Properties

Label 23.25.b.a.22.1
Level $23$
Weight $25$
Character 23.22
Self dual yes
Analytic conductor $83.942$
Analytic rank $0$
Dimension $1$
CM discriminant -23
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [23,25,Mod(22,23)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(23, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 25, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("23.22");
 
S:= CuspForms(chi, 25);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 23 \)
Weight: \( k \) \(=\) \( 25 \)
Character orbit: \([\chi]\) \(=\) 23.b (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: \(83.9424450193\)
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 22.1
Character \(\chi\) \(=\) 23.22

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1951.00 q^{2} -1.06269e6 q^{3} -1.29708e7 q^{4} +2.07330e9 q^{6} +5.80384e10 q^{8} +8.46872e11 q^{9} +O(q^{10})\) \(q-1951.00 q^{2} -1.06269e6 q^{3} -1.29708e7 q^{4} +2.07330e9 q^{6} +5.80384e10 q^{8} +8.46872e11 q^{9} +1.37839e13 q^{12} +2.53633e13 q^{13} +1.04381e14 q^{16} -1.65225e15 q^{18} +2.19146e16 q^{23} -6.16766e16 q^{24} +5.96046e16 q^{25} -4.94838e16 q^{26} -5.99825e17 q^{27} -6.47932e17 q^{29} +1.23709e18 q^{31} -1.17737e18 q^{32} -1.09846e19 q^{36} -2.69532e19 q^{39} +1.25765e19 q^{41} -4.27554e19 q^{46} +1.92262e20 q^{47} -1.10924e20 q^{48} +1.91581e20 q^{49} -1.16289e20 q^{50} -3.28983e20 q^{52} +1.17026e21 q^{54} +1.26412e21 q^{58} -3.16340e21 q^{59} -2.41356e21 q^{62} +5.45824e20 q^{64} -2.32884e22 q^{69} +2.86154e21 q^{71} +4.91511e22 q^{72} +3.93269e21 q^{73} -6.33410e22 q^{75} +5.25858e22 q^{78} +3.98244e23 q^{81} -2.45368e22 q^{82} +6.88549e23 q^{87} -2.84251e23 q^{92} -1.31464e24 q^{93} -3.75102e23 q^{94} +1.25118e24 q^{96} -3.73775e23 q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/23\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1951.00 −0.476318 −0.238159 0.971226i \(-0.576544\pi\)
−0.238159 + 0.971226i \(0.576544\pi\)
\(3\) −1.06269e6 −1.99963 −0.999816 0.0192035i \(-0.993887\pi\)
−0.999816 + 0.0192035i \(0.993887\pi\)
\(4\) −1.29708e7 −0.773121
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 2.07330e9 0.952461
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 5.80384e10 0.844570
\(9\) 8.46872e11 2.99852
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 1.37839e13 1.54596
\(13\) 2.53633e13 1.08864 0.544322 0.838876i \(-0.316787\pi\)
0.544322 + 0.838876i \(0.316787\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 1.04381e14 0.370837
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −1.65225e15 −1.42825
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.19146e16 1.00000
\(24\) −6.16766e16 −1.68883
\(25\) 5.96046e16 1.00000
\(26\) −4.94838e16 −0.518541
\(27\) −5.99825e17 −3.99631
\(28\) 0 0
\(29\) −6.47932e17 −1.83128 −0.915638 0.402004i \(-0.868314\pi\)
−0.915638 + 0.402004i \(0.868314\pi\)
\(30\) 0 0
\(31\) 1.23709e18 1.57058 0.785290 0.619128i \(-0.212514\pi\)
0.785290 + 0.619128i \(0.212514\pi\)
\(32\) −1.17737e18 −1.02121
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −1.09846e19 −2.31822
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −2.69532e19 −2.17689
\(40\) 0 0
\(41\) 1.25765e19 0.557384 0.278692 0.960381i \(-0.410099\pi\)
0.278692 + 0.960381i \(0.410099\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −4.27554e19 −0.476318
\(47\) 1.92262e20 1.65470 0.827348 0.561689i \(-0.189848\pi\)
0.827348 + 0.561689i \(0.189848\pi\)
\(48\) −1.10924e20 −0.741536
\(49\) 1.91581e20 1.00000
\(50\) −1.16289e20 −0.476318
\(51\) 0 0
\(52\) −3.28983e20 −0.841653
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 1.17026e21 1.90352
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 1.26412e21 0.872270
