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Properties

Label 72.10.a.c.1.1
Level $72$
Weight $10$
Character 72.1
Self dual yes
Analytic conductor $37.083$
Analytic rank $1$
Dimension $1$
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] = [72,10,Mod(1,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 72.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: \(37.0825802038\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 72.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-614.000 q^{5} +2184.00 q^{7} +O(q^{10})\) \(q-614.000 q^{5} +2184.00 q^{7} -4940.00 q^{11} +69934.0 q^{13} -376978. q^{17} +780884. q^{19} -1.76463e6 q^{23} -1.57613e6 q^{25} +3.21223e6 q^{29} -342880. q^{31} -1.34098e6 q^{35} -1.97445e7 q^{37} +1.58824e7 q^{41} -2.25758e7 q^{43} -4.89485e7 q^{47} -3.55838e7 q^{49} -5.23426e7 q^{53} +3.03316e6 q^{55} -7.70577e7 q^{59} +1.30457e7 q^{61} -4.29395e7 q^{65} -2.80727e8 q^{67} +8.85547e7 q^{71} -5.91057e7 q^{73} -1.07890e7 q^{77} -4.15337e8 q^{79} -4.28069e7 q^{83} +2.31464e8 q^{85} +8.03466e8 q^{89} +1.52736e8 q^{91} -4.79463e8 q^{95} +6.74418e8 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(5\) −614.000 −0.439343 −0.219671 − 0.975574i \(-0.570498\pi\)
−0.219671 + 0.975574i \(0.570498\pi\)
\(6\) 0 0
\(7\) 2184.00 0.343804 0.171902 − 0.985114i \(-0.445009\pi\)
0.171902 + 0.985114i \(0.445009\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −4940.00 −0.101733 −0.0508663 − 0.998705i \(-0.516198\pi\)
−0.0508663 + 0.998705i \(0.516198\pi\)
\(12\) 0 0
\(13\) 69934.0 0.679115 0.339557 − 0.940585i \(-0.389723\pi\)
0.339557 + 0.940585i \(0.389723\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −376978. −1.09470 −0.547351 − 0.836903i \(-0.684364\pi\)
−0.547351 + 0.836903i \(0.684364\pi\)
\(18\) 0 0
\(19\) 780884. 1.37466 0.687330 − 0.726345i \(-0.258783\pi\)
0.687330 + 0.726345i \(0.258783\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1.76463e6 −1.31486 −0.657429 − 0.753516i \(-0.728356\pi\)
−0.657429 + 0.753516i \(0.728356\pi\)
\(24\) 0 0
\(25\) −1.57613e6 −0.806978
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.21223e6 0.843364 0.421682 − 0.906744i \(-0.361440\pi\)
0.421682 + 0.906744i \(0.361440\pi\)
\(30\) 0 0
\(31\) −342880. −0.0666829 −0.0333415 − 0.999444i \(-0.510615\pi\)
−0.0333415 + 0.999444i \(0.510615\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.34098e6 −0.151048
\(36\) 0 0
\(37\) −1.97445e7 −1.73196 −0.865981 − 0.500076i \(-0.833305\pi\)
−0.865981 + 0.500076i \(0.833305\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.58824e7 0.877786 0.438893 − 0.898539i \(-0.355371\pi\)
0.438893 + 0.898539i \(0.355371\pi\)
\(42\) 0 0
\(43\) −2.25758e7 −1.00701 −0.503506 − 0.863992i \(-0.667957\pi\)
−0.503506 + 0.863992i \(0.667957\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.89485e7 −1.46319 −0.731593 − 0.681742i \(-0.761223\pi\)
−0.731593 + 0.681742i \(0.761223\pi\)
\(48\) 0 0
\(49\) −3.55838e7 −0.881799
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.23426e7 −0.911199 −0.455600 − 0.890185i \(-0.650575\pi\)
−0.455600 + 0.890185i \(0.650575\pi\)
\(54\) 0 0
\(55\) 3.03316e6 0.0446954
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.70577e7 −0.827907 −0.413954 − 0.910298i \(-0.635852\pi\)
−0.413954 + 0.910298i \(0.635852\pi\)
\(60\) 0 0
\(61\) 1.30457e7 0.120638 0.0603190 − 0.998179i \(-0.480788\pi\)
0.0603190 + 0.998179i \(0.480788\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.29395e7 −0.298364
\(66\) 0 0
\(67\) −2.80727e8 −1.70195 −0.850977 − 0.525203i \(-0.823989\pi\)
−0.850977 + 0.525203i \(0.823989\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.85547e7 0.413570 0.206785 − 0.978386i \(-0.433700\pi\)
0.206785 + 0.978386i \(0.433700\pi\)
\(72\) 0 0
\(73\) −5.91057e7 −0.243599 −0.121800 − 0.992555i \(-0.538867\pi\)
−0.121800 + 0.992555i \(0.538867\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.07890e7 −0.0349761
\(78\) 0 0
\(79\) −4.15337e8 −1.19972 −0.599859 − 0.800106i \(-0.704777\pi\)
−0.599859 + 0.800106i \(0.704777\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.28069e7 −0.0990063 −0.0495031 − 0.998774i \(-0.515764\pi\)
