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Properties

Label 56.5.h.a.13.1
Level $56$
Weight $5$
Character 56.13
Self dual yes
Analytic conductor $5.789$
Analytic rank $0$
Dimension $1$
CM discriminant -56
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [56,5,Mod(13,56)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(56, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("56.13");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 56.h (of order \(2\), degree \(1\), minimal)

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: \(5.78871793270\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 13.1
Character \(\chi\) \(=\) 56.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -10.0000 q^{3} +16.0000 q^{4} +22.0000 q^{5} -40.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +19.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -10.0000 q^{3} +16.0000 q^{4} +22.0000 q^{5} -40.0000 q^{6} +49.0000 q^{7} +64.0000 q^{8} +19.0000 q^{9} +88.0000 q^{10} -160.000 q^{12} +310.000 q^{13} +196.000 q^{14} -220.000 q^{15} +256.000 q^{16} +76.0000 q^{18} -650.000 q^{19} +352.000 q^{20} -490.000 q^{21} -958.000 q^{23} -640.000 q^{24} -141.000 q^{25} +1240.00 q^{26} +620.000 q^{27} +784.000 q^{28} -880.000 q^{30} +1024.00 q^{32} +1078.00 q^{35} +304.000 q^{36} -2600.00 q^{38} -3100.00 q^{39} +1408.00 q^{40} -1960.00 q^{42} +418.000 q^{45} -3832.00 q^{46} -2560.00 q^{48} +2401.00 q^{49} -564.000 q^{50} +4960.00 q^{52} +2480.00 q^{54} +3136.00 q^{56} +6500.00 q^{57} -1130.00 q^{59} -3520.00 q^{60} -7370.00 q^{61} +931.000 q^{63} +4096.00 q^{64} +6820.00 q^{65} +9580.00 q^{69} +4312.00 q^{70} +2018.00 q^{71} +1216.00 q^{72} +1410.00 q^{75} -10400.0 q^{76} -12400.0 q^{78} +4418.00 q^{79} +5632.00 q^{80} -7739.00 q^{81} -13130.0 q^{83} -7840.00 q^{84} +1672.00 q^{90} +15190.0 q^{91} -15328.0 q^{92} -14300.0 q^{95} -10240.0 q^{96} +9604.00 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/56\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(29\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 1.00000
\(3\) −10.0000 −1.11111 −0.555556 − 0.831479i \(-0.687494\pi\)
−0.555556 + 0.831479i \(0.687494\pi\)
\(4\) 16.0000 1.00000
\(5\) 22.0000 0.880000 0.440000 − 0.897998i \(-0.354978\pi\)
0.440000 + 0.897998i \(0.354978\pi\)
\(6\) −40.0000 −1.11111
\(7\) 49.0000 1.00000
\(8\) 64.0000 1.00000
\(9\) 19.0000 0.234568
\(10\) 88.0000 0.880000
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −160.000 −1.11111
\(13\) 310.000 1.83432 0.917160 − 0.398520i \(-0.130476\pi\)
0.917160 + 0.398520i \(0.130476\pi\)
\(14\) 196.000 1.00000
\(15\) −220.000 −0.977778
\(16\) 256.000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 76.0000 0.234568
\(19\) −650.000 −1.80055 −0.900277 − 0.435317i \(-0.856636\pi\)
−0.900277 + 0.435317i \(0.856636\pi\)
\(20\) 352.000 0.880000
\(21\) −490.000 −1.11111
\(22\) 0 0
\(23\) −958.000 −1.81096 −0.905482 − 0.424385i \(-0.860490\pi\)
−0.905482 + 0.424385i \(0.860490\pi\)
\(24\) −640.000 −1.11111
\(25\) −141.000 −0.225600
\(26\) 1240.00 1.83432
\(27\) 620.000 0.850480
\(28\) 784.000 1.00000
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) −880.000 −0.977778
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 1024.00 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) 1078.00 0.880000
\(36\) 304.000 0.234568
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) −2600.00 −1.80055
\(39\) −3100.00 −2.03813
\(40\) 1408.00 0.880000
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) −1960.00 −1.11111
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 418.000 0.206420
\(46\) −3832.00 −1.81096