\(59\) −3.16340e21 −1.77799 −0.888996 0.457915i \(-0.848596\pi\)
−0.888996 + 0.457915i \(0.848596\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −2.41356e21 −0.748096
\(63\) 0 0
\(64\) 5.45824e20 0.115583
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) −2.32884e22 −1.99963
\(70\) 0 0
\(71\) 2.86154e21 0.174381 0.0871905 0.996192i \(-0.472211\pi\)
0.0871905 + 0.996192i \(0.472211\pi\)
\(72\) 4.91511e22 2.53246
\(73\) 3.93269e21 0.171718 0.0858588 0.996307i \(-0.472637\pi\)
0.0858588 + 0.996307i \(0.472637\pi\)
\(74\) 0 0
\(75\) −6.33410e22 −1.99963
\(76\) 0 0
\(77\) 0 0
\(78\) 5.25858e22 1.03689
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 3.98244e23 4.99263
\(82\) −2.45368e22 −0.265492
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.88549e23 3.66188
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −2.84251e23 −0.773121
\(93\) −1.31464e24 −3.14058
\(94\) −3.75102e23 −0.788162
\(95\) 0 0
\(96\) 1.25118e24 2.04204
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −3.73775e23 −0.476318
\(99\) 0 0
\(100\) −7.73121e23 −0.773121
\(101\) 1.23280e24 1.09405 0.547025 0.837116i \(-0.315760\pi\)
0.547025 + 0.837116i \(0.315760\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 1.47205e24 0.919436
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 7.78022e24 3.08963
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 8.40421e24 1.41580
\(117\) 2.14795e25 3.26433
\(118\) 6.17179e24 0.846890
\(119\) 0 0
\(120\) 0 0
\(121\) 9.84973e24 1.00000
\(122\) 0 0
\(123\) −1.33649e25 −1.11456
\(124\) −1.60460e25 −1.21425
\(125\) 0 0
\(126\) 0 0
\(127\) −3.40545e25 −1.93433 −0.967163 0.254157i \(-0.918202\pi\)
−0.967163 + 0.254157i \(0.918202\pi\)
\(128\) 1.86881e25 0.966152
\(129\) 0 0
\(130\) 0 0
\(131\) −5.02301e25 −1.96657 −0.983283 0.182082i \(-0.941716\pi\)
−0.983283 + 0.182082i \(0.941716\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 4.54356e25 0.952461
\(139\) 5.54863e25 1.06662 0.533308 0.845921i \(-0.320949\pi\)
0.533308 + 0.845921i \(0.320949\pi\)
\(140\) 0 0
\(141\) −2.04314e26 −3.30878
\(142\) −5.58286e24 −0.0830609
\(143\) 0 0
\(144\) 8.83975e25 1.11196
\(145\) 0 0
\(146\) −7.67267e24 −0.0817923
\(147\) −2.03591e26 −1.99963
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 1.23578e26 0.952461
\(151\) −6.77900e25 −0.482439 −0.241219 0.970471i \(-0.577547\pi\)
−0.241219 + 0.970471i \(0.577547\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 3.49606e26 1.68300
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −7.76974e26 −2.37808
\(163\) −7.00670e26 −1.99188 −0.995938 0.0900446i \(-0.971299\pi\)
−0.995938 + 0.0900446i \(0.971299\pi\)
\(164\) −1.63128e26 −0.430925
\(165\) 0 0
\(166\) 0 0
\(167\) −8.99102e26 −1.91078 −0.955392 0.295341i \(-0.904567\pi\)
−0.955392 + 0.295341i \(0.904567\pi\)
\(168\) 0 0
\(169\) 1.00497e26 0.185146
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.03840e27 1.44481 0.722406 0.691469i \(-0.243036\pi\)
0.722406 + 0.691469i \(0.243036\pi\)
\(174\) −1.34336e27 −1.74422
\(175\) 0 0
\(176\) 0 0
\(177\) 3.36170e27 3.55533
\(178\) 0 0
\(179\) 3.78968e26 0.350241 0.175120 0.984547i \(-0.443969\pi\)
0.175120 + 0.984547i \(0.443969\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.27189e27 0.844570
\(185\) 0 0
\(186\) 2.56485e27 1.49592
\(187\) 0 0
\(188\) −2.49379e27 −1.27928
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −5.80039e26 −0.231123
\(193\) 2.30211e27 0.861862 0.430931 0.902385i \(-0.358185\pi\)
0.430931 + 0.902385i \(0.358185\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.48496e27 −0.773121
\(197\) 4.73049e27 1.38456 0.692279 0.721630i \(-0.256607\pi\)
0.692279 + 0.721630i \(0.256607\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 3.45936e27 0.844570
