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Properties

Label 55.5.d.b.54.1
Level $55$
Weight $5$
Character 55.54
Self dual yes
Analytic conductor $5.685$
Analytic rank $0$
Dimension $1$
CM discriminant -55
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] = [55,5,Mod(54,55)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(55, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("55.54");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 55 = 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 55.d (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: \(5.68534796961\)
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 54.1
Character \(\chi\) \(=\) 55.54

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{2} -7.00000 q^{4} +25.0000 q^{5} +78.0000 q^{7} -69.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+3.00000 q^{2} -7.00000 q^{4} +25.0000 q^{5} +78.0000 q^{7} -69.0000 q^{8} +81.0000 q^{9} +75.0000 q^{10} +121.000 q^{11} -162.000 q^{13} +234.000 q^{14} -95.0000 q^{16} -402.000 q^{17} +243.000 q^{18} -175.000 q^{20} +363.000 q^{22} +625.000 q^{25} -486.000 q^{26} -546.000 q^{28} -1598.00 q^{31} +819.000 q^{32} -1206.00 q^{34} +1950.00 q^{35} -567.000 q^{36} -1725.00 q^{40} -3522.00 q^{43} -847.000 q^{44} +2025.00 q^{45} +3683.00 q^{49} +1875.00 q^{50} +1134.00 q^{52} +3025.00 q^{55} -5382.00 q^{56} +3442.00 q^{59} -4794.00 q^{62} +6318.00 q^{63} +3977.00 q^{64} -4050.00 q^{65} +2814.00 q^{68} +5850.00 q^{70} -3998.00 q^{71} -5589.00 q^{72} +10638.0 q^{73} +9438.00 q^{77} -2375.00 q^{80} +6561.00 q^{81} -13602.0 q^{83} -10050.0 q^{85} -10566.0 q^{86} -8349.00 q^{88} -15838.0 q^{89} +6075.00 q^{90} -12636.0 q^{91} +11049.0 q^{98} +9801.00 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/55\mathbb{Z}\right)^\times\).

\(n\) \(12\) \(46\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.00000 0.750000 0.375000 − 0.927025i \(-0.377643\pi\)
0.375000 + 0.927025i \(0.377643\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) −7.00000 −0.437500
\(5\) 25.0000 1.00000
\(6\) 0 0
\(7\) 78.0000 1.59184 0.795918 − 0.605404i \(-0.206988\pi\)
0.795918 + 0.605404i \(0.206988\pi\)
\(8\) −69.0000 −1.07812
\(9\) 81.0000 1.00000
\(10\) 75.0000 0.750000
\(11\) 121.000 1.00000
\(12\) 0 0
\(13\) −162.000 −0.958580 −0.479290 − 0.877657i \(-0.659106\pi\)
−0.479290 + 0.877657i \(0.659106\pi\)
\(14\) 234.000 1.19388
\(15\) 0 0
\(16\) −95.0000 −0.371094
\(17\) −402.000 −1.39100 −0.695502 − 0.718524i \(-0.744818\pi\)
−0.695502 + 0.718524i \(0.744818\pi\)
\(18\) 243.000 0.750000
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −175.000 −0.437500
\(21\) 0 0
\(22\) 363.000 0.750000
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 625.000 1.00000
\(26\) −486.000 −0.718935
\(27\) 0 0
\(28\) −546.000 −0.696429
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1598.00 −1.66285 −0.831426 − 0.555636i \(-0.812475\pi\)
−0.831426 + 0.555636i \(0.812475\pi\)
\(32\) 819.000 0.799805
\(33\) 0 0
\(34\) −1206.00 −1.04325
\(35\) 1950.00 1.59184
\(36\) −567.000 −0.437500
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1725.00 −1.07812
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −3522.00 −1.90481 −0.952407 − 0.304830i \(-0.901400\pi\)
−0.952407 + 0.304830i \(0.901400\pi\)
\(44\) −847.000 −0.437500
\(45\) 2025.00 1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 3683.00 1.53394
