Properties

Label 9800.2.a.bw.1.1
Level $9800$
Weight $2$
Character 9800.1
Self dual yes
Analytic conductor $78.253$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9800,2,Mod(1,9800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9800, 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("9800.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9800 = 2^{3} \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9800.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: \(78.2533939809\)
Analytic rank: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 392)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9800.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.82843 q^{3} +5.00000 q^{9} +O(q^{10})\) \(q-2.82843 q^{3} +5.00000 q^{9} -4.00000 q^{11} +2.82843 q^{13} +5.65685 q^{17} -2.82843 q^{19} -5.65685 q^{27} +2.00000 q^{29} -5.65685 q^{31} +11.3137 q^{33} -10.0000 q^{37} -8.00000 q^{39} +5.65685 q^{41} +4.00000 q^{43} -5.65685 q^{47} -16.0000 q^{51} -6.00000 q^{53} +8.00000 q^{57} -2.82843 q^{59} +14.1421 q^{61} -12.0000 q^{67} +8.00000 q^{79} +1.00000 q^{81} +14.1421 q^{83} -5.65685 q^{87} +16.0000 q^{93} -5.65685 q^{97} -20.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 10 q^{9} - 8 q^{11} + 4 q^{29} - 20 q^{37} - 16 q^{39} + 8 q^{43} - 32 q^{51} - 12 q^{53} + 16 q^{57} - 24 q^{67} + 16 q^{79} + 2 q^{81} + 32 q^{93} - 40 q^{99}+O(q^{100}) \) 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
\(5\) 0 0
\(6\) 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 \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 2.82843 0.784465 0.392232 0.919866i \(-0.371703\pi\)
0.392232 + 0.919866i \(0.371703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.65685 1.37199 0.685994 0.727607i \(-0.259367\pi\)
0.685994 + 0.727607i \(0.259367\pi\)
\(18\) 0 0
\(19\) −2.82843 −0.648886 −0.324443 0.945905i \(-0.605177\pi\)
−0.324443 + 0.945905i \(0.605177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 2.00000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) 0 0
\(33\) 11.3137 1.96946
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −10.0000 −1.64399 −0.821995 0.569495i \(-0.807139\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) −8.00000 −1.28103
\(40\) 0 0
\(41\) 5.65685 0.883452 0.441726 0.897150i \(-0.354366\pi\)
0.441726 + 0.897150i \(0.354366\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 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
\(51\) −16.0000 −2.24045
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) −2.82843 −0.368230 −0.184115 0.982905i \(-0.558942\pi\)
−0.184115 + 0.982905i \(0.558942\pi\)
\(60\) 0 0
\(61\) 14.1421 1.81071 0.905357 0.424650i \(-0.139603\pi\)
0.905357 + 0.424650i \(0.139603\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −12.0000 −1.46603 −0.733017 0.680211i \(-0.761888\pi\)
−0.733017 + 0.680211i \(0.761888\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 14.1421 1.55230 0.776151 0.630548i \(-0.217170\pi\)
0.776151 + 0.630548i \(0.217170\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −5.65685 −0.606478
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 16.0000 1.65912
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.65685 −0.574367 −0.287183 0.957876i \(-0.592719\pi\)
−0.287183 + 0.957876i \(0.592719\pi\)
\(98\) 0 0
\(99\) −20.0000 −2.01008
\(100\) 0 0
\(101\) −8.48528 −0.844317 −0.422159 0.906522i \(-0.638727\pi\)
−0.422159 + 0.906522i \(0.638727\pi\)
\(102\) 0 0
\(103\) 16.9706 1.67216 0.836080 0.548608i \(-0.184842\pi\)
