Properties

Label 252.12.a.h.1.6
Level $252$
Weight $12$
Character 252.1
Self dual yes
Analytic conductor $193.622$
Analytic rank $1$
Dimension $6$
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] = [252,12,Mod(1,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 252.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: \(193.622481501\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 7264685 x^{4} + 1525918007 x^{3} + 9975646206208 x^{2} + 929027169420686 x - 10\!\cdots\!80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{19}\cdot 3^{8}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(290.010\) of defining polynomial
Character \(\chi\) \(=\) 252.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10562.7 q^{5} -16807.0 q^{7} +O(q^{10})\) \(q+10562.7 q^{5} -16807.0 q^{7} -487870. q^{11} -29411.3 q^{13} +2.33196e6 q^{17} -1.94204e6 q^{19} +1.87855e7 q^{23} +6.27431e7 q^{25} +1.99877e7 q^{29} -2.43126e8 q^{31} -1.77528e8 q^{35} +5.32206e8 q^{37} -8.90556e8 q^{41} +5.15347e8 q^{43} -2.38113e9 q^{47} +2.82475e8 q^{49} -3.59935e9 q^{53} -5.15324e9 q^{55} -4.83221e9 q^{59} +2.98288e9 q^{61} -3.10664e8 q^{65} +1.87198e10 q^{67} -5.13149e9 q^{71} -3.28191e10 q^{73} +8.19964e9 q^{77} -2.03185e10 q^{79} -1.70301e10 q^{83} +2.46318e10 q^{85} +4.06379e10 q^{89} +4.94316e8 q^{91} -2.05133e10 q^{95} +9.26360e10 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 100842 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 100842 q^{7} - 2302212 q^{13} + 25404408 q^{19} + 94198506 q^{25} + 251989848 q^{31} - 35389524 q^{37} + 720466200 q^{43} + 1694851494 q^{49} + 8043243432 q^{55} - 5272981644 q^{61} - 13406069736 q^{67} - 55724067180 q^{73} - 2111676432 q^{79} - 51331653912 q^{85} + 38693277084 q^{91} + 151487901012 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 10562.7 1.51161 0.755807 0.654794i \(-0.227245\pi\)
0.755807 + 0.654794i \(0.227245\pi\)
\(6\) 0 0
\(7\) −16807.0 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −487870. −0.913366 −0.456683 0.889630i \(-0.650963\pi\)
−0.456683 + 0.889630i \(0.650963\pi\)
\(12\) 0 0
\(13\) −29411.3 −0.0219698 −0.0109849 0.999940i \(-0.503497\pi\)
−0.0109849 + 0.999940i \(0.503497\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.33196e6 0.398338 0.199169 0.979965i \(-0.436176\pi\)
0.199169 + 0.979965i \(0.436176\pi\)
\(18\) 0 0
\(19\) −1.94204e6 −0.179934 −0.0899671 0.995945i \(-0.528676\pi\)
−0.0899671 + 0.995945i \(0.528676\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.87855e7 0.608583 0.304291 0.952579i \(-0.401580\pi\)
0.304291 + 0.952579i \(0.401580\pi\)
\(24\) 0 0
\(25\) 6.27431e7 1.28498
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.99877e7 0.180956 0.0904780 0.995898i \(-0.471161\pi\)
0.0904780 + 0.995898i \(0.471161\pi\)
\(30\) 0 0
\(31\) −2.43126e8 −1.52525 −0.762625 0.646841i \(-0.776090\pi\)
−0.762625 + 0.646841i \(0.776090\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.77528e8 −0.571337
\(36\) 0 0
\(37\) 5.32206e8 1.26174 0.630871 0.775888i \(-0.282698\pi\)
0.630871 + 0.775888i \(0.282698\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.90556e8 −1.20047 −0.600234 0.799825i \(-0.704926\pi\)
−0.600234 + 0.799825i \(0.704926\pi\)
\(42\) 0 0
\(43\) 5.15347e8 0.534593 0.267297 0.963614i \(-0.413870\pi\)
0.267297 + 0.963614i \(0.413870\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.38113e9 −1.51442 −0.757208 0.653174i \(-0.773437\pi\)
−0.757208 + 0.653174i \(0.773437\pi\)
\(48\) 0 0
\(49\) 2.82475e8 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.59935e9 −1.18224 −0.591122 0.806582i \(-0.701315\pi\)
−0.591122 + 0.806582i \(0.701315\pi\)
\(54\) 0 0
\(55\) −5.15324e9 −1.38066
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.83221e9 −0.879953 −0.439977 0.898009i \(-0.645013\pi\)
−0.439977 + 0.898009i \(0.645013\pi\)
\(60\) 0 0
\(61\) 2.98288e9 0.452191 0.226095 0.974105i \(-0.427404\pi\)
0.226095 + 0.974105i \(0.427404\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.10664e8 −0.0332098
\(66\) 0 0
\(67\) 1.87198e10 1.69390 0.846952 0.531669i \(-0.178435\pi\)