\(50\) 1875.00 0.750000
\(51\) 0 0
\(52\) 1134.00 0.419379
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 3025.00 1.00000
\(56\) −5382.00 −1.71620
\(57\) 0 0
\(58\) 0 0
\(59\) 3442.00 0.988796 0.494398 − 0.869236i \(-0.335388\pi\)
0.494398 + 0.869236i \(0.335388\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −4794.00 −1.24714
\(63\) 6318.00 1.59184
\(64\) 3977.00 0.970947
\(65\) −4050.00 −0.958580
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 2814.00 0.608564
\(69\) 0 0
\(70\) 5850.00 1.19388
\(71\) −3998.00 −0.793097 −0.396548 − 0.918014i \(-0.629792\pi\)
−0.396548 + 0.918014i \(0.629792\pi\)
\(72\) −5589.00 −1.07812
\(73\) 10638.0 1.99625 0.998123 − 0.0612334i \(-0.0195034\pi\)
0.998123 + 0.0612334i \(0.0195034\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 9438.00 1.59184
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −2375.00 −0.371094
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) −13602.0 −1.97445 −0.987226 − 0.159326i \(-0.949068\pi\)
−0.987226 + 0.159326i \(0.949068\pi\)
\(84\) 0 0
\(85\) −10050.0 −1.39100
\(86\) −10566.0 −1.42861
\(87\) 0 0
\(88\) −8349.00 −1.07812
\(89\) −15838.0 −1.99950 −0.999748 − 0.0224705i \(-0.992847\pi\)
−0.999748 + 0.0224705i \(0.992847\pi\)
\(90\) 6075.00 0.750000
\(91\) −12636.0 −1.52590
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 11049.0 1.15046
\(99\) 9801.00 1.00000
\(100\) −4375.00 −0.437500
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 11178.0 1.03347
\(105\) 0 0
\(106\) 0 0
\(107\) −1602.00 −0.139925 −0.0699624 − 0.997550i \(-0.522288\pi\)
−0.0699624 + 0.997550i \(0.522288\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 9075.00 0.750000
\(111\) 0 0
\(112\) −7410.00 −0.590721
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −13122.0 −0.958580
\(118\) 10326.0 0.741597
\(119\) −31356.0 −2.21425
\(120\) 0 0
\(121\) 14641.0 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 11186.0 0.727497
\(125\) 15625.0 1.00000
\(126\) 18954.0 1.19388
\(127\) 31278.0 1.93924 0.969620 − 0.244616i \(-0.0786620\pi\)
0.969620 + 0.244616i \(0.0786620\pi\)
\(128\) −1173.00 −0.0715942
\(129\) 0 0
\(130\) −12150.0 −0.718935
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 27738.0 1.49968
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −13650.0 −0.696429
\(141\) 0 0
\(142\) −11994.0 −0.594822
\(143\) −19602.0 −0.958580
\(144\) −7695.00 −0.371094
\(145\) 0 0
\(146\) 31914.0 1.49719
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) −32562.0 −1.39100
\(154\) 28314.0 1.19388
\(155\) −39950.0 −1.66285
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 20475.0 0.799805
\(161\) 0 0
\(162\) 19683.0 0.750000
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −40806.0 −1.48084
\(167\) 7758.00 0.278174 0.139087 − 0.990280i \(-0.455583\pi\)
0.139087 + 0.990280i \(0.455583\pi\)
\(168\) 0 0
\(169\) −2317.00 −0.0811246
\(170\) −30150.0 −1.04325
\(171\) 0 0
\(172\) 24654.0 0.833356
\(173\) 3678.00 0.122891 0.0614454 − 0.998110i \(-0.480429\pi\)
0.0614454 + 0.998110i \(0.480429\pi\)
\(174\) 0 0
\(175\) 48750.0 1.59184
\(176\) −11495.0 −0.371094
\(177\) 0 0
\(178\) −47514.0 −1.49962
\(179\) −62638.0 −1.95493 −0.977466 − 0.211091i \(-0.932298\pi\)
−0.977466 + 0.211091i \(0.932298\pi\)
\(180\) −14175.0 −0.437500
\(181\) 51442.0 1.57022 0.785110 − 0.619356i \(-0.212606\pi\)