−0.0495031 + 0.998774i \(0.515764\pi\)
\(84\) 0 0
\(85\) 2.31464e8 0.480949
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.03466e8 1.35741 0.678707 − 0.734409i \(-0.262541\pi\)
0.678707 + 0.734409i \(0.262541\pi\)
\(90\) 0 0
\(91\) 1.52736e8 0.233483
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.79463e8 −0.603947
\(96\) 0 0
\(97\) 6.74418e8 0.773493 0.386746 − 0.922186i \(-0.373599\pi\)
0.386746 + 0.922186i \(0.373599\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 3.46839e8 0.331652 0.165826 − 0.986155i \(-0.446971\pi\)
0.165826 + 0.986155i \(0.446971\pi\)
\(102\) 0 0
\(103\) 1.30280e9 1.14054 0.570268 − 0.821459i \(-0.306839\pi\)
0.570268 + 0.821459i \(0.306839\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.02396e9 −1.49271 −0.746356 − 0.665547i \(-0.768198\pi\)
−0.746356 + 0.665547i \(0.768198\pi\)
\(108\) 0 0
\(109\) 2.69987e9 1.83199 0.915996 − 0.401187i \(-0.131402\pi\)
0.915996 + 0.401187i \(0.131402\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.86022e9 1.65024 0.825118 − 0.564960i \(-0.191108\pi\)
0.825118 + 0.564960i \(0.191108\pi\)
\(114\) 0 0
\(115\) 1.08348e9 0.577673
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −8.23320e8 −0.376363
\(120\) 0 0
\(121\) −2.33354e9 −0.989650
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.16696e9 0.793882
\(126\) 0 0
\(127\) 3.04278e8 0.103790 0.0518948 − 0.998653i \(-0.483474\pi\)
0.0518948 + 0.998653i \(0.483474\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.13499e8 0.0336721 0.0168360 − 0.999858i \(-0.494641\pi\)
0.0168360 + 0.999858i \(0.494641\pi\)
\(132\) 0 0
\(133\) 1.70545e9 0.472614
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 5.27408e9 1.27910 0.639550 − 0.768750i \(-0.279121\pi\)
0.639550 + 0.768750i \(0.279121\pi\)
\(138\) 0 0
\(139\) −3.28693e9 −0.746834 −0.373417 − 0.927663i \(-0.621814\pi\)
−0.373417 + 0.927663i \(0.621814\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −3.45474e8 −0.0690881
\(144\) 0 0
\(145\) −1.97231e9 −0.370526
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.47524e9 −1.24247 −0.621237 − 0.783623i \(-0.713369\pi\)
−0.621237 + 0.783623i \(0.713369\pi\)
\(150\) 0 0
\(151\) −5.77805e9 −0.904450 −0.452225 − 0.891904i \(-0.649370\pi\)
−0.452225 + 0.891904i \(0.649370\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.10528e8 0.0292966
\(156\) 0 0
\(157\) 9.95607e9 1.30779 0.653897 − 0.756583i \(-0.273133\pi\)
0.653897 + 0.756583i \(0.273133\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.85396e9 −0.452054
\(162\) 0 0
\(163\) −6.78825e9 −0.753205 −0.376603 − 0.926375i \(-0.622908\pi\)
−0.376603 + 0.926375i \(0.622908\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.21803e9 −0.817605 −0.408802 − 0.912623i \(-0.634054\pi\)
−0.408802 + 0.912623i \(0.634054\pi\)
\(168\) 0 0
\(169\) −5.71374e9 −0.538803
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.76914e10 −1.50160 −0.750799 − 0.660530i \(-0.770331\pi\)
−0.750799 + 0.660530i \(0.770331\pi\)
\(174\) 0 0
\(175\) −3.44227e9 −0.277443
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.49444e9 −0.181608 −0.0908039 − 0.995869i \(-0.528944\pi\)
−0.0908039 + 0.995869i \(0.528944\pi\)
\(180\) 0 0
\(181\) −8.72556e9 −0.604282 −0.302141 − 0.953263i \(-0.597701\pi\)
−0.302141 + 0.953263i \(0.597701\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.21231e10 0.760925
\(186\) 0 0
\(187\) 1.86227e9 0.111367
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.01873e10 1.09756 0.548781 − 0.835966i \(-0.315092\pi\)
0.548781 + 0.835966i \(0.315092\pi\)
\(192\) 0 0
\(193\) −2.87739e10 −1.49276 −0.746381 − 0.665519i \(-0.768210\pi\)
−0.746381 + 0.665519i \(0.768210\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 4.62409e9 0.218740 0.109370 − 0.994001i \(-0.465117\pi\)
0.109370 + 0.994001i \(0.465117\pi\)
\(198\) 0 0
\(199\) 7.47248e8 0.0337774 0.0168887 − 0.999857i \(-0.494624\pi\)
0.0168887 + 0.999857i \(0.494624\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.01550e9 0.289952
\(204\) 0 0
\(205\) −9.75179e9 −0.385649