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −2560.00 −1.11111
\(49\) 2401.00 1.00000
\(50\) −564.000 −0.225600
\(51\) 0 0
\(52\) 4960.00 1.83432
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 2480.00 0.850480
\(55\) 0 0
\(56\) 3136.00 1.00000
\(57\) 6500.00 2.00062
\(58\) 0 0
\(59\) −1130.00 −0.324619 −0.162310 − 0.986740i \(-0.551894\pi\)
−0.162310 + 0.986740i \(0.551894\pi\)
\(60\) −3520.00 −0.977778
\(61\) −7370.00 −1.98065 −0.990325 − 0.138766i \(-0.955686\pi\)
−0.990325 + 0.138766i \(0.955686\pi\)
\(62\) 0 0
\(63\) 931.000 0.234568
\(64\) 4096.00 1.00000
\(65\) 6820.00 1.61420
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 9580.00 2.01218
\(70\) 4312.00 0.880000
\(71\) 2018.00 0.400317 0.200159 − 0.979763i \(-0.435854\pi\)
0.200159 + 0.979763i \(0.435854\pi\)
\(72\) 1216.00 0.234568
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1410.00 0.250667
\(76\) −10400.0 −1.80055
\(77\) 0 0
\(78\) −12400.0 −2.03813
\(79\) 4418.00 0.707899 0.353950 − 0.935264i \(-0.384838\pi\)
0.353950 + 0.935264i \(0.384838\pi\)
\(80\) 5632.00 0.880000
\(81\) −7739.00 −1.17955
\(82\) 0 0
\(83\) −13130.0 −1.90594 −0.952969 − 0.303069i \(-0.901989\pi\)
−0.952969 + 0.303069i \(0.901989\pi\)
\(84\) −7840.00 −1.11111
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 1672.00 0.206420
\(91\) 15190.0 1.83432
\(92\) −15328.0 −1.81096
\(93\) 0 0
\(94\) 0 0
\(95\) −14300.0 −1.58449
\(96\) −10240.0 −1.11111
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 9604.00 1.00000
\(99\) 0 0
\(100\) −2256.00 −0.225600
\(101\) 5590.00 0.547985 0.273993 − 0.961732i \(-0.411656\pi\)
0.273993 + 0.961732i \(0.411656\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 19840.0 1.83432
\(105\) −10780.0 −0.977778
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 9920.00 0.850480
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 12544.0 1.00000
\(113\) −24862.0 −1.94706 −0.973530 − 0.228561i \(-0.926598\pi\)
−0.973530 + 0.228561i \(0.926598\pi\)
\(114\) 26000.0 2.00062
\(115\) −21076.0 −1.59365
\(116\) 0 0
\(117\) 5890.00 0.430272
\(118\) −4520.00 −0.324619
\(119\) 0 0
\(120\) −14080.0 −0.977778
\(121\) 14641.0 1.00000
\(122\) −29480.0 −1.98065
\(123\) 0 0
\(124\) 0 0
\(125\) −16852.0 −1.07853
\(126\) 3724.00 0.234568
\(127\) 30242.0 1.87501 0.937504 − 0.347975i \(-0.113131\pi\)
0.937504 + 0.347975i \(0.113131\pi\)
\(128\) 16384.0 1.00000
\(129\) 0 0
\(130\) 27280.0 1.61420
\(131\) 26230.0 1.52847 0.764233 − 0.644940i \(-0.223118\pi\)
0.764233 + 0.644940i \(0.223118\pi\)
\(132\) 0 0
\(133\) −31850.0 −1.80055
\(134\) 0 0
\(135\) 13640.0 0.748422
\(136\) 0 0
\(137\) −35038.0 −1.86680 −0.933401 − 0.358835i \(-0.883174\pi\)
−0.933401 + 0.358835i \(0.883174\pi\)
\(138\) 38320.0 2.01218
\(139\) 30550.0 1.58118 0.790591 − 0.612345i \(-0.209774\pi\)
0.790591 + 0.612345i \(0.209774\pi\)
\(140\) 17248.0 0.880000
\(141\) 0 0
\(142\) 8072.00 0.400317
\(143\) 0 0
\(144\) 4864.00 0.234568
\(145\) 0 0
\(146\) 0 0
\(147\) −24010.0 −1.11111
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 5640.00 0.250667
\(151\) −4798.00 −0.210429 −0.105215 − 0.994450i \(-0.533553\pi\)
−0.105215 + 0.994450i \(0.533553\pi\)
\(152\) −41600.0 −1.80055
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −49600.0 −2.03813
\(157\) 49270.0 1.99886 0.999432 − 0.0336991i \(-0.0107288\pi\)
0.999432 + 0.0336991i \(0.0107288\pi\)
\(158\) 17672.0 0.707899
\(159\) 0 0
\(160\) 22528.0 0.880000
\(161\) −46942.0 −1.81096
\(162\) −30956.0 −1.17955
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −52520.0 −1.90594
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) −31360.0 −1.11111
\(169\) 67539.0 2.36473
\(170\) 0 0
\(171\) −12350.0 −0.422352