0.785110 + 0.619356i \(0.212606\pi\)
\(182\) −37908.0 −1.14443
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −48642.0 −1.39100
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 41282.0 1.13160 0.565801 − 0.824542i \(-0.308567\pi\)
0.565801 + 0.824542i \(0.308567\pi\)
\(192\) 0 0
\(193\) 73518.0 1.97369 0.986845 − 0.161668i \(-0.0516872\pi\)
0.986845 + 0.161668i \(0.0516872\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −25781.0 −0.671101
\(197\) 70398.0 1.81396 0.906980 − 0.421173i \(-0.138381\pi\)
0.906980 + 0.421173i \(0.138381\pi\)
\(198\) 29403.0 0.750000
\(199\) −47518.0 −1.19992 −0.599960 − 0.800030i \(-0.704817\pi\)
−0.599960 + 0.800030i \(0.704817\pi\)
\(200\) −43125.0 −1.07812
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 15390.0 0.355723
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −4806.00 −0.104944
\(215\) −88050.0 −1.90481
\(216\) 0 0
\(217\) −124644. −2.64699
\(218\) 0 0
\(219\) 0 0
\(220\) −21175.0 −0.437500
\(221\) 65124.0 1.33339
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 63882.0 1.27316
\(225\) 50625.0 1.00000
\(226\) 0 0
\(227\) 66078.0 1.28235 0.641173 − 0.767396i \(-0.278448\pi\)
0.641173 + 0.767396i \(0.278448\pi\)
\(228\) 0 0
\(229\) 73202.0 1.39589 0.697946 − 0.716150i \(-0.254097\pi\)
0.697946 + 0.716150i \(0.254097\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 60558.0 1.11547 0.557737 − 0.830018i \(-0.311670\pi\)
0.557737 + 0.830018i \(0.311670\pi\)
\(234\) −39366.0 −0.718935
\(235\) 0 0
\(236\) −24094.0 −0.432598
\(237\) 0 0
\(238\) −94068.0 −1.66069
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 43923.0 0.750000
\(243\) 0 0
\(244\) 0 0
\(245\) 92075.0 1.53394
\(246\) 0 0
\(247\) 0 0
\(248\) 110262. 1.79276
\(249\) 0 0
\(250\) 46875.0 0.750000
\(251\) −46478.0 −0.737734 −0.368867 − 0.929482i \(-0.620254\pi\)
−0.368867 + 0.929482i \(0.620254\pi\)
\(252\) −44226.0 −0.696429
\(253\) 0 0
\(254\) 93834.0 1.45443
\(255\) 0 0
\(256\) −67151.0 −1.02464
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 28350.0 0.419379
\(261\) 0 0
\(262\) 0 0
\(263\) −117042. −1.69212 −0.846058 − 0.533091i \(-0.821031\pi\)
−0.846058 + 0.533091i \(0.821031\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −27758.0 −0.383604 −0.191802 − 0.981434i \(-0.561433\pi\)
−0.191802 + 0.981434i \(0.561433\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 38190.0 0.516193
\(273\) 0 0
\(274\) 0 0
\(275\) 75625.0 1.00000
\(276\) 0 0
\(277\) 15678.0 0.204330 0.102165 − 0.994767i \(-0.467423\pi\)
0.102165 + 0.994767i \(0.467423\pi\)
\(278\) 0 0
\(279\) −129438. −1.66285
\(280\) −134550. −1.71620
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 135678. 1.69409 0.847045 − 0.531521i \(-0.178379\pi\)
0.847045 + 0.531521i \(0.178379\pi\)
\(284\) 27986.0 0.346980
\(285\) 0 0
\(286\) −58806.0 −0.718935
\(287\) 0 0
\(288\) 66339.0 0.799805
\(289\) 78083.0 0.934891
\(290\) 0 0
\(291\) 0 0
\(292\) −74466.0 −0.873358
\(293\) −171522. −1.99795 −0.998975 − 0.0452665i \(-0.985586\pi\)
−0.998975 + 0.0452665i \(0.985586\pi\)
\(294\) 0 0
\(295\) 86050.0 0.988796
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −274716. −3.03215
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −97686.0 −1.04325
\(307\) −40482.0 −0.429522 −0.214761 − 0.976667i \(-0.568897\pi\)
−0.214761 + 0.976667i \(0.568897\pi\)
\(308\) −66066.0 −0.696429
\(309\) 0 0