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.85757e9 −0.139848
\(210\) 0 0
\(211\) 3.25965e10 1.13214 0.566069 − 0.824358i \(-0.308464\pi\)
0.566069 + 0.824358i \(0.308464\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.38615e10 0.442423
\(216\) 0 0
\(217\) −7.48850e8 −0.0229259
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.63636e10 −0.743428
\(222\) 0 0
\(223\) 3.34280e10 0.905187 0.452594 − 0.891717i \(-0.350499\pi\)
0.452594 + 0.891717i \(0.350499\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −3.46895e9 −0.0867124 −0.0433562 − 0.999060i \(-0.513805\pi\)
−0.0433562 + 0.999060i \(0.513805\pi\)
\(228\) 0 0
\(229\) 2.27546e10 0.546776 0.273388 − 0.961904i \(-0.411856\pi\)
0.273388 + 0.961904i \(0.411856\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5.59285e9 −0.124317 −0.0621586 − 0.998066i \(-0.519798\pi\)
−0.0621586 + 0.998066i \(0.519798\pi\)
\(234\) 0 0
\(235\) 3.00544e10 0.642840
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.24283e10 0.444637 0.222319 − 0.974974i \(-0.428637\pi\)
0.222319 + 0.974974i \(0.428637\pi\)
\(240\) 0 0
\(241\) 6.86817e10 1.31149 0.655744 − 0.754983i \(-0.272355\pi\)
0.655744 + 0.754983i \(0.272355\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.18484e10 0.387412
\(246\) 0 0
\(247\) 5.46103e10 0.933552
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.06168e10 0.168835 0.0844175 − 0.996430i \(-0.473097\pi\)
0.0844175 + 0.996430i \(0.473097\pi\)
\(252\) 0 0
\(253\) 8.71728e9 0.133764
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.53205e10 −1.21998 −0.609992 − 0.792407i \(-0.708828\pi\)
−0.609992 + 0.792407i \(0.708828\pi\)
\(258\) 0 0
\(259\) −4.31220e10 −0.595456
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.35617e11 1.74788 0.873942 − 0.486030i \(-0.161555\pi\)
0.873942 + 0.486030i \(0.161555\pi\)
\(264\) 0 0
\(265\) 3.21383e10 0.400329
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.76465e10 0.438369 0.219184 − 0.975683i \(-0.429660\pi\)
0.219184 + 0.975683i \(0.429660\pi\)
\(270\) 0 0
\(271\) −1.66830e10 −0.187894 −0.0939468 − 0.995577i \(-0.529948\pi\)
−0.0939468 + 0.995577i \(0.529948\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 7.78608e9 0.0820959
\(276\) 0 0
\(277\) −1.19925e11 −1.22392 −0.611958 − 0.790890i \(-0.709618\pi\)
−0.611958 + 0.790890i \(0.709618\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.97478e10 −0.667347 −0.333674 − 0.942689i \(-0.608288\pi\)
−0.333674 + 0.942689i \(0.608288\pi\)
\(282\) 0 0
\(283\) −2.04570e11 −1.89584 −0.947922 − 0.318504i \(-0.896820\pi\)
−0.947922 + 0.318504i \(0.896820\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.46871e10 0.301787
\(288\) 0 0
\(289\) 2.35245e10 0.198372
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.36328e10 0.425135 0.212567 − 0.977146i \(-0.431818\pi\)
0.212567 + 0.977146i \(0.431818\pi\)
\(294\) 0 0
\(295\) 4.73134e10 0.363735
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.23408e11 −0.892940
\(300\) 0 0
\(301\) −4.93055e10 −0.346215
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.01008e9 −0.0530014
\(306\) 0 0
\(307\) 4.31259e10 0.277086 0.138543 − 0.990356i \(-0.455758\pi\)
0.138543 + 0.990356i \(0.455758\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.27994e11 −1.98813 −0.994063 − 0.108810i \(-0.965296\pi\)
−0.994063 + 0.108810i \(0.965296\pi\)
\(312\) 0 0
\(313\) 3.63824e9 0.0214260 0.0107130 − 0.999943i \(-0.496590\pi\)
0.0107130 + 0.999943i \(0.496590\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.41046e11 1.34070 0.670352 − 0.742044i \(-0.266143\pi\)
0.670352 + 0.742044i \(0.266143\pi\)
\(318\) 0 0
\(319\) −1.58684e10 −0.0857976
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.94376e11 −1.50484
\(324\) 0 0
\(325\) −1.10225e11 −0.548031
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.06904e11 −0.503050
\(330\) 0 0
\(331\) −5.88628e10 −0.269535 −0.134767 − 0.990877i \(-0.543029\pi\)
−0.134767 + 0.990877i \(0.543029\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.72366e11 0.747741
\(336\) 0 0
\(337\) −2.11238e10 −0.0892149 −0.0446074 − 0.999005i \(-0.514204\pi\)