0.836080 + 0.548608i \(0.184842\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 28.2843 2.68462
\(112\) 0 0
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 14.1421 1.30744
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −16.0000 −1.44267
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) −11.3137 −0.996116
\(130\) 0 0
\(131\) −14.1421 −1.23560 −0.617802 0.786334i \(-0.711977\pi\)
−0.617802 + 0.786334i \(0.711977\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 10.0000 0.854358 0.427179 0.904167i \(-0.359507\pi\)
0.427179 + 0.904167i \(0.359507\pi\)
\(138\) 0 0
\(139\) 14.1421 1.19952 0.599760 0.800180i \(-0.295263\pi\)
0.599760 + 0.800180i \(0.295263\pi\)
\(140\) 0 0
\(141\) 16.0000 1.34744
\(142\) 0 0
\(143\) −11.3137 −0.946100
\(144\) 0 0
\(145\) 0 0
\(146\) 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
\(151\) 16.0000 1.30206 0.651031 0.759051i \(-0.274337\pi\)
0.651031 + 0.759051i \(0.274337\pi\)
\(152\) 0 0
\(153\) 28.2843 2.28665
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 14.1421 1.12867 0.564333 0.825547i \(-0.309134\pi\)
0.564333 + 0.825547i \(0.309134\pi\)
\(158\) 0 0
\(159\) 16.9706 1.34585
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −20.0000 −1.56652 −0.783260 0.621694i \(-0.786445\pi\)
−0.783260 + 0.621694i \(0.786445\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 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
\(173\) 2.82843 0.215041 0.107521 0.994203i \(-0.465709\pi\)
0.107521 + 0.994203i \(0.465709\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 8.00000 0.601317
\(178\) 0 0
\(179\) −20.0000 −1.49487 −0.747435 0.664335i \(-0.768715\pi\)
−0.747435 + 0.664335i \(0.768715\pi\)
\(180\) 0 0
\(181\) 8.48528 0.630706 0.315353 0.948974i \(-0.397877\pi\)
0.315353 + 0.948974i \(0.397877\pi\)
\(182\) 0 0
\(183\) −40.0000 −2.95689
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −22.6274 −1.65468
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −22.0000 −1.56744 −0.783718 0.621117i \(-0.786679\pi\)
−0.783718 + 0.621117i \(0.786679\pi\)
\(198\) 0 0
\(199\) 16.9706 1.20301 0.601506 0.798869i \(-0.294568\pi\)
0.601506 + 0.798869i \(0.294568\pi\)
\(200\) 0 0
\(201\) 33.9411 2.39402
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 11.3137 0.782586
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 16.0000 1.07628
\(222\) 0 0
\(223\) 11.3137 0.757622 0.378811 0.925474i \(-0.376333\pi\)
0.378811 + 0.925474i \(0.376333\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −8.48528 −0.563188 −0.281594 0.959534i \(-0.590863\pi\)
−0.281594 + 0.959534i \(0.590863\pi\)
\(228\) 0 0
\(229\) −25.4558 −1.68217 −0.841085 0.540903i \(-0.818082\pi\)
−0.841085 + 0.540903i \(0.818082\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.0000 0.655122 0.327561 0.944830i \(-0.393773\pi\)
0.327561 + 0.944830i \(0.393773\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −22.6274 −1.46981
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 28.2843 1.82195 0.910975 0.412461i \(-0.135331\pi\)
0.910975 + 0.412461i \(0.135331\pi\)
\(242\) 0 0
\(243\) 14.1421 0.907218
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.00000 −0.509028
\(248\) 0 0
\(249\) −40.0000 −2.53490
\(250\) 0 0
\(251\) 14.1421 0.892644 0.446322 0.894873i \(-0.352734\pi\)
0.446322 + 0.894873i \(0.352734\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 10.0000 0.618984
\(262\) 0 0
\(263\) 16.0000 0.986602 0.493301 0.869859i \(-0.335790\pi\)
0.493301 + 0.869859i \(0.335790\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.1421 −0.862261 −0.431131 0.902290i \(-0.641885\pi\)