\(172\) 0 0
\(173\) −7370.00 −0.246249 −0.123125 − 0.992391i \(-0.539292\pi\)
−0.123125 + 0.992391i \(0.539292\pi\)
\(174\) 0 0
\(175\) −6909.00 −0.225600
\(176\) 0 0
\(177\) 11300.0 0.360688
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 6688.00 0.206420
\(181\) 3670.00 0.112023 0.0560117 − 0.998430i \(-0.482162\pi\)
0.0560117 + 0.998430i \(0.482162\pi\)
\(182\) 60760.0 1.83432
\(183\) 73700.0 2.20072
\(184\) −61312.0 −1.81096
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 30380.0 0.850480
\(190\) −57200.0 −1.58449
\(191\) −56062.0 −1.53675 −0.768373 − 0.640003i \(-0.778933\pi\)
−0.768373 + 0.640003i \(0.778933\pi\)
\(192\) −40960.0 −1.11111
\(193\) 24098.0 0.646944 0.323472 − 0.946238i \(-0.395150\pi\)
0.323472 + 0.946238i \(0.395150\pi\)
\(194\) 0 0
\(195\) −68200.0 −1.79356
\(196\) 38416.0 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −9024.00 −0.225600
\(201\) 0 0
\(202\) 22360.0 0.547985
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −18202.0 −0.424794
\(208\) 79360.0 1.83432
\(209\) 0 0
\(210\) −43120.0 −0.977778
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) −20180.0 −0.444797
\(214\) 0 0
\(215\) 0 0
\(216\) 39680.0 0.850480
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 50176.0 1.00000
\(225\) −2679.00 −0.0529185
\(226\) −99448.0 −1.94706
\(227\) 55990.0 1.08657 0.543286 − 0.839547i \(-0.317180\pi\)
0.543286 + 0.839547i \(0.317180\pi\)
\(228\) 104000. 2.00062
\(229\) −44330.0 −0.845331 −0.422665 − 0.906286i \(-0.638905\pi\)
−0.422665 + 0.906286i \(0.638905\pi\)
\(230\) −84304.0 −1.59365
\(231\) 0 0
\(232\) 0 0
\(233\) 76322.0 1.40585 0.702923 − 0.711266i \(-0.251878\pi\)
0.702923 + 0.711266i \(0.251878\pi\)
\(234\) 23560.0 0.430272
\(235\) 0 0
\(236\) −18080.0 −0.324619
\(237\) −44180.0 −0.786555
\(238\) 0 0
\(239\) 63842.0 1.11766 0.558831 − 0.829281i \(-0.311250\pi\)
0.558831 + 0.829281i \(0.311250\pi\)
\(240\) −56320.0 −0.977778
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 58564.0 1.00000
\(243\) 27170.0 0.460126
\(244\) −117920. −1.98065
\(245\) 52822.0 0.880000
\(246\) 0 0
\(247\) −201500. −3.30279
\(248\) 0 0
\(249\) 131300. 2.11771
\(250\) −67408.0 −1.07853
\(251\) 124630. 1.97822 0.989111 − 0.147170i \(-0.0470163\pi\)
0.989111 + 0.147170i \(0.0470163\pi\)
\(252\) 14896.0 0.234568
\(253\) 0 0
\(254\) 120968. 1.87501
\(255\) 0 0
\(256\) 65536.0 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 109120. 1.61420
\(261\) 0 0
\(262\) 104920. 1.52847
\(263\) −63262.0 −0.914600 −0.457300 − 0.889312i \(-0.651184\pi\)
−0.457300 + 0.889312i \(0.651184\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −127400. −1.80055
\(267\) 0 0
\(268\) 0 0
\(269\) −118730. −1.64080 −0.820400 − 0.571789i \(-0.806249\pi\)
−0.820400 + 0.571789i \(0.806249\pi\)
\(270\) 54560.0 0.748422
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) −151900. −2.03813
\(274\) −140152. −1.86680
\(275\) 0 0
\(276\) 153280. 2.01218
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 122200. 1.58118
\(279\) 0 0
\(280\) 68992.0 0.880000
\(281\) −43678.0 −0.553159 −0.276580 − 0.960991i \(-0.589201\pi\)
−0.276580 + 0.960991i \(0.589201\pi\)
\(282\) 0 0
\(283\) −61610.0 −0.769269 −0.384635 − 0.923069i \(-0.625673\pi\)
−0.384635 + 0.923069i \(0.625673\pi\)
\(284\) 32288.0 0.400317
\(285\) 143000. 1.76054
\(286\) 0 0
\(287\) 0 0
\(288\) 19456.0 0.234568
\(289\) 83521.0 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 30550.0 0.355857 0.177929 − 0.984043i \(-0.443060\pi\)
0.177929 + 0.984043i \(0.443060\pi\)
\(294\) −96040.0 −1.11111
\(295\) −24860.0 −0.285665
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −296980. −3.32189
\(300\) 22560.0 0.250667
