Properties

Label 1080.6.a.a.1.1
Level $1080$
Weight $6$
Character 1080.1
Self dual yes
Analytic conductor $173.215$
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] = [1080,6,Mod(1,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 1080.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: \(173.214525398\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-25.0000 q^{5} -234.000 q^{7} +O(q^{10})\) \(q-25.0000 q^{5} -234.000 q^{7} -347.000 q^{11} -33.0000 q^{13} -237.000 q^{17} +1496.00 q^{19} +2811.00 q^{23} +625.000 q^{25} +5513.00 q^{29} +2911.00 q^{31} +5850.00 q^{35} +5602.00 q^{37} -4716.00 q^{41} +10479.0 q^{43} -5963.00 q^{47} +37949.0 q^{49} +17964.0 q^{53} +8675.00 q^{55} -30372.0 q^{59} -35530.0 q^{61} +825.000 q^{65} -12476.0 q^{67} -7520.00 q^{71} +36378.0 q^{73} +81198.0 q^{77} -22727.0 q^{79} -46254.0 q^{83} +5925.00 q^{85} +58832.0 q^{89} +7722.00 q^{91} -37400.0 q^{95} -145906. q^{97} +O(q^{100})\)

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
\(5\) −25.0000 −0.447214
\(6\) 0 0
\(7\) −234.000 −1.80497 −0.902487 0.430718i \(-0.858260\pi\)
−0.902487 + 0.430718i \(0.858260\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −347.000 −0.864665 −0.432332 0.901714i \(-0.642309\pi\)
−0.432332 + 0.901714i \(0.642309\pi\)
\(12\) 0 0
\(13\) −33.0000 −0.0541571 −0.0270786 0.999633i \(-0.508620\pi\)
−0.0270786 + 0.999633i \(0.508620\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −237.000 −0.198896 −0.0994480 0.995043i \(-0.531708\pi\)
−0.0994480 + 0.995043i \(0.531708\pi\)
\(18\) 0 0
\(19\) 1496.00 0.950709 0.475354 0.879794i \(-0.342320\pi\)
0.475354 + 0.879794i \(0.342320\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2811.00 1.10800 0.554002 0.832515i \(-0.313100\pi\)
0.554002 + 0.832515i \(0.313100\pi\)
\(24\) 0 0
\(25\) 625.000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5513.00 1.21729 0.608644 0.793444i \(-0.291714\pi\)
0.608644 + 0.793444i \(0.291714\pi\)
\(30\) 0 0
\(31\) 2911.00 0.544049 0.272024 0.962290i \(-0.412307\pi\)
0.272024 + 0.962290i \(0.412307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5850.00 0.807209
\(36\) 0 0
\(37\) 5602.00 0.672727 0.336363 0.941732i \(-0.390803\pi\)
0.336363 + 0.941732i \(0.390803\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4716.00 −0.438141 −0.219071 0.975709i \(-0.570303\pi\)
−0.219071 + 0.975709i \(0.570303\pi\)
\(42\) 0 0
\(43\) 10479.0 0.864269 0.432134 0.901809i \(-0.357761\pi\)
0.432134 + 0.901809i \(0.357761\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −5963.00 −0.393750 −0.196875 0.980429i \(-0.563079\pi\)
−0.196875 + 0.980429i \(0.563079\pi\)
\(48\) 0 0
\(49\) 37949.0 2.25793
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 17964.0 0.878443 0.439221 0.898379i \(-0.355254\pi\)
0.439221 + 0.898379i \(0.355254\pi\)
\(54\) 0 0
\(55\) 8675.00 0.386690
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −30372.0 −1.13591 −0.567954 0.823060i \(-0.692265\pi\)
−0.567954 + 0.823060i \(0.692265\pi\)
\(60\) 0 0
\(61\) −35530.0 −1.22256 −0.611281 0.791414i \(-0.709345\pi\)
−0.611281 + 0.791414i \(0.709345\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 825.000 0.0242198
\(66\) 0 0
\(67\) −12476.0 −0.339538 −0.169769 0.985484i \(-0.554302\pi\)
−0.169769 + 0.985484i \(0.554302\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7520.00 −0.177040 −0.0885201 0.996074i \(-0.528214\pi\)
−0.0885201 + 0.996074i \(0.528214\pi\)
\(72\) 0 0
\(73\) 36378.0 0.798972 0.399486 0.916739i \(-0.369189\pi\)
0.399486 + 0.916739i \(0.369189\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 81198.0 1.56070
\(78\) 0 0
\(79\) −22727.0 −0.409708 −0.204854 0.978793i \(-0.565672\pi\)
−0.204854 + 0.978793i \(0.565672\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −46254.0 −0.736977 −0.368489 0.929632i \(-0.620125\pi\)
−0.368489 + 0.929632i \(0.620125\pi\)
\(84\) 0 0
\(85\) 5925.00 0.0889490
