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Properties

Label 72.10.a.b.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)
 
\(f(q)\) \(=\) \(q-830.000 q^{5} +672.000 q^{7} +O(q^{10})\) \(q-830.000 q^{5} +672.000 q^{7} +73468.0 q^{11} -78242.0 q^{13} +161726. q^{17} -653572. q^{19} +1.06670e6 q^{23} -1.26422e6 q^{25} -3.82484e6 q^{29} -1.57948e6 q^{31} -557760. q^{35} +1.60156e7 q^{37} -2.62683e7 q^{41} -4.44952e7 q^{43} -1.43242e7 q^{47} -3.99020e7 q^{49} +2.43860e7 q^{53} -6.09784e7 q^{55} -1.19421e7 q^{59} -1.89740e8 q^{61} +6.49409e7 q^{65} -1.06710e8 q^{67} -3.02754e8 q^{71} +8.17695e7 q^{73} +4.93705e7 q^{77} +3.15315e8 q^{79} -7.52833e8 q^{83} -1.34233e8 q^{85} +4.33284e8 q^{89} -5.25786e7 q^{91} +5.42465e8 q^{95} +1.28250e9 q^{97} +O(q^{100})\)

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
\(5\) −830.000 −0.593900 −0.296950 − 0.954893i \(-0.595969\pi\)
−0.296950 + 0.954893i \(0.595969\pi\)
\(6\) 0 0
\(7\) 672.000 0.105786 0.0528930 − 0.998600i \(-0.483156\pi\)
0.0528930 + 0.998600i \(0.483156\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 73468.0 1.51297 0.756486 − 0.654009i \(-0.226914\pi\)
0.756486 + 0.654009i \(0.226914\pi\)
\(12\) 0 0
\(13\) −78242.0 −0.759792 −0.379896 − 0.925029i \(-0.624040\pi\)
−0.379896 + 0.925029i \(0.624040\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 161726. 0.469634 0.234817 − 0.972040i \(-0.424551\pi\)
0.234817 + 0.972040i \(0.424551\pi\)
\(18\) 0 0
\(19\) −653572. −1.15054 −0.575271 − 0.817963i \(-0.695103\pi\)
−0.575271 + 0.817963i \(0.695103\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.06670e6 0.794814 0.397407 − 0.917642i \(-0.369910\pi\)
0.397407 + 0.917642i \(0.369910\pi\)
\(24\) 0 0
\(25\) −1.26422e6 −0.647283
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.82484e6 −1.00420 −0.502102 − 0.864808i \(-0.667440\pi\)
−0.502102 + 0.864808i \(0.667440\pi\)
\(30\) 0 0
\(31\) −1.57948e6 −0.307175 −0.153588 − 0.988135i \(-0.549083\pi\)
−0.153588 + 0.988135i \(0.549083\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −557760. −0.0628263
\(36\) 0 0
\(37\) 1.60156e7 1.40487 0.702433 − 0.711749i \(-0.252097\pi\)
0.702433 + 0.711749i \(0.252097\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.62683e7 −1.45179 −0.725896 − 0.687805i \(-0.758575\pi\)
−0.725896 + 0.687805i \(0.758575\pi\)
\(42\) 0 0
\(43\) −4.44952e7 −1.98475 −0.992374 − 0.123263i \(-0.960664\pi\)
−0.992374 + 0.123263i \(0.960664\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.43242e7 −0.428182 −0.214091 − 0.976814i \(-0.568679\pi\)
−0.214091 + 0.976814i \(0.568679\pi\)
\(48\) 0 0
\(49\) −3.99020e7 −0.988809
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.43860e7 0.424522 0.212261 − 0.977213i \(-0.431917\pi\)
0.212261 + 0.977213i \(0.431917\pi\)
\(54\) 0 0
\(55\) −6.09784e7 −0.898554
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.19421e7 −0.128306 −0.0641529 − 0.997940i \(-0.520435\pi\)
−0.0641529 + 0.997940i \(0.520435\pi\)
\(60\) 0 0
\(61\) −1.89740e8 −1.75459 −0.877294 − 0.479953i \(-0.840654\pi\)
−0.877294 + 0.479953i \(0.840654\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 6.49409e7 0.451240
\(66\) 0 0
\(67\) −1.06710e8 −0.646944 −0.323472 − 0.946238i \(-0.604850\pi\)
−0.323472 + 0.946238i \(0.604850\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −3.02754e8 −1.41393 −0.706965 − 0.707249i \(-0.749936\pi\)
−0.706965 + 0.707249i \(0.749936\pi\)
\(72\) 0 0
\(73\) 8.17695e7 0.337007 0.168503 − 0.985701i \(-0.446107\pi\)
0.168503 + 0.985701i \(0.446107\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.93705e7 0.160051
\(78\) 0 0
\(79\) 3.15315e8 0.910800 0.455400 − 0.890287i \(-0.349496\pi\)
0.455400 + 0.890287i \(0.349496\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.52833e8 −1.74119 −0.870597 − 0.491996i \(-0.836267\pi\)
−0.870597 + 0.491996i \(0.836267\pi\)
\(84\) 0 0
\(85\) −1.34233e8 −0.278916
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.33284e8 0.732012 0.366006 − 0.930613i \(-0.380725\pi\)
0.366006 + 0.930613i \(0.380725\pi\)
\(90\) 0 0