−0.431131 + 0.902290i \(0.641885\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) −28.2843 −1.69334
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 2.82843 0.168133 0.0840663 0.996460i \(-0.473209\pi\)
0.0840663 + 0.996460i \(0.473209\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 16.0000 0.937937
\(292\) 0 0
\(293\) −2.82843 −0.165238 −0.0826192 0.996581i \(-0.526329\pi\)
−0.0826192 + 0.996581i \(0.526329\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 22.6274 1.31298
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 24.0000 1.37876
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −8.48528 −0.484281 −0.242140 0.970241i \(-0.577849\pi\)
−0.242140 + 0.970241i \(0.577849\pi\)
\(308\) 0 0
\(309\) −48.0000 −2.73062
\(310\) 0 0
\(311\) −22.6274 −1.28308 −0.641542 0.767088i \(-0.721705\pi\)
−0.641542 + 0.767088i \(0.721705\pi\)
\(312\) 0 0
\(313\) 16.9706 0.959233 0.479616 0.877478i \(-0.340776\pi\)
0.479616 + 0.877478i \(0.340776\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) −8.00000 −0.447914
\(320\) 0 0
\(321\) −33.9411 −1.89441
\(322\) 0 0
\(323\) −16.0000 −0.890264
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −5.65685 −0.312825
\(328\) 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
\(333\) −50.0000 −2.73998
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 0 0
\(339\) 28.2843 1.53619
\(340\) 0 0
\(341\) 22.6274 1.22534
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.0000 1.07366 0.536828 0.843692i \(-0.319622\pi\)
0.536828 + 0.843692i \(0.319622\pi\)
\(348\) 0 0
\(349\) −25.4558 −1.36262 −0.681310 0.731995i \(-0.738589\pi\)
−0.681310 + 0.731995i \(0.738589\pi\)
\(350\) 0 0
\(351\) −16.0000 −0.854017
\(352\) 0 0
\(353\) −11.3137 −0.602168 −0.301084 0.953598i \(-0.597348\pi\)
−0.301084 + 0.953598i \(0.597348\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) −14.1421 −0.742270
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −33.9411 −1.77171 −0.885856 0.463960i \(-0.846428\pi\)
−0.885856 + 0.463960i \(0.846428\pi\)
\(368\) 0 0
\(369\) 28.2843 1.47242
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.65685 0.291343
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) 0 0
\(381\) 22.6274 1.15924
\(382\) 0 0
\(383\) −28.2843 −1.44526 −0.722629 0.691236i \(-0.757067\pi\)
−0.722629 + 0.691236i \(0.757067\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 20.0000 1.01666
\(388\) 0 0
\(389\) −22.0000 −1.11544 −0.557722 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 40.0000 2.01773
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −8.48528 −0.425864 −0.212932 0.977067i \(-0.568301\pi\)
−0.212932 + 0.977067i \(0.568301\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) −16.0000 −0.797017
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 40.0000 1.98273
\(408\) 0 0
\(409\) −16.9706 −0.839140 −0.419570 0.907723i \(-0.637819\pi\)
−0.419570 + 0.907723i \(0.637819\pi\)
\(410\) 0 0
\(411\) −28.2843 −1.39516
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −40.0000 −1.95881
\(418\) 0 0
\(419\) 2.82843 0.138178 0.0690889 0.997611i \(-0.477991\pi\)
0.0690889 + 0.997611i \(0.477991\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) −28.2843 −1.37523
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 32.0000 1.54497
\(430\) 0 0
\(431\) 24.0000 1.15604 0.578020 0.816023i \(-0.303826\pi\)
0.578020 + 0.816023i \(0.303826\pi\)
\(432\) 0 0
\(433\) −5.65685 −0.271851 −0.135926 0.990719i \(-0.543401\pi\)
−0.135926 + 0.990719i \(0.543401\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 28.2843 1.33780
