022\pi\)
\(972\) 0 0
\(973\) 1.23607 0.0396265
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 12.8328 0.410558 0.205279 0.978703i \(-0.434190\pi\)
0.205279 + 0.978703i \(0.434190\pi\)
\(978\) 0 0
\(979\) 38.8328 1.24110
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 5.52786 0.176311 0.0881557 0.996107i \(-0.471903\pi\)
0.0881557 + 0.996107i \(0.471903\pi\)
\(984\) 0 0
\(985\) 35.4164 1.12846
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −25.8885 −0.823208
\(990\) 0 0
\(991\) −24.0000 −0.762385 −0.381193 0.924496i \(-0.624487\pi\)
−0.381193 + 0.924496i \(0.624487\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −49.8885 −1.58157
\(996\) 0 0
\(997\) 18.0689 0.572247 0.286124 0.958193i \(-0.407633\pi\)
0.286124 + 0.958193i \(0.407633\pi\)
\(998\) 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 2016.2.a.r.1.1 2
3.2 odd 2 224.2.a.c.1.2 2
4.3 odd 2 2016.2.a.o.1.1 2
8.3 odd 2 4032.2.a.bv.1.2 2
8.5 even 2 4032.2.a.bw.1.2 2
12.11 even 2 224.2.a.d.1.1 yes 2
15.14 odd 2 5600.2.a.bk.1.1 2
21.2 odd 6 1568.2.i.v.1537.1 4
21.5 even 6 1568.2.i.n.1537.2 4
21.11 odd 6 1568.2.i.v.961.1 4
21.17 even 6 1568.2.i.n.961.2 4
21.20 even 2 1568.2.a.v.1.1 2
24.5 odd 2 448.2.a.j.1.1 2
24.11 even 2 448.2.a.i.1.2 2
48.5 odd 4 1792.2.b.k.897.2 4
48.11 even 4 1792.2.b.m.897.3 4
48.29 odd 4 1792.2.b.k.897.3 4
48.35 even 4 1792.2.b.m.897.2 4
60.59 even 2 5600.2.a.z.1.2 2
84.11 even 6 1568.2.i.m.961.2 4
84.23 even 6 1568.2.i.m.1537.2 4
84.47 odd 6 1568.2.i.w.1537.1 4
84.59 odd 6 1568.2.i.w.961.1 4
84.83 odd 2 1568.2.a.k.1.2 2
168.83 odd 2 3136.2.a.by.1.1 2
168.125 even 2 3136.2.a.bf.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.a.c.1.2 2 3.2 odd 2
224.2.a.d.1.1 yes 2 12.11 even 2
448.2.a.i.1.2 2 24.11 even 2
448.2.a.j.1.1 2 24.5 odd 2
1568.2.a.k.1.2 2 84.83 odd 2
1568.2.a.v.1.1 2 21.20 even 2
1568.2.i.m.961.2 4 84.11 even 6
1568.2.i.m.1537.2 4 84.23 even 6
1568.2.i.n.961.2 4 21.17 even 6
1568.2.i.n.1537.2 4 21.5 even 6
1568.2.i.v.961.1 4 21.11 odd 6
1568.2.i.v.1537.1 4 21.2 odd 6
1568.2.i.w.961.1 4 84.59 odd 6
1568.2.i.w.1537.1 4 84.47 odd 6
1792.2.b.k.897.2 4 48.5 odd 4
1792.2.b.k.897.3 4 48.29 odd 4
1792.2.b.m.897.2 4 48.35 even 4
1792.2.b.m.897.3 4 48.11 even 4
2016.2.a.o.1.1 2 4.3 odd 2
2016.2.a.r.1.1 2 1.1 even 1 trivial
3136.2.a.bf.1.2 2 168.125 even 2
3136.2.a.by.1.1 2 168.83 odd 2
4032.2.a.bv.1.2 2 8.3 odd 2
4032.2.a.bw.1.2 2 8.5 even 2
5600.2.a.z.1.2 2 60.59 even 2
5600.2.a.bk.1.1 2 15.14 odd 2