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 58832.0 0.787297 0.393648 0.919261i \(-0.371213\pi\)
0.393648 + 0.919261i \(0.371213\pi\)
\(90\) 0 0
\(91\) 7722.00 0.0977521
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −37400.0 −0.425170
\(96\) 0 0
\(97\) −145906. −1.57450 −0.787252 0.616631i \(-0.788497\pi\)
−0.787252 + 0.616631i \(0.788497\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −89421.0 −0.872240 −0.436120 0.899888i \(-0.643648\pi\)
−0.436120 + 0.899888i \(0.643648\pi\)
\(102\) 0 0
\(103\) −134718. −1.25122 −0.625608 0.780137i \(-0.715149\pi\)
−0.625608 + 0.780137i \(0.715149\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −66002.0 −0.557311 −0.278656 0.960391i \(-0.589889\pi\)
−0.278656 + 0.960391i \(0.589889\pi\)
\(108\) 0 0
\(109\) −8460.00 −0.0682031 −0.0341016 0.999418i \(-0.510857\pi\)
−0.0341016 + 0.999418i \(0.510857\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 255183. 1.87999 0.939995 0.341188i \(-0.110829\pi\)
0.939995 + 0.341188i \(0.110829\pi\)
\(114\) 0 0
\(115\) −70275.0 −0.495514
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 55458.0 0.359002
\(120\) 0 0
\(121\) −40642.0 −0.252355
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −15625.0 −0.0894427
\(126\) 0 0
\(127\) 63790.0 0.350948 0.175474 0.984484i \(-0.443854\pi\)
0.175474 + 0.984484i \(0.443854\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −237541. −1.20937 −0.604687 0.796464i \(-0.706702\pi\)
−0.604687 + 0.796464i \(0.706702\pi\)
\(132\) 0 0
\(133\) −350064. −1.71600
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 137062. 0.623901 0.311950 0.950098i \(-0.399018\pi\)
0.311950 + 0.950098i \(0.399018\pi\)
\(138\) 0 0
\(139\) 189754. 0.833017 0.416509 0.909132i \(-0.363254\pi\)
0.416509 + 0.909132i \(0.363254\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11451.0 0.0468278
\(144\) 0 0
\(145\) −137825. −0.544387
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 50689.0 0.187046 0.0935229 0.995617i \(-0.470187\pi\)
0.0935229 + 0.995617i \(0.470187\pi\)
\(150\) 0 0
\(151\) −322103. −1.14961 −0.574807 0.818289i \(-0.694923\pi\)
−0.574807 + 0.818289i \(0.694923\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −72775.0 −0.243306
\(156\) 0 0
\(157\) −513807. −1.66361 −0.831804 0.555070i \(-0.812692\pi\)
−0.831804 + 0.555070i \(0.812692\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −657774. −1.99992
\(162\) 0 0
\(163\) 410491. 1.21014 0.605069 0.796173i \(-0.293146\pi\)
0.605069 + 0.796173i \(0.293146\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 362204. 1.00499 0.502495 0.864580i \(-0.332415\pi\)
0.502495 + 0.864580i \(0.332415\pi\)
\(168\) 0 0
\(169\) −370204. −0.997067
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 4690.00 0.0119140 0.00595700 0.999982i \(-0.498104\pi\)
0.00595700 + 0.999982i \(0.498104\pi\)
\(174\) 0 0
\(175\) −146250. −0.360995
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 280980. 0.655455 0.327727 0.944772i \(-0.393717\pi\)
0.327727 + 0.944772i \(0.393717\pi\)
\(180\) 0 0
\(181\) 721880. 1.63783 0.818915 0.573915i \(-0.194576\pi\)
0.818915 + 0.573915i \(0.194576\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −140050. −0.300853
\(186\) 0 0
\(187\) 82239.0 0.171978
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 104538. 0.207344 0.103672 0.994612i \(-0.466941\pi\)
0.103672 + 0.994612i \(0.466941\pi\)
\(192\) 0 0
\(193\) 695302. 1.34363 0.671816 0.740718i \(-0.265515\pi\)
0.671816 + 0.740718i \(0.265515\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 369268. 0.677916 0.338958 0.940801i \(-0.389925\pi\)
0.338958 + 0.940801i \(0.389925\pi\)
\(198\) 0 0
\(199\) −398589. −0.713498 −0.356749 0.934200i \(-0.616115\pi\)
−0.356749 + 0.934200i \(0.616115\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1.29004e6 −2.19717
\(204\) 0 0
\(205\) 117900. 0.195943
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −519112. −0.822045
\(210\) 0 0
\(211\) −142182. −0.219856 −0.109928 0.993940i \(-0.535062\pi\)