\(448\) 0 0
\(449\) 2.00000 0.0943858 0.0471929 0.998886i \(-0.484972\pi\)
0.0471929 + 0.998886i \(0.484972\pi\)
\(450\) 0 0
\(451\) −22.6274 −1.06548
\(452\) 0 0
\(453\) −45.2548 −2.12626
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −18.0000 −0.842004 −0.421002 0.907060i \(-0.638322\pi\)
−0.421002 + 0.907060i \(0.638322\pi\)
\(458\) 0 0
\(459\) −32.0000 −1.49363
\(460\) 0 0
\(461\) −14.1421 −0.658665 −0.329332 0.944214i \(-0.606824\pi\)
−0.329332 + 0.944214i \(0.606824\pi\)
\(462\) 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
\(467\) −8.48528 −0.392652 −0.196326 0.980539i \(-0.562901\pi\)
−0.196326 + 0.980539i \(0.562901\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −40.0000 −1.84310
\(472\) 0 0
\(473\) −16.0000 −0.735681
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −30.0000 −1.37361
\(478\) 0 0
\(479\) −28.2843 −1.29234 −0.646171 0.763193i \(-0.723631\pi\)
−0.646171 + 0.763193i \(0.723631\pi\)
\(480\) 0 0
\(481\) −28.2843 −1.28965
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 56.5685 2.55812
\(490\) 0 0
\(491\) 4.00000 0.180517 0.0902587 0.995918i \(-0.471231\pi\)
0.0902587 + 0.995918i \(0.471231\pi\)
\(492\) 0 0
\(493\) 11.3137 0.509544
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) −16.0000 −0.714827
\(502\) 0 0
\(503\) −11.3137 −0.504453 −0.252227 0.967668i \(-0.581163\pi\)
−0.252227 + 0.967668i \(0.581163\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 14.1421 0.628074
\(508\) 0 0
\(509\) −14.1421 −0.626839 −0.313420 0.949615i \(-0.601475\pi\)
−0.313420 + 0.949615i \(0.601475\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 16.0000 0.706417
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 22.6274 0.995153
\(518\) 0 0
\(519\) −8.00000 −0.351161
\(520\) 0 0
\(521\) −5.65685 −0.247831 −0.123916 0.992293i \(-0.539545\pi\)
−0.123916 + 0.992293i \(0.539545\pi\)
\(522\) 0 0
\(523\) −19.7990 −0.865749 −0.432875 0.901454i \(-0.642501\pi\)
−0.432875 + 0.901454i \(0.642501\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −32.0000 −1.39394
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −14.1421 −0.613716
\(532\) 0 0
\(533\) 16.0000 0.693037
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 56.5685 2.44111
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) −24.0000 −1.02994
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 0 0
\(549\) 70.7107 3.01786
\(550\) 0 0
\(551\) −5.65685 −0.240990
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −30.0000 −1.27114 −0.635570 0.772043i \(-0.719235\pi\)
−0.635570 + 0.772043i \(0.719235\pi\)
\(558\) 0 0
\(559\) 11.3137 0.478519
\(560\) 0 0
\(561\) 64.0000 2.70208
\(562\) 0 0
\(563\) 42.4264 1.78806 0.894030 0.448007i \(-0.147866\pi\)
0.894030 + 0.448007i \(0.147866\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) −67.8823 −2.83582
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −33.9411 −1.41299 −0.706494 0.707719i \(-0.749724\pi\)
−0.706494 + 0.707719i \(0.749724\pi\)
\(578\) 0 0
\(579\) 28.2843 1.17545
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 24.0000 0.993978
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 42.4264 1.75113 0.875563 0.483105i \(-0.160491\pi\)
0.875563 + 0.483105i \(0.160491\pi\)
\(588\) 0 0
\(589\) 16.0000 0.659269
\(590\) 0 0
\(591\) 62.2254 2.55961
\(592\) 0 0
\(593\) 11.3137 0.464598 0.232299 0.972644i \(-0.425375\pi\)
0.232299 + 0.972644i \(0.425375\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −48.0000 −1.96451
\(598\) 0 0
\(599\) 32.0000 1.30748 0.653742 0.756717i \(-0.273198\pi\)
