Properties

Label 3136.2.a.bs.1.1
Level $3136$
Weight $2$
Character 3136.1
Self dual yes
Analytic conductor $25.041$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3136,2,Mod(1,3136)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3136, 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("3136.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3136 = 2^{6} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3136.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: \(25.0410860739\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 196)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3136.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.82843 q^{3} -1.41421 q^{5} +5.00000 q^{9} +O(q^{10})\) \(q-2.82843 q^{3} -1.41421 q^{5} +5.00000 q^{9} +4.00000 q^{11} -4.24264 q^{13} +4.00000 q^{15} +1.41421 q^{17} +2.82843 q^{19} +4.00000 q^{23} -3.00000 q^{25} -5.65685 q^{27} -8.00000 q^{29} -11.3137 q^{33} +8.00000 q^{37} +12.0000 q^{39} -7.07107 q^{41} -4.00000 q^{43} -7.07107 q^{45} -5.65685 q^{47} -4.00000 q^{51} -10.0000 q^{53} -5.65685 q^{55} -8.00000 q^{57} +14.1421 q^{59} +7.07107 q^{61} +6.00000 q^{65} -11.3137 q^{69} -7.07107 q^{73} +8.48528 q^{75} -8.00000 q^{79} +1.00000 q^{81} -14.1421 q^{83} -2.00000 q^{85} +22.6274 q^{87} +7.07107 q^{89} -4.00000 q^{95} -1.41421 q^{97} +20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{9} + 8 q^{11} + 8 q^{15} + 8 q^{23} - 6 q^{25} - 16 q^{29} + 16 q^{37} + 24 q^{39} - 8 q^{43} - 8 q^{51} - 20 q^{53} - 16 q^{57} + 12 q^{65} - 16 q^{79} + 2 q^{81} - 4 q^{85} - 8 q^{95} + 40 q^{99}+O(q^{100}) \) 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
\(3\) −2.82843 −1.63299 −0.816497 0.577350i \(-0.804087\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) −1.41421 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 4.00000 1.03280
\(16\) 0 0
\(17\) 1.41421 0.342997 0.171499 0.985184i \(-0.445139\pi\)
0.171499 + 0.985184i \(0.445139\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.00000 0.834058 0.417029 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) −8.00000 −1.48556 −0.742781 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −11.3137 −1.96946
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 0 0
\(39\) 12.0000 1.92154
\(40\) 0 0
\(41\) −7.07107 −1.10432 −0.552158 0.833740i \(-0.686195\pi\)
−0.552158 + 0.833740i \(0.686195\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −7.07107 −1.05409
\(46\) 0 0
\(47\) −5.65685 −0.825137 −0.412568 0.910927i \(-0.635368\pi\)
−0.412568 + 0.910927i \(0.635368\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) −5.65685 −0.762770
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) 14.1421 1.84115 0.920575 0.390567i \(-0.127721\pi\)
0.920575 + 0.390567i \(0.127721\pi\)
\(60\) 0 0
\(61\) 7.07107 0.905357 0.452679 0.891674i \(-0.350468\pi\)
0.452679 + 0.891674i \(0.350468\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.00000 0.744208
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) −11.3137 −1.36201
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −7.07107 −0.827606 −0.413803 0.910366i \(-0.635800\pi\)
−0.413803 + 0.910366i \(0.635800\pi\)
\(74\) 0 0
\(75\) 8.48528 0.979796
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.1421 −1.55230 −0.776151 0.630548i \(-0.782830\pi\)
−0.776151 + 0.630548i \(0.782830\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 22.6274 2.42591
\(88\) 0 0
\(89\) 7.07107 0.749532 0.374766 0.927119i \(-0.377723\pi\)