−0.109928 + 0.993940i \(0.535062\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −261975. −0.386513
\(216\) 0 0
\(217\) −681174. −0.981994
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7821.00 0.0107716
\(222\) 0 0
\(223\) −287548. −0.387211 −0.193606 0.981079i \(-0.562018\pi\)
−0.193606 + 0.981079i \(0.562018\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.07446e6 1.38397 0.691983 0.721914i \(-0.256737\pi\)
0.691983 + 0.721914i \(0.256737\pi\)
\(228\) 0 0
\(229\) 832856. 1.04950 0.524749 0.851257i \(-0.324159\pi\)
0.524749 + 0.851257i \(0.324159\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −249566. −0.301159 −0.150579 0.988598i \(-0.548114\pi\)
−0.150579 + 0.988598i \(0.548114\pi\)
\(234\) 0 0
\(235\) 149075. 0.176090
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.43083e6 1.62029 0.810144 0.586231i \(-0.199389\pi\)
0.810144 + 0.586231i \(0.199389\pi\)
\(240\) 0 0
\(241\) −854747. −0.947971 −0.473985 0.880533i \(-0.657185\pi\)
−0.473985 + 0.880533i \(0.657185\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −948725. −1.00978
\(246\) 0 0
\(247\) −49368.0 −0.0514877
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.77159e6 −1.77492 −0.887461 0.460883i \(-0.847533\pi\)
−0.887461 + 0.460883i \(0.847533\pi\)
\(252\) 0 0
\(253\) −975417. −0.958052
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.50216e6 1.41868 0.709338 0.704869i \(-0.248994\pi\)
0.709338 + 0.704869i \(0.248994\pi\)
\(258\) 0 0
\(259\) −1.31087e6 −1.21425
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.09514e6 1.86777 0.933886 0.357572i \(-0.116395\pi\)
0.933886 + 0.357572i \(0.116395\pi\)
\(264\) 0 0
\(265\) −449100. −0.392851
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1035.00 0.000872087 0 0.000436043 1.00000i \(-0.499861\pi\)
0.000436043 1.00000i \(0.499861\pi\)
\(270\) 0 0
\(271\) −1.46544e6 −1.21212 −0.606059 0.795420i \(-0.707250\pi\)
−0.606059 + 0.795420i \(0.707250\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −216875. −0.172933
\(276\) 0 0
\(277\) −911462. −0.713739 −0.356869 0.934154i \(-0.616156\pi\)
−0.356869 + 0.934154i \(0.616156\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −333494. −0.251955 −0.125977 0.992033i \(-0.540207\pi\)
−0.125977 + 0.992033i \(0.540207\pi\)
\(282\) 0 0
\(283\) −2.27522e6 −1.68872 −0.844358 0.535780i \(-0.820018\pi\)
−0.844358 + 0.535780i \(0.820018\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.10354e6 0.790833
\(288\) 0 0
\(289\) −1.36369e6 −0.960440
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.14585e6 0.779753 0.389877 0.920867i \(-0.372518\pi\)
0.389877 + 0.920867i \(0.372518\pi\)
\(294\) 0 0
\(295\) 759300. 0.507994
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −92763.0 −0.0600063
\(300\) 0 0
\(301\) −2.45209e6 −1.55998
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 888250. 0.546746
\(306\) 0 0
\(307\) −1.61298e6 −0.976750 −0.488375 0.872634i \(-0.662410\pi\)
−0.488375 + 0.872634i \(0.662410\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2.42846e6 −1.42374 −0.711869 0.702312i \(-0.752151\pi\)
−0.711869 + 0.702312i \(0.752151\pi\)
\(312\) 0 0
\(313\) −336764. −0.194296 −0.0971482 0.995270i \(-0.530972\pi\)
−0.0971482 + 0.995270i \(0.530972\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 743008. 0.415284 0.207642 0.978205i \(-0.433421\pi\)
0.207642 + 0.978205i \(0.433421\pi\)
\(318\) 0 0
\(319\) −1.91301e6 −1.05255
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −354552. −0.189092
\(324\) 0 0
\(325\) −20625.0 −0.0108314
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.39534e6 0.710708
\(330\) 0 0
\(331\) −200254. −0.100464 −0.0502321 0.998738i \(-0.515996\pi\)
−0.0502321 + 0.998738i \(0.515996\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 311900. 0.151846
\(336\) 0 0
\(337\) −1.14969e6 −0.551449 −0.275724 0.961237i \(-0.588918\pi\)
−0.275724 + 0.961237i \(0.588918\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.01012e6 −0.470420