0.653742 + 0.756717i \(0.273198\pi\)
\(600\) 0 0
\(601\) −22.6274 −0.922992 −0.461496 0.887142i \(-0.652687\pi\)
−0.461496 + 0.887142i \(0.652687\pi\)
\(602\) 0 0
\(603\) −60.0000 −2.44339
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −22.6274 −0.918419 −0.459209 0.888328i \(-0.651867\pi\)
−0.459209 + 0.888328i \(0.651867\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −16.0000 −0.647291
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.0000 −1.20775 −0.603877 0.797077i \(-0.706378\pi\)
−0.603877 + 0.797077i \(0.706378\pi\)
\(618\) 0 0
\(619\) −14.1421 −0.568420 −0.284210 0.958762i \(-0.591731\pi\)
−0.284210 + 0.958762i \(0.591731\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −32.0000 −1.27796
\(628\) 0 0
\(629\) −56.5685 −2.25554
\(630\) 0 0
\(631\) −16.0000 −0.636950 −0.318475 0.947931i \(-0.603171\pi\)
−0.318475 + 0.947931i \(0.603171\pi\)
\(632\) 0 0
\(633\) 11.3137 0.449680
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −10.0000 −0.394976 −0.197488 0.980305i \(-0.563278\pi\)
−0.197488 + 0.980305i \(0.563278\pi\)
\(642\) 0 0
\(643\) −25.4558 −1.00388 −0.501940 0.864902i \(-0.667380\pi\)
−0.501940 + 0.864902i \(0.667380\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −16.9706 −0.667182 −0.333591 0.942718i \(-0.608260\pi\)
−0.333591 + 0.942718i \(0.608260\pi\)
\(648\) 0 0
\(649\) 11.3137 0.444102
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 14.0000 0.547862 0.273931 0.961749i \(-0.411676\pi\)
0.273931 + 0.961749i \(0.411676\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 14.1421 0.550065 0.275033 0.961435i \(-0.411311\pi\)
0.275033 + 0.961435i \(0.411311\pi\)
\(662\) 0 0
\(663\) −45.2548 −1.75755
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −32.0000 −1.23719
\(670\) 0 0
\(671\) −56.5685 −2.18380
\(672\) 0 0
\(673\) −30.0000 −1.15642 −0.578208 0.815890i \(-0.696248\pi\)
−0.578208 + 0.815890i \(0.696248\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.48528 −0.326116 −0.163058 0.986616i \(-0.552136\pi\)
−0.163058 + 0.986616i \(0.552136\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 24.0000 0.919682
\(682\) 0 0
\(683\) −20.0000 −0.765279 −0.382639 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 72.0000 2.74697
\(688\) 0 0
\(689\) −16.9706 −0.646527
\(690\) 0 0
\(691\) −14.1421 −0.537992 −0.268996 0.963141i \(-0.586692\pi\)
−0.268996 + 0.963141i \(0.586692\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 32.0000 1.21209
\(698\) 0 0
\(699\) −28.2843 −1.06981
\(700\) 0 0
\(701\) 34.0000 1.28416 0.642081 0.766637i \(-0.278071\pi\)
0.642081 + 0.766637i \(0.278071\pi\)
\(702\) 0 0
\(703\) 28.2843 1.06676
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 40.0000 1.50012
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 22.6274 0.845036
\(718\) 0 0
\(719\) −28.2843 −1.05483 −0.527413 0.849609i \(-0.676838\pi\)
−0.527413 + 0.849609i \(0.676838\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −80.0000 −2.97523
\(724\) 0 0
\(725\) 0 0
\(726\) 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
\(731\) 22.6274 0.836905
\(732\) 0 0
\(733\) −25.4558 −0.940233 −0.470117 0.882604i \(-0.655788\pi\)
−0.470117 + 0.882604i \(0.655788\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 48.0000 1.76810
\(738\) 0 0
\(739\) −12.0000 −0.441427 −0.220714 0.975339i \(-0.570839\pi\)
−0.220714 + 0.975339i \(0.570839\pi\)
\(740\) 0 0
\(741\) 22.6274 0.831239
\(742\) 0 0
\(743\) 16.0000 0.586983 0.293492 0.955962i \(-0.405183\pi\)
0.293492 + 0.955962i \(0.405183\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 70.7107 2.58717