0.374766 + 0.927119i \(0.377723\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.00000 −0.410391
\(96\) 0 0
\(97\) −1.41421 −0.143592 −0.0717958 0.997419i \(-0.522873\pi\)
−0.0717958 + 0.997419i \(0.522873\pi\)
\(98\) 0 0
\(99\) 20.0000 2.01008
\(100\) 0 0
\(101\) 12.7279 1.26648 0.633238 0.773957i \(-0.281726\pi\)
0.633238 + 0.773957i \(0.281726\pi\)
\(102\) 0 0
\(103\) 11.3137 1.11477 0.557386 0.830253i \(-0.311804\pi\)
0.557386 + 0.830253i \(0.311804\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) −22.6274 −2.14770
\(112\) 0 0
\(113\) 6.00000 0.564433 0.282216 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −5.65685 −0.527504
\(116\) 0 0
\(117\) −21.2132 −1.96116
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) 20.0000 1.80334
\(124\) 0 0
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 11.3137 0.996116
\(130\) 0 0
\(131\) −8.48528 −0.741362 −0.370681 0.928760i \(-0.620876\pi\)
−0.370681 + 0.928760i \(0.620876\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 8.00000 0.688530
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 2.82843 0.239904 0.119952 0.992780i \(-0.461726\pi\)
0.119952 + 0.992780i \(0.461726\pi\)
\(140\) 0 0
\(141\) 16.0000 1.34744
\(142\) 0 0
\(143\) −16.9706 −1.41915
\(144\) 0 0
\(145\) 11.3137 0.939552
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 7.07107 0.571662
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −7.07107 −0.564333 −0.282166 0.959366i \(-0.591053\pi\)
−0.282166 + 0.959366i \(0.591053\pi\)
\(158\) 0 0
\(159\) 28.2843 2.24309
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 16.0000 1.24560
\(166\) 0 0
\(167\) 5.65685 0.437741 0.218870 0.975754i \(-0.429763\pi\)
0.218870 + 0.975754i \(0.429763\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 14.1421 1.08148
\(172\) 0 0
\(173\) 4.24264 0.322562 0.161281 0.986909i \(-0.448437\pi\)
0.161281 + 0.986909i \(0.448437\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −40.0000 −3.00658
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 21.2132 1.57676 0.788382 0.615185i \(-0.210919\pi\)
0.788382 + 0.615185i \(0.210919\pi\)
\(182\) 0 0
\(183\) −20.0000 −1.47844
\(184\) 0 0
\(185\) −11.3137 −0.831800
\(186\) 0 0
\(187\) 5.65685 0.413670
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 0 0
\(195\) −16.9706 −1.21529
\(196\) 0 0
\(197\) 10.0000 0.712470 0.356235 0.934396i \(-0.384060\pi\)
0.356235 + 0.934396i \(0.384060\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 10.0000 0.698430
\(206\) 0 0
\(207\) 20.0000 1.39010
\(208\) 0 0
\(209\) 11.3137 0.782586
\(210\) 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
\(215\) 5.65685 0.385794
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 20.0000 1.35147
\(220\) 0 0
\(221\) −6.00000 −0.403604
\(222\) 0 0
\(223\) −16.9706 −1.13643 −0.568216 0.822879i \(-0.692366\pi\)
−0.568216 + 0.822879i \(0.692366\pi\)
\(224\) 0 0
\(225\) −15.0000 −1.00000
\(226\) 0 0
\(227\) 8.48528 0.563188 0.281594 0.959534i \(-0.409137\pi\)
0.281594 + 0.959534i \(0.409137\pi\)
\(228\) 0 0
\(229\) 21.2132 1.40181 0.700904 0.713256i \(-0.252780\pi\)
0.700904 + 0.713256i \(0.252780\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 8.00000 0.521862
\(236\) 0 0
\(237\) 22.6274 1.46981
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 12.7279 0.819878 0.409939 0.912113i \(-0.365550\pi\)
0.409939 + 0.912113i \(0.365550\pi\)
\(242\) 0 0
\(243\) 14.1421 0.907218
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −12.0000 −0.763542
\(248\) 0 0
\(249\) 40.0000 2.53490
\(250\) 0 0