\(342\) 0 0
\(343\) −4.94723e6 −2.27053
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.35551e6 −0.604336 −0.302168 0.953255i \(-0.597710\pi\)
−0.302168 + 0.953255i \(0.597710\pi\)
\(348\) 0 0
\(349\) 1.24421e6 0.546802 0.273401 0.961900i \(-0.411851\pi\)
0.273401 + 0.961900i \(0.411851\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −492467. −0.210349 −0.105174 0.994454i \(-0.533540\pi\)
−0.105174 + 0.994454i \(0.533540\pi\)
\(354\) 0 0
\(355\) 188000. 0.0791748
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 524400. 0.214747 0.107373 0.994219i \(-0.465756\pi\)
0.107373 + 0.994219i \(0.465756\pi\)
\(360\) 0 0
\(361\) −238083. −0.0961525
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −909450. −0.357311
\(366\) 0 0
\(367\) 3.15695e6 1.22349 0.611747 0.791053i \(-0.290467\pi\)
0.611747 + 0.791053i \(0.290467\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −4.20358e6 −1.58557
\(372\) 0 0
\(373\) 416509. 0.155007 0.0775037 0.996992i \(-0.475305\pi\)
0.0775037 + 0.996992i \(0.475305\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −181929. −0.0659248
\(378\) 0 0
\(379\) −3.96054e6 −1.41630 −0.708151 0.706061i \(-0.750470\pi\)
−0.708151 + 0.706061i \(0.750470\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.38757e6 −1.18003 −0.590013 0.807394i \(-0.700877\pi\)
−0.590013 + 0.807394i \(0.700877\pi\)
\(384\) 0 0
\(385\) −2.02995e6 −0.697965
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −248475. −0.0832547 −0.0416273 0.999133i \(-0.513254\pi\)
−0.0416273 + 0.999133i \(0.513254\pi\)
\(390\) 0 0
\(391\) −666207. −0.220378
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 568175. 0.183227
\(396\) 0 0
\(397\) −3.40110e6 −1.08304 −0.541518 0.840689i \(-0.682150\pi\)
−0.541518 + 0.840689i \(0.682150\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.22917e6 1.93450 0.967251 0.253820i \(-0.0816871\pi\)
0.967251 + 0.253820i \(0.0816871\pi\)
\(402\) 0 0
\(403\) −96063.0 −0.0294641
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.94389e6 −0.581683
\(408\) 0 0
\(409\) 510359. 0.150858 0.0754289 0.997151i \(-0.475967\pi\)
0.0754289 + 0.997151i \(0.475967\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.10705e6 2.05028
\(414\) 0 0
\(415\) 1.15635e6 0.329586
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −6.40576e6 −1.78253 −0.891263 0.453486i \(-0.850180\pi\)
−0.891263 + 0.453486i \(0.850180\pi\)
\(420\) 0 0
\(421\) 1.01408e6 0.278848 0.139424 0.990233i \(-0.455475\pi\)
0.139424 + 0.990233i \(0.455475\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −148125. −0.0397792
\(426\) 0 0
\(427\) 8.31402e6 2.20669
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 144184. 0.0373873 0.0186936 0.999825i \(-0.494049\pi\)
0.0186936 + 0.999825i \(0.494049\pi\)
\(432\) 0 0
\(433\) −501426. −0.128525 −0.0642624 0.997933i \(-0.520469\pi\)
−0.0642624 + 0.997933i \(0.520469\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.20526e6 1.05339
\(438\) 0 0
\(439\) −4.62831e6 −1.14620 −0.573101 0.819485i \(-0.694260\pi\)
−0.573101 + 0.819485i \(0.694260\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.89143e6 −1.42630 −0.713151 0.701010i \(-0.752733\pi\)
−0.713151 + 0.701010i \(0.752733\pi\)
\(444\) 0 0
\(445\) −1.47080e6 −0.352090
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 3.08619e6 0.722448 0.361224 0.932479i \(-0.382359\pi\)
0.361224 + 0.932479i \(0.382359\pi\)
\(450\) 0 0
\(451\) 1.63645e6 0.378845
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −193050. −0.0437161
\(456\) 0 0
\(457\) −5.26306e6 −1.17882 −0.589410 0.807834i \(-0.700640\pi\)
−0.589410 + 0.807834i \(0.700640\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 4.92047e6 1.07834 0.539168 0.842198i \(-0.318739\pi\)
0.539168 + 0.842198i \(0.318739\pi\)
\(462\) 0 0
\(463\) 3.85093e6 0.834860 0.417430 0.908709i \(-0.362931\pi\)
0.417430 + 0.908709i \(0.362931\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −7.75105e6 −1.64463 −0.822315 0.569033i \(-0.807318\pi\)