−0.0446074 + 0.999005i \(0.514204\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.69383e9 0.00678382
\(342\) 0 0
\(343\) −1.65847e11 −0.646971
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.34453e11 −1.60865 −0.804323 − 0.594193i \(-0.797472\pi\)
−0.804323 + 0.594193i \(0.797472\pi\)
\(348\) 0 0
\(349\) −3.08865e10 −0.111443 −0.0557216 − 0.998446i \(-0.517746\pi\)
−0.0557216 + 0.998446i \(0.517746\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.57147e11 1.56700 0.783502 − 0.621390i \(-0.213432\pi\)
0.783502 + 0.621390i \(0.213432\pi\)
\(354\) 0 0
\(355\) −5.43726e10 −0.181699
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.89953e11 0.921302 0.460651 − 0.887581i \(-0.347616\pi\)
0.460651 + 0.887581i \(0.347616\pi\)
\(360\) 0 0
\(361\) 2.87092e11 0.889690
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.62909e10 0.107024
\(366\) 0 0
\(367\) −3.53888e10 −0.101828 −0.0509142 − 0.998703i \(-0.516214\pi\)
−0.0509142 + 0.998703i \(0.516214\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.14316e11 −0.313274
\(372\) 0 0
\(373\) −1.02453e11 −0.274053 −0.137026 − 0.990567i \(-0.543755\pi\)
−0.137026 + 0.990567i \(0.543755\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.24644e11 0.572741
\(378\) 0 0
\(379\) −1.51545e11 −0.377282 −0.188641 − 0.982046i \(-0.560408\pi\)
−0.188641 + 0.982046i \(0.560408\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 5.45078e11 1.29439 0.647193 − 0.762326i \(-0.275943\pi\)
0.647193 + 0.762326i \(0.275943\pi\)
\(384\) 0 0
\(385\) 6.62442e9 0.0153665
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.13365e11 0.251018 0.125509 − 0.992092i \(-0.459944\pi\)
0.125509 + 0.992092i \(0.459944\pi\)
\(390\) 0 0
\(391\) 6.65227e11 1.43938
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.55017e11 0.527087
\(396\) 0 0
\(397\) 6.88512e11 1.39109 0.695543 − 0.718484i \(-0.255164\pi\)
0.695543 + 0.718484i \(0.255164\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.35503e11 −1.42048 −0.710239 − 0.703960i \(-0.751413\pi\)
−0.710239 + 0.703960i \(0.751413\pi\)
\(402\) 0 0
\(403\) −2.39790e10 −0.0452854
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.75379e10 0.176197
\(408\) 0 0
\(409\) 5.59969e11 0.989485 0.494742 − 0.869040i \(-0.335262\pi\)
0.494742 + 0.869040i \(0.335262\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.68294e11 −0.284638
\(414\) 0 0
\(415\) 2.62835e10 0.0434977
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −1.07971e12 −1.71137 −0.855687 − 0.517493i \(-0.826865\pi\)
−0.855687 + 0.517493i \(0.826865\pi\)
\(420\) 0 0
\(421\) 1.12290e12 1.74209 0.871047 − 0.491200i \(-0.163442\pi\)
0.871047 + 0.491200i \(0.163442\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.94166e11 0.883400
\(426\) 0 0
\(427\) 2.84919e10 0.0414759
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.10912e11 0.992357 0.496178 − 0.868221i \(-0.334736\pi\)
0.496178 + 0.868221i \(0.334736\pi\)
\(432\) 0 0
\(433\) −1.11868e12 −1.52936 −0.764682 − 0.644407i \(-0.777104\pi\)
−0.764682 + 0.644407i \(0.777104\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.37797e12 −1.80748
\(438\) 0 0
\(439\) 7.09728e11 0.912014 0.456007 − 0.889976i \(-0.349279\pi\)
0.456007 + 0.889976i \(0.349279\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.29172e12 1.59349 0.796746 − 0.604314i \(-0.206553\pi\)
0.796746 + 0.604314i \(0.206553\pi\)
\(444\) 0 0
\(445\) −4.93328e11 −0.596370
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.16414e11 0.599639 0.299820 − 0.953996i \(-0.403074\pi\)
0.299820 + 0.953996i \(0.403074\pi\)
\(450\) 0 0
\(451\) −7.84590e10 −0.0892994
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −9.37798e10 −0.102579
\(456\) 0 0
\(457\) −1.15459e12 −1.23824 −0.619118 − 0.785298i \(-0.712510\pi\)
−0.619118 + 0.785298i \(0.712510\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.08222e11 0.317841 0.158920 − 0.987291i \(-0.449199\pi\)
0.158920 + 0.987291i \(0.449199\pi\)
\(462\) 0 0
\(463\) −2.60120e11 −0.263062 −0.131531 − 0.991312i \(-0.541989\pi\)
−0.131531 + 0.991312i \(0.541989\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −1.17196e12 −1.14021 −0.570106 − 0.821571i \(-0.693098\pi\)