\(310\) −119850. −1.24714
\(311\) −31838.0 −0.329174 −0.164587 − 0.986363i \(-0.552629\pi\)
−0.164587 + 0.986363i \(0.552629\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 157950. 1.59184
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 99425.0 0.970947
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −45927.0 −0.437500
\(325\) −101250. −0.958580
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 162802. 1.48595 0.742974 − 0.669320i \(-0.233414\pi\)
0.742974 + 0.669320i \(0.233414\pi\)
\(332\) 95214.0 0.863823
\(333\) 0 0
\(334\) 23274.0 0.208631
\(335\) 0 0
\(336\) 0 0
\(337\) 68718.0 0.605077 0.302539 − 0.953137i \(-0.402166\pi\)
0.302539 + 0.953137i \(0.402166\pi\)
\(338\) −6951.00 −0.0608435
\(339\) 0 0
\(340\) 70350.0 0.608564
\(341\) −193358. −1.66285
\(342\) 0 0
\(343\) 99996.0 0.849952
\(344\) 243018. 2.05363
\(345\) 0 0
\(346\) 11034.0 0.0921681
\(347\) −71682.0 −0.595321 −0.297660 − 0.954672i \(-0.596206\pi\)
−0.297660 + 0.954672i \(0.596206\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 146250. 1.19388
\(351\) 0 0
\(352\) 99099.0 0.799805
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −99950.0 −0.793097
\(356\) 110866. 0.874779
\(357\) 0 0
\(358\) −187914. −1.46620
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) −139725. −1.07812
\(361\) 130321. 1.00000
\(362\) 154326. 1.17767
\(363\) 0 0
\(364\) 88452.0 0.667582
\(365\) 265950. 1.99625
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −214722. −1.54333 −0.771665 − 0.636029i \(-0.780576\pi\)
−0.771665 + 0.636029i \(0.780576\pi\)
\(374\) −145926. −1.04325
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 2162.00 0.0150514 0.00752571 − 0.999972i \(-0.497604\pi\)
0.00752571 + 0.999972i \(0.497604\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 123846. 0.848702
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 235950. 1.59184
\(386\) 220554. 1.48027
\(387\) −285282. −1.90481
\(388\) 0 0
\(389\) −292238. −1.93125 −0.965623 − 0.259948i \(-0.916295\pi\)
−0.965623 + 0.259948i \(0.916295\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −254127. −1.65378
\(393\) 0 0
\(394\) 211194. 1.36047
\(395\) 0 0
\(396\) −68607.0 −0.437500
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −142554. −0.899939
\(399\) 0 0
\(400\) −59375.0 −0.371094
\(401\) 289922. 1.80299 0.901493 − 0.432793i \(-0.142472\pi\)
0.901493 + 0.432793i \(0.142472\pi\)
\(402\) 0 0
\(403\) 258876. 1.59398
\(404\) 0 0
\(405\) 164025. 1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 268476. 1.57400
\(414\) 0 0
\(415\) −340050. −1.97445
\(416\) −132678. −0.766677
\(417\) 0 0
\(418\) 0 0
\(419\) −155758. −0.887202 −0.443601 − 0.896224i \(-0.646299\pi\)
−0.443601 + 0.896224i \(0.646299\pi\)
\(420\) 0 0
\(421\) −335438. −1.89255 −0.946277 − 0.323358i \(-0.895188\pi\)
−0.946277 + 0.323358i \(0.895188\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −251250. −1.39100
\(426\) 0 0
\(427\) 0 0
\(428\) 11214.0 0.0612171
\(429\) 0 0
\(430\) −264150. −1.42861
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) −373932. −1.98524
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) −208725. −1.07812
\(441\) 298323. 1.53394
\(442\) 195372. 1.00004
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −395950. −1.99950
\(446\) 0 0
\(447\) 0 0
\(448\) 310206. 1.54559
\(449\) 118082. 0.585721 0.292861 − 0.956155i \(-0.405393\pi\)
