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Properties

Label 56.5.h.b.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 \(-0.312506\pi\)
0.555556 + 0.831479i \(0.312506\pi\)
\(4\) 16.0000 1.00000
\(5\) −22.0000 −0.880000 −0.440000 − 0.897998i \(-0.645022\pi\)
−0.440000 + 0.897998i \(0.645022\pi\)
\(6\) 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 \(-0.869524\pi\)
−0.917160 + 0.398520i \(0.869524\pi\)
\(14\) 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 \(-0.143364\pi\)
0.900277 + 0.435317i \(0.143364\pi\)
\(20\) −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 \(-0.448106\pi\)
0.162310 + 0.986740i \(0.448106\pi\)
\(60\) −3520.00 −0.977778
\(61\) 7370.00 1.98065 0.990325 − 0.138766i \(-0.0443136\pi\)
0.990325 + 0.138766i \(0.0443136\pi\)
\(62\) 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 \(-0.0980114\pi\)
0.952969 + 0.303069i \(0.0980114\pi\)
\(84\) 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 \(-0.588344\pi\)
−0.273993 + 0.961732i \(0.588344\pi\)
\(102\) 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 \(-0.776882\pi\)
−0.764233 + 0.644940i \(0.776882\pi\)
\(132\) 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 \(-0.790226\pi\)
−0.790591 + 0.612345i \(0.790226\pi\)
\(140\) −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 \(-0.989271\pi\)
−0.999432 + 0.0336991i \(0.989271\pi\)
\(158\) 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 \(-0.460708\pi\)
0.123125 + 0.992391i \(0.460708\pi\)
\(174\) 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 \(-0.517838\pi\)
−0.0560117 + 0.998430i \(0.517838\pi\)
\(182\) −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 \(-0.682820\pi\)
−0.543286 + 0.839547i \(0.682820\pi\)
\(228\) 104000. 2.00062
\(229\) 44330.0 0.845331 0.422665 − 0.906286i \(-0.361095\pi\)
0.422665 + 0.906286i \(0.361095\pi\)
\(230\) 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 \(-0.952984\pi\)
−0.989111 + 0.147170i \(0.952984\pi\)
\(252\) 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 \(-0.193751\pi\)
0.820400 + 0.571789i \(0.193751\pi\)
\(270\) 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 \(-0.374327\pi\)
0.384635 + 0.923069i \(0.374327\pi\)
\(284\) 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 \(-0.556940\pi\)
−0.177929 + 0.984043i \(0.556940\pi\)
\(294\) 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 \(-0.523240\pi\)
−0.0729451 + 0.997336i \(0.523240\pi\)
\(308\) 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 \(-0.888428\pi\)
−0.939196 + 0.343382i \(0.888428\pi\)
\(350\) −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 \(-0.416974\pi\)
0.257885 + 0.966176i \(0.416974\pi\)
\(398\) 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 \(-0.407609\pi\)
0.286197 + 0.958171i \(0.407609\pi\)
\(420\) −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 \(-0.184184\pi\)
0.837211 + 0.546879i \(0.184184\pi\)
\(462\) 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 \(-0.346526\pi\)
0.463687 + 0.885999i \(0.346526\pi\)
\(468\) −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 \(-0.752227\pi\)
−0.712036 + 0.702143i \(0.752227\pi\)
\(510\) 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 \(-0.738267\pi\)
−0.680568 + 0.732685i \(0.738267\pi\)
\(524\) −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 \(-0.230481\pi\)
0.749111 + 0.662444i \(0.230481\pi\)
\(564\) 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 \(-0.910757\pi\)
−0.960954 + 0.276708i \(0.910757\pi\)
\(588\) 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 \(-0.709782\pi\)
−0.612367 + 0.790574i \(0.709782\pi\)
\(620\) 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 \(-0.389252\pi\)
0.340949 + 0.940082i \(0.389252\pi\)
\(644\) −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 \(-0.811230\pi\)
−0.829246 + 0.558884i \(0.811230\pi\)
\(662\) 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 \(-0.232543\pi\)
0.744803 + 0.667284i \(0.232543\pi\)
\(678\) −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 \(-0.113962\pi\)
0.936592 + 0.350421i \(0.113962\pi\)
\(692\) 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 \(-0.835011\pi\)
−0.868648 + 0.495430i \(0.835011\pi\)
\(734\) 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 \(-0.382494\pi\)
0.360828 + 0.932633i \(0.382494\pi\)
\(774\) 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 \(-0.806559\pi\)
−0.820956 + 0.570991i \(0.806559\pi\)
\(788\) 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 \(-0.182084\pi\)
0.840802 + 0.541343i \(0.182084\pi\)
\(798\) 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 \(-0.705028\pi\)
−0.600490 + 0.799632i \(0.705028\pi\)
\(812\) 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 \(-0.0293331\pi\)
0.995757 + 0.0920223i \(0.0293331\pi\)
\(830\) −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 \(-0.169827\pi\)
0.861019 + 0.508573i \(0.169827\pi\)
\(854\) 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 \(-0.490437\pi\)
0.0300387 + 0.999549i \(0.490437\pi\)
\(860\) 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 \(-0.402244\pi\)
0.302305 + 0.953211i \(0.402244\pi\)
\(942\) −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 \(-0.00278162\pi\)
0.999962 + 0.00873862i \(0.00278162\pi\)
\(972\) −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 \(-0.916984\pi\)
−0.966183 + 0.257856i \(0.916984\pi\)
\(998\) 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.b.13.1 yes 1
4.3 odd 2 224.5.h.a.209.1 1
7.6 odd 2 56.5.h.a.13.1 ✓ 1
8.3 odd 2 224.5.h.b.209.1 1
8.5 even 2 56.5.h.a.13.1 ✓ 1
28.27 even 2 224.5.h.b.209.1 1
56.13 odd 2 CM 56.5.h.b.13.1 yes 1
56.27 even 2 224.5.h.a.209.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.5.h.a.13.1 ✓ 1 7.6 odd 2
56.5.h.a.13.1 ✓ 1 8.5 even 2
56.5.h.b.13.1 yes 1 1.1 even 1 trivial
56.5.h.b.13.1 yes 1 56.13 odd 2 CM
224.5.h.a.209.1 1 4.3 odd 2
224.5.h.a.209.1 1 56.27 even 2
224.5.h.b.209.1 1 8.3 odd 2
224.5.h.b.209.1 1 28.27 even 2