Properties

Label 87.7.d.b.86.1
Level $87$
Weight $7$
Character 87.86
Self dual yes
Analytic conductor $20.015$
Analytic rank $0$
Dimension $1$
CM discriminant -87
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] = [87,7,Mod(86,87)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(87, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("87.86");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 87.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: \(20.0147052749\)
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 86.1
Character \(\chi\) \(=\) 87.86

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+13.0000 q^{2} -27.0000 q^{3} +105.000 q^{4} -351.000 q^{6} +338.000 q^{7} +533.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+13.0000 q^{2} -27.0000 q^{3} +105.000 q^{4} -351.000 q^{6} +338.000 q^{7} +533.000 q^{8} +729.000 q^{9} -806.000 q^{11} -2835.00 q^{12} +3002.00 q^{13} +4394.00 q^{14} +209.000 q^{16} +9778.00 q^{17} +9477.00 q^{18} -9126.00 q^{21} -10478.0 q^{22} -14391.0 q^{24} +15625.0 q^{25} +39026.0 q^{26} -19683.0 q^{27} +35490.0 q^{28} +24389.0 q^{29} -31395.0 q^{32} +21762.0 q^{33} +127114. q^{34} +76545.0 q^{36} -81054.0 q^{39} -132158. q^{41} -118638. q^{42} -84630.0 q^{44} -151502. q^{47} -5643.00 q^{48} -3405.00 q^{49} +203125. q^{50} -264006. q^{51} +315210. q^{52} -255879. q^{54} +180154. q^{56} +317057. q^{58} +246402. q^{63} -421511. q^{64} +282906. q^{66} +267098. q^{67} +1.02669e6 q^{68} +388557. q^{72} -421875. q^{75} -272428. q^{77} -1.05370e6 q^{78} +531441. q^{81} -1.71805e6 q^{82} -958230. q^{84} -658503. q^{87} -429598. q^{88} +71266.0 q^{89} +1.01468e6 q^{91} -1.96953e6 q^{94} +847665. q^{96} -44265.0 q^{98} -587574. q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/87\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(59\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 13.0000 1.62500 0.812500 0.582961i \(-0.198106\pi\)
0.812500 + 0.582961i \(0.198106\pi\)
\(3\) −27.0000 −1.00000
\(4\) 105.000 1.64062
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) −351.000 −1.62500
\(7\) 338.000 0.985423 0.492711 0.870193i \(-0.336006\pi\)
0.492711 + 0.870193i \(0.336006\pi\)
\(8\) 533.000 1.04102
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) −806.000 −0.605560 −0.302780 0.953061i \(-0.597915\pi\)
−0.302780 + 0.953061i \(0.597915\pi\)
\(12\) −2835.00 −1.64062
\(13\) 3002.00 1.36641 0.683204 0.730227i \(-0.260586\pi\)
0.683204 + 0.730227i \(0.260586\pi\)
\(14\) 4394.00 1.60131
\(15\) 0 0
\(16\) 209.000 0.0510254
\(17\) 9778.00 1.99023 0.995115 0.0987225i \(-0.0314756\pi\)
0.995115 + 0.0987225i \(0.0314756\pi\)
\(18\) 9477.00 1.62500
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −9126.00 −0.985423
\(22\) −10478.0 −0.984035
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) −14391.0 −1.04102
\(25\) 15625.0 1.00000
\(26\) 39026.0 2.22041
\(27\) −19683.0 −1.00000
\(28\) 35490.0 1.61671
\(29\) 24389.0 1.00000
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −31395.0 −0.958099
\(33\) 21762.0 0.605560
\(34\) 127114. 3.23412
\(35\) 0 0
\(36\) 76545.0 1.64062
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −81054.0 −1.36641
\(40\) 0 0
\(41\) −132158. −1.91753 −0.958764 0.284202i \(-0.908271\pi\)
−0.958764 + 0.284202i \(0.908271\pi\)
\(42\) −118638. −1.60131
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −84630.0 −0.993496
\(45\) 0 0
\(46\) 0 0
\(47\) −151502. −1.45923 −0.729617 0.683856i \(-0.760302\pi\)
−0.729617 + 0.683856i \(0.760302\pi\)
\(48\) −5643.00 −0.0510254
\(49\) −3405.00 −0.0289420