\(91\) −5.25786e7 −0.0803754
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.42465e8 0.683306
\(96\) 0 0
\(97\) 1.28250e9 1.47090 0.735450 − 0.677578i \(-0.236971\pi\)
0.735450 + 0.677578i \(0.236971\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.96115e9 1.87528 0.937639 − 0.347611i \(-0.113007\pi\)
0.937639 + 0.347611i \(0.113007\pi\)
\(102\) 0 0
\(103\) −9.48061e8 −0.829982 −0.414991 − 0.909825i \(-0.636215\pi\)
−0.414991 + 0.909825i \(0.636215\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.46613e8 0.181881 0.0909407 − 0.995856i \(-0.471013\pi\)
0.0909407 + 0.995856i \(0.471013\pi\)
\(108\) 0 0
\(109\) 1.47308e9 0.999556 0.499778 − 0.866154i \(-0.333415\pi\)
0.499778 + 0.866154i \(0.333415\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.66579e8 −0.442287 −0.221143 − 0.975241i \(-0.570979\pi\)
−0.221143 + 0.975241i \(0.570979\pi\)
\(114\) 0 0
\(115\) −8.85358e8 −0.472040
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.08680e8 0.0496807
\(120\) 0 0
\(121\) 3.03960e9 1.28909
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.67040e9 0.978321
\(126\) 0 0
\(127\) 3.08546e9 1.05246 0.526228 − 0.850344i \(-0.323606\pi\)
0.526228 + 0.850344i \(0.323606\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.76317e9 −0.819759 −0.409880 − 0.912140i \(-0.634429\pi\)
−0.409880 + 0.912140i \(0.634429\pi\)
\(132\) 0 0
\(133\) −4.39200e8 −0.121711
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.75041e9 −0.667044 −0.333522 − 0.942742i \(-0.608237\pi\)
−0.333522 + 0.942742i \(0.608237\pi\)
\(138\) 0 0
\(139\) 1.58704e9 0.360597 0.180299 − 0.983612i \(-0.442294\pi\)
0.180299 + 0.983612i \(0.442294\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.74828e9 −1.14955
\(144\) 0 0
\(145\) 3.17462e9 0.596397
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.16328e9 1.52305 0.761523 − 0.648138i \(-0.224452\pi\)
0.761523 + 0.648138i \(0.224452\pi\)
\(150\) 0 0
\(151\) −6.46073e9 −1.01131 −0.505656 − 0.862735i \(-0.668750\pi\)
−0.505656 + 0.862735i \(0.668750\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.31097e9 0.182431
\(156\) 0 0
\(157\) −1.34930e10 −1.77239 −0.886194 − 0.463315i \(-0.846660\pi\)
−0.886194 + 0.463315i \(0.846660\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.16820e8 0.0840802
\(162\) 0 0
\(163\) −9.11445e9 −1.01131 −0.505657 − 0.862734i \(-0.668750\pi\)
−0.505657 + 0.862734i \(0.668750\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.28739e9 −0.725016 −0.362508 − 0.931981i \(-0.618079\pi\)
−0.362508 + 0.931981i \(0.618079\pi\)
\(168\) 0 0
\(169\) −4.48269e9 −0.422716
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.79271e9 −0.152161 −0.0760804 − 0.997102i \(-0.524241\pi\)
−0.0760804 + 0.997102i \(0.524241\pi\)
\(174\) 0 0
\(175\) −8.49559e8 −0.0684735
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.58921e9 −0.261313 −0.130656 − 0.991428i \(-0.541708\pi\)
−0.130656 + 0.991428i \(0.541708\pi\)
\(180\) 0 0
\(181\) −2.34516e10 −1.62412 −0.812060 − 0.583574i \(-0.801654\pi\)
−0.812060 + 0.583574i \(0.801654\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.32929e10 −0.834350
\(186\) 0 0
\(187\) 1.18817e10 0.710544
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.13072e10 1.15845 0.579224 − 0.815169i \(-0.303356\pi\)
0.579224 + 0.815169i \(0.303356\pi\)
\(192\) 0 0
\(193\) 2.02497e10 1.05054 0.525269 − 0.850936i \(-0.323965\pi\)
0.525269 + 0.850936i \(0.323965\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.27339e10 −0.602372 −0.301186 − 0.953565i \(-0.597382\pi\)
−0.301186 + 0.953565i \(0.597382\pi\)
\(198\) 0 0
\(199\) −5.66200e9 −0.255936 −0.127968 − 0.991778i \(-0.540845\pi\)
−0.127968 + 0.991778i \(0.540845\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.57029e9 −0.106231
\(204\) 0 0
\(205\) 2.18027e10 0.862219
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −4.80166e10 −1.74074
\(210\) 0 0
\(211\) 5.34254e9 0.185557 0.0927783 − 0.995687i \(-0.470425\pi\)
0.0927783 + 0.995687i \(0.470425\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.69310e10 1.17874