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) −40.0000 −1.45768
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −28.2843 −1.02530 −0.512652 0.858596i \(-0.671337\pi\)
−0.512652 + 0.858596i \(0.671337\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) 16.9706 0.611974 0.305987 0.952036i \(-0.401014\pi\)
0.305987 + 0.952036i \(0.401014\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 53.7401 1.93290 0.966449 0.256859i \(-0.0826877\pi\)
0.966449 + 0.256859i \(0.0826877\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −16.0000 −0.573259
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −11.3137 −0.404319
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −8.48528 −0.302468 −0.151234 0.988498i \(-0.548325\pi\)
−0.151234 + 0.988498i \(0.548325\pi\)
\(788\) 0 0
\(789\) −45.2548 −1.61111
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −8.48528 −0.300564 −0.150282 0.988643i \(-0.548018\pi\)
−0.150282 + 0.988643i \(0.548018\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 40.0000 1.40807
\(808\) 0 0
\(809\) 18.0000 0.632846 0.316423 0.948618i \(-0.397518\pi\)
0.316423 + 0.948618i \(0.397518\pi\)
\(810\) 0 0
\(811\) −14.1421 −0.496598 −0.248299 0.968683i \(-0.579871\pi\)
−0.248299 + 0.968683i \(0.579871\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −11.3137 −0.395817
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 54.0000 1.88461 0.942306 0.334751i \(-0.108652\pi\)
0.942306 + 0.334751i \(0.108652\pi\)
\(822\) 0 0
\(823\) 16.0000 0.557725 0.278862 0.960331i \(-0.410043\pi\)
0.278862 + 0.960331i \(0.410043\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −20.0000 −0.695468 −0.347734 0.937593i \(-0.613049\pi\)
−0.347734 + 0.937593i \(0.613049\pi\)
\(828\) 0 0
\(829\) −31.1127 −1.08059 −0.540294 0.841476i \(-0.681687\pi\)
−0.540294 + 0.841476i \(0.681687\pi\)
\(830\) 0 0
\(831\) −28.2843 −0.981170
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 32.0000 1.10608
\(838\) 0 0
\(839\) 28.2843 0.976481 0.488241 0.872709i \(-0.337639\pi\)
0.488241 + 0.872709i \(0.337639\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −16.9706 −0.584497
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −8.00000 −0.274559
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −42.4264 −1.45265 −0.726326 0.687350i \(-0.758774\pi\)
−0.726326 + 0.687350i \(0.758774\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −28.2843 −0.966172 −0.483086 0.875573i \(-0.660484\pi\)
−0.483086 + 0.875573i \(0.660484\pi\)
\(858\) 0 0
\(859\) −42.4264 −1.44757 −0.723785 0.690025i \(-0.757599\pi\)
−0.723785 + 0.690025i \(0.757599\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −24.0000 −0.816970 −0.408485 0.912765i \(-0.633943\pi\)
−0.408485 + 0.912765i \(0.633943\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −42.4264 −1.44088
\(868\) 0 0
\(869\) −32.0000 −1.08553
\(870\) 0 0
\(871\) −33.9411 −1.15005
\(872\) 0 0
\(873\) −28.2843 −0.957278
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2.00000 −0.0675352 −0.0337676 0.999430i \(-0.510751\pi\)
−0.0337676 + 0.999430i \(0.510751\pi\)
\(878\) 0 0
\(879\) 8.00000 0.269833
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 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
\(897\) 0 0
\(898\) 0 0
\(899\) −11.3137 −0.377333
\(900\) 0 0
\(901\) −33.9411 −1.13074
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −20.0000 −0.664089 −0.332045 0.943264i \(-0.607738\pi\)
−0.332045 + 0.943264i \(0.607738\pi\)
\(908\) 0 0
\(909\) −42.4264 −1.40720
\(910\) 0 0