−0.822315 + 0.569033i \(0.807318\pi\)
\(468\) 0 0
\(469\) 2.91938e6 0.612857
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −3.63621e6 −0.747303
\(474\) 0 0
\(475\) 935000. 0.190142
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −603352. −0.120152 −0.0600761 0.998194i \(-0.519134\pi\)
−0.0600761 + 0.998194i \(0.519134\pi\)
\(480\) 0 0
\(481\) −184866. −0.0364330
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.64765e6 0.704140
\(486\) 0 0
\(487\) 1.12268e6 0.214502 0.107251 0.994232i \(-0.465795\pi\)
0.107251 + 0.994232i \(0.465795\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.26616e6 1.36019 0.680097 0.733122i \(-0.261938\pi\)
0.680097 + 0.733122i \(0.261938\pi\)
\(492\) 0 0
\(493\) −1.30658e6 −0.242114
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.75968e6 0.319553
\(498\) 0 0
\(499\) −6.28754e6 −1.13039 −0.565196 0.824956i \(-0.691200\pi\)
−0.565196 + 0.824956i \(0.691200\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 6.62421e6 1.16739 0.583693 0.811975i \(-0.301607\pi\)
0.583693 + 0.811975i \(0.301607\pi\)
\(504\) 0 0
\(505\) 2.23552e6 0.390078
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7.55534e6 −1.29259 −0.646293 0.763089i \(-0.723682\pi\)
−0.646293 + 0.763089i \(0.723682\pi\)
\(510\) 0 0
\(511\) −8.51245e6 −1.44212
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 3.36795e6 0.559561
\(516\) 0 0
\(517\) 2.06916e6 0.340461
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −7.75865e6 −1.25225 −0.626126 0.779722i \(-0.715360\pi\)
−0.626126 + 0.779722i \(0.715360\pi\)
\(522\) 0 0
\(523\) 407729. 0.0651805 0.0325902 0.999469i \(-0.489624\pi\)
0.0325902 + 0.999469i \(0.489624\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −689907. −0.108209
\(528\) 0 0
\(529\) 1.46538e6 0.227672
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 155628. 0.0237285
\(534\) 0 0
\(535\) 1.65005e6 0.249237
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.31683e7 −1.95235
\(540\) 0 0
\(541\) 53682.0 0.00788561 0.00394281 0.999992i \(-0.498745\pi\)
0.00394281 + 0.999992i \(0.498745\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 211500. 0.0305014
\(546\) 0 0
\(547\) −1.21855e7 −1.74131 −0.870656 0.491893i \(-0.836305\pi\)
−0.870656 + 0.491893i \(0.836305\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.24745e6 1.15729
\(552\) 0 0
\(553\) 5.31812e6 0.739512
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.83593e6 0.250737 0.125368 0.992110i \(-0.459989\pi\)
0.125368 + 0.992110i \(0.459989\pi\)
\(558\) 0 0
\(559\) −345807. −0.0468063
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −1.14405e7 −1.52116 −0.760581 0.649243i \(-0.775086\pi\)
−0.760581 + 0.649243i \(0.775086\pi\)
\(564\) 0 0
\(565\) −6.37958e6 −0.840757
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.89543e6 0.763370 0.381685 0.924293i \(-0.375344\pi\)
0.381685 + 0.924293i \(0.375344\pi\)
\(570\) 0 0
\(571\) 2.71925e6 0.349027 0.174514 0.984655i \(-0.444165\pi\)
0.174514 + 0.984655i \(0.444165\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.75688e6 0.221601
\(576\) 0 0
\(577\) −1.29749e7 −1.62243 −0.811215 0.584748i \(-0.801193\pi\)
−0.811215 + 0.584748i \(0.801193\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.08234e7 1.33022
\(582\) 0 0
\(583\) −6.23351e6 −0.759558
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.13612e6 0.136091 0.0680455 0.997682i \(-0.478324\pi\)
0.0680455 + 0.997682i \(0.478324\pi\)
\(588\) 0 0
\(589\) 4.35486e6 0.517232
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 6.91577e6 0.807614 0.403807 0.914844i \(-0.367687\pi\)
0.403807 + 0.914844i \(0.367687\pi\)
\(594\) 0 0
\(595\) −1.38645e6 −0.160551
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.17157e7 1.33414 0.667072 0.744993i \(-0.267547\pi\)
0.667072 + 0.744993i \(0.267547\pi\)
\(600\) 0 0
\(601\) −1.45503e7 −1.64318 −0.821589 0.570080i \(-0.806912\pi\)