\(216\) 0 0
\(217\) −1.06141e9 −0.0324949
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.26538e10 −0.356824
\(222\) 0 0
\(223\) 1.53127e10 0.414649 0.207325 − 0.978272i \(-0.433524\pi\)
0.207325 + 0.978272i \(0.433524\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.24911e10 1.31211 0.656053 − 0.754714i \(-0.272225\pi\)
0.656053 + 0.754714i \(0.272225\pi\)
\(228\) 0 0
\(229\) 4.27719e9 0.102778 0.0513888 − 0.998679i \(-0.483635\pi\)
0.0513888 + 0.998679i \(0.483635\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4.40992e10 0.980232 0.490116 − 0.871657i \(-0.336954\pi\)
0.490116 + 0.871657i \(0.336954\pi\)
\(234\) 0 0
\(235\) 1.18891e10 0.254297
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.95413e10 0.783899 0.391949 − 0.919987i \(-0.371801\pi\)
0.391949 + 0.919987i \(0.371801\pi\)
\(240\) 0 0
\(241\) −4.42570e10 −0.845094 −0.422547 − 0.906341i \(-0.638864\pi\)
−0.422547 + 0.906341i \(0.638864\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.31187e10 0.587254
\(246\) 0 0
\(247\) 5.11368e10 0.874172
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.93264e10 −0.466366 −0.233183 − 0.972433i \(-0.574914\pi\)
−0.233183 + 0.972433i \(0.574914\pi\)
\(252\) 0 0
\(253\) 7.83680e10 1.20253
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.75455e9 0.0679846 0.0339923 − 0.999422i \(-0.489178\pi\)
0.0339923 + 0.999422i \(0.489178\pi\)
\(258\) 0 0
\(259\) 1.07625e10 0.148615
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.32650e7 0.000557617 0 0.000278809 − 1.00000i \(-0.499911\pi\)
0.000278809 1.00000i \(0.499911\pi\)
\(264\) 0 0
\(265\) −2.02404e10 −0.252123
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.74801e10 −0.785761 −0.392881 − 0.919590i \(-0.628521\pi\)
−0.392881 + 0.919590i \(0.628521\pi\)
\(270\) 0 0
\(271\) 1.17456e11 1.32286 0.661431 − 0.750006i \(-0.269950\pi\)
0.661431 + 0.750006i \(0.269950\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −9.28801e10 −0.979322
\(276\) 0 0
\(277\) −4.12220e10 −0.420698 −0.210349 − 0.977626i \(-0.567460\pi\)
−0.210349 + 0.977626i \(0.567460\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.48899e10 0.812227 0.406114 − 0.913823i \(-0.366884\pi\)
0.406114 + 0.913823i \(0.366884\pi\)
\(282\) 0 0
\(283\) −3.45640e10 −0.320321 −0.160160 − 0.987091i \(-0.551201\pi\)
−0.160160 + 0.987091i \(0.551201\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −1.76523e10 −0.153579
\(288\) 0 0
\(289\) −9.24326e10 −0.779444
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.21053e11 −1.75223 −0.876116 − 0.482100i \(-0.839874\pi\)
−0.876116 + 0.482100i \(0.839874\pi\)
\(294\) 0 0
\(295\) 9.91193e9 0.0762007
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −8.34604e10 −0.603893
\(300\) 0 0
\(301\) −2.99008e10 −0.209959
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.57484e11 1.04205
\(306\) 0 0
\(307\) −2.52457e11 −1.62205 −0.811024 − 0.585012i \(-0.801090\pi\)
−0.811024 + 0.585012i \(0.801090\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.13082e10 0.553462 0.276731 − 0.960947i \(-0.410749\pi\)
0.276731 + 0.960947i \(0.410749\pi\)
\(312\) 0 0
\(313\) 4.06665e10 0.239490 0.119745 − 0.992805i \(-0.461792\pi\)
0.119745 + 0.992805i \(0.461792\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.10345e11 1.16995 0.584974 − 0.811052i \(-0.301105\pi\)
0.584974 + 0.811052i \(0.301105\pi\)
\(318\) 0 0
\(319\) −2.81003e11 −1.51933
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.05700e11 −0.540334
\(324\) 0 0
\(325\) 9.89155e10 0.491801
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.62584e9 −0.0452957
\(330\) 0 0
\(331\) −3.07269e11 −1.40700 −0.703499 − 0.710697i \(-0.748380\pi\)
−0.703499 + 0.710697i \(0.748380\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.85689e10 0.384220
\(336\) 0 0
\(337\) −1.81037e11 −0.764596 −0.382298 − 0.924039i \(-0.624867\pi\)
−0.382298 + 0.924039i \(0.624867\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.16041e11 −0.464748
\(342\) 0 0
\(343\) −5.39318e10 −0.210388