\(50\) 203125. 1.62500
\(51\) −264006. −1.99023
\(52\) 315210. 2.24176
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) −255879. −1.62500
\(55\) 0 0
\(56\) 180154. 1.02584
\(57\) 0 0
\(58\) 317057. 1.62500
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 246402. 0.985423
\(64\) −421511. −1.60794
\(65\) 0 0
\(66\) 282906. 0.984035
\(67\) 267098. 0.888068 0.444034 0.896010i \(-0.353547\pi\)
0.444034 + 0.896010i \(0.353547\pi\)
\(68\) 1.02669e6 3.26522
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 388557. 1.04102
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) −421875. −1.00000
\(76\) 0 0
\(77\) −272428. −0.596732
\(78\) −1.05370e6 −2.22041
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 531441. 1.00000
\(82\) −1.71805e6 −3.11598
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −958230. −1.61671
\(85\) 0 0
\(86\) 0 0
\(87\) −658503. −1.00000
\(88\) −429598. −0.630397
\(89\) 71266.0 0.101091 0.0505455 0.998722i \(-0.483904\pi\)
0.0505455 + 0.998722i \(0.483904\pi\)
\(90\) 0 0
\(91\) 1.01468e6 1.34649
\(92\) 0 0
\(93\) 0 0
\(94\) −1.96953e6 −2.37125
\(95\) 0 0
\(96\) 847665. 0.958099
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −44265.0 −0.0470308
\(99\) −587574. −0.605560
\(100\) 1.64062e6 1.64062
\(101\) −1.86561e6 −1.81074 −0.905369 0.424625i \(-0.860406\pi\)
−0.905369 + 0.424625i \(0.860406\pi\)
\(102\) −3.43208e6 −3.23412
\(103\) 1.65615e6 1.51561 0.757804 0.652482i \(-0.226272\pi\)
0.757804 + 0.652482i \(0.226272\pi\)
\(104\) 1.60007e6 1.42245
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −2.06672e6 −1.64062
\(109\) −2.58957e6 −1.99963 −0.999813 0.0193314i \(-0.993846\pi\)
−0.999813 + 0.0193314i \(0.993846\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 70642.0 0.0502816
\(113\) 2.77976e6 1.92651 0.963257 0.268580i \(-0.0865545\pi\)
0.963257 + 0.268580i \(0.0865545\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.56084e6 1.64062
\(117\) 2.18846e6 1.36641
\(118\) 0 0
\(119\) 3.30496e6 1.96122
\(120\) 0 0
\(121\) −1.12192e6 −0.633297
\(122\) 0 0
\(123\) 3.56827e6 1.91753
\(124\) 0 0
\(125\) 0 0
\(126\) 3.20323e6 1.60131
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −3.47036e6 −1.65480
\(129\) 0 0
\(130\) 0 0
\(131\) −3.24153e6 −1.44190 −0.720951 0.692986i \(-0.756295\pi\)
−0.720951 + 0.692986i \(0.756295\pi\)
\(132\) 2.28501e6 0.993496
\(133\) 0 0
\(134\) 3.47227e6 1.44311
\(135\) 0 0
\(136\) 5.21167e6 2.07186
\(137\) 4.95219e6 1.92591 0.962955 0.269662i \(-0.0869121\pi\)
0.962955 + 0.269662i \(0.0869121\pi\)
\(138\) 0 0
\(139\) −5.04405e6 −1.87817 −0.939086 0.343682i \(-0.888326\pi\)
−0.939086 + 0.343682i \(0.888326\pi\)
\(140\) 0 0
\(141\) 4.09055e6 1.45923
\(142\) 0 0
\(143\) −2.41961e6 −0.827442
\(144\) 152361. 0.0510254
\(145\) 0 0
\(146\) 0 0
\(147\) 91935.0 0.0289420
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) −5.48438e6 −1.62500
\(151\) 3.04920e6 0.885636 0.442818 0.896611i \(-0.353979\pi\)
0.442818 + 0.896611i \(0.353979\pi\)
\(152\) 0 0
\(153\) 7.12816e6 1.99023
\(154\) −3.54156e6 −0.969690
\(155\) 0 0
\(156\) −8.51067e6 −2.24176
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 6.90873e6 1.62500
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −1.38766e7 −3.14595
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) −4.86416e6 −1.02584
\(169\) 4.18520e6 0.867073
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) −8.56054e6 −1.62500
\(175\) 5.28125e6 0.985423
\(176\) −168454. −0.0308989
\(177\) 0 0
\(178\) 926458. 0.164273
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 1.16242e7 1.96033 0.980164 0.198190i \(-0.0635061\pi\)