\(251\) 19.7990 1.24970 0.624851 0.780744i \(-0.285160\pi\)
0.624851 + 0.780744i \(0.285160\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 5.65685 0.354246
\(256\) 0 0
\(257\) 21.2132 1.32324 0.661622 0.749838i \(-0.269869\pi\)
0.661622 + 0.749838i \(0.269869\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −40.0000 −2.47594
\(262\) 0 0
\(263\) 24.0000 1.47990 0.739952 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 14.1421 0.868744
\(266\) 0 0
\(267\) −20.0000 −1.22398
\(268\) 0 0
\(269\) 18.3848 1.12094 0.560470 0.828175i \(-0.310621\pi\)
0.560470 + 0.828175i \(0.310621\pi\)
\(270\) 0 0
\(271\) −28.2843 −1.71815 −0.859074 0.511852i \(-0.828960\pi\)
−0.859074 + 0.511852i \(0.828960\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −12.0000 −0.723627
\(276\) 0 0
\(277\) −22.0000 −1.32185 −0.660926 0.750451i \(-0.729836\pi\)
−0.660926 + 0.750451i \(0.729836\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.0000 0.954480 0.477240 0.878773i \(-0.341637\pi\)
0.477240 + 0.878773i \(0.341637\pi\)
\(282\) 0 0
\(283\) −2.82843 −0.168133 −0.0840663 0.996460i \(-0.526791\pi\)
−0.0840663 + 0.996460i \(0.526791\pi\)
\(284\) 0 0
\(285\) 11.3137 0.670166
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −15.0000 −0.882353
\(290\) 0 0
\(291\) 4.00000 0.234484
\(292\) 0 0
\(293\) −32.5269 −1.90024 −0.950121 0.311881i \(-0.899041\pi\)
−0.950121 + 0.311881i \(0.899041\pi\)
\(294\) 0 0
\(295\) −20.0000 −1.16445
\(296\) 0 0
\(297\) −22.6274 −1.31298
\(298\) 0 0
\(299\) −16.9706 −0.981433
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −36.0000 −2.06815
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −19.7990 −1.12999 −0.564994 0.825095i \(-0.691122\pi\)
−0.564994 + 0.825095i \(0.691122\pi\)
\(308\) 0 0
\(309\) −32.0000 −1.82042
\(310\) 0 0
\(311\) 22.6274 1.28308 0.641542 0.767088i \(-0.278295\pi\)
0.641542 + 0.767088i \(0.278295\pi\)
\(312\) 0 0
\(313\) 4.24264 0.239808 0.119904 0.992785i \(-0.461741\pi\)
0.119904 + 0.992785i \(0.461741\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −32.0000 −1.79166
\(320\) 0 0
\(321\) −22.6274 −1.26294
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) 12.7279 0.706018
\(326\) 0 0
\(327\) −22.6274 −1.25130
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 0 0
\(333\) 40.0000 2.19199
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) −16.9706 −0.921714
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 16.0000 0.861411
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 0 0
\(349\) 4.24264 0.227103 0.113552 0.993532i \(-0.463777\pi\)
0.113552 + 0.993532i \(0.463777\pi\)
\(350\) 0 0
\(351\) 24.0000 1.28103
\(352\) 0 0
\(353\) 9.89949 0.526897 0.263448 0.964673i \(-0.415140\pi\)
0.263448 + 0.964673i \(0.415140\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) −14.1421 −0.742270
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −5.65685 −0.295285 −0.147643 0.989041i \(-0.547169\pi\)
−0.147643 + 0.989041i \(0.547169\pi\)
\(368\) 0 0
\(369\) −35.3553 −1.84053
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 0 0
\(375\) −32.0000 −1.65247
\(376\) 0 0
\(377\) 33.9411 1.74806
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) −56.5685 −2.89809
\(382\) 0 0
\(383\) 5.65685 0.289052 0.144526 0.989501i \(-0.453834\pi\)
0.144526 + 0.989501i \(0.453834\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −20.0000 −1.01666
\(388\) 0 0
\(389\) −8.00000 −0.405616 −0.202808 0.979219i \(-0.565007\pi\)
−0.202808 + 0.979219i \(0.565007\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) 24.0000 1.21064