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5.22927e11 −1.93624 −0.968119 − 0.250491i \(-0.919408\pi\)
−0.968119 + 0.250491i \(0.919408\pi\)
\(348\) 0 0
\(349\) 3.04567e11 1.09893 0.549463 − 0.835518i \(-0.314832\pi\)
0.549463 + 0.835518i \(0.314832\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −7.04187e10 −0.241380 −0.120690 − 0.992690i \(-0.538511\pi\)
−0.120690 + 0.992690i \(0.538511\pi\)
\(354\) 0 0
\(355\) 2.51286e11 0.839732
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.91400e11 0.925900 0.462950 − 0.886384i \(-0.346791\pi\)
0.462950 + 0.886384i \(0.346791\pi\)
\(360\) 0 0
\(361\) 1.04469e11 0.323745
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.78687e10 −0.200148
\(366\) 0 0
\(367\) 2.65943e11 0.765230 0.382615 − 0.923908i \(-0.375024\pi\)
0.382615 + 0.923908i \(0.375024\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.63874e10 0.0449085
\(372\) 0 0
\(373\) 3.21656e11 0.860403 0.430201 − 0.902733i \(-0.358443\pi\)
0.430201 + 0.902733i \(0.358443\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.99263e11 0.762987
\(378\) 0 0
\(379\) 2.83177e11 0.704988 0.352494 − 0.935814i \(-0.385334\pi\)
0.352494 + 0.935814i \(0.385334\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −6.70279e11 −1.59170 −0.795850 − 0.605494i \(-0.792975\pi\)
−0.795850 + 0.605494i \(0.792975\pi\)
\(384\) 0 0
\(385\) −4.09775e10 −0.0950544
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.67505e11 1.92087 0.960437 − 0.278498i \(-0.0898366\pi\)
0.960437 + 0.278498i \(0.0898366\pi\)
\(390\) 0 0
\(391\) 1.72512e11 0.373272
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.61712e11 −0.540924
\(396\) 0 0
\(397\) −3.14512e11 −0.635449 −0.317724 − 0.948183i \(-0.602919\pi\)
−0.317724 + 0.948183i \(0.602919\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.95667e11 0.377892 0.188946 − 0.981987i \(-0.439493\pi\)
0.188946 + 0.981987i \(0.439493\pi\)
\(402\) 0 0
\(403\) 1.23582e11 0.233390
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.17663e12 2.12553
\(408\) 0 0
\(409\) 3.63692e11 0.642656 0.321328 − 0.946968i \(-0.395871\pi\)
0.321328 + 0.946968i \(0.395871\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −8.02508e9 −0.0135729
\(414\) 0 0
\(415\) 6.24852e11 1.03410
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.20311e11 −0.349199 −0.174600 − 0.984640i \(-0.555863\pi\)
−0.174600 + 0.984640i \(0.555863\pi\)
\(420\) 0 0
\(421\) −9.96326e11 −1.54572 −0.772862 − 0.634574i \(-0.781175\pi\)
−0.772862 + 0.634574i \(0.781175\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.04458e11 −0.303986
\(426\) 0 0
\(427\) −1.27505e11 −0.185611
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.96850e11 1.39150 0.695748 − 0.718286i \(-0.255073\pi\)
0.695748 + 0.718286i \(0.255073\pi\)
\(432\) 0 0
\(433\) 8.11011e11 1.10874 0.554372 − 0.832269i \(-0.312959\pi\)
0.554372 + 0.832269i \(0.312959\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.97163e11 −0.914466
\(438\) 0 0
\(439\) −1.24677e11 −0.160212 −0.0801062 − 0.996786i \(-0.525526\pi\)
−0.0801062 + 0.996786i \(0.525526\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.13417e11 −0.633364 −0.316682 − 0.948532i \(-0.602569\pi\)
−0.316682 + 0.948532i \(0.602569\pi\)
\(444\) 0 0
\(445\) −3.59626e11 −0.434741
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7.29234e11 −0.846756 −0.423378 − 0.905953i \(-0.639156\pi\)
−0.423378 + 0.905953i \(0.639156\pi\)
\(450\) 0 0
\(451\) −1.92988e12 −2.19652
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.36403e10 0.0477349
\(456\) 0 0
\(457\) 1.10073e12 1.18048 0.590240 − 0.807227i \(-0.299033\pi\)
0.590240 + 0.807227i \(0.299033\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.26400e12 1.30345 0.651723 − 0.758457i \(-0.274047\pi\)
0.651723 + 0.758457i \(0.274047\pi\)
\(462\) 0 0
\(463\) 1.82843e12 1.84911 0.924557 − 0.381045i \(-0.124436\pi\)
0.924557 + 0.381045i \(0.124436\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.95640e10 0.0968671 0.0484336 − 0.998826i \(-0.484577\pi\)
0.0484336 + 0.998826i \(0.484577\pi\)
\(468\) 0 0
\(469\) −7.17088e10 −0.0684376