−0.821589 + 0.570080i \(0.806912\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.01605e6 0.112857
\(606\) 0 0
\(607\) 7.53972e6 0.830584 0.415292 0.909688i \(-0.363679\pi\)
0.415292 + 0.909688i \(0.363679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 196779. 0.0213243
\(612\) 0 0
\(613\) 8.90986e6 0.957679 0.478839 0.877903i \(-0.341058\pi\)
0.478839 + 0.877903i \(0.341058\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −8.62134e6 −0.911721 −0.455860 0.890051i \(-0.650668\pi\)
−0.455860 + 0.890051i \(0.650668\pi\)
\(618\) 0 0
\(619\) −2.89049e6 −0.303211 −0.151605 0.988441i \(-0.548444\pi\)
−0.151605 + 0.988441i \(0.548444\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.37667e7 −1.42105
\(624\) 0 0
\(625\) 390625. 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.32767e6 −0.133803
\(630\) 0 0
\(631\) 7.04252e6 0.704133 0.352066 0.935975i \(-0.385479\pi\)
0.352066 + 0.935975i \(0.385479\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.59475e6 −0.156949
\(636\) 0 0
\(637\) −1.25232e6 −0.122283
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.58361e7 1.52231 0.761157 0.648568i \(-0.224632\pi\)
0.761157 + 0.648568i \(0.224632\pi\)
\(642\) 0 0
\(643\) −7.62712e6 −0.727500 −0.363750 0.931497i \(-0.618504\pi\)
−0.363750 + 0.931497i \(0.618504\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.11038e7 −1.04282 −0.521412 0.853305i \(-0.674595\pi\)
−0.521412 + 0.853305i \(0.674595\pi\)
\(648\) 0 0
\(649\) 1.05391e7 0.982180
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.84407e7 −1.69237 −0.846183 0.532892i \(-0.821105\pi\)
−0.846183 + 0.532892i \(0.821105\pi\)
\(654\) 0 0
\(655\) 5.93852e6 0.540848
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1.82622e7 −1.63810 −0.819050 0.573722i \(-0.805499\pi\)
−0.819050 + 0.573722i \(0.805499\pi\)
\(660\) 0 0
\(661\) 1.44528e7 1.28661 0.643306 0.765609i \(-0.277562\pi\)
0.643306 + 0.765609i \(0.277562\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 8.75160e6 0.767420
\(666\) 0 0
\(667\) 1.54970e7 1.34876
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.23289e7 1.05711
\(672\) 0 0
\(673\) −1.24675e7 −1.06107 −0.530533 0.847665i \(-0.678008\pi\)
−0.530533 + 0.847665i \(0.678008\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.51168e6 −0.546036 −0.273018 0.962009i \(-0.588022\pi\)
−0.273018 + 0.962009i \(0.588022\pi\)
\(678\) 0 0
\(679\) 3.41420e7 2.84194
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.69933e7 1.39388 0.696940 0.717129i \(-0.254544\pi\)
0.696940 + 0.717129i \(0.254544\pi\)
\(684\) 0 0
\(685\) −3.42655e6 −0.279017
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −592812. −0.0475739
\(690\) 0 0
\(691\) 1.66330e7 1.32518 0.662591 0.748982i \(-0.269457\pi\)
0.662591 + 0.748982i \(0.269457\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.74385e6 −0.372537
\(696\) 0 0
\(697\) 1.11769e6 0.0871445
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.65476e7 1.27187 0.635933 0.771744i \(-0.280616\pi\)
0.635933 + 0.771744i \(0.280616\pi\)
\(702\) 0 0
\(703\) 8.38059e6 0.639567
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.09245e7 1.57437
\(708\) 0 0
\(709\) −837868. −0.0625979 −0.0312990 0.999510i \(-0.509964\pi\)
−0.0312990 + 0.999510i \(0.509964\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.18282e6 0.602808
\(714\) 0 0
\(715\) −286275. −0.0209420
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −2.14252e7 −1.54562 −0.772809 0.634639i \(-0.781149\pi\)
−0.772809 + 0.634639i \(0.781149\pi\)
\(720\) 0 0
\(721\) 3.15240e7 2.25841
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.44562e6 0.243457
\(726\) 0 0
\(727\) −4.96740e6 −0.348572 −0.174286 0.984695i \(-0.555762\pi\)
−0.174286 + 0.984695i \(0.555762\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.48352e6 −0.171900
\(732\) 0 0
\(733\) 2.37177e7 1.63047 0.815233 0.579133i \(-0.196609\pi\)
0.815233 + 0.579133i \(0.196609\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.32917e6 0.293587