\(394\) 0 0
\(395\) 11.3137 0.569254
\(396\) 0 0
\(397\) −15.5563 −0.780751 −0.390375 0.920656i \(-0.627655\pi\)
−0.390375 + 0.920656i \(0.627655\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.41421 −0.0702728
\(406\) 0 0
\(407\) 32.0000 1.58618
\(408\) 0 0
\(409\) 38.1838 1.88807 0.944033 0.329851i \(-0.106999\pi\)
0.944033 + 0.329851i \(0.106999\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 20.0000 0.981761
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) 14.1421 0.690889 0.345444 0.938439i \(-0.387728\pi\)
0.345444 + 0.938439i \(0.387728\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) −28.2843 −1.37523
\(424\) 0 0
\(425\) −4.24264 −0.205798
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 48.0000 2.31746
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 0 0
\(433\) −21.2132 −1.01944 −0.509721 0.860340i \(-0.670251\pi\)
−0.509721 + 0.860340i \(0.670251\pi\)
\(434\) 0 0
\(435\) −32.0000 −1.53428
\(436\) 0 0
\(437\) 11.3137 0.541208
\(438\) 0 0
\(439\) 16.9706 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −16.0000 −0.760183 −0.380091 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 0 0
\(445\) −10.0000 −0.474045
\(446\) 0 0
\(447\) 28.2843 1.33780
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −28.2843 −1.33185
\(452\) 0 0
\(453\) −11.3137 −0.531564
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 30.0000 1.40334 0.701670 0.712502i \(-0.252438\pi\)
0.701670 + 0.712502i \(0.252438\pi\)
\(458\) 0 0
\(459\) −8.00000 −0.373408
\(460\) 0 0
\(461\) −7.07107 −0.329332 −0.164666 0.986349i \(-0.552655\pi\)
−0.164666 + 0.986349i \(0.552655\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −19.7990 −0.916188 −0.458094 0.888904i \(-0.651468\pi\)
−0.458094 + 0.888904i \(0.651468\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) −8.48528 −0.389331
\(476\) 0 0
\(477\) −50.0000 −2.28934
\(478\) 0 0
\(479\) 11.3137 0.516937 0.258468 0.966020i \(-0.416782\pi\)
0.258468 + 0.966020i \(0.416782\pi\)
\(480\) 0 0
\(481\) −33.9411 −1.54758
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) 12.0000 0.543772 0.271886 0.962329i \(-0.412353\pi\)
0.271886 + 0.962329i \(0.412353\pi\)
\(488\) 0 0
\(489\) −11.3137 −0.511624
\(490\) 0 0
\(491\) 24.0000 1.08310 0.541552 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) −11.3137 −0.509544
\(494\) 0 0
\(495\) −28.2843 −1.27128
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 32.0000 1.43252 0.716258 0.697835i \(-0.245853\pi\)
0.716258 + 0.697835i \(0.245853\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 0 0
\(503\) 39.5980 1.76559 0.882793 0.469762i \(-0.155660\pi\)
0.882793 + 0.469762i \(0.155660\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) −14.1421 −0.628074
\(508\) 0 0
\(509\) 18.3848 0.814891 0.407445 0.913230i \(-0.366420\pi\)
0.407445 + 0.913230i \(0.366420\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −16.0000 −0.706417
\(514\) 0 0
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) −22.6274 −0.995153
\(518\) 0 0
\(519\) −12.0000 −0.526742
\(520\) 0 0
\(521\) −41.0122 −1.79678 −0.898388 0.439202i \(-0.855261\pi\)
−0.898388 + 0.439202i \(0.855261\pi\)
\(522\) 0 0
\(523\) 42.4264 1.85518 0.927589 0.373603i \(-0.121878\pi\)
0.927589 + 0.373603i \(0.121878\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 70.7107 3.06858
\(532\) 0 0
\(533\) 30.0000 1.29944
\(534\) 0 0
\(535\) −11.3137 −0.489134
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 14.0000 0.601907 0.300954 0.953639i \(-0.402695\pi\)