\(911\) −40.0000 −1.32526 −0.662630 0.748947i \(-0.730560\pi\)
−0.662630 + 0.748947i \(0.730560\pi\)
\(912\) 0 0
\(913\) −56.5685 −1.87215
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 24.0000 0.790827
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 84.8528 2.78693
\(928\) 0 0
\(929\) 39.5980 1.29917 0.649584 0.760290i \(-0.274943\pi\)
0.649584 + 0.760290i \(0.274943\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 64.0000 2.09527
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 22.6274 0.739205 0.369603 0.929190i \(-0.379494\pi\)
0.369603 + 0.929190i \(0.379494\pi\)
\(938\) 0 0
\(939\) −48.0000 −1.56642
\(940\) 0 0
\(941\) 14.1421 0.461020 0.230510 0.973070i \(-0.425960\pi\)
0.230510 + 0.973070i \(0.425960\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 12.0000 0.389948 0.194974 0.980808i \(-0.437538\pi\)
0.194974 + 0.980808i \(0.437538\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −50.9117 −1.65092
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 22.6274 0.731441
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 60.0000 1.93347
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −48.0000 −1.54358 −0.771788 0.635880i \(-0.780637\pi\)
−0.771788 + 0.635880i \(0.780637\pi\)
\(968\) 0 0
\(969\) 45.2548 1.45379
\(970\) 0 0
\(971\) 19.7990 0.635380 0.317690 0.948195i \(-0.397093\pi\)
0.317690 + 0.948195i \(0.397093\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 10.0000 0.319275
\(982\) 0 0
\(983\) 39.5980 1.26298 0.631490 0.775384i \(-0.282444\pi\)
0.631490 + 0.775384i \(0.282444\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) 0 0
\(993\) 56.5685 1.79515
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −36.7696 −1.16450 −0.582252 0.813009i \(-0.697828\pi\)
−0.582252 + 0.813009i \(0.697828\pi\)
\(998\) 0 0
\(999\) 56.5685 1.78975
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 9800.2.a.bw.1.1 2
5.4 even 2 392.2.a.h.1.2 yes 2
7.6 odd 2 inner 9800.2.a.bw.1.2 2
15.14 odd 2 3528.2.a.bj.1.1 2
20.19 odd 2 784.2.a.n.1.1 2
35.4 even 6 392.2.i.g.177.1 4
35.9 even 6 392.2.i.g.361.1 4
35.19 odd 6 392.2.i.g.361.2 4
35.24 odd 6 392.2.i.g.177.2 4
35.34 odd 2 392.2.a.h.1.1 2
40.19 odd 2 3136.2.a.bq.1.2 2
40.29 even 2 3136.2.a.bt.1.1 2
60.59 even 2 7056.2.a.cj.1.1 2
105.44 odd 6 3528.2.s.be.361.2 4
105.59 even 6 3528.2.s.be.3313.1 4
105.74 odd 6 3528.2.s.be.3313.2 4
105.89 even 6 3528.2.s.be.361.1 4
105.104 even 2 3528.2.a.bj.1.2 2
140.19 even 6 784.2.i.k.753.1 4
140.39 odd 6 784.2.i.k.177.2 4
140.59 even 6 784.2.i.k.177.1 4
140.79 odd 6 784.2.i.k.753.2 4
140.139 even 2 784.2.a.n.1.2 2
280.69 odd 2 3136.2.a.bt.1.2 2
280.139 even 2 3136.2.a.bq.1.1 2
420.419 odd 2 7056.2.a.cj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
392.2.a.h.1.1 2 35.34 odd 2
392.2.a.h.1.2 yes 2 5.4 even 2
392.2.i.g.177.1 4 35.4 even 6
392.2.i.g.177.2 4 35.24 odd 6
392.2.i.g.361.1 4 35.9 even 6
392.2.i.g.361.2 4 35.19 odd 6
784.2.a.n.1.1 2 20.19 odd 2
784.2.a.n.1.2 2 140.139 even 2
784.2.i.k.177.1 4 140.59 even 6
784.2.i.k.177.2 4 140.39 odd 6
784.2.i.k.753.1 4 140.19 even 6
784.2.i.k.753.2 4 140.79 odd 6
3136.2.a.bq.1.1 2 280.139 even 2
3136.2.a.bq.1.2 2 40.19 odd 2
3136.2.a.bt.1.1 2 40.29 even 2
3136.2.a.bt.1.2 2 280.69 odd 2
3528.2.a.bj.1.1 2 15.14 odd 2
3528.2.a.bj.1.2 2 105.104 even 2
3528.2.s.be.361.1 4 105.89 even 6
3528.2.s.be.361.2 4 105.44 odd 6
3528.2.s.be.3313.1 4 105.59 even 6
3528.2.s.be.3313.2 4 105.74 odd 6
7056.2.a.cj.1.1 2 60.59 even 2
7056.2.a.cj.1.2 2 420.419 odd 2
9800.2.a.bw.1.1 2 1.1 even 1 trivial
9800.2.a.bw.1.2 2 7.6 odd 2 inner