\(201\) 0 0
\(202\) −2.40520e27 −0.521116
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.85589e28 2.99852
\(208\) 2.64745e27 0.403709
\(209\) 0 0
\(210\) 0 0
\(211\) −1.53411e28 −1.97001 −0.985005 0.172527i \(-0.944807\pi\)
−0.985005 + 0.172527i \(0.944807\pi\)
\(212\) 0 0
\(213\) −3.04092e27 −0.348698
\(214\) 0 0
\(215\) 0 0
\(216\) −3.48129e28 −3.37517
\(217\) 0 0
\(218\) 0 0
\(219\) −4.17921e27 −0.343372
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.94856e28 1.94962 0.974812 0.223027i \(-0.0715937\pi\)
0.974812 + 0.223027i \(0.0715937\pi\)
\(224\) 0 0
\(225\) 5.04775e28 2.99852
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3.76050e28 −1.54664
\(233\) 1.58866e28 0.620525 0.310262 0.950651i \(-0.399583\pi\)
0.310262 + 0.950651i \(0.399583\pi\)
\(234\) −4.19065e28 −1.55486
\(235\) 0 0
\(236\) 4.10318e28 1.37460
\(237\) 0 0
\(238\) 0 0
\(239\) 5.68125e28 1.63557 0.817784 0.575525i \(-0.195202\pi\)
0.817784 + 0.575525i \(0.195202\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −1.92168e28 −0.476318
\(243\) −2.53800e29 −5.98710
\(244\) 0 0
\(245\) 0 0
\(246\) 2.60749e28 0.530887
\(247\) 0 0
\(248\) 7.17986e28 1.32647
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 6.64403e28 0.921355
\(255\) 0 0
\(256\) −4.56179e28 −0.575779
\(257\) −1.64519e29 −1.98161 −0.990807 0.135282i \(-0.956806\pi\)
−0.990807 + 0.135282i \(0.956806\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −5.48716e29 −5.49113
\(262\) 9.79990e28 0.936712
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.76123e29 1.22684 0.613422 0.789755i \(-0.289793\pi\)
0.613422 + 0.789755i \(0.289793\pi\)
\(270\) 0 0
\(271\) 2.90581e29 1.85198 0.925988 0.377553i \(-0.123234\pi\)
0.925988 + 0.377553i \(0.123234\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 3.02069e29 1.54596
\(277\) −1.41523e28 −0.0693537 −0.0346768 0.999399i \(-0.511040\pi\)
−0.0346768 + 0.999399i \(0.511040\pi\)
\(278\) −1.08254e29 −0.508049
\(279\) 1.04766e30 4.70942
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 3.98616e29 1.57603
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −3.71165e28 −0.134818
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −9.97082e29 −3.06211
\(289\) 3.39449e29 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −5.10101e28 −0.132759
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 3.97205e29 0.952461
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.55828e29 1.08864
\(300\) 8.21585e29 1.54596
\(301\) 0 0
\(302\) 1.32258e29 0.229794
\(303\) −1.31008e30 −2.18770
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −5.97478e29 −0.852435 −0.426218 0.904621i \(-0.640154\pi\)
−0.426218 + 0.904621i \(0.640154\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.36656e29 −0.411208 −0.205604 0.978635i \(-0.565916\pi\)
−0.205604 + 0.978635i \(0.565916\pi\)
\(312\) −1.56432e30 −1.83853
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.56676e29 0.929081 0.464541 0.885552i \(-0.346219\pi\)
0.464541 + 0.885552i \(0.346219\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −5.16555e30 −3.85990
\(325\) 1.51177e30 1.08864
\(326\) 1.36701e30 0.948767
\(327\) 0 0
\(328\) 7.29922e29 0.470750
\(329\) 0 0
\(330\) 0 0
\(331\) 3.45849e30 1.99963 0.999817 0.0191120i \(-0.00608392\pi\)
0.999817 + 0.0191120i \(0.00608392\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 1.75415e30 0.910141
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −1.96070e29 −0.0881883
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −2.02592e30 −0.688190
\(347\) 6.09163e30 1.99885 0.999425 0.0339107i \(-0.0107962\pi\)
0.999425 + 0.0339107i \(0.0107962\pi\)
\(348\) −8.93104e30 −2.83107
\(349\) 5.88613e30 1.80271 0.901353 0.433085i \(-0.142575\pi\)