0.300954 + 0.953639i \(0.402695\pi\)
\(542\) 0 0
\(543\) −60.0000 −2.57485
\(544\) 0 0
\(545\) −11.3137 −0.484626
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 35.3553 1.50893
\(550\) 0 0
\(551\) −22.6274 −0.963960
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 32.0000 1.35832
\(556\) 0 0
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 16.9706 0.717778
\(560\) 0 0
\(561\) −16.0000 −0.675521
\(562\) 0 0
\(563\) 14.1421 0.596020 0.298010 0.954563i \(-0.403677\pi\)
0.298010 + 0.954563i \(0.403677\pi\)
\(564\) 0 0
\(565\) −8.48528 −0.356978
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 40.0000 1.67689 0.838444 0.544988i \(-0.183466\pi\)
0.838444 + 0.544988i \(0.183466\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) −45.2548 −1.89055
\(574\) 0 0
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 12.7279 0.529870 0.264935 0.964266i \(-0.414649\pi\)
0.264935 + 0.964266i \(0.414649\pi\)
\(578\) 0 0
\(579\) 28.2843 1.17545
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −40.0000 −1.65663
\(584\) 0 0
\(585\) 30.0000 1.24035
\(586\) 0 0
\(587\) 25.4558 1.05068 0.525338 0.850894i \(-0.323939\pi\)
0.525338 + 0.850894i \(0.323939\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −28.2843 −1.16346
\(592\) 0 0
\(593\) −9.89949 −0.406524 −0.203262 0.979124i \(-0.565154\pi\)
−0.203262 + 0.979124i \(0.565154\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.00000 −0.326871 −0.163436 0.986554i \(-0.552258\pi\)
−0.163436 + 0.986554i \(0.552258\pi\)
\(600\) 0 0
\(601\) −29.6985 −1.21143 −0.605713 0.795683i \(-0.707112\pi\)
−0.605713 + 0.795683i \(0.707112\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.07107 −0.287480
\(606\) 0 0
\(607\) 33.9411 1.37763 0.688814 0.724938i \(-0.258132\pi\)
0.688814 + 0.724938i \(0.258132\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 0 0
\(613\) −24.0000 −0.969351 −0.484675 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) 0 0
\(615\) −28.2843 −1.14053
\(616\) 0 0
\(617\) −8.00000 −0.322068 −0.161034 0.986949i \(-0.551483\pi\)
−0.161034 + 0.986949i \(0.551483\pi\)
\(618\) 0 0
\(619\) −31.1127 −1.25052 −0.625262 0.780415i \(-0.715008\pi\)
−0.625262 + 0.780415i \(0.715008\pi\)
\(620\) 0 0
\(621\) −22.6274 −0.908007
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) −32.0000 −1.27796
\(628\) 0 0
\(629\) 11.3137 0.451107
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) −67.8823 −2.69808
\(634\) 0 0
\(635\) −28.2843 −1.12243
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 8.00000 0.315981 0.157991 0.987441i \(-0.449498\pi\)
0.157991 + 0.987441i \(0.449498\pi\)
\(642\) 0 0
\(643\) 2.82843 0.111542 0.0557711 0.998444i \(-0.482238\pi\)
0.0557711 + 0.998444i \(0.482238\pi\)
\(644\) 0 0
\(645\) −16.0000 −0.629999
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 56.5685 2.22051
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.0000 0.939193 0.469596 0.882881i \(-0.344399\pi\)
0.469596 + 0.882881i \(0.344399\pi\)
\(654\) 0 0
\(655\) 12.0000 0.468879
\(656\) 0 0
\(657\) −35.3553 −1.37934
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) 21.2132 0.825098 0.412549 0.910935i \(-0.364639\pi\)
0.412549 + 0.910935i \(0.364639\pi\)
\(662\) 0 0
\(663\) 16.9706 0.659082
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −32.0000 −1.23904
\(668\) 0 0
\(669\) 48.0000 1.85579
\(670\) 0 0
\(671\) 28.2843 1.09190
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 16.9706 0.653197
\(676\) 0 0