\(301\) 0 0
\(302\) −19192.0 −0.210429
\(303\) −55900.0 −0.608873
\(304\) −166400. −1.80055
\(305\) −162140. −1.74297
\(306\) 0 0
\(307\) 13750.0 0.145890 0.0729451 − 0.997336i \(-0.476760\pi\)
0.0729451 + 0.997336i \(0.476760\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) −198400. −2.03813
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 197080. 1.99886
\(315\) 20482.0 0.206420
\(316\) 70688.0 0.707899
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 90112.0 0.880000
\(321\) 0 0
\(322\) −187768. −1.81096
\(323\) 0 0
\(324\) −123824. −1.17955
\(325\) −43710.0 −0.413822
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −210080. −1.90594
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −125440. −1.11111
\(337\) −226462. −1.99405 −0.997024 − 0.0770939i \(-0.975436\pi\)
−0.997024 + 0.0770939i \(0.975436\pi\)
\(338\) 270156. 2.36473
\(339\) 248620. 2.16340
\(340\) 0 0
\(341\) 0 0
\(342\) −49400.0 −0.422352
\(343\) 117649. 1.00000
\(344\) 0 0
\(345\) 210760. 1.77072
\(346\) −29480.0 −0.246249
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 228790. 1.87839 0.939196 − 0.343382i \(-0.111572\pi\)
0.939196 + 0.343382i \(0.111572\pi\)
\(350\) −27636.0 −0.225600
\(351\) 192200. 1.56005
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 45200.0 0.360688
\(355\) 44396.0 0.352279
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 207362. 1.60894 0.804471 − 0.593992i \(-0.202449\pi\)
0.804471 + 0.593992i \(0.202449\pi\)
\(360\) 26752.0 0.206420
\(361\) 292179. 2.24199
\(362\) 14680.0 0.112023
\(363\) −146410. −1.11111
\(364\) 243040. 1.83432
\(365\) 0 0
\(366\) 294800. 2.20072
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −245248. −1.81096
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 168520. 1.19836
\(376\) 0 0
\(377\) 0 0
\(378\) 121520. 0.850480
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) −228800. −1.58449
\(381\) −302420. −2.08334
\(382\) −224248. −1.53675
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −163840. −1.11111
\(385\) 0 0
\(386\) 96392.0 0.646944
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) −272800. −1.79356
\(391\) 0 0
\(392\) 153664. 1.00000
\(393\) −262300. −1.69830
\(394\) 0 0
\(395\) 97196.0 0.622951
\(396\) 0 0
\(397\) −81290.0 −0.515770 −0.257885 − 0.966176i \(-0.583026\pi\)
−0.257885 + 0.966176i \(0.583026\pi\)
\(398\) 0 0
\(399\) 318500. 2.00062
\(400\) −36096.0 −0.225600
\(401\) −19102.0 −0.118793 −0.0593964 − 0.998234i \(-0.518918\pi\)
−0.0593964 + 0.998234i \(0.518918\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 89440.0 0.547985
\(405\) −170258. −1.03800
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 350380. 2.07422
\(412\) 0 0
\(413\) −55370.0 −0.324619
\(414\) −72808.0 −0.424794
\(415\) −288860. −1.67722
\(416\) 317440. 1.83432
\(417\) −305500. −1.75687
\(418\) 0 0
\(419\) −100490. −0.572394 −0.286197 − 0.958171i \(-0.592391\pi\)
−0.286197 + 0.958171i \(0.592391\pi\)
\(420\) −172480. −0.977778
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) −80720.0 −0.444797
\(427\) −361130. −1.98065
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −82078.0 −0.441847 −0.220924 − 0.975291i \(-0.570907\pi\)
−0.220924 + 0.975291i \(0.570907\pi\)
\(432\) 158720. 0.850480
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 622700. 3.26074
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 45619.0 0.234568
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 200704. 1.00000
\(449\) −403198. −1.99998 −0.999990 − 0.00445433i \(-0.998582\pi\)
−0.999990 + 0.00445433i \(0.998582\pi\)
\(450\) −10716.0 −0.0529185
\(451\) 0 0
\(452\) −397792. −1.94706
\(453\) 47980.0 0.233810