0.846952 + 0.531669i \(0.178435\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.13149e9 −0.337538 −0.168769 0.985656i \(-0.553979\pi\)
−0.168769 + 0.985656i \(0.553979\pi\)
\(72\) 0 0
\(73\) −3.28191e10 −1.85290 −0.926449 0.376421i \(-0.877155\pi\)
−0.926449 + 0.376421i \(0.877155\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.19964e9 0.345220
\(78\) 0 0
\(79\) −2.03185e10 −0.742920 −0.371460 0.928449i \(-0.621143\pi\)
−0.371460 + 0.928449i \(0.621143\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.70301e10 −0.474557 −0.237278 0.971442i \(-0.576255\pi\)
−0.237278 + 0.971442i \(0.576255\pi\)
\(84\) 0 0
\(85\) 2.46318e10 0.602133
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.06379e10 0.771412 0.385706 0.922622i \(-0.373958\pi\)
0.385706 + 0.922622i \(0.373958\pi\)
\(90\) 0 0
\(91\) 4.94316e8 0.00830380
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.05133e10 −0.271991
\(96\) 0 0
\(97\) 9.26360e10 1.09531 0.547653 0.836706i \(-0.315522\pi\)
0.547653 + 0.836706i \(0.315522\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.55048e11 1.46791 0.733953 0.679200i \(-0.237673\pi\)
0.733953 + 0.679200i \(0.237673\pi\)
\(102\) 0 0
\(103\) −1.00875e11 −0.857393 −0.428697 0.903448i \(-0.641027\pi\)
−0.428697 + 0.903448i \(0.641027\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.05859e11 0.729655 0.364828 0.931075i \(-0.381128\pi\)
0.364828 + 0.931075i \(0.381128\pi\)
\(108\) 0 0
\(109\) −1.06411e11 −0.662431 −0.331215 0.943555i \(-0.607459\pi\)
−0.331215 + 0.943555i \(0.607459\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.46301e11 0.746990 0.373495 0.927632i \(-0.378159\pi\)
0.373495 + 0.927632i \(0.378159\pi\)
\(114\) 0 0
\(115\) 1.98426e11 0.919942
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.91932e10 −0.150558
\(120\) 0 0
\(121\) −4.72941e10 −0.165763
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.46980e11 0.430778
\(126\) 0 0
\(127\) −5.82386e10 −0.156419 −0.0782096 0.996937i \(-0.524920\pi\)
−0.0782096 + 0.996937i \(0.524920\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.70533e11 −0.386203 −0.193102 0.981179i \(-0.561855\pi\)
−0.193102 + 0.981179i \(0.561855\pi\)
\(132\) 0 0
\(133\) 3.26399e10 0.0680087
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.50175e11 −1.15098 −0.575489 0.817810i \(-0.695188\pi\)
−0.575489 + 0.817810i \(0.695188\pi\)
\(138\) 0 0
\(139\) −7.50597e11 −1.22695 −0.613473 0.789715i \(-0.710228\pi\)
−0.613473 + 0.789715i \(0.710228\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.43489e10 0.0200664
\(144\) 0 0
\(145\) 2.11124e11 0.273536
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.55556e11 −0.173525 −0.0867627 0.996229i \(-0.527652\pi\)
−0.0867627 + 0.996229i \(0.527652\pi\)
\(150\) 0 0
\(151\) −7.73262e11 −0.801592 −0.400796 0.916167i \(-0.631266\pi\)
−0.400796 + 0.916167i \(0.631266\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.56807e12 −2.30559
\(156\) 0 0
\(157\) −3.08932e11 −0.258473 −0.129236 0.991614i \(-0.541253\pi\)
−0.129236 + 0.991614i \(0.541253\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.15728e11 −0.230023
\(162\) 0 0
\(163\) 7.52668e11 0.512356 0.256178 0.966630i \(-0.417537\pi\)
0.256178 + 0.966630i \(0.417537\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.64430e11 −0.455404 −0.227702 0.973731i \(-0.573121\pi\)
−0.227702 + 0.973731i \(0.573121\pi\)
\(168\) 0 0
\(169\) −1.79130e12 −0.999517
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.24546e12 0.611047 0.305524 0.952185i \(-0.401169\pi\)
0.305524 + 0.952185i \(0.401169\pi\)
\(174\) 0 0
\(175\) −1.05452e12 −0.485676
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.51610e11 −0.102338 −0.0511690 0.998690i \(-0.516295\pi\)
−0.0511690 + 0.998690i \(0.516295\pi\)
\(180\) 0 0
\(181\) −3.73790e11 −0.143020 −0.0715098 0.997440i \(-0.522782\pi\)
−0.0715098 + 0.997440i \(0.522782\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.62155e12 1.90727
\(186\) 0 0
\(187\) −1.13769e12 −0.363828
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.55209e12 0.726461 0.363230 0.931699i \(-0.381674\pi\)
0.363230 + 0.931699i \(0.381674\pi\)
\(192\) 0 0