0.980164 + 0.198190i \(0.0635061\pi\)
\(182\) 1.31908e7 2.18805
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −7.88107e6 −1.20520
\(188\) −1.59077e7 −2.39405
\(189\) −6.65285e6 −0.985423
\(190\) 0 0
\(191\) −8.42296e6 −1.20883 −0.604414 0.796670i \(-0.706593\pi\)
−0.604414 + 0.796670i \(0.706593\pi\)
\(192\) 1.13808e7 1.60794
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −357525. −0.0474830
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −7.63846e6 −0.984035
\(199\) −5.46993e6 −0.694101 −0.347051 0.937846i \(-0.612817\pi\)
−0.347051 + 0.937846i \(0.612817\pi\)
\(200\) 8.32812e6 1.04102
\(201\) −7.21165e6 −0.888068
\(202\) −2.42529e7 −2.94245
\(203\) 8.24348e6 0.985423
\(204\) −2.77206e7 −3.26522
\(205\) 0 0
\(206\) 2.15299e7 2.46286
\(207\) 0 0
\(208\) 627418. 0.0697215
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) −1.04910e7 −1.04102
\(217\) 0 0
\(218\) −3.36645e7 −3.24939
\(219\) 0 0
\(220\) 0 0
\(221\) 2.93536e7 2.71947
\(222\) 0 0
\(223\) −1.27910e7 −1.15343 −0.576715 0.816945i \(-0.695666\pi\)
−0.576715 + 0.816945i \(0.695666\pi\)
\(224\) −1.06115e7 −0.944133
\(225\) 1.13906e7 1.00000
\(226\) 3.61369e7 3.13059
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 7.35556e6 0.596732
\(232\) 1.29993e7 1.04102
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 2.84500e7 2.22041
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 4.29645e7 3.18698
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −2.12038e7 −1.51483 −0.757413 0.652936i \(-0.773537\pi\)
−0.757413 + 0.652936i \(0.773537\pi\)
\(242\) −1.45850e7 −1.02911
\(243\) −1.43489e7 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 4.63875e7 3.11598
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.03718e7 −1.92066 −0.960329 0.278870i \(-0.910040\pi\)
−0.960329 + 0.278870i \(0.910040\pi\)
\(252\) 2.58722e7 1.61671
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −1.81380e7 −1.08111
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.77796e7 1.00000
\(262\) −4.21398e7 −2.34309
\(263\) 1.26271e7 0.694123 0.347062 0.937842i \(-0.387179\pi\)
0.347062 + 0.937842i \(0.387179\pi\)
\(264\) 1.15991e7 0.630397
\(265\) 0 0
\(266\) 0 0
\(267\) −1.92418e6 −0.101091
\(268\) 2.80453e7 1.45699
\(269\) −7.92469e6 −0.407123 −0.203562 0.979062i \(-0.565252\pi\)
−0.203562 + 0.979062i \(0.565252\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 2.04360e6 0.101552
\(273\) −2.73963e7 −1.34649
\(274\) 6.43785e7 3.12960
\(275\) −1.25938e7 −0.605560
\(276\) 0 0
\(277\) −4.24167e7 −1.99571 −0.997854 0.0654718i \(-0.979145\pi\)
−0.997854 + 0.0654718i \(0.979145\pi\)
\(278\) −6.55727e7 −3.05203
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 5.31772e7 2.37125
\(283\) −4.20180e7 −1.85386 −0.926928 0.375240i \(-0.877560\pi\)
−0.926928 + 0.375240i \(0.877560\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −3.14550e7 −1.34459
\(287\) −4.46694e7 −1.88958
\(288\) −2.28870e7 −0.958099
\(289\) 7.14717e7 2.96102
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.17371e7 1.26172 0.630862 0.775895i \(-0.282702\pi\)
0.630862 + 0.775895i \(0.282702\pi\)
\(294\) 1.19516e6 0.0470308
\(295\) 0 0
\(296\) 0 0
\(297\) 1.58645e7 0.605560
\(298\) 0 0
\(299\) 0 0
\(300\) −4.42969e7 −1.64062
\(301\) 0 0
\(302\) 3.96396e7 1.43916
\(303\) 5.03714e7 1.81074
\(304\) 0 0
\(305\) 0 0
\(306\) 9.26661e7 3.23412
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) −2.86049e7 −0.979014
\(309\) −4.47159e7 −1.51561
\(310\) 0 0
\(311\) 5.26143e7 1.74913 0.874566 0.484907i \(-0.161147\pi\)