\(454\) 223960. 1.08657
\(455\) 334180. 1.61420
\(456\) 416000. 2.00062
\(457\) 367298. 1.75868 0.879339 − 0.476197i \(-0.157985\pi\)
0.879339 + 0.476197i \(0.157985\pi\)
\(458\) −177320. −0.845331
\(459\) 0 0
\(460\) −337216. −1.59365
\(461\) −355850. −1.67442 −0.837211 − 0.546879i \(-0.815816\pi\)
−0.837211 + 0.546879i \(0.815816\pi\)
\(462\) 0 0
\(463\) −377662. −1.76174 −0.880869 − 0.473360i \(-0.843041\pi\)
−0.880869 + 0.473360i \(0.843041\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 305288. 1.40585
\(467\) −202250. −0.927374 −0.463687 − 0.885999i \(-0.653474\pi\)
−0.463687 + 0.885999i \(0.653474\pi\)
\(468\) 94240.0 0.430272
\(469\) 0 0
\(470\) 0 0
\(471\) −492700. −2.22096
\(472\) −72320.0 −0.324619
\(473\) 0 0
\(474\) −176720. −0.786555
\(475\) 91650.0 0.406205
\(476\) 0 0
\(477\) 0 0
\(478\) 255368. 1.11766
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) −225280. −0.977778
\(481\) 0 0
\(482\) 0 0
\(483\) 469420. 2.01218
\(484\) 234256. 1.00000
\(485\) 0 0
\(486\) 108680. 0.460126
\(487\) −253438. −1.06860 −0.534298 − 0.845296i \(-0.679424\pi\)
−0.534298 + 0.845296i \(0.679424\pi\)
\(488\) −471680. −1.98065
\(489\) 0 0
\(490\) 211288. 0.880000
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −806000. −3.30279
\(495\) 0 0
\(496\) 0 0
\(497\) 98882.0 0.400317
\(498\) 525200. 2.11771
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −269632. −1.07853
\(501\) 0 0
\(502\) 498520. 1.97822
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 59584.0 0.234568
\(505\) 122980. 0.482227
\(506\) 0 0
\(507\) −675390. −2.62748
\(508\) 483872. 1.87501
\(509\) 368950. 1.42407 0.712036 − 0.702143i \(-0.247773\pi\)
0.712036 + 0.702143i \(0.247773\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 262144. 1.00000
\(513\) −403000. −1.53134
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 73700.0 0.273611
\(520\) 436480. 1.61420
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 372310. 1.36114 0.680568 − 0.732685i \(-0.261733\pi\)
0.680568 + 0.732685i \(0.261733\pi\)
\(524\) 419680. 1.52847
\(525\) 69090.0 0.250667
\(526\) −253048. −0.914600
\(527\) 0 0
\(528\) 0 0
\(529\) 637923. 2.27959
\(530\) 0 0
\(531\) −21470.0 −0.0761453
\(532\) −509600. −1.80055
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −474920. −1.64080
\(539\) 0 0
\(540\) 218240. 0.748422
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −36700.0 −0.124470
\(544\) 0 0
\(545\) 0 0
\(546\) −607600. −2.03813
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −560608. −1.86680
\(549\) −140030. −0.464597
\(550\) 0 0
\(551\) 0 0
\(552\) 613120. 2.01218
\(553\) 216482. 0.707899
\(554\) 0 0
\(555\) 0 0
\(556\) 488800. 1.58118
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 275968. 0.880000
\(561\) 0 0
\(562\) −174712. −0.553159
\(563\) −474890. −1.49822 −0.749111 − 0.662444i \(-0.769519\pi\)
−0.749111 + 0.662444i \(0.769519\pi\)
\(564\) 0 0
\(565\) −546964. −1.71341
\(566\) −246440. −0.769269
\(567\) −379211. −1.17955
\(568\) 129152. 0.400317
\(569\) 629378. 1.94396 0.971979 − 0.235066i \(-0.0755307\pi\)
0.971979 + 0.235066i \(0.0755307\pi\)
\(570\) 572000. 1.76054
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 560620. 1.70749
\(574\) 0 0
\(575\) 135078. 0.408553
\(576\) 77824.0 0.234568
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 334084. 1.00000
\(579\) −240980. −0.718826
\(580\) 0 0
\(581\) −643370. −1.90594
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 129580. 0.378640
\(586\) 122200. 0.355857
\(587\) 662230. 1.92191 0.960954 − 0.276708i \(-0.0892434\pi\)
0.960954 + 0.276708i \(0.0892434\pi\)
\(588\) −384160. −1.11111
\(589\) 0 0
\(590\) −99440.0 −0.285665