−0.570106 + 0.821571i \(0.693098\pi\)
\(468\) 0 0
\(469\) −6.13108e11 −0.585139
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.11524e11 0.102446
\(474\) 0 0
\(475\) −1.23077e12 −1.10932
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.74551e12 1.51500 0.757498 − 0.652838i \(-0.226422\pi\)
0.757498 + 0.652838i \(0.226422\pi\)
\(480\) 0 0
\(481\) −1.38081e12 −1.17620
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4.14093e11 −0.339828
\(486\) 0 0
\(487\) 1.33263e12 1.07356 0.536782 − 0.843721i \(-0.319640\pi\)
0.536782 + 0.843721i \(0.319640\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.06229e12 −1.60134 −0.800670 − 0.599106i \(-0.795523\pi\)
−0.800670 + 0.599106i \(0.795523\pi\)
\(492\) 0 0
\(493\) −1.21094e12 −0.923232
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.93403e11 0.142187
\(498\) 0 0
\(499\) 5.85740e11 0.422914 0.211457 − 0.977387i \(-0.432179\pi\)
0.211457 + 0.977387i \(0.432179\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.44066e12 1.00347 0.501735 − 0.865021i \(-0.332695\pi\)
0.501735 + 0.865021i \(0.332695\pi\)
\(504\) 0 0
\(505\) −2.12959e11 −0.145709
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2.60061e12 −1.71730 −0.858650 − 0.512563i \(-0.828696\pi\)
−0.858650 + 0.512563i \(0.828696\pi\)
\(510\) 0 0
\(511\) −1.29087e11 −0.0837505
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.99917e11 −0.501086
\(516\) 0 0
\(517\) 2.41806e11 0.148854
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.29064e12 −1.36203 −0.681016 − 0.732268i \(-0.738462\pi\)
−0.681016 + 0.732268i \(0.738462\pi\)
\(522\) 0 0
\(523\) −2.89279e12 −1.69067 −0.845337 − 0.534233i \(-0.820600\pi\)
−0.845337 + 0.534233i \(0.820600\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.29258e11 0.0729979
\(528\) 0 0
\(529\) 1.31277e12 0.728852
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.11072e12 0.596117
\(534\) 0 0
\(535\) 1.24271e12 0.655812
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.75784e11 0.0897076
\(540\) 0 0
\(541\) 3.09295e12 1.55233 0.776167 − 0.630527i \(-0.217161\pi\)
0.776167 + 0.630527i \(0.217161\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.65772e12 −0.804872
\(546\) 0 0
\(547\) 1.77386e12 0.847183 0.423592 − 0.905853i \(-0.360769\pi\)
0.423592 + 0.905853i \(0.360769\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.50838e12 1.15934
\(552\) 0 0
\(553\) −9.07097e11 −0.412468
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.47718e12 0.650258 0.325129 − 0.945670i \(-0.394592\pi\)
0.325129 + 0.945670i \(0.394592\pi\)
\(558\) 0 0
\(559\) −1.57881e12 −0.683877
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.11744e12 −0.888224 −0.444112 − 0.895971i \(-0.646481\pi\)
−0.444112 + 0.895971i \(0.646481\pi\)
\(564\) 0 0
\(565\) −1.75617e12 −0.725019
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.62190e12 1.44854 0.724272 − 0.689515i \(-0.242176\pi\)
0.724272 + 0.689515i \(0.242176\pi\)
\(570\) 0 0
\(571\) −1.52737e12 −0.601287 −0.300644 − 0.953737i \(-0.597201\pi\)
−0.300644 + 0.953737i \(0.597201\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 2.78129e12 1.06106
\(576\) 0 0
\(577\) 3.80010e12 1.42726 0.713631 − 0.700522i \(-0.247050\pi\)
0.713631 + 0.700522i \(0.247050\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −9.34903e10 −0.0340388
\(582\) 0 0
\(583\) 2.58572e11 0.0926986
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.03100e12 1.05369 0.526846 − 0.849961i \(-0.323374\pi\)
0.526846 + 0.849961i \(0.323374\pi\)
\(588\) 0 0
\(589\) −2.67750e11 −0.0916663
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.03554e12 1.00807 0.504034 − 0.863684i \(-0.331848\pi\)
0.504034 + 0.863684i \(0.331848\pi\)
\(594\) 0 0
\(595\) 5.05518e11 0.165352
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −5.64232e12 −1.79076 −0.895379 − 0.445304i \(-0.853096\pi\)
−0.895379 + 0.445304i \(0.853096\pi\)
\(600\) 0 0
\(601\) 3.07442e12 0.961231 0.480616 − 0.876931i \(-0.340413\pi\)
0.480616 + 0.876931i \(0.340413\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.43280e12 0.434796