\(193\) 3.17515e12 0.853490 0.426745 0.904372i \(-0.359660\pi\)
0.426745 + 0.904372i \(0.359660\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.90882e11 0.213922 0.106961 0.994263i \(-0.465888\pi\)
0.106961 + 0.994263i \(0.465888\pi\)
\(198\) 0 0
\(199\) −5.79997e12 −1.31745 −0.658725 0.752384i \(-0.728904\pi\)
−0.658725 + 0.752384i \(0.728904\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.35933e11 −0.0683950
\(204\) 0 0
\(205\) −9.40671e12 −1.81464
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.47465e11 0.164346
\(210\) 0 0
\(211\) −7.07644e11 −0.116483 −0.0582413 0.998303i \(-0.518549\pi\)
−0.0582413 + 0.998303i \(0.518549\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.44347e12 0.808099
\(216\) 0 0
\(217\) 4.08621e12 0.576491
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −6.85859e10 −0.00875139
\(222\) 0 0
\(223\) −7.96653e10 −0.00967370 −0.00483685 0.999988i \(-0.501540\pi\)
−0.00483685 + 0.999988i \(0.501540\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.07241e13 −1.18092 −0.590460 0.807067i \(-0.701053\pi\)
−0.590460 + 0.807067i \(0.701053\pi\)
\(228\) 0 0
\(229\) −1.48664e13 −1.55995 −0.779974 0.625812i \(-0.784768\pi\)
−0.779974 + 0.625812i \(0.784768\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.50195e13 1.43284 0.716421 0.697668i \(-0.245779\pi\)
0.716421 + 0.697668i \(0.245779\pi\)
\(234\) 0 0
\(235\) −2.51512e13 −2.28921
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.98552e13 −1.64697 −0.823486 0.567336i \(-0.807974\pi\)
−0.823486 + 0.567336i \(0.807974\pi\)
\(240\) 0 0
\(241\) −5.51467e12 −0.436944 −0.218472 0.975843i \(-0.570107\pi\)
−0.218472 + 0.975843i \(0.570107\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.98371e12 0.215945
\(246\) 0 0
\(247\) 5.71180e10 0.00395311
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.70789e11 0.0488349 0.0244174 0.999702i \(-0.492227\pi\)
0.0244174 + 0.999702i \(0.492227\pi\)
\(252\) 0 0
\(253\) −9.16489e12 −0.555858
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.35879e13 0.755996 0.377998 0.925806i \(-0.376613\pi\)
0.377998 + 0.925806i \(0.376613\pi\)
\(258\) 0 0
\(259\) −8.94479e12 −0.476894
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2.30882e13 −1.13145 −0.565723 0.824595i \(-0.691403\pi\)
−0.565723 + 0.824595i \(0.691403\pi\)
\(264\) 0 0
\(265\) −3.80190e13 −1.78710
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −9.20663e12 −0.398532 −0.199266 0.979945i \(-0.563856\pi\)
−0.199266 + 0.979945i \(0.563856\pi\)
\(270\) 0 0
\(271\) 3.03206e12 0.126010 0.0630052 0.998013i \(-0.479932\pi\)
0.0630052 + 0.998013i \(0.479932\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.06105e13 −1.17366
\(276\) 0 0
\(277\) −2.01089e12 −0.0740882 −0.0370441 0.999314i \(-0.511794\pi\)
−0.0370441 + 0.999314i \(0.511794\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −3.74313e13 −1.27453 −0.637265 0.770645i \(-0.719934\pi\)
−0.637265 + 0.770645i \(0.719934\pi\)
\(282\) 0 0
\(283\) −1.78019e13 −0.582963 −0.291482 0.956576i \(-0.594148\pi\)
−0.291482 + 0.956576i \(0.594148\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.49676e13 0.453734
\(288\) 0 0
\(289\) −2.88339e13 −0.841327
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.73563e13 1.01063 0.505315 0.862935i \(-0.331376\pi\)
0.505315 + 0.862935i \(0.331376\pi\)
\(294\) 0 0
\(295\) −5.10413e13 −1.33015
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −5.52506e11 −0.0133704
\(300\) 0 0
\(301\) −8.66144e12 −0.202057
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.15074e13 0.683538
\(306\) 0 0
\(307\) −2.02676e13 −0.424172 −0.212086 0.977251i \(-0.568026\pi\)
−0.212086 + 0.977251i \(0.568026\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −6.66339e13 −1.29871 −0.649357 0.760484i \(-0.724962\pi\)
−0.649357 + 0.760484i \(0.724962\pi\)
\(312\) 0 0
\(313\) −5.07089e13 −0.954092 −0.477046 0.878878i \(-0.658293\pi\)
−0.477046 + 0.878878i \(0.658293\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.79440e13 0.665760 0.332880 0.942969i \(-0.391980\pi\)
0.332880 + 0.942969i \(0.391980\pi\)
\(318\) 0 0