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −1.18792e6 −3.32189
\(599\) 427298. 1.19091 0.595453 − 0.803390i \(-0.296973\pi\)
0.595453 + 0.803390i \(0.296973\pi\)
\(600\) 90240.0 0.250667
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −76768.0 −0.210429
\(605\) 322102. 0.880000
\(606\) −223600. −0.608873
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −665600. −1.80055
\(609\) 0 0
\(610\) −648560. −1.74297
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 55000.0 0.145890
\(615\) 0 0
\(616\) 0 0
\(617\) 307778. 0.808476 0.404238 − 0.914654i \(-0.367537\pi\)
0.404238 + 0.914654i \(0.367537\pi\)
\(618\) 0 0
\(619\) 469270. 1.22473 0.612367 − 0.790574i \(-0.290218\pi\)
0.612367 + 0.790574i \(0.290218\pi\)
\(620\) 0 0
\(621\) −593960. −1.54019
\(622\) 0 0
\(623\) 0 0
\(624\) −793600. −2.03813
\(625\) −282619. −0.723505
\(626\) 0 0
\(627\) 0 0
\(628\) 788320. 1.99886
\(629\) 0 0
\(630\) 81928.0 0.206420
\(631\) −784222. −1.96961 −0.984805 − 0.173663i \(-0.944440\pi\)
−0.984805 + 0.173663i \(0.944440\pi\)
\(632\) 282752. 0.707899
\(633\) 0 0
\(634\) 0 0
\(635\) 665324. 1.65001
\(636\) 0 0
\(637\) 744310. 1.83432
\(638\) 0 0
\(639\) 38342.0 0.0939016
\(640\) 360448. 0.880000
\(641\) 239138. 0.582013 0.291006 − 0.956721i \(-0.406010\pi\)
0.291006 + 0.956721i \(0.406010\pi\)
\(642\) 0 0
\(643\) −281930. −0.681898 −0.340949 − 0.940082i \(-0.610748\pi\)
−0.340949 + 0.940082i \(0.610748\pi\)
\(644\) −751072. −1.81096
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −495296. −1.17955
\(649\) 0 0
\(650\) −174840. −0.413822
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 577060. 1.34505
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 724630. 1.65849 0.829246 − 0.558884i \(-0.188770\pi\)
0.829246 + 0.558884i \(0.188770\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −840320. −1.90594
\(665\) −700700. −1.58449
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) −501760. −1.11111
\(673\) −456958. −1.00890 −0.504448 − 0.863442i \(-0.668304\pi\)
−0.504448 + 0.863442i \(0.668304\pi\)
\(674\) −905848. −1.99405
\(675\) −87420.0 −0.191868
\(676\) 1.08062e6 2.36473
\(677\) −682730. −1.48961 −0.744803 − 0.667284i \(-0.767457\pi\)
−0.744803 + 0.667284i \(0.767457\pi\)
\(678\) 994480. 2.16340
\(679\) 0 0
\(680\) 0 0
\(681\) −559900. −1.20730
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −197600. −0.422352
\(685\) −770836. −1.64279
\(686\) 470596. 1.00000
\(687\) 443300. 0.939257
\(688\) 0 0
\(689\) 0 0
\(690\) 843040. 1.77072
\(691\) −894410. −1.87318 −0.936592 − 0.350421i \(-0.886038\pi\)
−0.936592 + 0.350421i \(0.886038\pi\)
\(692\) −117920. −0.246249
\(693\) 0 0
\(694\) 0 0
\(695\) 672100. 1.39144
\(696\) 0 0
\(697\) 0 0
\(698\) 915160. 1.87839
\(699\) −763220. −1.56205
\(700\) −110544. −0.225600
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 768800. 1.56005
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 273910. 0.547985
\(708\) 180800. 0.360688
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 177584. 0.352279
\(711\) 83942.0 0.166050
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −638420. −1.24185
\(718\) 829448. 1.60894
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 107008. 0.206420
\(721\) 0 0
\(722\) 1.16872e6 2.24199
\(723\) 0 0
\(724\) 58720.0 0.112023
\(725\) 0 0
\(726\) −585640. −1.11111
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 972160. 1.83432
\(729\) 355159. 0.668294
\(730\) 0 0
\(731\) 0 0
\(732\) 1.17920e6 2.20072
\(733\) 933430. 1.73730 0.868648 − 0.495430i \(-0.164989\pi\)
0.868648 + 0.495430i \(0.164989\pi\)
\(734\) 0 0
\(735\) −528220. −0.977778
\(736\) −980992. −1.81096