0.874566 + 0.484907i \(0.161147\pi\)
\(312\) −4.32018e7 −1.42245
\(313\) −5.25314e7 −1.71311 −0.856557 0.516052i \(-0.827401\pi\)
−0.856557 + 0.516052i \(0.827401\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.13678e6 0.255432 0.127716 0.991811i \(-0.459235\pi\)
0.127716 + 0.991811i \(0.459235\pi\)
\(318\) 0 0
\(319\) −1.96575e7 −0.605560
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 5.58013e7 1.64062
\(325\) 4.69062e7 1.36641
\(326\) 0 0
\(327\) 6.99185e7 1.99963
\(328\) −7.04402e7 −1.99618
\(329\) −5.12077e7 −1.43796
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −1.90733e6 −0.0502816
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 5.44075e7 1.40899
\(339\) −7.50536e7 −1.92651
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −4.09163e7 −1.01394
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) −6.91428e7 −1.64062
\(349\) −8.06657e7 −1.89763 −0.948817 0.315825i \(-0.897719\pi\)
−0.948817 + 0.315825i \(0.897719\pi\)
\(350\) 6.86562e7 1.60131
\(351\) −5.90884e7 −1.36641
\(352\) 2.53044e7 0.580186
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 7.48293e6 0.165852
\(357\) −8.92340e7 −1.96122
\(358\) 0 0
\(359\) 8.18203e7 1.76839 0.884194 0.467120i \(-0.154708\pi\)
0.884194 + 0.467120i \(0.154708\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 1.51115e8 3.18553
\(363\) 3.02920e7 0.633297
\(364\) 1.06541e8 2.20909
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) −9.63432e7 −1.91753
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 45866.0 0.000883821 0 0.000441911 1.00000i \(-0.499859\pi\)
0.000441911 1.00000i \(0.499859\pi\)
\(374\) −1.02454e8 −1.95846
\(375\) 0 0
\(376\) −8.07506e7 −1.51908
\(377\) 7.32158e7 1.36641
\(378\) −8.64871e7 −1.60131
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.09498e8 −1.96435
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 9.36998e7 1.65480
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7.00237e6 −0.118959 −0.0594794 0.998230i \(-0.518944\pi\)
−0.0594794 + 0.998230i \(0.518944\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −1.81486e6 −0.0301291
\(393\) 8.75212e7 1.44190
\(394\) 0 0
\(395\) 0 0
\(396\) −6.16953e7 −0.993496
\(397\) 6.89034e7 1.10121 0.550603 0.834767i \(-0.314398\pi\)
0.550603 + 0.834767i \(0.314398\pi\)
\(398\) −7.11091e7 −1.12791
\(399\) 0 0
\(400\) 3.26562e6 0.0510254
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) −9.37514e7 −1.44311
\(403\) 0 0
\(404\) −1.95889e8 −2.97074
\(405\) 0 0
\(406\) 1.07165e8 1.60131
\(407\) 0 0
\(408\) −1.40715e8 −2.07186
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) −1.33709e8 −1.92591
\(412\) 1.73895e8 2.48654
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −9.42478e7 −1.30916
\(417\) 1.36189e8 1.87817
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) −1.10445e8 −1.45923
\(424\) 0 0
\(425\) 1.52781e8 1.99023
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 6.53295e7 0.827442
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −4.11375e6 −0.0510254
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.71905e8 −3.28064
\(437\) 0 0
\(438\) 0 0
\(439\) 1.18161e8 1.39663 0.698315 0.715790i \(-0.253933\pi\)
0.698315 + 0.715790i \(0.253933\pi\)
\(440\) 0 0
\(441\) −2.48224e6 −0.0289420
\(442\) 3.81596e8 4.41913
\(443\) 1.14561e7 0.131773 0.0658865 0.997827i \(-0.479012\pi\)
0.0658865 + 0.997827i \(0.479012\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1.66283e8 −1.87432
\(447\) 0 0
\(448\) −1.42471e8 −1.58450
\(449\) −1.67070e8 −1.84570 −0.922848 0.385165i \(-0.874145\pi\)
−0.922848 + 0.385165i \(0.874145\pi\)