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.26898e12 −3.00287
\(474\) 0 0
\(475\) 8.26262e11 0.744726
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1.32763e12 1.15230 0.576150 − 0.817344i \(-0.304554\pi\)
0.576150 + 0.817344i \(0.304554\pi\)
\(480\) 0 0
\(481\) −1.25309e12 −1.06741
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.06447e12 −0.873568
\(486\) 0 0
\(487\) −1.37834e12 −1.11039 −0.555197 − 0.831719i \(-0.687357\pi\)
−0.555197 + 0.831719i \(0.687357\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.13645e12 −0.882439 −0.441219 − 0.897399i \(-0.645454\pi\)
−0.441219 + 0.897399i \(0.645454\pi\)
\(492\) 0 0
\(493\) −6.18576e11 −0.471609
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.03451e11 −0.149574
\(498\) 0 0
\(499\) −1.91132e12 −1.38001 −0.690004 − 0.723806i \(-0.742391\pi\)
−0.690004 + 0.723806i \(0.742391\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.79892e12 1.25302 0.626509 − 0.779414i \(-0.284483\pi\)
0.626509 + 0.779414i \(0.284483\pi\)
\(504\) 0 0
\(505\) −1.62776e12 −1.11373
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.65330e12 −1.09174 −0.545872 − 0.837869i \(-0.683801\pi\)
−0.545872 + 0.837869i \(0.683801\pi\)
\(510\) 0 0
\(511\) 5.49491e10 0.0356506
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.86891e11 0.492926
\(516\) 0 0
\(517\) −1.05237e12 −0.647828
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.43255e12 1.44641 0.723207 − 0.690632i \(-0.242667\pi\)
0.723207 + 0.690632i \(0.242667\pi\)
\(522\) 0 0
\(523\) 1.35462e12 0.791697 0.395849 − 0.918316i \(-0.370451\pi\)
0.395849 + 0.918316i \(0.370451\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.55443e11 −0.144260
\(528\) 0 0
\(529\) −6.63312e11 −0.368271
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.05528e12 1.10306
\(534\) 0 0
\(535\) −2.04689e11 −0.108019
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.93152e12 −1.49604
\(540\) 0 0
\(541\) 7.25578e11 0.364163 0.182082 − 0.983283i \(-0.441716\pi\)
0.182082 + 0.983283i \(0.441716\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.22266e12 −0.593636
\(546\) 0 0
\(547\) −2.32193e12 −1.10893 −0.554467 − 0.832206i \(-0.687078\pi\)
−0.554467 + 0.832206i \(0.687078\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.49981e12 1.15538
\(552\) 0 0
\(553\) 2.11892e11 0.0963499
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.15050e12 0.506452 0.253226 − 0.967407i \(-0.418508\pi\)
0.253226 + 0.967407i \(0.418508\pi\)
\(558\) 0 0
\(559\) 3.48140e12 1.50800
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3.26347e12 −1.36897 −0.684483 − 0.729029i \(-0.739972\pi\)
−0.684483 + 0.729029i \(0.739972\pi\)
\(564\) 0 0
\(565\) 6.36261e11 0.262674
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.14577e12 1.25812 0.629059 − 0.777357i \(-0.283440\pi\)
0.629059 + 0.777357i \(0.283440\pi\)
\(570\) 0 0
\(571\) 2.86591e12 1.12824 0.564119 − 0.825694i \(-0.309216\pi\)
0.564119 + 0.825694i \(0.309216\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.34854e12 −0.514470
\(576\) 0 0
\(577\) 3.40532e12 1.27899 0.639494 − 0.768796i \(-0.279144\pi\)
0.639494 + 0.768796i \(0.279144\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5.05904e11 −0.184194
\(582\) 0 0
\(583\) 1.79159e12 0.642290
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.19783e11 0.145933 0.0729665 − 0.997334i \(-0.476753\pi\)
0.0729665 + 0.997334i \(0.476753\pi\)
\(588\) 0 0
\(589\) 1.03230e12 0.353418
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.91495e9 0.00130011 0.000650055 − 1.00000i \(-0.499793\pi\)
0.000650055 1.00000i \(0.499793\pi\)
\(594\) 0 0
\(595\) −9.02043e10 −0.0295054
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −9.79764e11 −0.310957 −0.155479 − 0.987839i \(-0.549692\pi\)
−0.155479 + 0.987839i \(0.549692\pi\)
\(600\) 0 0
\(601\) −3.20416e11 −0.100179 −0.0500897 − 0.998745i \(-0.515951\pi\)
−0.0500897 + 0.998745i \(0.515951\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −2.52287e12 −0.765588
\(606\) 0 0
\(607\) 4.50671e11 0.134744 0.0673721 − 0.997728i \(-0.478539\pi\)