\(738\) 0 0
\(739\) 274304. 0.0184766 0.00923828 0.999957i \(-0.497059\pi\)
0.00923828 + 0.999957i \(0.497059\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −7.63188e6 −0.507177 −0.253588 0.967312i \(-0.581611\pi\)
−0.253588 + 0.967312i \(0.581611\pi\)
\(744\) 0 0
\(745\) −1.26722e6 −0.0836494
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1.54445e7 1.00593
\(750\) 0 0
\(751\) −4.28073e6 −0.276961 −0.138480 0.990365i \(-0.544222\pi\)
−0.138480 + 0.990365i \(0.544222\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.05258e6 0.514123
\(756\) 0 0
\(757\) −4.74502e6 −0.300952 −0.150476 0.988614i \(-0.548081\pi\)
−0.150476 + 0.988614i \(0.548081\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.08597e7 −0.679758 −0.339879 0.940469i \(-0.610386\pi\)
−0.339879 + 0.940469i \(0.610386\pi\)
\(762\) 0 0
\(763\) 1.97964e6 0.123105
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.00228e6 0.0615175
\(768\) 0 0
\(769\) 2.60429e7 1.58808 0.794042 0.607863i \(-0.207973\pi\)
0.794042 + 0.607863i \(0.207973\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.60052e7 1.56535 0.782676 0.622430i \(-0.213854\pi\)
0.782676 + 0.622430i \(0.213854\pi\)
\(774\) 0 0
\(775\) 1.81938e6 0.108810
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −7.05514e6 −0.416545
\(780\) 0 0
\(781\) 2.60944e6 0.153080
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.28452e7 0.743988
\(786\) 0 0
\(787\) 3.33737e7 1.92074 0.960369 0.278733i \(-0.0899144\pi\)
0.960369 + 0.278733i \(0.0899144\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.97128e7 −3.39333
\(792\) 0 0
\(793\) 1.17249e6 0.0662104
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.51931e7 1.40487 0.702434 0.711749i \(-0.252097\pi\)
0.702434 + 0.711749i \(0.252097\pi\)
\(798\) 0 0
\(799\) 1.41323e6 0.0783152
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.26232e7 −0.690843
\(804\) 0 0
\(805\) 1.64444e7 0.894390
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.18255e7 −1.17245 −0.586224 0.810149i \(-0.699386\pi\)
−0.586224 + 0.810149i \(0.699386\pi\)
\(810\) 0 0
\(811\) −3.44158e7 −1.83741 −0.918704 0.394946i \(-0.870763\pi\)
−0.918704 + 0.394946i \(0.870763\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −1.02623e7 −0.541190
\(816\) 0 0
\(817\) 1.56766e7 0.821668
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.53637e7 −1.31327 −0.656637 0.754207i \(-0.728022\pi\)
−0.656637 + 0.754207i \(0.728022\pi\)
\(822\) 0 0
\(823\) 2.28931e7 1.17816 0.589080 0.808075i \(-0.299490\pi\)
0.589080 + 0.808075i \(0.299490\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.21487e7 −1.12612 −0.563060 0.826416i \(-0.690376\pi\)
−0.563060 + 0.826416i \(0.690376\pi\)
\(828\) 0 0
\(829\) −2.61213e7 −1.32010 −0.660051 0.751220i \(-0.729466\pi\)
−0.660051 + 0.751220i \(0.729466\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −8.99391e6 −0.449093
\(834\) 0 0
\(835\) −9.05510e6 −0.449446
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −2.72008e7 −1.33407 −0.667033 0.745028i \(-0.732436\pi\)
−0.667033 + 0.745028i \(0.732436\pi\)
\(840\) 0 0
\(841\) 9.88202e6 0.481788
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.25510e6 0.445902
\(846\) 0 0
\(847\) 9.51023e6 0.455494
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.57472e7 0.745384
\(852\) 0 0
\(853\) 1.92152e7 0.904217 0.452109 0.891963i \(-0.350672\pi\)
0.452109 + 0.891963i \(0.350672\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.29237e7 1.53129 0.765644 0.643264i \(-0.222420\pi\)
0.765644 + 0.643264i \(0.222420\pi\)
\(858\) 0 0
\(859\) −2.72165e6 −0.125849 −0.0629244 0.998018i \(-0.520043\pi\)
−0.0629244 + 0.998018i \(0.520043\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.94282e6 0.225917 0.112958 0.993600i \(-0.463967\pi\)
0.112958 + 0.993600i \(0.463967\pi\)
\(864\) 0 0
\(865\) −117250. −0.00532810
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 7.88627e6 0.354260