\(319\) −9.75139e12 −0.165279
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −4.52876e12 −0.0716746
\(324\) 0 0
\(325\) −1.84536e12 −0.0282307
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.00197e13 0.572395
\(330\) 0 0
\(331\) −5.71251e13 −0.790266 −0.395133 0.918624i \(-0.629301\pi\)
−0.395133 + 0.918624i \(0.629301\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.97732e14 2.56053
\(336\) 0 0
\(337\) −7.13875e13 −0.894659 −0.447330 0.894369i \(-0.647625\pi\)
−0.447330 + 0.894369i \(0.647625\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 1.18614e14 1.39311
\(342\) 0 0
\(343\) −4.74756e12 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 2.75474e12 0.0293946 0.0146973 0.999892i \(-0.495322\pi\)
0.0146973 + 0.999892i \(0.495322\pi\)
\(348\) 0 0
\(349\) −1.66638e13 −0.172279 −0.0861396 0.996283i \(-0.527453\pi\)
−0.0861396 + 0.996283i \(0.527453\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.35893e14 −1.31958 −0.659792 0.751448i \(-0.729356\pi\)
−0.659792 + 0.751448i \(0.729356\pi\)
\(354\) 0 0
\(355\) −5.42026e13 −0.510228
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.49245e14 1.32093 0.660464 0.750857i \(-0.270359\pi\)
0.660464 + 0.750857i \(0.270359\pi\)
\(360\) 0 0
\(361\) −1.12719e14 −0.967624
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −3.46660e14 −2.80087
\(366\) 0 0
\(367\) −1.96772e14 −1.54277 −0.771384 0.636370i \(-0.780435\pi\)
−0.771384 + 0.636370i \(0.780435\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 6.04943e13 0.446846
\(372\) 0 0
\(373\) 4.25191e13 0.304920 0.152460 0.988310i \(-0.451281\pi\)
0.152460 + 0.988310i \(0.451281\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.87863e11 −0.00397556
\(378\) 0 0
\(379\) 2.41365e14 1.58547 0.792736 0.609565i \(-0.208656\pi\)
0.792736 + 0.609565i \(0.208656\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.74375e14 1.08116 0.540580 0.841293i \(-0.318205\pi\)
0.540580 + 0.841293i \(0.318205\pi\)
\(384\) 0 0
\(385\) 8.66106e13 0.521839
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.64981e14 −1.50831 −0.754157 0.656694i \(-0.771954\pi\)
−0.754157 + 0.656694i \(0.771954\pi\)
\(390\) 0 0
\(391\) 4.38070e13 0.242421
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.14619e14 −1.12301
\(396\) 0 0
\(397\) −2.99270e12 −0.0152305 −0.00761526 0.999971i \(-0.502424\pi\)
−0.00761526 + 0.999971i \(0.502424\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.45178e14 −0.699210 −0.349605 0.936897i \(-0.613684\pi\)
−0.349605 + 0.936897i \(0.613684\pi\)
\(402\) 0 0
\(403\) 7.15064e12 0.0335094
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.59648e14 −1.15243
\(408\) 0 0
\(409\) 1.63187e14 0.705028 0.352514 0.935807i \(-0.385327\pi\)
0.352514 + 0.935807i \(0.385327\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.12149e13 0.332591
\(414\) 0 0
\(415\) −1.79884e14 −0.717347
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.42018e13 −0.242868 −0.121434 0.992600i \(-0.538749\pi\)
−0.121434 + 0.992600i \(0.538749\pi\)
\(420\) 0 0
\(421\) −1.49920e14 −0.552471 −0.276236 0.961090i \(-0.589087\pi\)
−0.276236 + 0.961090i \(0.589087\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.46314e14 0.511856
\(426\) 0 0
\(427\) −5.01333e13 −0.170912
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.60524e14 0.519895 0.259948 0.965623i \(-0.416295\pi\)
0.259948 + 0.965623i \(0.416295\pi\)
\(432\) 0 0
\(433\) 9.26688e13 0.292584 0.146292 0.989241i \(-0.453266\pi\)
0.146292 + 0.989241i \(0.453266\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.64822e13 −0.109505
\(438\) 0 0
\(439\) 3.60669e14 1.05573 0.527866 0.849327i \(-0.322992\pi\)
0.527866 + 0.849327i \(0.322992\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4.41959e14 1.23073 0.615363 0.788244i \(-0.289009\pi\)
0.615363 + 0.788244i \(0.289009\pi\)
\(444\) 0 0
\(445\) 4.29247e14 1.16608
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −6.20805e14 −1.60546 −0.802731 0.596341i \(-0.796621\pi\)
−0.802731 + 0.596341i \(0.796621\pi\)
\(450\) 0 0
\(451\) 4.34476e14 1.09647
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 5.22132e12 0.0125521