\(450\) 1.48078e8 1.62500
\(451\) 1.06519e8 1.16118
\(452\) 2.91875e8 3.16069
\(453\) −8.23285e7 −0.885636
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.58727e8 1.66304 0.831520 0.555495i \(-0.187471\pi\)
0.831520 + 0.555495i \(0.187471\pi\)
\(458\) 0 0
\(459\) −1.92460e8 −1.99023
\(460\) 0 0
\(461\) −9.73084e6 −0.0993225 −0.0496612 0.998766i \(-0.515814\pi\)
−0.0496612 + 0.998766i \(0.515814\pi\)
\(462\) 9.56222e7 0.969690
\(463\) −1.97688e8 −1.99177 −0.995883 0.0906462i \(-0.971107\pi\)
−0.995883 + 0.0906462i \(0.971107\pi\)
\(464\) 5.09730e6 0.0510254
\(465\) 0 0
\(466\) 0 0
\(467\) 1.80543e8 1.77268 0.886341 0.463034i \(-0.153239\pi\)
0.886341 + 0.463034i \(0.153239\pi\)
\(468\) 2.29788e8 2.24176
\(469\) 9.02791e7 0.875122
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 3.47021e8 3.21762
\(477\) 0 0
\(478\) 0 0
\(479\) 2.19478e8 1.99703 0.998514 0.0545017i \(-0.0173570\pi\)
0.998514 + 0.0545017i \(0.0173570\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −2.75649e8 −2.46159
\(483\) 0 0
\(484\) −1.17802e8 −1.03900
\(485\) 0 0
\(486\) −1.86536e8 −1.62500
\(487\) −2.00089e8 −1.73235 −0.866176 0.499738i \(-0.833429\pi\)
−0.866176 + 0.499738i \(0.833429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.55132e7 0.300017 0.150008 0.988685i \(-0.452070\pi\)
0.150008 + 0.988685i \(0.452070\pi\)
\(492\) 3.74668e8 3.14595
\(493\) 2.38476e8 1.99023
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.46649e8 1.98507 0.992537 0.121941i \(-0.0389119\pi\)
0.992537 + 0.121941i \(0.0389119\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −3.94834e8 −3.12107
\(503\) 1.93831e8 1.52307 0.761534 0.648125i \(-0.224446\pi\)
0.761534 + 0.648125i \(0.224446\pi\)
\(504\) 1.31332e8 1.02584
\(505\) 0 0
\(506\) 0 0
\(507\) −1.13000e8 −0.867073
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.36910e7 −0.102006
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1.22111e8 0.883653
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 2.31135e8 1.62500
\(523\) −8.09344e7 −0.565754 −0.282877 0.959156i \(-0.591289\pi\)
−0.282877 + 0.959156i \(0.591289\pi\)
\(524\) −3.40360e8 −2.36562
\(525\) −1.42594e8 −0.985423
\(526\) 1.64152e8 1.12795
\(527\) 0 0
\(528\) 4.54826e6 0.0308989
\(529\) 1.48036e8 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3.96738e8 −2.62013
\(534\) −2.50144e7 −0.164273
\(535\) 0 0
\(536\) 1.42363e8 0.924493
\(537\) 0 0
\(538\) −1.03021e8 −0.661575
\(539\) 2.74443e6 0.0175261
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −3.13854e8 −1.96033
\(544\) −3.06980e8 −1.90684
\(545\) 0 0
\(546\) −3.56151e8 −2.18805
\(547\) −3.17273e8 −1.93852 −0.969262 0.246031i \(-0.920874\pi\)
−0.969262 + 0.246031i \(0.920874\pi\)
\(548\) 5.19980e8 3.15970
\(549\) 0 0
\(550\) −1.63719e8 −0.984035
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −5.51417e8 −3.24303
\(555\) 0 0
\(556\) −5.29626e8 −3.08138
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 2.12789e8 1.20520
\(562\) 0 0
\(563\) −3.56237e8 −1.99625 −0.998123 0.0612458i \(-0.980493\pi\)
−0.998123 + 0.0612458i \(0.980493\pi\)
\(564\) 4.29508e8 2.39405
\(565\) 0 0
\(566\) −5.46234e8 −3.01251
\(567\) 1.79627e8 0.985423
\(568\) 0 0
\(569\) −2.27313e8 −1.23392 −0.616962 0.786993i \(-0.711637\pi\)
−0.616962 + 0.786993i \(0.711637\pi\)
\(570\) 0 0
\(571\) −2.76063e8 −1.48286 −0.741431 0.671029i \(-0.765852\pi\)
−0.741431 + 0.671029i \(0.765852\pi\)
\(572\) −2.54059e8 −1.35752
\(573\) 2.27420e8 1.20883
\(574\) −5.80702e8 −3.07056
\(575\) 0 0
\(576\) −3.07282e8 −1.60794
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 9.29132e8 4.81165