\(606\) 0 0
\(607\) −7.10994e10 −0.0212577 −0.0106289 − 0.999944i \(-0.503383\pi\)
−0.0106289 + 0.999944i \(0.503383\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −3.42317e12 −0.993671
\(612\) 0 0
\(613\) 3.56214e9 0.00101892 0.000509458 − 1.00000i \(-0.499838\pi\)
0.000509458 1.00000i \(0.499838\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −2.06512e12 −0.573671 −0.286835 − 0.957980i \(-0.592603\pi\)
−0.286835 + 0.957980i \(0.592603\pi\)
\(618\) 0 0
\(619\) 1.79298e12 0.490871 0.245436 − 0.969413i \(-0.421069\pi\)
0.245436 + 0.969413i \(0.421069\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1.75477e12 0.466685
\(624\) 0 0
\(625\) 1.74786e12 0.458192
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 7.44324e12 1.89598
\(630\) 0 0
\(631\) 4.45644e12 1.11907 0.559533 − 0.828808i \(-0.310980\pi\)
0.559533 + 0.828808i \(0.310980\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.86827e11 −0.0455992
\(636\) 0 0
\(637\) −2.48851e12 −0.598843
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −2.78753e12 −0.652165 −0.326083 − 0.945341i \(-0.605729\pi\)
−0.326083 + 0.945341i \(0.605729\pi\)
\(642\) 0 0
\(643\) 3.57451e12 0.824646 0.412323 − 0.911038i \(-0.364718\pi\)
0.412323 + 0.911038i \(0.364718\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −7.08773e12 −1.59015 −0.795075 − 0.606511i \(-0.792569\pi\)
−0.795075 + 0.606511i \(0.792569\pi\)
\(648\) 0 0
\(649\) 3.80665e11 0.0842251
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.75706e12 0.593386 0.296693 − 0.954973i \(-0.404116\pi\)
0.296693 + 0.954973i \(0.404116\pi\)
\(654\) 0 0
\(655\) −6.96882e10 −0.0147936
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −5.22578e12 −1.07936 −0.539680 − 0.841870i \(-0.681455\pi\)
−0.539680 + 0.841870i \(0.681455\pi\)
\(660\) 0 0
\(661\) 2.45091e11 0.0499368 0.0249684 − 0.999688i \(-0.492051\pi\)
0.0249684 + 0.999688i \(0.492051\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.04715e12 −0.207640
\(666\) 0 0
\(667\) −5.66840e12 −1.10890
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −6.44459e10 −0.0122728
\(672\) 0 0
\(673\) 5.62685e11 0.105730 0.0528649 − 0.998602i \(-0.483165\pi\)
0.0528649 + 0.998602i \(0.483165\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.75744e12 1.05337 0.526684 − 0.850061i \(-0.323435\pi\)
0.526684 + 0.850061i \(0.323435\pi\)
\(678\) 0 0
\(679\) 1.47293e12 0.265930
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −7.47824e10 −0.0131494 −0.00657470 − 0.999978i \(-0.502093\pi\)
−0.00657470 + 0.999978i \(0.502093\pi\)
\(684\) 0 0
\(685\) −3.23829e12 −0.561963
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −3.66052e12 −0.618809
\(690\) 0 0
\(691\) −7.04040e12 −1.17475 −0.587376 − 0.809314i \(-0.699839\pi\)
−0.587376 + 0.809314i \(0.699839\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.01818e12 0.328116
\(696\) 0 0
\(697\) −5.98731e12 −0.960914
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −6.25776e12 −0.978786 −0.489393 − 0.872063i \(-0.662782\pi\)
−0.489393 + 0.872063i \(0.662782\pi\)
\(702\) 0 0
\(703\) −1.54182e13 −2.38086
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.57497e11 0.114023
\(708\) 0 0
\(709\) −2.51856e12 −0.374321 −0.187161 − 0.982329i \(-0.559929\pi\)
−0.187161 + 0.982329i \(0.559929\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 6.05057e11 0.0876786
\(714\) 0 0
\(715\) 2.12121e11 0.0303533
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9.14081e12 1.27557 0.637786 − 0.770214i \(-0.279851\pi\)
0.637786 + 0.770214i \(0.279851\pi\)
\(720\) 0 0
\(721\) 2.84531e12 0.392121
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.06288e12 −0.680576
\(726\) 0 0
\(727\) −2.60406e12 −0.345737 −0.172869 − 0.984945i \(-0.555304\pi\)
−0.172869 + 0.984945i \(0.555304\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.51057e12 1.10238
\(732\) 0 0
\(733\) −4.45281e12 −0.569726 −0.284863 − 0.958568i \(-0.591948\pi\)
−0.284863 + 0.958568i \(0.591948\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.38679e12 0.173144
\(738\) 0 0