\(677\) −12.7279 −0.489174 −0.244587 0.969627i \(-0.578652\pi\)
−0.244587 + 0.969627i \(0.578652\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −24.0000 −0.919682
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −60.0000 −2.28914
\(688\) 0 0
\(689\) 42.4264 1.61632
\(690\) 0 0
\(691\) −42.4264 −1.61398 −0.806988 0.590567i \(-0.798904\pi\)
−0.806988 + 0.590567i \(0.798904\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) −10.0000 −0.378777
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −24.0000 −0.906467 −0.453234 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 0 0
\(703\) 22.6274 0.853409
\(704\) 0 0
\(705\) −22.6274 −0.852198
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 8.00000 0.300446 0.150223 0.988652i \(-0.452001\pi\)
0.150223 + 0.988652i \(0.452001\pi\)
\(710\) 0 0
\(711\) −40.0000 −1.50012
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 24.0000 0.897549
\(716\) 0 0
\(717\) −33.9411 −1.26755
\(718\) 0 0
\(719\) −39.5980 −1.47676 −0.738378 0.674387i \(-0.764408\pi\)
−0.738378 + 0.674387i \(0.764408\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −36.0000 −1.33885
\(724\) 0 0
\(725\) 24.0000 0.891338
\(726\) 0 0
\(727\) −28.2843 −1.04901 −0.524503 0.851409i \(-0.675749\pi\)
−0.524503 + 0.851409i \(0.675749\pi\)
\(728\) 0 0
\(729\) −43.0000 −1.59259
\(730\) 0 0
\(731\) −5.65685 −0.209226
\(732\) 0 0
\(733\) 38.1838 1.41035 0.705175 0.709034i \(-0.250869\pi\)
0.705175 + 0.709034i \(0.250869\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 33.9411 1.24686
\(742\) 0 0
\(743\) 20.0000 0.733729 0.366864 0.930274i \(-0.380431\pi\)
0.366864 + 0.930274i \(0.380431\pi\)
\(744\) 0 0
\(745\) 14.1421 0.518128
\(746\) 0 0
\(747\) −70.7107 −2.58717
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4.00000 0.145962 0.0729810 0.997333i \(-0.476749\pi\)
0.0729810 + 0.997333i \(0.476749\pi\)
\(752\) 0 0
\(753\) −56.0000 −2.04075
\(754\) 0 0
\(755\) −5.65685 −0.205874
\(756\) 0 0
\(757\) 40.0000 1.45382 0.726912 0.686730i \(-0.240955\pi\)
0.726912 + 0.686730i \(0.240955\pi\)
\(758\) 0 0
\(759\) −45.2548 −1.64265
\(760\) 0 0
\(761\) 41.0122 1.48669 0.743345 0.668908i \(-0.233238\pi\)
0.743345 + 0.668908i \(0.233238\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −10.0000 −0.361551
\(766\) 0 0
\(767\) −60.0000 −2.16647
\(768\) 0 0
\(769\) 46.6690 1.68293 0.841464 0.540312i \(-0.181694\pi\)
0.841464 + 0.540312i \(0.181694\pi\)
\(770\) 0 0
\(771\) −60.0000 −2.16085
\(772\) 0 0
\(773\) −24.0416 −0.864717 −0.432359 0.901702i \(-0.642319\pi\)
−0.432359 + 0.901702i \(0.642319\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −20.0000 −0.716574
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 45.2548 1.61728
\(784\) 0 0
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) 48.0833 1.71398 0.856992 0.515330i \(-0.172331\pi\)
0.856992 + 0.515330i \(0.172331\pi\)
\(788\) 0 0
\(789\) −67.8823 −2.41667
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −30.0000 −1.06533
\(794\) 0 0
\(795\) −40.0000 −1.41865
\(796\) 0 0
\(797\) 29.6985 1.05197 0.525987 0.850493i \(-0.323696\pi\)
0.525987 + 0.850493i \(0.323696\pi\)
\(798\) 0 0
\(799\) −8.00000 −0.283020
\(800\) 0 0
\(801\) 35.3553 1.24922
\(802\) 0 0
\(803\) −28.2843 −0.998130
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −52.0000 −1.83049
\(808\) 0 0
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 8.48528 0.297959 0.148979 0.988840i \(-0.452401\pi\)
0.148979 + 0.988840i \(0.452401\pi\)
\(812\) 0 0
\(813\) 80.0000 2.80572
\(814\) 0 0