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 4.12582e8 2.05030
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 9.65318e6 0.0474830
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 2.06238e8 0.984035
\(595\) 0 0
\(596\) 0 0
\(597\) 1.47688e8 0.694101
\(598\) 0 0
\(599\) 3.79855e8 1.76741 0.883706 0.468042i \(-0.155040\pi\)
0.883706 + 0.468042i \(0.155040\pi\)
\(600\) −2.24859e8 −1.04102
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 1.94714e8 0.888068
\(604\) 3.20166e8 1.45300
\(605\) 0 0
\(606\) 6.54828e8 2.94245
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) −2.22574e8 −0.985423
\(610\) 0 0
\(611\) −4.54809e8 −1.99391
\(612\) 7.48457e8 3.26522
\(613\) 4.09911e8 1.77954 0.889771 0.456407i \(-0.150864\pi\)
0.889771 + 0.456407i \(0.150864\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −1.45204e8 −0.621208
\(617\) −1.20584e8 −0.513375 −0.256687 0.966494i \(-0.582631\pi\)
−0.256687 + 0.966494i \(0.582631\pi\)
\(618\) −5.81307e8 −2.46286
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 6.83986e8 2.84234
\(623\) 2.40879e7 0.0996173
\(624\) −1.69403e7 −0.0697215
\(625\) 2.44141e8 1.00000
\(626\) −6.82909e8 −2.78381
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −4.04012e7 −0.160807 −0.0804037 0.996762i \(-0.525621\pi\)
−0.0804037 + 0.996762i \(0.525621\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 1.05778e8 0.415076
\(635\) 0 0
\(636\) 0 0
\(637\) −1.02218e7 −0.0395466
\(638\) −2.55548e8 −0.984035
\(639\) 0 0
\(640\) 0 0
\(641\) 8.63571e7 0.327887 0.163943 0.986470i \(-0.447579\pi\)
0.163943 + 0.986470i \(0.447579\pi\)
\(642\) 0 0
\(643\) 4.09322e8 1.53968 0.769842 0.638234i \(-0.220335\pi\)
0.769842 + 0.638234i \(0.220335\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 2.83258e8 1.04102
\(649\) 0 0
\(650\) 6.09781e8 2.22041
\(651\) 0 0
\(652\) 0 0
\(653\) 5.51406e8 1.98031 0.990153 0.139990i \(-0.0447069\pi\)
0.990153 + 0.139990i \(0.0447069\pi\)
\(654\) 9.08940e8 3.24939
\(655\) 0 0
\(656\) −2.76210e7 −0.0978427
\(657\) 0 0
\(658\) −6.65700e8 −2.33669
\(659\) −1.02798e8 −0.359192 −0.179596 0.983740i \(-0.557479\pi\)
−0.179596 + 0.983740i \(0.557479\pi\)
\(660\) 0 0
\(661\) 4.76905e7 0.165130 0.0825652 0.996586i \(-0.473689\pi\)
0.0825652 + 0.996586i \(0.473689\pi\)
\(662\) 0 0
\(663\) −7.92546e8 −2.71947
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 3.45358e8 1.15343
\(670\) 0 0
\(671\) 0 0
\(672\) 2.86511e8 0.944133
\(673\) −1.81732e8 −0.596192 −0.298096 0.954536i \(-0.596351\pi\)
−0.298096 + 0.954536i \(0.596351\pi\)
\(674\) 0 0
\(675\) −3.07547e8 −1.00000
\(676\) 4.39445e8 1.42254
\(677\) 6.14494e8 1.98039 0.990197 0.139676i \(-0.0446060\pi\)
0.990197 + 0.139676i \(0.0446060\pi\)
\(678\) −9.75696e8 −3.13059
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −5.31911e8 −1.64766
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 5.54226e8 1.67978 0.839890 0.542757i \(-0.182619\pi\)
0.839890 + 0.542757i \(0.182619\pi\)
\(692\) 0 0
\(693\) −1.98600e8 −0.596732
\(694\) 0 0
\(695\) 0 0
\(696\) −3.50982e8 −1.04102
\(697\) −1.29224e9 −3.81632
\(698\) −1.04865e9 −3.08366
\(699\) 0 0
\(700\) 5.54531e8 1.61671
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) −7.68149e8 −2.22041
\(703\) 0 0
\(704\) 3.39738e8 0.973702
\(705\) 0 0
\(706\) 0 0
\(707\) −6.30575e8 −1.78434
\(708\) 0 0
\(709\) −3.91915e7 −0.109965 −0.0549824 0.998487i \(-0.517510\pi\)
−0.0549824 + 0.998487i \(0.517510\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 3.79848e7 0.105237
\(713\) 0 0
\(714\) −1.16004e9 −3.18698