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 2.01500e6 3.66977
\(742\) 0 0
\(743\) 940802. 1.70420 0.852100 − 0.523379i \(-0.175329\pi\)
0.852100 + 0.523379i \(0.175329\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −249470. −0.447072
\(748\) 0 0
\(749\) 0 0
\(750\) 674080. 1.19836
\(751\) −131998. −0.234039 −0.117019 − 0.993130i \(-0.537334\pi\)
−0.117019 + 0.993130i \(0.537334\pi\)
\(752\) 0 0
\(753\) −1.24630e6 −2.19803
\(754\) 0 0
\(755\) −105556. −0.185178
\(756\) 486080. 0.850480
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) −915200. −1.58449
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) −1.20968e6 −2.08334
\(763\) 0 0
\(764\) −896992. −1.53675
\(765\) 0 0
\(766\) 0 0
\(767\) −350300. −0.595456
\(768\) −655360. −1.11111
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 385568. 0.646944
\(773\) −431210. −0.721655 −0.360828 − 0.932633i \(-0.617506\pi\)
−0.360828 + 0.932633i \(0.617506\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) −1.09120e6 −1.79356
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 614656. 1.00000
\(785\) 1.08394e6 1.75900
\(786\) −1.04920e6 −1.69830
\(787\) 1.01695e6 1.64191 0.820956 − 0.570991i \(-0.193441\pi\)
0.820956 + 0.570991i \(0.193441\pi\)
\(788\) 0 0
\(789\) 632620. 1.01622
\(790\) 388784. 0.622951
\(791\) −1.21824e6 −1.94706
\(792\) 0 0
\(793\) −2.28470e6 −3.63315
\(794\) −325160. −0.515770
\(795\) 0 0
\(796\) 0 0
\(797\) −1.06817e6 −1.68160 −0.840802 − 0.541343i \(-0.817916\pi\)
−0.840802 + 0.541343i \(0.817916\pi\)
\(798\) 1.27400e6 2.00062
\(799\) 0 0
\(800\) −144384. −0.225600
\(801\) 0 0
\(802\) −76408.0 −0.118793
\(803\) 0 0
\(804\) 0 0
\(805\) −1.03272e6 −1.59365
\(806\) 0 0
\(807\) 1.18730e6 1.82311
\(808\) 357760. 0.547985
\(809\) −886462. −1.35445 −0.677225 − 0.735776i \(-0.736818\pi\)
−0.677225 + 0.735776i \(0.736818\pi\)
\(810\) −681032. −1.03800
\(811\) 789910. 1.20098 0.600490 − 0.799632i \(-0.294972\pi\)
0.600490 + 0.799632i \(0.294972\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 288610. 0.430272
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 1.40152e6 2.07422
\(823\) −459742. −0.678757 −0.339379 − 0.940650i \(-0.610217\pi\)
−0.339379 + 0.940650i \(0.610217\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −221480. −0.324619
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −291232. −0.424794
\(829\) −1.36865e6 −1.99151 −0.995757 − 0.0920223i \(-0.970667\pi\)
−0.995757 + 0.0920223i \(0.970667\pi\)
\(830\) −1.15544e6 −1.67722
\(831\) 0 0
\(832\) 1.26976e6 1.83432
\(833\) 0 0
\(834\) −1.22200e6 −1.75687
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −401960. −0.572394
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) −689920. −0.977778
\(841\) 707281. 1.00000
\(842\) 0 0
\(843\) 436780. 0.614621
\(844\) 0 0
\(845\) 1.48586e6 2.08096
\(846\) 0 0
\(847\) 717409. 1.00000
\(848\) 0 0
\(849\) 616100. 0.854744
\(850\) 0 0
\(851\) 0 0
\(852\) −322880. −0.444797
\(853\) −1.25297e6 −1.72204 −0.861019 − 0.508573i \(-0.830173\pi\)
−0.861019 + 0.508573i \(0.830173\pi\)
\(854\) −1.44452e6 −1.98065
\(855\) −271700. −0.371670
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −44330.0 −0.0600774 −0.0300387 − 0.999549i \(-0.509563\pi\)
−0.0300387 + 0.999549i \(0.509563\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −328312. −0.441847
\(863\) 683138. 0.917248 0.458624 − 0.888630i \(-0.348342\pi\)
0.458624 + 0.888630i \(0.348342\pi\)
\(864\) 634880. 0.850480
\(865\) −162140. −0.216700
\(866\) 0 0
\(867\) −835210. −1.11111
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 2.49080e6 3.26074
\(875\) −825748. −1.07853
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) −305500. −0.395397