\(815\) −5.65685 −0.198151
\(816\) 0 0
\(817\) −11.3137 −0.395817
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −6.00000 −0.209401 −0.104701 0.994504i \(-0.533388\pi\)
−0.104701 + 0.994504i \(0.533388\pi\)
\(822\) 0 0
\(823\) −40.0000 −1.39431 −0.697156 0.716919i \(-0.745552\pi\)
−0.697156 + 0.716919i \(0.745552\pi\)
\(824\) 0 0
\(825\) 33.9411 1.18168
\(826\) 0 0
\(827\) 48.0000 1.66912 0.834562 0.550914i \(-0.185721\pi\)
0.834562 + 0.550914i \(0.185721\pi\)
\(828\) 0 0
\(829\) 7.07107 0.245588 0.122794 0.992432i \(-0.460815\pi\)
0.122794 + 0.992432i \(0.460815\pi\)
\(830\) 0 0
\(831\) 62.2254 2.15858
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 11.3137 0.390593 0.195296 0.980744i \(-0.437433\pi\)
0.195296 + 0.980744i \(0.437433\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 0 0
\(843\) −45.2548 −1.55866
\(844\) 0 0
\(845\) −7.07107 −0.243252
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 8.00000 0.274559
\(850\) 0 0
\(851\) 32.0000 1.09695
\(852\) 0 0
\(853\) −12.7279 −0.435796 −0.217898 0.975972i \(-0.569920\pi\)
−0.217898 + 0.975972i \(0.569920\pi\)
\(854\) 0 0
\(855\) −20.0000 −0.683986
\(856\) 0 0
\(857\) 7.07107 0.241543 0.120772 0.992680i \(-0.461463\pi\)
0.120772 + 0.992680i \(0.461463\pi\)
\(858\) 0 0
\(859\) 14.1421 0.482523 0.241262 0.970460i \(-0.422439\pi\)
0.241262 + 0.970460i \(0.422439\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 40.0000 1.36162 0.680808 0.732462i \(-0.261629\pi\)
0.680808 + 0.732462i \(0.261629\pi\)
\(864\) 0 0
\(865\) −6.00000 −0.204006
\(866\) 0 0
\(867\) 42.4264 1.44088
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −7.07107 −0.239319
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) 0 0
\(879\) 92.0000 3.10308
\(880\) 0 0
\(881\) −21.2132 −0.714691 −0.357345 0.933972i \(-0.616318\pi\)
−0.357345 + 0.933972i \(0.616318\pi\)
\(882\) 0 0
\(883\) −40.0000 −1.34611 −0.673054 0.739594i \(-0.735018\pi\)
−0.673054 + 0.739594i \(0.735018\pi\)
\(884\) 0 0
\(885\) 56.5685 1.90153
\(886\) 0 0
\(887\) −28.2843 −0.949693 −0.474846 0.880069i \(-0.657496\pi\)
−0.474846 + 0.880069i \(0.657496\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 4.00000 0.134005
\(892\) 0 0
\(893\) −16.0000 −0.535420
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 48.0000 1.60267
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −14.1421 −0.471143
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −30.0000 −0.997234
\(906\) 0 0
\(907\) −8.00000 −0.265636 −0.132818 0.991140i \(-0.542403\pi\)
−0.132818 + 0.991140i \(0.542403\pi\)
\(908\) 0 0
\(909\) 63.6396 2.11079
\(910\) 0 0
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) −56.5685 −1.87215
\(914\) 0 0
\(915\) 28.2843 0.935049
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8.00000 0.263896 0.131948 0.991257i \(-0.457877\pi\)
0.131948 + 0.991257i \(0.457877\pi\)
\(920\) 0 0
\(921\) 56.0000 1.84526
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −24.0000 −0.789115
\(926\) 0 0
\(927\) 56.5685 1.85795
\(928\) 0 0
\(929\) 35.3553 1.15997 0.579986 0.814627i \(-0.303058\pi\)
0.579986 + 0.814627i \(0.303058\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −64.0000 −2.09527
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) 15.5563 0.508204 0.254102 0.967177i \(-0.418220\pi\)
0.254102 + 0.967177i \(0.418220\pi\)
\(938\) 0 0
\(939\) −12.0000 −0.391605
\(940\) 0 0
\(941\) 7.07107 0.230510 0.115255 0.993336i \(-0.463231\pi\)
0.115255 + 0.993336i \(0.463231\pi\)
\(942\) 0 0
\(943\) −28.2843 −0.921063