0.0673721 + 0.997728i \(0.478539\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.12075e12 0.325330
\(612\) 0 0
\(613\) 4.91821e12 1.40681 0.703404 − 0.710790i \(-0.251662\pi\)
0.703404 + 0.710790i \(0.251662\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 3.02063e12 0.839102 0.419551 − 0.907732i \(-0.362188\pi\)
0.419551 + 0.907732i \(0.362188\pi\)
\(618\) 0 0
\(619\) −1.29184e12 −0.353672 −0.176836 − 0.984240i \(-0.556586\pi\)
−0.176836 + 0.984240i \(0.556586\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.91167e11 0.0774366
\(624\) 0 0
\(625\) 2.52757e11 0.0662587
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.59014e12 0.659774
\(630\) 0 0
\(631\) −7.29259e12 −1.83126 −0.915629 − 0.402024i \(-0.868307\pi\)
−0.915629 + 0.402024i \(0.868307\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −2.56093e12 −0.625053
\(636\) 0 0
\(637\) 3.12201e12 0.751290
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.42622e12 1.50347 0.751734 − 0.659466i \(-0.229218\pi\)
0.751734 + 0.659466i \(0.229218\pi\)
\(642\) 0 0
\(643\) −1.85597e12 −0.428175 −0.214087 − 0.976814i \(-0.568678\pi\)
−0.214087 + 0.976814i \(0.568678\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6.59916e12 1.48054 0.740269 − 0.672311i \(-0.234698\pi\)
0.740269 + 0.672311i \(0.234698\pi\)
\(648\) 0 0
\(649\) −8.77361e11 −0.194123
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −3.05288e12 −0.657054 −0.328527 − 0.944495i \(-0.606552\pi\)
−0.328527 + 0.944495i \(0.606552\pi\)
\(654\) 0 0
\(655\) 2.29343e12 0.486855
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.04254e12 1.86770 0.933848 − 0.357669i \(-0.116429\pi\)
0.933848 + 0.357669i \(0.116429\pi\)
\(660\) 0 0
\(661\) −2.42286e12 −0.493652 −0.246826 − 0.969060i \(-0.579388\pi\)
−0.246826 + 0.969060i \(0.579388\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.64536e11 0.0722842
\(666\) 0 0
\(667\) −4.07994e12 −0.798155
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.39398e13 −2.65464
\(672\) 0 0
\(673\) −5.42076e12 −1.01857 −0.509287 − 0.860597i \(-0.670091\pi\)
−0.509287 + 0.860597i \(0.670091\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −5.70581e12 −1.04392 −0.521961 − 0.852969i \(-0.674799\pi\)
−0.521961 + 0.852969i \(0.674799\pi\)
\(678\) 0 0
\(679\) 8.61838e11 0.155601
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −9.01096e12 −1.58445 −0.792224 − 0.610231i \(-0.791077\pi\)
−0.792224 + 0.610231i \(0.791077\pi\)
\(684\) 0 0
\(685\) 2.28284e12 0.396157
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.90801e12 −0.322548
\(690\) 0 0
\(691\) 3.93638e12 0.656819 0.328410 − 0.944535i \(-0.393487\pi\)
0.328410 + 0.944535i \(0.393487\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.31725e12 −0.214159
\(696\) 0 0
\(697\) −4.24826e12 −0.681811
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 2.61506e12 0.409026 0.204513 − 0.978864i \(-0.434439\pi\)
0.204513 + 0.978864i \(0.434439\pi\)
\(702\) 0 0
\(703\) −1.04673e13 −1.61636
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.31790e12 0.198378
\(708\) 0 0
\(709\) 6.29669e12 0.935845 0.467923 − 0.883769i \(-0.345003\pi\)
0.467923 + 0.883769i \(0.345003\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.68482e12 −0.244147
\(714\) 0 0
\(715\) 4.77108e12 0.682714
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.23756e12 −0.451791 −0.225895 − 0.974152i \(-0.572531\pi\)
−0.225895 + 0.974152i \(0.572531\pi\)
\(720\) 0 0
\(721\) −6.37097e11 −0.0878005
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.83546e12 0.650005
\(726\) 0 0
\(727\) −3.73928e12 −0.496459 −0.248230 − 0.968701i \(-0.579849\pi\)
−0.248230 + 0.968701i \(0.579849\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7.19604e12 −0.932106
\(732\) 0 0
\(733\) −8.14796e12 −1.04251 −0.521256 − 0.853400i \(-0.674536\pi\)
−0.521256 + 0.853400i \(0.674536\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −7.83974e12 −0.978809
\(738\) 0 0
\(739\) −1.90382e12 −0.234815 −0.117408 − 0.993084i \(-0.537458\pi\)