\(456\) 0 0
\(457\) 7.38797e14 1.73375 0.866874 0.498528i \(-0.166126\pi\)
0.866874 + 0.498528i \(0.166126\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.86457e14 1.08815 0.544076 0.839036i \(-0.316880\pi\)
0.544076 + 0.839036i \(0.316880\pi\)
\(462\) 0 0
\(463\) −3.98986e14 −0.871490 −0.435745 0.900070i \(-0.643515\pi\)
−0.435745 + 0.900070i \(0.643515\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.15156e14 1.90657 0.953284 0.302077i \(-0.0976799\pi\)
0.953284 + 0.302077i \(0.0976799\pi\)
\(468\) 0 0
\(469\) −3.14623e14 −0.640236
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2.51423e14 −0.488279
\(474\) 0 0
\(475\) −1.21850e14 −0.231212
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 7.92244e13 0.143553 0.0717767 0.997421i \(-0.477133\pi\)
0.0717767 + 0.997421i \(0.477133\pi\)
\(480\) 0 0
\(481\) −1.56529e13 −0.0277202
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.78489e14 1.65568
\(486\) 0 0
\(487\) 4.18996e14 0.693108 0.346554 0.938030i \(-0.387352\pi\)
0.346554 + 0.938030i \(0.387352\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.06855e15 −1.68984 −0.844922 0.534890i \(-0.820353\pi\)
−0.844922 + 0.534890i \(0.820353\pi\)
\(492\) 0 0
\(493\) 4.66104e13 0.0720816
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.62450e13 0.127577
\(498\) 0 0
\(499\) 2.81718e14 0.407626 0.203813 0.979010i \(-0.434667\pi\)
0.203813 + 0.979010i \(0.434667\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.62474e14 1.33280 0.666400 0.745594i \(-0.267834\pi\)
0.666400 + 0.745594i \(0.267834\pi\)
\(504\) 0 0
\(505\) 1.63773e15 2.21891
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −3.47837e14 −0.451261 −0.225630 0.974213i \(-0.572444\pi\)
−0.225630 + 0.974213i \(0.572444\pi\)
\(510\) 0 0
\(511\) 5.51591e14 0.700329
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.06552e15 −1.29605
\(516\) 0 0
\(517\) 1.16168e15 1.38322
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.00284e15 1.14452 0.572260 0.820072i \(-0.306067\pi\)
0.572260 + 0.820072i \(0.306067\pi\)
\(522\) 0 0
\(523\) 4.13233e14 0.461781 0.230891 0.972980i \(-0.425836\pi\)
0.230891 + 0.972980i \(0.425836\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −5.66958e14 −0.607565
\(528\) 0 0
\(529\) −5.99915e14 −0.629627
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.61924e13 0.0263740
\(534\) 0 0
\(535\) 1.11816e15 1.10296
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.37811e14 −0.130481
\(540\) 0 0
\(541\) −5.18938e14 −0.481427 −0.240714 0.970596i \(-0.577381\pi\)
−0.240714 + 0.970596i \(0.577381\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.12399e15 −1.00134
\(546\) 0 0
\(547\) 3.07863e14 0.268799 0.134399 0.990927i \(-0.457090\pi\)
0.134399 + 0.990927i \(0.457090\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.88169e13 −0.0325602
\(552\) 0 0
\(553\) 3.41493e14 0.280797
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.54386e15 1.22012 0.610061 0.792354i \(-0.291145\pi\)
0.610061 + 0.792354i \(0.291145\pi\)
\(558\) 0 0
\(559\) −1.51570e13 −0.0117449
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.19701e15 −0.891872 −0.445936 0.895065i \(-0.647129\pi\)
−0.445936 + 0.895065i \(0.647129\pi\)
\(564\) 0 0
\(565\) 1.54533e15 1.12916
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.82173e15 −1.28046 −0.640231 0.768182i \(-0.721161\pi\)
−0.640231 + 0.768182i \(0.721161\pi\)
\(570\) 0 0
\(571\) 1.09310e15 0.753639 0.376819 0.926287i \(-0.377018\pi\)
0.376819 + 0.926287i \(0.377018\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.17866e15 0.782016
\(576\) 0 0
\(577\) −7.41570e14 −0.482709 −0.241354 0.970437i \(-0.577592\pi\)
−0.241354 + 0.970437i \(0.577592\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 2.86225e14 0.179366
\(582\) 0 0
\(583\) 1.75602e15 1.07982
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.43620e15 −0.850561 −0.425280 0.905062i \(-0.639824\pi\)
−0.425280 + 0.905062i \(0.639824\pi\)
\(588\) 0 0
\(589\) 4.72160e14 0.274445
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.33066e15 −0.745189 −0.372595 0.927994i \(-0.621532\pi\)