\(870\) 0 0
\(871\) 411708. 0.0183884
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.65625e6 0.161442
\(876\) 0 0
\(877\) 5.57096e6 0.244586 0.122293 0.992494i \(-0.460975\pi\)
0.122293 + 0.992494i \(0.460975\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.21235e7 −0.526243 −0.263122 0.964763i \(-0.584752\pi\)
−0.263122 + 0.964763i \(0.584752\pi\)
\(882\) 0 0
\(883\) 1.21668e7 0.525141 0.262570 0.964913i \(-0.415430\pi\)
0.262570 + 0.964913i \(0.415430\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1.76495e6 −0.0753223 −0.0376612 0.999291i \(-0.511991\pi\)
−0.0376612 + 0.999291i \(0.511991\pi\)
\(888\) 0 0
\(889\) −1.49269e7 −0.633453
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −8.92065e6 −0.374341
\(894\) 0 0
\(895\) −7.02450e6 −0.293128
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.60483e7 0.662264
\(900\) 0 0
\(901\) −4.25747e6 −0.174719
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.80470e7 −0.732459
\(906\) 0 0
\(907\) −7.44128e6 −0.300351 −0.150176 0.988659i \(-0.547984\pi\)
−0.150176 + 0.988659i \(0.547984\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.44782e7 −1.37641 −0.688206 0.725515i \(-0.741602\pi\)
−0.688206 + 0.725515i \(0.741602\pi\)
\(912\) 0 0
\(913\) 1.60501e7 0.637238
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.55846e7 2.18289
\(918\) 0 0
\(919\) −4.55258e6 −0.177815 −0.0889076 0.996040i \(-0.528338\pi\)
−0.0889076 + 0.996040i \(0.528338\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 248160. 0.00958799
\(924\) 0 0
\(925\) 3.50125e6 0.134545
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 2.51258e7 0.955170 0.477585 0.878585i \(-0.341512\pi\)
0.477585 + 0.878585i \(0.341512\pi\)
\(930\) 0 0
\(931\) 5.67717e7 2.14663
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2.05598e6 −0.0769111
\(936\) 0 0
\(937\) 1.49058e7 0.554635 0.277318 0.960778i \(-0.410555\pi\)
0.277318 + 0.960778i \(0.410555\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 3.96462e7 1.45958 0.729789 0.683673i \(-0.239619\pi\)
0.729789 + 0.683673i \(0.239619\pi\)
\(942\) 0 0
\(943\) −1.32567e7 −0.485462
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.81296e7 1.38161 0.690807 0.723039i \(-0.257255\pi\)
0.690807 + 0.723039i \(0.257255\pi\)
\(948\) 0 0
\(949\) −1.20047e6 −0.0432700
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.95418e7 −0.697000 −0.348500 0.937309i \(-0.613309\pi\)
−0.348500 + 0.937309i \(0.613309\pi\)
\(954\) 0 0
\(955\) −2.61345e6 −0.0927269
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −3.20725e7 −1.12612
\(960\) 0 0
\(961\) −2.01552e7 −0.704011
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.73826e7 −0.600890
\(966\) 0 0
\(967\) −5.27096e7 −1.81269 −0.906345 0.422539i \(-0.861139\pi\)
−0.906345 + 0.422539i \(0.861139\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −2.41631e7 −0.822440 −0.411220 0.911536i \(-0.634897\pi\)
−0.411220 + 0.911536i \(0.634897\pi\)
\(972\) 0 0
\(973\) −4.44024e7 −1.50357
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 4.87535e7 1.63407 0.817033 0.576591i \(-0.195618\pi\)
0.817033 + 0.576591i \(0.195618\pi\)
\(978\) 0 0
\(979\) −2.04147e7 −0.680748
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.79029e7 −0.590936 −0.295468 0.955353i \(-0.595476\pi\)
−0.295468 + 0.955353i \(0.595476\pi\)
\(984\) 0 0
\(985\) −9.23170e6 −0.303173
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.94565e7 0.957613
\(990\) 0 0
\(991\) −5.11153e7 −1.65336 −0.826679 0.562674i \(-0.809773\pi\)
−0.826679 + 0.562674i \(0.809773\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 9.96472e6 0.319086
\(996\) 0 0
\(997\) 4.95876e7 1.57992 0.789961 0.613157i \(-0.210101\pi\)
0.789961 + 0.613157i \(0.210101\pi\)
\(998\) 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 1080.6.a.a.1.1 1
3.2 odd 2 1080.6.a.b.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.6.a.a.1.1 1 1.1 even 1 trivial
1080.6.a.b.1.1 yes 1 3.2 odd 2