0.292861 + 0.956155i \(0.405393\pi\)
\(450\) 151875. 0.750000
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 198234. 0.961759
\(455\) −315900. −1.52590
\(456\) 0 0
\(457\) 237198. 1.13574 0.567870 − 0.823119i \(-0.307768\pi\)
0.567870 + 0.823119i \(0.307768\pi\)
\(458\) 219606. 1.04692
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 181674. 0.836606
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 91854.0 0.419379
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −237498. −1.06605
\(473\) −426162. −1.90481
\(474\) 0 0
\(475\) 0 0
\(476\) 219492. 0.968735
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −102487. −0.437500
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 276225. 1.15046
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 245025. 1.00000
\(496\) 151810. 0.617074
\(497\) −311844. −1.26248
\(498\) 0 0
\(499\) 466322. 1.87277 0.936386 − 0.350972i \(-0.114149\pi\)
0.936386 + 0.350972i \(0.114149\pi\)
\(500\) −109375. −0.437500
\(501\) 0 0
\(502\) −139434. −0.553301
\(503\) 381198. 1.50666 0.753329 − 0.657644i \(-0.228447\pi\)
0.753329 + 0.657644i \(0.228447\pi\)
\(504\) −435942. −1.71620
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −218946. −0.848417
\(509\) −499118. −1.92649 −0.963247 − 0.268617i \(-0.913433\pi\)
−0.963247 + 0.268617i \(0.913433\pi\)
\(510\) 0 0
\(511\) 829764. 3.17770
\(512\) −182685. −0.696888
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 279450. 1.03347
\(521\) −249118. −0.917761 −0.458881 − 0.888498i \(-0.651749\pi\)
−0.458881 + 0.888498i \(0.651749\pi\)
\(522\) 0 0
\(523\) −483522. −1.76772 −0.883859 − 0.467754i \(-0.845063\pi\)
−0.883859 + 0.467754i \(0.845063\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −351126. −1.26909
\(527\) 642396. 2.31303
\(528\) 0 0
\(529\) 279841. 1.00000
\(530\) 0 0
\(531\) 278802. 0.988796
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −40050.0 −0.139925
\(536\) 0 0
\(537\) 0 0
\(538\) −83274.0 −0.287703
\(539\) 445643. 1.53394
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −329238. −1.11253
\(545\) 0 0
\(546\) 0 0
\(547\) −450402. −1.50531 −0.752654 − 0.658416i \(-0.771227\pi\)
−0.752654 + 0.658416i \(0.771227\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 226875. 0.750000
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 47034.0 0.153247
\(555\) 0 0
\(556\) 0 0
\(557\) 266718. 0.859690 0.429845 − 0.902903i \(-0.358568\pi\)
0.429845 + 0.902903i \(0.358568\pi\)
\(558\) −388314. −1.24714
\(559\) 570564. 1.82592
\(560\) −185250. −0.590721
\(561\) 0 0
\(562\) 0 0
\(563\) −21282.0 −0.0671422 −0.0335711 − 0.999436i \(-0.510688\pi\)
−0.0335711 + 0.999436i \(0.510688\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 407034. 1.27057
\(567\) 511758. 1.59184
\(568\) 275862. 0.855057
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 137214. 0.419379
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 322137. 0.970947
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 234249. 0.701168
\(579\) 0 0
\(580\) 0 0
\(581\) −1.06096e6 −3.14301
\(582\) 0 0
\(583\) 0 0
\(584\) −734022. −2.15220
\(585\) −328050. −0.958580
\(586\) −514566. −1.49846
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 258150. 0.741597
\(591\) 0 0
\(592\) 0 0
\(593\) 690798. 1.96445 0.982227 − 0.187699i \(-0.0601029\pi\)
0.982227 + 0.187699i \(0.0601029\pi\)
\(594\) 0 0