−0.372595 + 0.927994i \(0.621532\pi\)
\(594\) 0 0
\(595\) −4.13987e14 −0.227585
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −2.13669e15 −1.13212 −0.566062 0.824362i \(-0.691534\pi\)
−0.566062 + 0.824362i \(0.691534\pi\)
\(600\) 0 0
\(601\) 8.37107e14 0.435483 0.217742 0.976006i \(-0.430131\pi\)
0.217742 + 0.976006i \(0.430131\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.99555e14 −0.250570
\(606\) 0 0
\(607\) −3.23824e15 −1.59504 −0.797519 0.603293i \(-0.793855\pi\)
−0.797519 + 0.603293i \(0.793855\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.00322e13 0.0332714
\(612\) 0 0
\(613\) 2.00991e15 0.937871 0.468935 0.883232i \(-0.344638\pi\)
0.468935 + 0.883232i \(0.344638\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.29759e15 1.03444 0.517219 0.855853i \(-0.326967\pi\)
0.517219 + 0.855853i \(0.326967\pi\)
\(618\) 0 0
\(619\) −1.32881e15 −0.587713 −0.293856 0.955850i \(-0.594939\pi\)
−0.293856 + 0.955850i \(0.594939\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.83001e14 −0.291566
\(624\) 0 0
\(625\) −1.51112e15 −0.633808
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.24108e15 0.502600
\(630\) 0 0
\(631\) −1.69942e15 −0.676301 −0.338150 0.941092i \(-0.609801\pi\)
−0.338150 + 0.941092i \(0.609801\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.15158e14 −0.236446
\(636\) 0 0
\(637\) −8.30796e12 −0.00313854
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 3.41351e15 1.24590 0.622949 0.782262i \(-0.285934\pi\)
0.622949 + 0.782262i \(0.285934\pi\)
\(642\) 0 0
\(643\) 5.41457e14 0.194269 0.0971345 0.995271i \(-0.469032\pi\)
0.0971345 + 0.995271i \(0.469032\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.42396e15 0.493770 0.246885 0.969045i \(-0.420593\pi\)
0.246885 + 0.969045i \(0.420593\pi\)
\(648\) 0 0
\(649\) 2.35749e15 0.803719
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 3.53010e15 1.16349 0.581747 0.813370i \(-0.302369\pi\)
0.581747 + 0.813370i \(0.302369\pi\)
\(654\) 0 0
\(655\) −1.80129e15 −0.583791
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 5.83382e15 1.82845 0.914225 0.405207i \(-0.132800\pi\)
0.914225 + 0.405207i \(0.132800\pi\)
\(660\) 0 0
\(661\) 2.74739e15 0.846861 0.423431 0.905929i \(-0.360826\pi\)
0.423431 + 0.905929i \(0.360826\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.44766e14 0.102803
\(666\) 0 0
\(667\) 3.75478e14 0.110127
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.45526e15 −0.413016
\(672\) 0 0
\(673\) −3.72553e15 −1.04017 −0.520086 0.854114i \(-0.674100\pi\)
−0.520086 + 0.854114i \(0.674100\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2.03218e15 −0.549193 −0.274597 0.961559i \(-0.588544\pi\)
−0.274597 + 0.961559i \(0.588544\pi\)
\(678\) 0 0
\(679\) −1.55693e15 −0.413986
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.40117e15 1.39051 0.695255 0.718763i \(-0.255291\pi\)
0.695255 + 0.718763i \(0.255291\pi\)
\(684\) 0 0
\(685\) −6.86762e15 −1.73983
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.05862e14 0.0259736
\(690\) 0 0
\(691\) 4.49176e14 0.108464 0.0542322 0.998528i \(-0.482729\pi\)
0.0542322 + 0.998528i \(0.482729\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −7.92836e15 −1.85467
\(696\) 0 0
\(697\) −2.07674e15 −0.478191
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.88883e15 1.53708 0.768539 0.639803i \(-0.220984\pi\)
0.768539 + 0.639803i \(0.220984\pi\)
\(702\) 0 0
\(703\) −1.03357e15 −0.227031
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.60589e15 −0.554817
\(708\) 0 0
\(709\) 1.44603e15 0.303125 0.151562 0.988448i \(-0.451570\pi\)
0.151562 + 0.988448i \(0.451570\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.56723e15 −0.928241
\(714\) 0 0
\(715\) 1.51564e14 0.0303327
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −6.17402e15 −1.19828 −0.599141 0.800643i \(-0.704491\pi\)
−0.599141 + 0.800643i \(0.704491\pi\)
\(720\) 0 0
\(721\) 1.69541e15 0.324064
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.25409e15 0.232525
\(726\) 0 0
\(727\) −3.67029e15 −0.670287 −0.335144 0.942167i \(-0.608785\pi\)