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 60.0000 1.94974 0.974869 0.222779i \(-0.0715128\pi\)
0.974869 + 0.222779i \(0.0715128\pi\)
\(948\) 0 0
\(949\) 30.0000 0.973841
\(950\) 0 0
\(951\) −5.65685 −0.183436
\(952\) 0 0
\(953\) 10.0000 0.323932 0.161966 0.986796i \(-0.448217\pi\)
0.161966 + 0.986796i \(0.448217\pi\)
\(954\) 0 0
\(955\) −22.6274 −0.732206
\(956\) 0 0
\(957\) 90.5097 2.92576
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 40.0000 1.28898
\(964\) 0 0
\(965\) 14.1421 0.455251
\(966\) 0 0
\(967\) −12.0000 −0.385894 −0.192947 0.981209i \(-0.561805\pi\)
−0.192947 + 0.981209i \(0.561805\pi\)
\(968\) 0 0
\(969\) −11.3137 −0.363449
\(970\) 0 0
\(971\) 14.1421 0.453843 0.226921 0.973913i \(-0.427134\pi\)
0.226921 + 0.973913i \(0.427134\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −36.0000 −1.15292
\(976\) 0 0
\(977\) 48.0000 1.53566 0.767828 0.640656i \(-0.221338\pi\)
0.767828 + 0.640656i \(0.221338\pi\)
\(978\) 0 0
\(979\) 28.2843 0.903969
\(980\) 0 0
\(981\) 40.0000 1.27710
\(982\) 0 0
\(983\) −45.2548 −1.44341 −0.721703 0.692203i \(-0.756640\pi\)
−0.721703 + 0.692203i \(0.756640\pi\)
\(984\) 0 0
\(985\) −14.1421 −0.450606
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) 0 0
\(991\) 40.0000 1.27064 0.635321 0.772248i \(-0.280868\pi\)
0.635321 + 0.772248i \(0.280868\pi\)
\(992\) 0 0
\(993\) 56.5685 1.79515
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.41421 −0.0447886 −0.0223943 0.999749i \(-0.507129\pi\)
−0.0223943 + 0.999749i \(0.507129\pi\)
\(998\) 0 0
\(999\) −45.2548 −1.43180
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 3136.2.a.bs.1.1 2
4.3 odd 2 3136.2.a.br.1.2 2
7.6 odd 2 inner 3136.2.a.bs.1.2 2
8.3 odd 2 196.2.a.c.1.1 2
8.5 even 2 784.2.a.m.1.2 2
24.5 odd 2 7056.2.a.cr.1.1 2
24.11 even 2 1764.2.a.l.1.1 2
28.27 even 2 3136.2.a.br.1.1 2
40.3 even 4 4900.2.e.p.2549.1 4
40.19 odd 2 4900.2.a.y.1.2 2
40.27 even 4 4900.2.e.p.2549.3 4
56.3 even 6 196.2.e.b.177.1 4
56.5 odd 6 784.2.i.l.753.2 4
56.11 odd 6 196.2.e.b.177.2 4
56.13 odd 2 784.2.a.m.1.1 2
56.19 even 6 196.2.e.b.165.1 4
56.27 even 2 196.2.a.c.1.2 yes 2
56.37 even 6 784.2.i.l.753.1 4
56.45 odd 6 784.2.i.l.177.2 4
56.51 odd 6 196.2.e.b.165.2 4
56.53 even 6 784.2.i.l.177.1 4
168.11 even 6 1764.2.k.l.1549.2 4
168.59 odd 6 1764.2.k.l.1549.1 4
168.83 odd 2 1764.2.a.l.1.2 2
168.107 even 6 1764.2.k.l.361.2 4
168.125 even 2 7056.2.a.cr.1.2 2
168.131 odd 6 1764.2.k.l.361.1 4
280.27 odd 4 4900.2.e.p.2549.2 4
280.83 odd 4 4900.2.e.p.2549.4 4
280.139 even 2 4900.2.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
196.2.a.c.1.1 2 8.3 odd 2
196.2.a.c.1.2 yes 2 56.27 even 2
196.2.e.b.165.1 4 56.19 even 6
196.2.e.b.165.2 4 56.51 odd 6
196.2.e.b.177.1 4 56.3 even 6
196.2.e.b.177.2 4 56.11 odd 6
784.2.a.m.1.1 2 56.13 odd 2
784.2.a.m.1.2 2 8.5 even 2
784.2.i.l.177.1 4 56.53 even 6
784.2.i.l.177.2 4 56.45 odd 6
784.2.i.l.753.1 4 56.37 even 6
784.2.i.l.753.2 4 56.5 odd 6
1764.2.a.l.1.1 2 24.11 even 2
1764.2.a.l.1.2 2 168.83 odd 2
1764.2.k.l.361.1 4 168.131 odd 6
1764.2.k.l.361.2 4 168.107 even 6
1764.2.k.l.1549.1 4 168.59 odd 6
1764.2.k.l.1549.2 4 168.11 even 6
3136.2.a.br.1.1 2 28.27 even 2
3136.2.a.br.1.2 2 4.3 odd 2
3136.2.a.bs.1.1 2 1.1 even 1 trivial
3136.2.a.bs.1.2 2 7.6 odd 2 inner
4900.2.a.y.1.1 2 280.139 even 2
4900.2.a.y.1.2 2 40.19 odd 2
4900.2.e.p.2549.1 4 40.3 even 4
4900.2.e.p.2549.2 4 280.27 odd 4
4900.2.e.p.2549.3 4 40.27 even 4
4900.2.e.p.2549.4 4 280.83 odd 4
7056.2.a.cr.1.1 2 24.5 odd 2
7056.2.a.cr.1.2 2 168.125 even 2