\(595\) −783900. −2.21425
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −74398.0 −0.207352 −0.103676 − 0.994611i \(-0.533060\pi\)
−0.103676 + 0.994611i \(0.533060\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) −824148. −2.27411
\(603\) 0 0
\(604\) 0 0
\(605\) 366025. 1.00000
\(606\) 0 0
\(607\) 361518. 0.981189 0.490594 − 0.871388i \(-0.336780\pi\)
0.490594 + 0.871388i \(0.336780\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 227934. 0.608564
\(613\) −138882. −0.369594 −0.184797 − 0.982777i \(-0.559163\pi\)
−0.184797 + 0.982777i \(0.559163\pi\)
\(614\) −121446. −0.322141
\(615\) 0 0
\(616\) −651222. −1.71620
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 76402.0 0.199399 0.0996996 − 0.995018i \(-0.468212\pi\)
0.0996996 + 0.995018i \(0.468212\pi\)
\(620\) 279650. 0.727497
\(621\) 0 0
\(622\) −95514.0 −0.246880
\(623\) −1.23536e6 −3.18287
\(624\) 0 0
\(625\) 390625. 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 473850. 1.19388
\(631\) 289442. 0.726947 0.363474 − 0.931605i \(-0.381591\pi\)
0.363474 + 0.931605i \(0.381591\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 781950. 1.93924
\(636\) 0 0
\(637\) −596646. −1.47041
\(638\) 0 0
\(639\) −323838. −0.793097
\(640\) −29325.0 −0.0715942
\(641\) −730558. −1.77803 −0.889014 − 0.457880i \(-0.848609\pi\)
−0.889014 + 0.457880i \(0.848609\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) −452709. −1.07812
\(649\) 416482. 0.988796
\(650\) −303750. −0.718935
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 861678. 1.99625
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −678478. −1.55286 −0.776431 − 0.630202i \(-0.782972\pi\)
−0.776431 + 0.630202i \(0.782972\pi\)
\(662\) 488406. 1.11446
\(663\) 0 0
\(664\) 938538. 2.12871
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −54306.0 −0.121701
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −18642.0 −0.0411588 −0.0205794 − 0.999788i \(-0.506551\pi\)
−0.0205794 + 0.999788i \(0.506551\pi\)
\(674\) 206154. 0.453808
\(675\) 0 0
\(676\) 16219.0 0.0354920
\(677\) −895362. −1.95354 −0.976768 − 0.214300i \(-0.931253\pi\)
−0.976768 + 0.214300i \(0.931253\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 693450. 1.49968
\(681\) 0 0
\(682\) −580074. −1.24714
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 299988. 0.637464
\(687\) 0 0
\(688\) 334590. 0.706864
\(689\) 0 0
\(690\) 0 0
\(691\) −597358. −1.25106 −0.625531 − 0.780200i \(-0.715117\pi\)
−0.625531 + 0.780200i \(0.715117\pi\)
\(692\) −25746.0 −0.0537647
\(693\) 764478. 1.59184
\(694\) −215046. −0.446491
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −341250. −0.696429
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 481217. 0.970947
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 949042. 1.88796 0.943980 − 0.330002i \(-0.107049\pi\)
0.943980 + 0.330002i \(0.107049\pi\)
\(710\) −299850. −0.594822
\(711\) 0 0
\(712\) 1.09282e6 2.15571
\(713\) 0 0
\(714\) 0 0
\(715\) −490050. −0.958580
\(716\) 438466. 0.855283
\(717\) 0 0
\(718\) 0 0
\(719\) −518398. −1.00278 −0.501390 − 0.865221i \(-0.667178\pi\)
−0.501390 + 0.865221i \(0.667178\pi\)
\(720\) −192375. −0.371094
\(721\) 0 0
\(722\) 390963. 0.750000
\(723\) 0 0
\(724\) −360094. −0.686972
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 871884. 1.64511
\(729\) 531441. 1.00000
\(730\) 797850. 1.49719
\(731\) 1.41584e6 2.64960