−0.335144 + 0.942167i \(0.608785\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.20177e15 0.212949
\(732\) 0 0
\(733\) −5.92773e15 −1.03470 −0.517352 0.855773i \(-0.673082\pi\)
−0.517352 + 0.855773i \(0.673082\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.13282e15 −1.54715
\(738\) 0 0
\(739\) −9.79229e15 −1.63433 −0.817165 0.576403i \(-0.804456\pi\)
−0.817165 + 0.576403i \(0.804456\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 5.62273e15 0.910980 0.455490 0.890241i \(-0.349464\pi\)
0.455490 + 0.890241i \(0.349464\pi\)
\(744\) 0 0
\(745\) −1.64310e15 −0.262303
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.77918e15 −0.275784
\(750\) 0 0
\(751\) −9.92348e15 −1.51581 −0.757904 0.652366i \(-0.773777\pi\)
−0.757904 + 0.652366i \(0.773777\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.16776e15 −1.21170
\(756\) 0 0
\(757\) 2.43579e15 0.356134 0.178067 0.984018i \(-0.443016\pi\)
0.178067 + 0.984018i \(0.443016\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.18798e16 −1.68731 −0.843655 0.536886i \(-0.819601\pi\)
−0.843655 + 0.536886i \(0.819601\pi\)
\(762\) 0 0
\(763\) 1.78845e15 0.250375
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.42122e14 0.0193324
\(768\) 0 0
\(769\) 8.87658e15 1.19028 0.595142 0.803621i \(-0.297096\pi\)
0.595142 + 0.803621i \(0.297096\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.63183e15 1.12491 0.562453 0.826830i \(-0.309858\pi\)
0.562453 + 0.826830i \(0.309858\pi\)
\(774\) 0 0
\(775\) −1.52544e16 −1.95991
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.72950e15 0.216005
\(780\) 0 0
\(781\) 2.50350e15 0.308296
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −3.26316e15 −0.390711
\(786\) 0 0
\(787\) −1.60940e16 −1.90021 −0.950107 0.311923i \(-0.899027\pi\)
−0.950107 + 0.311923i \(0.899027\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.45888e15 −0.282336
\(792\) 0 0
\(793\) −8.77304e13 −0.00993454
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.12583e15 0.124008 0.0620042 0.998076i \(-0.480251\pi\)
0.0620042 + 0.998076i \(0.480251\pi\)
\(798\) 0 0
\(799\) −5.55270e15 −0.603249
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.60115e16 1.69237
\(804\) 0 0
\(805\) −3.33495e15 −0.347705
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1.65264e16 −1.67673 −0.838363 0.545112i \(-0.816487\pi\)
−0.838363 + 0.545112i \(0.816487\pi\)
\(810\) 0 0
\(811\) −5.72460e15 −0.572968 −0.286484 0.958085i \(-0.592487\pi\)
−0.286484 + 0.958085i \(0.592487\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 7.95023e15 0.774484
\(816\) 0 0
\(817\) −1.00083e15 −0.0961916
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.36110e16 1.27351 0.636756 0.771066i \(-0.280276\pi\)
0.636756 + 0.771066i \(0.280276\pi\)
\(822\) 0 0
\(823\) 9.12527e15 0.842455 0.421228 0.906955i \(-0.361599\pi\)
0.421228 + 0.906955i \(0.361599\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 7.09672e15 0.637936 0.318968 0.947765i \(-0.396664\pi\)
0.318968 + 0.947765i \(0.396664\pi\)
\(828\) 0 0
\(829\) 7.15670e15 0.634838 0.317419 0.948285i \(-0.397184\pi\)
0.317419 + 0.948285i \(0.397184\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6.58720e14 0.0569054
\(834\) 0 0
\(835\) −8.07446e15 −0.688396
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 6.67038e15 0.553937 0.276968 0.960879i \(-0.410670\pi\)
0.276968 + 0.960879i \(0.410670\pi\)
\(840\) 0 0
\(841\) −1.18010e16 −0.967255
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.89210e16 −1.51089
\(846\) 0 0
\(847\) 7.94872e14 0.0626525
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 9.99776e15 0.767874
\(852\) 0 0
\(853\) −1.37743e16 −1.04436 −0.522179 0.852836i \(-0.674881\pi\)
−0.522179 + 0.852836i \(0.674881\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 8.39602e15 0.620410 0.310205 0.950670i \(-0.399602\pi\)
0.310205 + 0.950670i \(0.399602\pi\)
\(858\) 0 0
\(859\) 1.86860e16 1.36318 0.681590 0.731735i \(-0.261289\pi\)
0.681590 + 0.731735i \(0.261289\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.08199e15 0.219165 0.109583 0.993978i \(-0.465049\pi\)
0.109583 + 0.993978i \(0.465049\pi\)