\(739\) 4.72266e12 0.582487 0.291244 − 0.956649i \(-0.405931\pi\)
0.291244 + 0.956649i \(0.405931\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.15236e13 −1.38719 −0.693597 − 0.720363i \(-0.743975\pi\)
−0.693597 + 0.720363i \(0.743975\pi\)
\(744\) 0 0
\(745\) 4.58980e12 0.545871
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −4.42034e12 −0.513201
\(750\) 0 0
\(751\) −2.49119e12 −0.285777 −0.142888 − 0.989739i \(-0.545639\pi\)
−0.142888 + 0.989739i \(0.545639\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.54772e12 0.397364
\(756\) 0 0
\(757\) 1.01135e13 1.11936 0.559679 − 0.828709i \(-0.310924\pi\)
0.559679 + 0.828709i \(0.310924\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.95119e12 −0.210896 −0.105448 − 0.994425i \(-0.533628\pi\)
−0.105448 + 0.994425i \(0.533628\pi\)
\(762\) 0 0
\(763\) 5.89652e12 0.629847
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −5.38895e12 −0.562244
\(768\) 0 0
\(769\) −9.75660e12 −1.00607 −0.503037 − 0.864265i \(-0.667784\pi\)
−0.503037 + 0.864265i \(0.667784\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.72765e13 −1.74040 −0.870200 − 0.492698i \(-0.836011\pi\)
−0.870200 + 0.492698i \(0.836011\pi\)
\(774\) 0 0
\(775\) 5.40423e11 0.0538116
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.24023e13 1.20666
\(780\) 0 0
\(781\) −4.37460e11 −0.0420735
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −6.11303e12 −0.574570
\(786\) 0 0
\(787\) 2.09355e13 1.94535 0.972674 − 0.232177i \(-0.0745849\pi\)
0.972674 + 0.232177i \(0.0745849\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.24672e12 0.567359
\(792\) 0 0
\(793\) 9.12340e11 0.0819270
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 5.44377e12 0.477900 0.238950 − 0.971032i \(-0.423197\pi\)
0.238950 + 0.971032i \(0.423197\pi\)
\(798\) 0 0
\(799\) 1.84525e13 1.60175
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.91982e11 0.0247820
\(804\) 0 0
\(805\) 2.36633e12 0.198607
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 7.14141e12 0.586160 0.293080 − 0.956088i \(-0.405320\pi\)
0.293080 + 0.956088i \(0.405320\pi\)
\(810\) 0 0
\(811\) −1.90226e13 −1.54410 −0.772052 − 0.635560i \(-0.780769\pi\)
−0.772052 + 0.635560i \(0.780769\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 4.16798e12 0.330915
\(816\) 0 0
\(817\) −1.76291e13 −1.38430
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.33222e12 0.102337 0.0511685 − 0.998690i \(-0.483705\pi\)
0.0511685 + 0.998690i \(0.483705\pi\)
\(822\) 0 0
\(823\) 1.66745e13 1.26694 0.633468 − 0.773769i \(-0.281631\pi\)
0.633468 + 0.773769i \(0.281631\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.47735e12 −0.184167 −0.0920836 − 0.995751i \(-0.529353\pi\)
−0.0920836 + 0.995751i \(0.529353\pi\)
\(828\) 0 0
\(829\) 1.89792e13 1.39567 0.697833 − 0.716261i \(-0.254148\pi\)
0.697833 + 0.716261i \(0.254148\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.34143e13 0.965306
\(834\) 0 0
\(835\) 5.04587e12 0.359209
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −7.04560e12 −0.490896 −0.245448 − 0.969410i \(-0.578935\pi\)
−0.245448 + 0.969410i \(0.578935\pi\)
\(840\) 0 0
\(841\) −4.18875e12 −0.288737
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.50823e12 0.236719
\(846\) 0 0
\(847\) −5.09646e12 −0.340246
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.48418e13 2.27729
\(852\) 0 0
\(853\) 2.68082e12 0.173379 0.0866895 − 0.996235i \(-0.472371\pi\)
0.0866895 + 0.996235i \(0.472371\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.16824e13 0.739808 0.369904 − 0.929070i \(-0.379391\pi\)
0.369904 + 0.929070i \(0.379391\pi\)
\(858\) 0 0
\(859\) 1.00736e13 0.631272 0.315636 − 0.948880i \(-0.397782\pi\)
0.315636 + 0.948880i \(0.397782\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.20088e13 −0.736970 −0.368485 − 0.929634i \(-0.620123\pi\)
−0.368485 + 0.929634i \(0.620123\pi\)
\(864\) 0 0
\(865\) 1.08625e13 0.659716
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.05177e12 0.122050
\(870\) 0 0
\(871\) −1.96324e13 −1.15582
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 4.73264e12 0.272940
\(876\) 0 0