0.901353 + 0.433085i \(0.142575\pi\)
\(350\) 0 0
\(351\) −1.52136e31 −4.35056
\(352\) 0 0
\(353\) −3.71482e30 −0.992297 −0.496148 0.868238i \(-0.665253\pi\)
−0.496148 + 0.868238i \(0.665253\pi\)
\(354\) −6.55867e30 −1.69347
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −7.39367e29 −0.166826
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 4.89876e30 1.00000
\(362\) 0 0
\(363\) −1.04672e31 −1.99963
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 2.28748e30 0.370837
\(369\) 1.06507e31 1.67133
\(370\) 0 0
\(371\) 0 0
\(372\) 1.70519e31 2.42805
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 1.11586e31 1.39751
\(377\) −1.64337e31 −1.99361
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 3.61892e31 3.86794
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) −1.98596e31 −1.93195
\(385\) 0 0
\(386\) −4.49141e30 −0.410521
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1.11191e31 0.844570
\(393\) 5.33788e31 3.93241
\(394\) −9.22918e30 −0.659491
\(395\) 0 0
\(396\) 0 0
\(397\) −2.36852e31 −1.54522 −0.772611 0.634880i \(-0.781049\pi\)
−0.772611 + 0.634880i \(0.781049\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6.22161e30 0.370837
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 3.13767e31 1.70980
\(404\) −1.59905e31 −0.845833
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −3.39594e31 −1.54982 −0.774908 0.632074i \(-0.782204\pi\)
−0.774908 + 0.632074i \(0.782204\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) −3.62084e31 −1.42825
\(415\) 0 0
\(416\) −2.98620e31 −1.11173
\(417\) −5.89645e31 −2.13284
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 2.99306e31 0.938352
\(423\) 1.62821e32 4.96165
\(424\) 0 0
\(425\) 0 0
\(426\) 5.93283e30 0.166091
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −6.26105e31 −1.48198
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 8.15364e30 0.163554
\(439\) −1.01770e32 −1.98629 −0.993147 0.116874i \(-0.962713\pi\)
−0.993147 + 0.116874i \(0.962713\pi\)
\(440\) 0 0
\(441\) 1.62245e32 2.99852
\(442\) 0 0
\(443\) −8.63627e31 −1.51176 −0.755878 0.654713i \(-0.772790\pi\)
−0.755878 + 0.654713i \(0.772790\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −5.75263e31 −0.928642
\(447\) 0 0
\(448\) 0 0
\(449\) −6.05083e31 −0.901278 −0.450639 0.892706i \(-0.648804\pi\)
−0.450639 + 0.892706i \(0.648804\pi\)
\(450\) −9.84816e31 −1.42825
\(451\) 0 0
\(452\) 0 0
\(453\) 7.20395e31 0.964699
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 5.48431e31 0.595266 0.297633 0.954680i \(-0.403803\pi\)
0.297633 + 0.954680i \(0.403803\pi\)
\(462\) 0 0
\(463\) 5.70671e31 0.588050 0.294025 0.955798i \(-0.405005\pi\)
0.294025 + 0.955798i \(0.405005\pi\)
\(464\) −6.76320e31 −0.679104
\(465\) 0 0
\(466\) −3.09947e31 −0.295567
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −2.78606e32 −2.52372
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −1.83599e32 −1.50164
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −1.10841e32 −0.779051
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −1.27759e32 −0.773121
\(485\) 0 0
\(486\) 4.95164e32 2.85177
\(487\) −3.50419e32 −1.96898 −0.984488 0.175454i \(-0.943861\pi\)
−0.984488 + 0.175454i \(0.943861\pi\)
\(488\) 0 0
\(489\) 7.44592e32 3.98302
\(490\) 0 0
\(491\) −3.22690e32 −1.64364 −0.821821 0.569746i \(-0.807042\pi\)
−0.821821 + 0.569746i \(0.807042\pi\)
\(492\) 1.73354e32 0.861691
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 1.29129e32 0.582429
\(497\) 0 0
\(498\) 0 0
\(499\) 4.76655e32 1.99985 0.999926 0.0121999i \(-0.00388345\pi\)
0.999926 + 0.0121999i \(0.00388345\pi\)
\(500\) 0 0
\(501\) 9.55464e32 3.82086
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.06797e32 −0.370223
\(508\) 4.41714e32 1.49547