\(864\) 0 0
\(865\) 1.31554e16 0.923668
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 9.91278e15 0.678558
\(870\) 0 0
\(871\) −5.50572e14 −0.0372147
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.47030e15 −0.162819
\(876\) 0 0
\(877\) 1.60620e16 1.04545 0.522723 0.852503i \(-0.324916\pi\)
0.522723 + 0.852503i \(0.324916\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.82581e16 −1.15901 −0.579507 0.814967i \(-0.696755\pi\)
−0.579507 + 0.814967i \(0.696755\pi\)
\(882\) 0 0
\(883\) −8.43785e15 −0.528990 −0.264495 0.964387i \(-0.585205\pi\)
−0.264495 + 0.964387i \(0.585205\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.34139e16 1.43184 0.715920 0.698182i \(-0.246007\pi\)
0.715920 + 0.698182i \(0.246007\pi\)
\(888\) 0 0
\(889\) 9.78816e14 0.0591209
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.62426e15 0.272495
\(894\) 0 0
\(895\) −2.65769e15 −0.154696
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.85951e15 −0.276003
\(900\) 0 0
\(901\) −8.39354e15 −0.470933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.94824e15 −0.216191
\(906\) 0 0
\(907\) −2.56483e16 −1.38745 −0.693726 0.720239i \(-0.744032\pi\)
−0.693726 + 0.720239i \(0.744032\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −8.30075e15 −0.438295 −0.219147 0.975692i \(-0.570328\pi\)
−0.219147 + 0.975692i \(0.570328\pi\)
\(912\) 0 0
\(913\) 8.30849e15 0.433444
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.86615e15 0.145971
\(918\) 0 0
\(919\) 6.44710e15 0.324436 0.162218 0.986755i \(-0.448135\pi\)
0.162218 + 0.986755i \(0.448135\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.50924e14 0.00741564
\(924\) 0 0
\(925\) 3.33923e16 1.62131
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.04782e15 −0.0496822 −0.0248411 0.999691i \(-0.507908\pi\)
−0.0248411 + 0.999691i \(0.507908\pi\)
\(930\) 0 0
\(931\) −5.48579e14 −0.0257049
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.20171e16 −0.549968
\(936\) 0 0
\(937\) 2.72463e16 1.23237 0.616184 0.787602i \(-0.288678\pi\)
0.616184 + 0.787602i \(0.288678\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1.22733e16 0.542271 0.271136 0.962541i \(-0.412601\pi\)
0.271136 + 0.962541i \(0.412601\pi\)
\(942\) 0 0
\(943\) −1.67295e16 −0.730583
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −8.45969e15 −0.360936 −0.180468 0.983581i \(-0.557761\pi\)
−0.180468 + 0.983581i \(0.557761\pi\)
\(948\) 0 0
\(949\) 9.65253e14 0.0407077
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −4.57712e16 −1.88617 −0.943086 0.332549i \(-0.892091\pi\)
−0.943086 + 0.332549i \(0.892091\pi\)
\(954\) 0 0
\(955\) 2.69570e16 1.09813
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.09275e16 0.435029
\(960\) 0 0
\(961\) 3.37015e16 1.32639
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 3.35382e16 1.29015
\(966\) 0 0
\(967\) −1.46356e16 −0.556629 −0.278314 0.960490i \(-0.589776\pi\)
−0.278314 + 0.960490i \(0.589776\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −5.30470e15 −0.197222 −0.0986109 0.995126i \(-0.531440\pi\)
−0.0986109 + 0.995126i \(0.531440\pi\)
\(972\) 0 0
\(973\) 1.26153e16 0.463742
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.46828e16 −0.527701 −0.263850 0.964564i \(-0.584993\pi\)
−0.263850 + 0.964564i \(0.584993\pi\)
\(978\) 0 0
\(979\) −1.98260e16 −0.704581
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −5.59170e16 −1.94312 −0.971560 0.236794i \(-0.923904\pi\)
−0.971560 + 0.236794i \(0.923904\pi\)
\(984\) 0 0
\(985\) 9.41014e15 0.323368
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.68105e15 0.325344
\(990\) 0 0
\(991\) 2.28712e16 0.760122 0.380061 0.924961i \(-0.375903\pi\)
0.380061 + 0.924961i \(0.375903\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.12635e16 −1.99148
\(996\) 0 0
\(997\) 2.80843e16 0.902901 0.451450 0.892296i \(-0.350907\pi\)
0.451450 + 0.892296i \(0.350907\pi\)
\(998\) 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 252.12.a.h.1.6 yes 6
3.2 odd 2 inner 252.12.a.h.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.12.a.h.1.1 6 3.2 odd 2 inner
252.12.a.h.1.6 yes 6 1.1 even 1 trivial