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 1.06366e9 2.87363
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 5.59777e8 1.49351
\(722\) 6.11596e8 1.62500
\(723\) 5.72503e8 1.51483
\(724\) 1.22054e9 3.21616
\(725\) 3.81078e8 1.00000
\(726\) 3.93796e8 1.02911
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 5.40822e8 1.40172
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.15281e8 −0.537778
\(738\) −1.25246e9 −3.11598
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.55700e8 1.84240 0.921198 0.389093i \(-0.127212\pi\)
0.921198 + 0.389093i \(0.127212\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 596258. 0.00143621
\(747\) 0 0
\(748\) −8.27512e8 −1.97729
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) −3.16639e7 −0.0744580
\(753\) 8.20040e8 1.92066
\(754\) 9.51805e8 2.22041
\(755\) 0 0
\(756\) −6.98550e8 −1.61671
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −8.75276e8 −1.97048
\(764\) −8.84411e8 −1.98323
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 4.89726e8 1.08111
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 5.51471e7 0.119394 0.0596972 0.998217i \(-0.480986\pi\)
0.0596972 + 0.998217i \(0.480986\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −9.10309e7 −0.193308
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −4.80049e8 −1.00000
\(784\) −711645. −0.00147678
\(785\) 0 0
\(786\) 1.13778e9 2.34309
\(787\) −8.86902e8 −1.81950 −0.909749 0.415158i \(-0.863726\pi\)
−0.909749 + 0.415158i \(0.863726\pi\)
\(788\) 0 0
\(789\) −3.40932e8 −0.694123
\(790\) 0 0
\(791\) 9.39560e8 1.89843
\(792\) −3.13177e8 −0.630397
\(793\) 0 0
\(794\) 8.95744e8 1.78946
\(795\) 0 0
\(796\) −5.74343e8 −1.13876
\(797\) −4.12151e8 −0.814106 −0.407053 0.913405i \(-0.633444\pi\)
−0.407053 + 0.913405i \(0.633444\pi\)
\(798\) 0 0
\(799\) −1.48139e9 −2.90421
\(800\) −4.90547e8 −0.958099
\(801\) 5.19529e7 0.101091
\(802\) 0 0
\(803\) 0 0
\(804\) −7.57223e8 −1.45699
\(805\) 0 0
\(806\) 0 0
\(807\) 2.13967e8 0.407123
\(808\) −9.94368e8 −1.88501
\(809\) 3.59615e8 0.679191 0.339596 0.940572i \(-0.389710\pi\)
0.339596 + 0.940572i \(0.389710\pi\)
\(810\) 0 0
\(811\) −8.89508e8 −1.66758 −0.833791 0.552080i \(-0.813834\pi\)
−0.833791 + 0.552080i \(0.813834\pi\)
\(812\) 8.65566e8 1.61671
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) −5.51773e7 −0.101552
\(817\) 0 0
\(818\) 0 0
\(819\) 7.39699e8 1.34649
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) −1.73822e9 −3.12960
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 8.82726e8 1.57777
\(825\) 3.40031e8 0.605560
\(826\) 0 0
\(827\) −4.06219e8 −0.718196 −0.359098 0.933300i \(-0.616916\pi\)
−0.359098 + 0.933300i \(0.616916\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 1.14525e9 1.99571
\(832\) −1.26538e9 −2.19710
\(833\) −3.32941e7 −0.0576013
\(834\) 1.77046e9 3.05203
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.07472e9 −1.81974 −0.909869 0.414895i \(-0.863818\pi\)
−0.909869 + 0.414895i \(0.863818\pi\)
\(840\) 0 0
\(841\) 5.94823e8 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) −1.43578e9 −2.37125
\(847\) −3.79211e8 −0.624066
\(848\) 0 0
\(849\) 1.13449e9 1.85386
\(850\) 1.98616e9 3.23412
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 8.49284e8 1.34459
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 1.20607e9 1.88958
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 6.17948e8 0.958099
\(865\) 0 0
\(866\) 0 0
\(867\) −1.92974e9 −2.96102
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 8.01828e8 1.21346
\(872\) −1.38024e9 −2.08164
