Properties

Label 1680.2.k.b.209.2
Level $1680$
Weight $2$
Character 1680.209
Analytic conductor $13.415$
Analytic rank $0$
Dimension $4$
CM discriminant -35
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,2,Mod(209,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.209");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1680.k (of order \(2\), degree \(1\), not 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: no
Analytic conductor: \(13.4148675396\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 209.2
Root \(1.32288 + 1.11803i\) of defining polynomial
Character \(\chi\) \(=\) 1680.209
Dual form 1680.2.k.b.209.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.32288 + 1.11803i) q^{3} -2.23607i q^{5} -2.64575 q^{7} +(0.500000 - 2.95804i) q^{9} +O(q^{10})\) \(q+(-1.32288 + 1.11803i) q^{3} -2.23607i q^{5} -2.64575 q^{7} +(0.500000 - 2.95804i) q^{9} -5.91608i q^{11} -2.64575 q^{13} +(2.50000 + 2.95804i) q^{15} +2.23607i q^{17} +(3.50000 - 2.95804i) q^{21} -5.00000 q^{25} +(2.64575 + 4.47214i) q^{27} +5.91608i q^{29} +(6.61438 + 7.82624i) q^{33} +5.91608i q^{35} +(3.50000 - 2.95804i) q^{39} +(-6.61438 - 1.11803i) q^{45} +11.1803i q^{47} +7.00000 q^{49} +(-2.50000 - 2.95804i) q^{51} -13.2288 q^{55} +(-1.32288 + 7.82624i) q^{63} +5.91608i q^{65} +11.8322i q^{71} -10.5830 q^{73} +(6.61438 - 5.59017i) q^{75} +15.6525i q^{77} +1.00000 q^{79} +(-8.50000 - 2.95804i) q^{81} -8.94427i q^{83} +5.00000 q^{85} +(-6.61438 - 7.82624i) q^{87} +7.00000 q^{91} -18.5203 q^{97} +(-17.5000 - 2.95804i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{9} + 10 q^{15} + 14 q^{21} - 20 q^{25} + 14 q^{39} + 28 q^{49} - 10 q^{51} + 4 q^{79} - 34 q^{81} + 20 q^{85} + 28 q^{91} - 70 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(421\) \(1121\) \(1471\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.32288 + 1.11803i −0.763763 + 0.645497i
\(4\) 0 0
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) −2.64575 −1.00000
\(8\) 0 0
\(9\) 0.500000 2.95804i 0.166667 0.986013i
\(10\) 0 0
\(11\) 5.91608i 1.78377i −0.452267 0.891883i \(-0.649385\pi\)
0.452267 0.891883i \(-0.350615\pi\)
\(12\) 0 0
\(13\) −2.64575 −0.733799 −0.366900 0.930261i \(-0.619581\pi\)
−0.366900 + 0.930261i \(0.619581\pi\)
\(14\) 0 0
\(15\) 2.50000 + 2.95804i 0.645497 + 0.763763i
\(16\) 0 0
\(17\) 2.23607i 0.542326i 0.962533 + 0.271163i \(0.0874083\pi\)
−0.962533 + 0.271163i \(0.912592\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 3.50000 2.95804i 0.763763 0.645497i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 2.64575 + 4.47214i 0.509175 + 0.860663i
\(28\) 0 0
\(29\) 5.91608i 1.09859i 0.835629 + 0.549294i \(0.185103\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 6.61438 + 7.82624i 1.15142 + 1.36237i
\(34\) 0 0
\(35\) 5.91608i 1.00000i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 3.50000 2.95804i 0.560449 0.473665i
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) −6.61438 1.11803i −0.986013 0.166667i
\(46\) 0 0
\(47\) 11.1803i 1.63082i 0.578884 + 0.815410i \(0.303489\pi\)
−0.578884 + 0.815410i \(0.696511\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) −2.50000 2.95804i −0.350070 0.414208i
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −13.2288 −1.78377
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −1.32288 + 7.82624i −0.166667 + 0.986013i
\(64\) 0 0
\(65\) 5.91608i 0.733799i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.8322i 1.40422i 0.712069 + 0.702109i \(0.247758\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −10.5830 −1.23865 −0.619324 0.785136i \(-0.712593\pi\)
−0.619324 + 0.785136i \(0.712593\pi\)
\(74\) 0 0
\(75\) 6.61438 5.59017i 0.763763 0.645497i
\(76\) 0 0
\(77\) 15.6525i 1.78377i
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) −8.50000 2.95804i −0.944444 0.328671i
\(82\) 0 0
\(83\) 8.94427i 0.981761i −0.871227 0.490881i \(-0.836675\pi\)
0.871227 0.490881i \(-0.163325\pi\)
\(84\) 0 0
\(85\) 5.00000 0.542326
\(86\) 0 0
\(87\) −6.61438 7.82624i −0.709136 0.839061i