−0.117408 + 0.993084i \(0.537458\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.95774e12 0.957945 0.478972 − 0.877830i \(-0.341009\pi\)
0.478972 + 0.877830i \(0.341009\pi\)
\(744\) 0 0
\(745\) −7.60552e12 −0.904536
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.65724e11 0.0192405
\(750\) 0 0
\(751\) 1.55077e13 1.77897 0.889483 − 0.456969i \(-0.151065\pi\)
0.889483 + 0.456969i \(0.151065\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.36241e12 0.600618
\(756\) 0 0
\(757\) −5.49750e12 −0.608463 −0.304231 − 0.952598i \(-0.598400\pi\)
−0.304231 + 0.952598i \(0.598400\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.81389e12 −0.412228 −0.206114 − 0.978528i \(-0.566082\pi\)
−0.206114 + 0.978528i \(0.566082\pi\)
\(762\) 0 0
\(763\) 9.89910e11 0.105739
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 9.34373e11 0.0974857
\(768\) 0 0
\(769\) 1.01006e12 0.104155 0.0520775 − 0.998643i \(-0.483416\pi\)
0.0520775 + 0.998643i \(0.483416\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.41222e12 0.645953 0.322976 − 0.946407i \(-0.395317\pi\)
0.322976 + 0.946407i \(0.395317\pi\)
\(774\) 0 0
\(775\) 1.99682e12 0.198830
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.71682e13 1.67035
\(780\) 0 0
\(781\) −2.22428e13 −2.13924
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.11992e13 1.05262
\(786\) 0 0
\(787\) −1.11599e12 −0.103699 −0.0518494 − 0.998655i \(-0.516512\pi\)
−0.0518494 + 0.998655i \(0.516512\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.15141e11 −0.0467877
\(792\) 0 0
\(793\) 1.48457e13 1.33312
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.36680e13 −1.19989 −0.599947 − 0.800040i \(-0.704812\pi\)
−0.599947 + 0.800040i \(0.704812\pi\)
\(798\) 0 0
\(799\) −2.31659e12 −0.201089
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.00745e12 0.509882
\(804\) 0 0
\(805\) −5.94960e11 −0.0499352
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.54771e13 −1.27035 −0.635174 − 0.772369i \(-0.719071\pi\)
−0.635174 + 0.772369i \(0.719071\pi\)
\(810\) 0 0
\(811\) 8.55212e12 0.694192 0.347096 − 0.937830i \(-0.387168\pi\)
0.347096 + 0.937830i \(0.387168\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 7.56499e12 0.600619
\(816\) 0 0
\(817\) 2.90808e13 2.28353
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −8.29284e12 −0.637029 −0.318515 − 0.947918i \(-0.603184\pi\)
−0.318515 + 0.947918i \(0.603184\pi\)
\(822\) 0 0
\(823\) −6.55958e12 −0.498399 −0.249199 − 0.968452i \(-0.580167\pi\)
−0.249199 + 0.968452i \(0.580167\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.63826e13 1.21789 0.608944 − 0.793214i \(-0.291594\pi\)
0.608944 + 0.793214i \(0.291594\pi\)
\(828\) 0 0
\(829\) 2.16258e13 1.59029 0.795144 − 0.606420i \(-0.207395\pi\)
0.795144 + 0.606420i \(0.207395\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.45319e12 −0.464379
\(834\) 0 0
\(835\) 6.04853e12 0.430587
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.73801e13 −1.21094 −0.605471 − 0.795868i \(-0.707015\pi\)
−0.605471 + 0.795868i \(0.707015\pi\)
\(840\) 0 0
\(841\) 1.22240e11 0.00842617
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.72063e12 0.251051
\(846\) 0 0
\(847\) 2.04261e12 0.136367
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.70838e13 1.11661
\(852\) 0 0
\(853\) 4.42386e12 0.286109 0.143054 − 0.989715i \(-0.454308\pi\)
0.143054 + 0.989715i \(0.454308\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.01532e13 −0.642966 −0.321483 − 0.946915i \(-0.604181\pi\)
−0.321483 + 0.946915i \(0.604181\pi\)
\(858\) 0 0
\(859\) 2.64047e13 1.65467 0.827337 − 0.561706i \(-0.189855\pi\)
0.827337 + 0.561706i \(0.189855\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.60032e12 0.0982104 0.0491052 − 0.998794i \(-0.484363\pi\)
0.0491052 + 0.998794i \(0.484363\pi\)
\(864\) 0 0
\(865\) 1.48795e12 0.0903683
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 2.31656e13 1.37802
\(870\) 0 0
\(871\) 8.34917e12 0.491543
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.79451e12 0.103493
\(876\) 0 0
\(877\) −1.44970e13 −0.827525 −0.413763 − 0.910385i \(-0.635786\pi\)