\(732\) 0 0
\(733\) 720798. 1.34155 0.670773 − 0.741663i \(-0.265962\pi\)
0.670773 + 0.741663i \(0.265962\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −216882. −0.392867 −0.196434 − 0.980517i \(-0.562936\pi\)
−0.196434 + 0.980517i \(0.562936\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −644166. −1.15750
\(747\) −1.10176e6 −1.97445
\(748\) 340494. 0.608564
\(749\) −124956. −0.222738
\(750\) 0 0
\(751\) −899518. −1.59489 −0.797444 − 0.603393i \(-0.793815\pi\)
−0.797444 + 0.603393i \(0.793815\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 6486.00 0.0112886
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −288974. −0.495076
\(765\) −814050. −1.39100
\(766\) 0 0
\(767\) −557604. −0.947840
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 707850. 1.19388
\(771\) 0 0
\(772\) −514626. −0.863490
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) −855846. −1.42861
\(775\) −998750. −1.66285
\(776\) 0 0
\(777\) 0 0
\(778\) −876714. −1.44843
\(779\) 0 0
\(780\) 0 0
\(781\) −483758. −0.793097
\(782\) 0 0
\(783\) 0 0
\(784\) −349885. −0.569237
\(785\) 0 0
\(786\) 0 0
\(787\) 852318. 1.37611 0.688053 − 0.725660i \(-0.258465\pi\)
0.688053 + 0.725660i \(0.258465\pi\)
\(788\) −492786. −0.793608
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −676269. −1.07812
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 332626. 0.524965
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 511875. 0.799805
\(801\) −1.28288e6 −1.99950
\(802\) 869766. 1.35224
\(803\) 1.28720e6 1.99625
\(804\) 0 0
\(805\) 0 0
\(806\) 776628. 1.19548
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 492075. 0.750000
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −1.02352e6 −1.52590
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 805428. 1.18050
\(827\) 26238.0 0.0383636 0.0191818 − 0.999816i \(-0.493894\pi\)
0.0191818 + 0.999816i \(0.493894\pi\)
\(828\) 0 0
\(829\) −33518.0 −0.0487718 −0.0243859 − 0.999703i \(-0.507763\pi\)
−0.0243859 + 0.999703i \(0.507763\pi\)
\(830\) −1.02015e6 −1.48084
\(831\) 0 0
\(832\) −644274. −0.930731
\(833\) −1.48057e6 −2.13372
\(834\) 0 0
\(835\) 193950. 0.278174
\(836\) 0 0
\(837\) 0 0
\(838\) −467274. −0.665401
\(839\) 717922. 1.01989 0.509945 − 0.860207i \(-0.329666\pi\)
0.509945 + 0.860207i \(0.329666\pi\)
\(840\) 0 0
\(841\) 707281. 1.00000
\(842\) −1.00631e6 −1.41941
\(843\) 0 0
\(844\) 0 0
\(845\) −57925.0 −0.0811246
\(846\) 0 0
\(847\) 1.14200e6 1.59184
\(848\) 0 0
\(849\) 0 0
\(850\) −753750. −1.04325
\(851\) 0 0
\(852\) 0 0
\(853\) −356802. −0.490376 −0.245188 − 0.969476i \(-0.578850\pi\)
−0.245188 + 0.969476i \(0.578850\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 110538. 0.150857
\(857\) 326478. 0.444521 0.222260 − 0.974987i \(-0.428656\pi\)
0.222260 + 0.974987i \(0.428656\pi\)
\(858\) 0 0
\(859\) −1.28392e6 −1.74001 −0.870003 − 0.493046i \(-0.835884\pi\)
−0.870003 + 0.493046i \(0.835884\pi\)
\(860\) 616350. 0.833356
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 91950.0 0.122891
\(866\) 0 0
\(867\) 0 0
\(868\) 872508. 1.15806
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.21875e6 1.59184
\(876\) 0 0
\(877\) 1.48208e6 1.92696 0.963478 − 0.267787i \(-0.0862923\pi\)
0.963478 + 0.267787i \(0.0862923\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −287375. −0.371094
\(881\) 1.53824e6 1.98186 0.990930 − 0.134381i \(-0.0429046\pi\)