\(877\) −3.17089e13 −1.81002 −0.905010 − 0.425390i \(-0.860137\pi\)
−0.905010 + 0.425390i \(0.860137\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.76869e12 0.154840 0.0774200 − 0.996999i \(-0.475332\pi\)
0.0774200 + 0.996999i \(0.475332\pi\)
\(882\) 0 0
\(883\) 1.34936e13 0.746974 0.373487 − 0.927635i \(-0.378162\pi\)
0.373487 + 0.927635i \(0.378162\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −8.08601e12 −0.438610 −0.219305 − 0.975656i \(-0.570379\pi\)
−0.219305 + 0.975656i \(0.570379\pi\)
\(888\) 0 0
\(889\) 6.64543e11 0.0356833
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −3.82231e13 −2.01138
\(894\) 0 0
\(895\) 1.53159e12 0.0797881
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.10141e12 −0.0562380
\(900\) 0 0
\(901\) 1.97320e13 0.997492
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 5.35749e12 0.265487
\(906\) 0 0
\(907\) 1.84757e13 0.906503 0.453251 − 0.891383i \(-0.350264\pi\)
0.453251 + 0.891383i \(0.350264\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.22135e13 −1.54955 −0.774774 − 0.632239i \(-0.782136\pi\)
−0.774774 + 0.632239i \(0.782136\pi\)
\(912\) 0 0
\(913\) 2.11466e11 0.0100722
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.47881e11 0.0115766
\(918\) 0 0
\(919\) −2.14028e13 −0.989806 −0.494903 − 0.868948i \(-0.664796\pi\)
−0.494903 + 0.868948i \(0.664796\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.19298e12 0.280862
\(924\) 0 0
\(925\) 3.11199e13 1.39766
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.54729e13 −0.681556 −0.340778 − 0.940144i \(-0.610690\pi\)
−0.340778 + 0.940144i \(0.610690\pi\)
\(930\) 0 0
\(931\) −2.77868e13 −1.21217
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.14343e12 −0.0489282
\(936\) 0 0
\(937\) −2.20614e12 −0.0934986 −0.0467493 − 0.998907i \(-0.514886\pi\)
−0.0467493 + 0.998907i \(0.514886\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.94044e12 0.0806765 0.0403383 − 0.999186i \(-0.487156\pi\)
0.0403383 + 0.999186i \(0.487156\pi\)
\(942\) 0 0
\(943\) −2.80266e13 −1.15416
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.34471e12 −0.337160 −0.168580 − 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(948\) 0 0
\(949\) −4.13349e12 −0.165432
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.30168e13 −1.29663 −0.648317 − 0.761371i \(-0.724527\pi\)
−0.648317 + 0.761371i \(0.724527\pi\)
\(954\) 0 0
\(955\) −1.23950e13 −0.482205
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.15186e13 0.439760
\(960\) 0 0
\(961\) −2.63221e13 −0.995553
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.76672e13 0.655834
\(966\) 0 0
\(967\) −3.32794e12 −0.122393 −0.0611964 − 0.998126i \(-0.519492\pi\)
−0.0611964 + 0.998126i \(0.519492\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.29308e13 0.827815 0.413908 − 0.910319i \(-0.364164\pi\)
0.413908 + 0.910319i \(0.364164\pi\)
\(972\) 0 0
\(973\) −7.17866e12 −0.256765
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.28330e13 1.15288 0.576440 − 0.817139i \(-0.304441\pi\)
0.576440 + 0.817139i \(0.304441\pi\)
\(978\) 0 0
\(979\) −3.96912e12 −0.138093
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.72706e13 −1.61473 −0.807366 − 0.590050i \(-0.799108\pi\)
−0.807366 + 0.590050i \(0.799108\pi\)
\(984\) 0 0
\(985\) −2.83919e12 −0.0961019
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.98379e13 1.32408
\(990\) 0 0
\(991\) −5.82427e13 −1.91827 −0.959136 − 0.282944i \(-0.908689\pi\)
−0.959136 + 0.282944i \(0.908689\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.58810e11 −0.0148398
\(996\) 0 0
\(997\) −3.52621e13 −1.13026 −0.565132 − 0.825000i \(-0.691175\pi\)
−0.565132 + 0.825000i \(0.691175\pi\)
\(998\) 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 72.10.a.c.1.1 1
3.2 odd 2 24.10.a.c.1.1 ✓ 1
4.3 odd 2 144.10.a.f.1.1 1
12.11 even 2 48.10.a.c.1.1 1
24.5 odd 2 192.10.a.d.1.1 1
24.11 even 2 192.10.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.10.a.c.1.1 ✓ 1 3.2 odd 2
48.10.a.c.1.1 1 12.11 even 2
72.10.a.c.1.1 1 1.1 even 1 trivial
144.10.a.f.1.1 1 4.3 odd 2
192.10.a.d.1.1 1 24.5 odd 2
192.10.a.k.1.1 1 24.11 even 2