\(509\) −5.76115e32 −1.90500 −0.952502 0.304534i \(-0.901499\pi\)
−0.952502 + 0.304534i \(0.901499\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −2.24534e32 −0.691898
\(513\) 0 0
\(514\) 3.20977e32 0.943879
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −1.10349e33 −2.88909
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 1.07054e33 2.61552
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 6.51526e32 1.52039
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 4.80251e32 1.00000
\(530\) 0 0
\(531\) −2.67899e33 −5.33135
\(532\) 0 0
\(533\) 3.18982e32 0.606793
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −4.02724e32 −0.700352
\(538\) −3.43616e32 −0.584368
\(539\) 0 0
\(540\) 0 0
\(541\) 1.20215e33 1.91246 0.956228 0.292623i \(-0.0945281\pi\)
0.956228 + 0.292623i \(0.0945281\pi\)
\(542\) −5.66923e32 −0.882130
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.66521e31 −0.0650164 −0.0325082 0.999471i \(-0.510350\pi\)
−0.0325082 + 0.999471i \(0.510350\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) −1.35162e33 −1.68883
\(553\) 0 0
\(554\) 2.76111e31 0.0330344
\(555\) 0 0
\(556\) −7.19703e32 −0.824623
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) −2.04397e33 −2.24319
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 2.65012e33 2.55809
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 1.66079e32 0.147277
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.30621e33 1.00000
\(576\) 4.62243e32 0.346578
\(577\) 1.59163e33 1.16878 0.584388 0.811474i \(-0.301334\pi\)
0.584388 + 0.811474i \(0.301334\pi\)
\(578\) −6.62264e32 −0.476318
\(579\) −2.44642e33 −1.72341
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 2.28247e32 0.145028
\(585\) 0 0
\(586\) 0 0
\(587\) 2.75366e33 1.64533 0.822664 0.568528i \(-0.192487\pi\)
0.822664 + 0.568528i \(0.192487\pi\)
\(588\) 2.64074e33 1.54596
\(589\) 0 0
\(590\) 0 0
\(591\) −5.02702e33 −2.76861
\(592\) 0 0
\(593\) −3.69135e33 −1.95222 −0.976110 0.217279i \(-0.930282\pi\)
−0.976110 + 0.217279i \(0.930282\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −1.08442e33 −0.518541
\(599\) 9.00167e31 0.0421892 0.0210946 0.999777i \(-0.493285\pi\)
0.0210946 + 0.999777i \(0.493285\pi\)
\(600\) −3.67621e33 −1.68883
\(601\) 4.42405e33 1.99217 0.996086 0.0883909i \(-0.0281725\pi\)
0.996086 + 0.0883909i \(0.0281725\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 8.79291e32 0.372983
\(605\) 0 0
\(606\) 2.55597e33 1.04204
\(607\) −3.69072e33 −1.47519 −0.737594 0.675244i \(-0.764038\pi\)
−0.737594 + 0.675244i \(0.764038\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.87639e33 1.80138
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 1.16568e33 0.406031
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) −1.31449e34 −3.99631
\(622\) 6.56816e32 0.195866
\(623\) 0 0
\(624\) −2.81341e33 −0.807269
\(625\) 3.55271e33 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 1.63028e34 3.93929
\(634\) −1.86647e33 −0.442538
\(635\) 0 0
\(636\) 0 0
\(637\) 4.85914e33 1.08864
\(638\) 0 0
\(639\) 2.42336e33 0.522886
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −5.84421e33 −1.08611 −0.543055 0.839697i \(-0.682733\pi\)
−0.543055 + 0.839697i \(0.682733\pi\)
\(648\) 2.31135e34 4.21662
\(649\) 0 0
\(650\) −2.94947e33 −0.518541
\(651\) 0 0
\(652\) 9.08826e33 1.53996
\(653\) 1.04525e34 1.73884 0.869422 0.494071i \(-0.164492\pi\)
0.869422 + 0.494071i \(0.164492\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.31275e33 0.206698
\(657\) 3.33048e33 0.514900
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) −6.74751e33 −0.952463
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.41992e34 −1.83128
\(668\) 1.16621e34 1.47727
\(669\) −3.13339e34 −3.89853