0.990930 + 0.134381i \(0.0429046\pi\)
\(882\) 894969. 1.15046
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) −455868. −0.583357
\(885\) 0 0
\(886\) 0 0
\(887\) −918642. −1.16761 −0.583807 − 0.811893i \(-0.698437\pi\)
−0.583807 + 0.811893i \(0.698437\pi\)
\(888\) 0 0
\(889\) 2.43968e6 3.08695
\(890\) −1.18785e6 −1.49962
\(891\) 793881. 1.00000
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −1.56595e6 −1.95493
\(896\) −91494.0 −0.113966
\(897\) 0 0
\(898\) 354246. 0.439291
\(899\) 0 0
\(900\) −354375. −0.437500
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.28605e6 1.57022
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) −462546. −0.561026
\(909\) 0 0
\(910\) −947700. −1.14443
\(911\) 1.37472e6 1.65645 0.828225 − 0.560396i \(-0.189351\pi\)
0.828225 + 0.560396i \(0.189351\pi\)
\(912\) 0 0
\(913\) −1.64584e6 −1.97445
\(914\) 711594. 0.851804
\(915\) 0 0
\(916\) −512414. −0.610703
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 647676. 0.760246
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.03616e6 1.20059 0.600297 − 0.799777i \(-0.295049\pi\)
0.600297 + 0.799777i \(0.295049\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −423906. −0.488020
\(933\) 0 0
\(934\) 0 0
\(935\) −1.21605e6 −1.39100
\(936\) 905418. 1.03347
\(937\) −1.46008e6 −1.66302 −0.831511 − 0.555508i \(-0.812524\pi\)
−0.831511 + 0.555508i \(0.812524\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −326990. −0.366936
\(945\) 0 0
\(946\) −1.27849e6 −1.42861
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −1.72336e6 −1.91356
\(950\) 0 0
\(951\) 0 0
\(952\) 2.16356e6 2.38724
\(953\) 1.20392e6 1.32560 0.662798 − 0.748798i \(-0.269369\pi\)
0.662798 + 0.748798i \(0.269369\pi\)
\(954\) 0 0
\(955\) 1.03205e6 1.13160
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 1.63008e6 1.76507
\(962\) 0 0
\(963\) −129762. −0.139925
\(964\) 0 0
\(965\) 1.83795e6 1.97369
\(966\) 0 0
\(967\) −1.57432e6 −1.68361 −0.841803 − 0.539784i \(-0.818506\pi\)
−0.841803 + 0.539784i \(0.818506\pi\)
\(968\) −1.01023e6 −1.07812
\(969\) 0 0
\(970\) 0 0
\(971\) 745202. 0.790379 0.395190 − 0.918600i \(-0.370679\pi\)
0.395190 + 0.918600i \(0.370679\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) −1.91640e6 −1.99950
\(980\) −644525. −0.671101
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 1.75995e6 1.81396
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 735075. 0.750000
\(991\) 1.79168e6 1.82437 0.912186 − 0.409775i \(-0.134393\pi\)
0.912186 + 0.409775i \(0.134393\pi\)
\(992\) −1.30876e6 −1.32996
\(993\) 0 0
\(994\) −935532. −0.946860
\(995\) −1.18795e6 −1.19992
\(996\) 0 0
\(997\) 103038. 0.103659 0.0518295 − 0.998656i \(-0.483495\pi\)
0.0518295 + 0.998656i \(0.483495\pi\)
\(998\) 1.39897e6 1.40458
\(999\) 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 55.5.d.b.54.1 yes 1
5.2 odd 4 275.5.c.c.76.2 2
5.3 odd 4 275.5.c.c.76.1 2
5.4 even 2 55.5.d.a.54.1 ✓ 1
11.10 odd 2 55.5.d.a.54.1 ✓ 1
55.32 even 4 275.5.c.c.76.1 2
55.43 even 4 275.5.c.c.76.2 2
55.54 odd 2 CM 55.5.d.b.54.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.5.d.a.54.1 ✓ 1 5.4 even 2
55.5.d.a.54.1 ✓ 1 11.10 odd 2
55.5.d.b.54.1 yes 1 1.1 even 1 trivial
55.5.d.b.54.1 yes 1 55.54 odd 2 CM
275.5.c.c.76.1 2 5.3 odd 4
275.5.c.c.76.1 2 55.32 even 4
275.5.c.c.76.2 2 5.2 odd 4
275.5.c.c.76.2 2 55.43 even 4