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −7.87836e8 −1.16798 −0.583992 0.811759i \(-0.698510\pi\)
−0.583992 + 0.811759i \(0.698510\pi\)
\(878\) 1.53610e9 2.26952
\(879\) −8.56901e8 −1.26172
\(880\) 0 0
\(881\) −1.05140e9 −1.53759 −0.768796 0.639494i \(-0.779144\pi\)
−0.768796 + 0.639494i \(0.779144\pi\)
\(882\) −3.22692e7 −0.0470308
\(883\) 6.46270e8 0.938710 0.469355 0.883009i \(-0.344486\pi\)
0.469355 + 0.883009i \(0.344486\pi\)
\(884\) 3.08212e9 4.46163
\(885\) 0 0
\(886\) 1.48930e8 0.214131
\(887\) −1.38515e9 −1.98484 −0.992420 0.122893i \(-0.960783\pi\)
−0.992420 + 0.122893i \(0.960783\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −4.28341e8 −0.605560
\(892\) −1.34306e9 −1.89235
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −1.17298e9 −1.63068
\(897\) 0 0
\(898\) −2.17191e9 −2.99926
\(899\) 0 0
\(900\) 1.19602e9 1.64062
\(901\) 0 0
\(902\) 1.38475e9 1.88691
\(903\) 0 0
\(904\) 1.48161e9 2.00553
\(905\) 0 0
\(906\) −1.07027e9 −1.43916
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −1.36003e9 −1.81074
\(910\) 0 0
\(911\) −1.66419e8 −0.220114 −0.110057 0.993925i \(-0.535103\pi\)
−0.110057 + 0.993925i \(0.535103\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 2.06345e9 2.70244
\(915\) 0 0
\(916\) 0 0
\(917\) −1.09564e9 −1.42088
\(918\) −2.50198e9 −3.23412
\(919\) −4.23881e8 −0.546131 −0.273066 0.961995i \(-0.588038\pi\)
−0.273066 + 0.961995i \(0.588038\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.26501e8 −0.161399
\(923\) 0 0
\(924\) 7.72333e8 0.979014
\(925\) 0 0
\(926\) −2.56995e9 −3.23662
\(927\) 1.20733e9 1.51561
\(928\) −7.65693e8 −0.958099
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −1.42059e9 −1.74913
\(934\) 2.34706e9 2.88061
\(935\) 0 0
\(936\) 1.16645e9 1.42245
\(937\) 8.02759e8 0.975812 0.487906 0.872896i \(-0.337761\pi\)
0.487906 + 0.872896i \(0.337761\pi\)
\(938\) 1.17363e9 1.42207
\(939\) 1.41835e9 1.71311
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −9.43083e8 −1.11045 −0.555226 0.831700i \(-0.687368\pi\)
−0.555226 + 0.831700i \(0.687368\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −2.19693e8 −0.255432
\(952\) 1.76155e9 2.04166
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 5.30753e8 0.605560
\(958\) 2.85321e9 3.24517
\(959\) 1.67384e9 1.89784
\(960\) 0 0
\(961\) 8.87504e8 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) −2.22640e9 −2.48526
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) −5.97986e8 −0.659273
\(969\) 0 0
\(970\) 0 0
\(971\) 2.42617e8 0.265011 0.132505 0.991182i \(-0.457698\pi\)
0.132505 + 0.991182i \(0.457698\pi\)
\(972\) −1.50664e9 −1.64062
\(973\) −1.70489e9 −1.85079
\(974\) −2.60116e9 −2.81507
\(975\) −1.26647e9 −1.36641
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) −5.74404e7 −0.0612166
\(980\) 0 0
\(981\) −1.88780e9 −1.99963
\(982\) 4.61672e8 0.487528
\(983\) 9.65580e8 1.01655 0.508274 0.861195i \(-0.330284\pi\)
0.508274 + 0.861195i \(0.330284\pi\)
\(984\) 1.90189e9 1.99618
\(985\) 0 0
\(986\) 3.10018e9 3.23412
\(987\) 1.38261e9 1.43796
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.51849e9 1.56023 0.780117 0.625633i \(-0.215159\pi\)
0.780117 + 0.625633i \(0.215159\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 3.20643e9 3.22575
\(999\) 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 87.7.d.b.86.1 yes 1
3.2 odd 2 87.7.d.a.86.1 1
29.28 even 2 87.7.d.a.86.1 1
87.86 odd 2 CM 87.7.d.b.86.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.7.d.a.86.1 1 3.2 odd 2
87.7.d.a.86.1 1 29.28 even 2
87.7.d.b.86.1 yes 1 1.1 even 1 trivial
87.7.d.b.86.1 yes 1 87.86 odd 2 CM