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 182476. 0.234568
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 248600. 0.317406
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.48186e6 1.87501
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 802816. 1.00000
\(897\) 2.96980e6 3.69099
\(898\) −1.61279e6 −1.99998
\(899\) 0 0
\(900\) −42864.0 −0.0529185
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −1.59117e6 −1.94706
\(905\) 80740.0 0.0985806
\(906\) 191920. 0.233810
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 895840. 1.08657
\(909\) 106210. 0.128540
\(910\) 1.33672e6 1.61420
\(911\) −809758. −0.975705 −0.487852 − 0.872926i \(-0.662220\pi\)
−0.487852 + 0.872926i \(0.662220\pi\)
\(912\) 1.66400e6 2.00062
\(913\) 0 0
\(914\) 1.46919e6 1.75868
\(915\) 1.62140e6 1.93664
\(916\) −709280. −0.845331
\(917\) 1.28527e6 1.52847
\(918\) 0 0
\(919\) 713378. 0.844673 0.422337 − 0.906439i \(-0.361210\pi\)
0.422337 + 0.906439i \(0.361210\pi\)
\(920\) −1.34886e6 −1.59365
\(921\) −137500. −0.162100
\(922\) −1.42340e6 −1.67442
\(923\) 625580. 0.734310
\(924\) 0 0
\(925\) 0 0
\(926\) −1.51065e6 −1.76174
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −1.56065e6 −1.80055
\(932\) 1.22115e6 1.40585
\(933\) 0 0
\(934\) −809000. −0.927374
\(935\) 0 0
\(936\) 376960. 0.430272
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −535370. −0.604609 −0.302305 − 0.953211i \(-0.597756\pi\)
−0.302305 + 0.953211i \(0.597756\pi\)
\(942\) −1.97080e6 −2.22096
\(943\) 0 0
\(944\) −289280. −0.324619
\(945\) 668360. 0.748422
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) −706880. −0.786555
\(949\) 0 0
\(950\) 366600. 0.406205
\(951\) 0 0
\(952\) 0 0
\(953\) −796318. −0.876800 −0.438400 − 0.898780i \(-0.644455\pi\)
−0.438400 + 0.898780i \(0.644455\pi\)
\(954\) 0 0
\(955\) −1.23336e6 −1.35234
\(956\) 1.02147e6 1.11766
\(957\) 0 0
\(958\) 0 0
\(959\) −1.71686e6 −1.86680
\(960\) −901120. −0.977778
\(961\) 923521. 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 530156. 0.569310
\(966\) 1.87768e6 2.01218
\(967\) 174722. 0.186851 0.0934253 − 0.995626i \(-0.470218\pi\)
0.0934253 + 0.995626i \(0.470218\pi\)
\(968\) 937024. 1.00000
\(969\) 0 0
\(970\) 0 0
\(971\) −1.88561e6 −1.99992 −0.999962 − 0.00873862i \(-0.997218\pi\)
−0.999962 + 0.00873862i \(0.997218\pi\)
\(972\) 434720. 0.460126
\(973\) 1.49695e6 1.58118
\(974\) −1.01375e6 −1.06860
\(975\) 437100. 0.459803
\(976\) −1.88672e6 −1.98065
\(977\) 747842. 0.783467 0.391734 − 0.920079i \(-0.371876\pi\)
0.391734 + 0.920079i \(0.371876\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 845152. 0.880000
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −3.22400e6 −3.30279
\(989\) 0 0
\(990\) 0 0
\(991\) −1.59206e6 −1.62111 −0.810555 − 0.585662i \(-0.800835\pi\)
−0.810555 + 0.585662i \(0.800835\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 395528. 0.400317
\(995\) 0 0
\(996\) 2.10080e6 2.11771
\(997\) 1.92079e6 1.93237 0.966183 − 0.257856i \(-0.0830160\pi\)
0.966183 + 0.257856i \(0.0830160\pi\)
\(998\) 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 56.5.h.a.13.1 ✓ 1
4.3 odd 2 224.5.h.b.209.1 1
7.6 odd 2 56.5.h.b.13.1 yes 1
8.3 odd 2 224.5.h.a.209.1 1
8.5 even 2 56.5.h.b.13.1 yes 1
28.27 even 2 224.5.h.a.209.1 1
56.13 odd 2 CM 56.5.h.a.13.1 ✓ 1
56.27 even 2 224.5.h.b.209.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.5.h.a.13.1 ✓ 1 1.1 even 1 trivial
56.5.h.a.13.1 ✓ 1 56.13 odd 2 CM
56.5.h.b.13.1 yes 1 7.6 odd 2
56.5.h.b.13.1 yes 1 8.5 even 2
224.5.h.a.209.1 1 8.3 odd 2
224.5.h.a.209.1 1 28.27 even 2
224.5.h.b.209.1 1 4.3 odd 2
224.5.h.b.209.1 1 56.27 even 2