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.03972e34 −1.20430 −0.602152 0.798382i \(-0.705690\pi\)
−0.602152 + 0.798382i \(0.705690\pi\)
\(674\) 0 0
\(675\) −3.57524e34 −3.99631
\(676\) −1.30353e33 −0.143140
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 9.39455e33 0.911649 0.455824 0.890070i \(-0.349344\pi\)
0.455824 + 0.890070i \(0.349344\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 5.56894e33 0.469933 0.234967 0.972003i \(-0.424502\pi\)
0.234967 + 0.972003i \(0.424502\pi\)
\(692\) −1.34689e34 −1.11701
\(693\) 0 0
\(694\) −1.18848e34 −0.952089
\(695\) 0 0
\(696\) 3.99623e34 3.09271
\(697\) 0 0
\(698\) −1.14838e34 −0.858662
\(699\) −1.68824e34 −1.24082
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 2.96816e34 2.07225
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 7.24762e33 0.472649
\(707\) 0 0
\(708\) −4.36040e34 −2.74870
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.71103e34 1.57058
\(714\) 0 0
\(715\) 0 0
\(716\) −4.91553e33 −0.270778
\(717\) −6.03738e34 −3.27053
\(718\) 0 0
\(719\) 5.33879e33 0.279702 0.139851 0.990173i \(-0.455338\pi\)
0.139851 + 0.990173i \(0.455338\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −9.55749e33 −0.476318
\(723\) 0 0
\(724\) 0 0
\(725\) −3.86198e34 −1.83128
\(726\) 2.04215e34 0.952461
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 1.57234e35 6.97936
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −2.58016e34 −1.02121
\(737\) 0 0
\(738\) −2.07795e34 −0.796085
\(739\) 4.29660e34 1.61954 0.809769 0.586749i \(-0.199592\pi\)
0.809769 + 0.586749i \(0.199592\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) −7.62994e34 −2.65244
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 2.00685e34 0.613622
\(753\) 0 0
\(754\) 3.20622e34 0.949592
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −7.43391e34 −1.97061 −0.985306 0.170798i \(-0.945365\pi\)
−0.985306 + 0.170798i \(0.945365\pi\)
\(762\) −7.06052e34 −1.84237
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.02343e34 −1.93560
\(768\) 4.84775e34 1.15135
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 1.74832e35 3.96250
\(772\) −2.98602e34 −0.666324
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 7.37362e34 1.57058
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 3.88646e35 7.31835
\(784\) 1.99975e34 0.370837
\(785\) 0 0
\(786\) −1.04142e35 −1.87308
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −6.13583e34 −1.07043
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 4.62098e34 0.736017
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −7.01768e34 −1.02121
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −6.12159e34 −0.814411
\(807\) −1.87164e35 −2.45323
\(808\) 7.15499e34 0.924002
\(809\) 7.34901e34 0.935075 0.467538 0.883973i \(-0.345141\pi\)
0.467538 + 0.883973i \(0.345141\pi\)
\(810\) 0 0
\(811\) 1.35155e35 1.66948 0.834740 0.550644i \(-0.185618\pi\)
0.834740 + 0.550644i \(0.185618\pi\)
\(812\) 0 0
\(813\) −3.08796e35 −3.70327
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 6.62548e34 0.738206
\(819\) 0 0
\(820\) 0 0
\(821\) −6.22334e34 −0.663599 −0.331800 0.943350i \(-0.607656\pi\)
−0.331800 + 0.943350i \(0.607656\pi\)
\(822\) 0 0
\(823\) −1.92356e35 −1.99208 −0.996042 0.0888864i \(-0.971669\pi\)
−0.996042 + 0.0888864i \(0.971669\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −2.40724e35 −2.31822
\(829\) 1.41623e35 1.34425 0.672124 0.740438i \(-0.265382\pi\)
0.672124 + 0.740438i \(0.265382\pi\)
\(830\) 0 0
\(831\) 1.50394e34 0.138682
\(832\) 1.38439e34 0.125828
\(833\) 0 0
\(834\) 1.15040e35 1.01591
\(835\) 0 0
\(836\) 0 0
\(837\) −7.42036e35 −6.27653
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 2.94631e35 2.35357
\(842\) 0 0
\(843\) 0 0
\(844\) 1.98987e35 1.52306