−0.413763 + 0.910385i \(0.635786\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.26716e13 −0.708661 −0.354330 − 0.935120i \(-0.615291\pi\)
−0.354330 + 0.935120i \(0.615291\pi\)
\(882\) 0 0
\(883\) −3.35703e13 −1.85837 −0.929183 − 0.369619i \(-0.879488\pi\)
−0.929183 + 0.369619i \(0.879488\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.82723e13 −0.991145 −0.495573 − 0.868567i \(-0.665042\pi\)
−0.495573 + 0.868567i \(0.665042\pi\)
\(888\) 0 0
\(889\) 2.07343e12 0.111335
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 9.36187e12 0.492642
\(894\) 0 0
\(895\) 2.97905e12 0.155194
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.04126e12 0.308467
\(900\) 0 0
\(901\) 3.94386e12 0.199370
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.94648e13 0.964564
\(906\) 0 0
\(907\) 6.92303e12 0.339675 0.169837 − 0.985472i \(-0.445676\pi\)
0.169837 + 0.985472i \(0.445676\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.10020e13 −1.49127 −0.745636 − 0.666353i \(-0.767854\pi\)
−0.745636 + 0.666353i \(0.767854\pi\)
\(912\) 0 0
\(913\) −5.53092e13 −2.63438
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.85685e12 −0.0867191
\(918\) 0 0
\(919\) −2.81550e13 −1.30208 −0.651038 − 0.759045i \(-0.725666\pi\)
−0.651038 + 0.759045i \(0.725666\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.36881e13 1.07429
\(924\) 0 0
\(925\) −2.02473e13 −0.909347
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −2.28510e13 −1.00655 −0.503273 − 0.864127i \(-0.667871\pi\)
−0.503273 + 0.864127i \(0.667871\pi\)
\(930\) 0 0
\(931\) 2.60788e13 1.13767
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −9.86180e12 −0.421992
\(936\) 0 0
\(937\) 3.16648e13 1.34199 0.670995 − 0.741462i \(-0.265867\pi\)
0.670995 + 0.741462i \(0.265867\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4.10217e13 1.70553 0.852767 − 0.522292i \(-0.174923\pi\)
0.852767 + 0.522292i \(0.174923\pi\)
\(942\) 0 0
\(943\) −2.80203e13 −1.15390
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.26511e13 −0.915196 −0.457598 − 0.889159i \(-0.651290\pi\)
−0.457598 + 0.889159i \(0.651290\pi\)
\(948\) 0 0
\(949\) −6.39781e12 −0.256055
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.06846e13 1.20504 0.602521 − 0.798103i \(-0.294163\pi\)
0.602521 + 0.798103i \(0.294163\pi\)
\(954\) 0 0
\(955\) −1.76850e13 −0.688002
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.84827e12 −0.0705639
\(960\) 0 0
\(961\) −2.39449e13 −0.905643
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.68073e13 −0.623914
\(966\) 0 0
\(967\) −1.91408e13 −0.703949 −0.351975 − 0.936010i \(-0.614490\pi\)
−0.351975 + 0.936010i \(0.614490\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.98467e13 −1.79949 −0.899746 − 0.436414i \(-0.856248\pi\)
−0.899746 + 0.436414i \(0.856248\pi\)
\(972\) 0 0
\(973\) 1.06649e12 0.0381461
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.74066e13 0.611205 0.305603 − 0.952159i \(-0.401142\pi\)
0.305603 + 0.952159i \(0.401142\pi\)
\(978\) 0 0
\(979\) 3.18325e13 1.10751
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −2.09397e13 −0.715286 −0.357643 − 0.933858i \(-0.616420\pi\)
−0.357643 + 0.933858i \(0.616420\pi\)
\(984\) 0 0
\(985\) 1.05692e13 0.357748
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.74629e13 −1.57751
\(990\) 0 0
\(991\) −9.68052e12 −0.318836 −0.159418 − 0.987211i \(-0.550962\pi\)
−0.159418 + 0.987211i \(0.550962\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.69946e12 0.152000
\(996\) 0 0
\(997\) 4.45555e12 0.142815 0.0714074 − 0.997447i \(-0.477251\pi\)
0.0714074 + 0.997447i \(0.477251\pi\)
\(998\) 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.b.1.1 1
3.2 odd 2 24.10.a.a.1.1 ✓ 1
4.3 odd 2 144.10.a.e.1.1 1
12.11 even 2 48.10.a.f.1.1 1
24.5 odd 2 192.10.a.j.1.1 1
24.11 even 2 192.10.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.10.a.a.1.1 ✓ 1 3.2 odd 2
48.10.a.f.1.1 1 12.11 even 2
72.10.a.b.1.1 1 1.1 even 1 trivial
144.10.a.e.1.1 1 4.3 odd 2
192.10.a.c.1.1 1 24.11 even 2
192.10.a.j.1.1 1 24.5 odd 2