\(845\) 0 0
\(846\) −3.17664e35 −2.36332
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 3.94431e34 0.269585
\(853\) −1.63462e35 −1.10161 −0.550805 0.834634i \(-0.685679\pi\)
−0.550805 + 0.834634i \(0.685679\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.34135e34 −0.212889 −0.106444 0.994319i \(-0.533947\pi\)
−0.106444 + 0.994319i \(0.533947\pi\)
\(858\) 0 0
\(859\) 3.21028e35 1.98895 0.994476 0.104960i \(-0.0334716\pi\)
0.994476 + 0.104960i \(0.0334716\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.01696e35 0.595900 0.297950 0.954582i \(-0.403697\pi\)
0.297950 + 0.954582i \(0.403697\pi\)
\(864\) 7.06216e35 4.08106
\(865\) 0 0
\(866\) 0 0
\(867\) −3.60727e35 −1.99963
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 5.42078e34 0.265468
\(877\) 3.51655e35 1.69872 0.849359 0.527816i \(-0.176989\pi\)
0.849359 + 0.527816i \(0.176989\pi\)
\(878\) 1.98552e35 0.946108
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) −3.16540e35 −1.42825
\(883\) 4.07247e35 1.81272 0.906358 0.422512i \(-0.138851\pi\)
0.906358 + 0.422512i \(0.138851\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.68494e35 0.720077
\(887\) 4.45180e35 1.87695 0.938474 0.345351i \(-0.112240\pi\)
0.938474 + 0.345351i \(0.112240\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) −3.82452e35 −1.50730
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −5.90670e35 −2.17689
\(898\) 1.18052e35 0.429295
\(899\) −8.01549e35 −2.87617
\(900\) −6.54734e35 −2.31822
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) −1.40549e35 −0.459504
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 1.04403e36 3.28054
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 6.34932e35 1.70456
\(922\) −1.06999e35 −0.283536
\(923\) 7.25781e34 0.189839
\(924\) 0 0
\(925\) 0 0
\(926\) −1.11338e35 −0.280099
\(927\) 0 0
\(928\) 7.62857e35 1.87011
\(929\) −4.94976e35 −1.19783 −0.598915 0.800812i \(-0.704401\pi\)
−0.598915 + 0.800812i \(0.704401\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −2.06062e35 −0.479740
\(933\) 3.57760e35 0.822265
\(934\) 0 0
\(935\) 0 0
\(936\) 1.24664e36 2.75695
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 2.75610e35 0.557384
\(944\) −3.30199e35 −0.659344
\(945\) 0 0
\(946\) 0 0
\(947\) −9.34907e35 −1.79709 −0.898543 0.438886i \(-0.855373\pi\)
−0.898543 + 0.438886i \(0.855373\pi\)
\(948\) 0 0
\(949\) 9.97460e34 0.186939
\(950\) 0 0
\(951\) −1.01665e36 −1.85782
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −7.36904e35 −1.26449
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 9.09974e35 1.46672
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 5.63091e35 0.842289 0.421144 0.906994i \(-0.361629\pi\)
0.421144 + 0.906994i \(0.361629\pi\)
\(968\) 5.71663e35 0.844570
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 3.29199e36 4.62875
\(973\) 0 0
\(974\) 6.83668e35 0.937859
\(975\) −1.60654e36 −2.17689
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) −1.45270e36 −1.89718
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 6.29567e35 0.782897
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) −7.75678e35 −0.941326
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 6.92487e35 0.771841 0.385921 0.922532i \(-0.373884\pi\)
0.385921 + 0.922532i \(0.373884\pi\)
\(992\) −1.45651e36 −1.60389
\(993\) −3.67529e36 −3.99853
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.39934e36 −1.45071 −0.725357 0.688373i \(-0.758325\pi\)
−0.725357 + 0.688373i \(0.758325\pi\)
\(998\) −9.29954e35 −0.952566
\(999\) 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 23.25.b.a.22.1 1
23.22 odd 2 CM 23.25.b.a.22.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.25.b.a.22.1 1 1.1 even 1 trivial
23.25.b.a.22.1 1 23.22 odd 2 CM