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 7.00000 0.733799
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −18.5203 −1.88045 −0.940224 0.340557i \(-0.889384\pi\)
−0.940224 + 0.340557i \(0.889384\pi\)
\(98\) 0 0
\(99\) −17.5000 2.95804i −1.75882 0.297294i
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 2.64575 0.260694 0.130347 0.991468i \(-0.458391\pi\)
0.130347 + 0.991468i \(0.458391\pi\)
\(104\) 0 0
\(105\) −6.61438 7.82624i −0.645497 0.763763i
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.32288 + 7.82624i −0.122300 + 0.723536i
\(118\) 0 0
\(119\) 5.91608i 0.542326i
\(120\) 0 0
\(121\) −24.0000 −2.18182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1803i 1.00000i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 10.0000 5.91608i 0.860663 0.509175i
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) −12.5000 14.7902i −1.05269 1.24556i
\(142\) 0 0
\(143\) 15.6525i 1.30893i
\(144\) 0 0
\(145\) 13.2288 1.09859
\(146\) 0 0
\(147\) −9.26013 + 7.82624i −0.763763 + 0.645497i
\(148\) 0 0
\(149\) 23.6643i 1.93866i 0.245770 + 0.969328i \(0.420959\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) −17.0000 −1.38344 −0.691720 0.722166i \(-0.743147\pi\)
−0.691720 + 0.722166i \(0.743147\pi\)
\(152\) 0 0
\(153\) 6.61438 + 1.11803i 0.534741 + 0.0903877i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 21.1660 1.68923 0.844616 0.535373i \(-0.179829\pi\)
0.844616 + 0.535373i \(0.179829\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 17.5000 14.7902i 1.36237 1.15142i
\(166\) 0 0
\(167\) 24.5967i 1.90335i 0.307102 + 0.951677i \(0.400641\pi\)
−0.307102 + 0.951677i \(0.599359\pi\)
\(168\) 0 0
\(169\) −6.00000 −0.461538
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 11.1803i 0.850026i −0.905187 0.425013i \(-0.860270\pi\)
0.905187 0.425013i \(-0.139730\pi\)
\(174\) 0 0
\(175\) 13.2288 1.00000
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.8322i 0.884377i 0.896922 + 0.442189i \(0.145798\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 13.2288 0.967382
\(188\) 0 0
\(189\) −7.00000 11.8322i −0.509175 0.860663i
\(190\) 0 0
\(191\) 5.91608i 0.428073i −0.976826 0.214036i \(-0.931339\pi\)
0.976826 0.214036i \(-0.0686611\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) −6.61438 7.82624i −0.473665 0.560449i
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 15.6525i 1.09859i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −23.0000 −1.58339 −0.791693 0.610920i \(-0.790800\pi\)
−0.791693 + 0.610920i \(0.790800\pi\)
\(212\) 0 0
\(213\) −13.2288 15.6525i −0.906419 1.07249i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 14.0000 11.8322i 0.946032 0.799543i
\(220\) 0 0
\(221\) 5.91608i 0.397959i
\(222\) 0 0
\(223\) −29.1033 −1.94890 −0.974449 0.224607i \(-0.927890\pi\)
−0.974449 + 0.224607i \(0.927890\pi\)
\(224\) 0 0
\(225\) −2.50000 + 14.7902i −0.166667 + 0.986013i
\(226\) 0 0
\(227\) 29.0689i 1.92937i −0.263407 0.964685i \(-0.584846\pi\)
0.263407 0.964685i \(-0.415154\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) −17.5000 20.7063i −1.15142 1.36237i
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 25.0000 1.63082
\(236\) 0 0
\(237\) −1.32288 + 1.11803i −0.0859300 + 0.0726241i
\(238\) 0 0
\(239\) 29.5804i 1.91340i 0.291081 + 0.956698i \(0.405985\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 14.5516 5.59017i 0.933488 0.358610i
\(244\) 0 0
\(245\) 15.6525i 1.00000i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 10.0000 + 11.8322i 0.633724 + 0.749833i
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −6.61438 + 5.59017i −0.414208 + 0.350070i
\(256\) 0 0
\(257\) 4.47214i 0.278964i −0.990225 0.139482i \(-0.955456\pi\)
0.990225 0.139482i \(-0.0445438\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 17.5000 + 2.95804i 1.08322 + 0.183098i
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) −9.26013 + 7.82624i −0.560449 + 0.473665i
\(274\) 0 0
\(275\) 29.5804i 1.78377i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.91608i 0.352924i 0.984307 + 0.176462i \(0.0564652\pi\)
−0.984307 + 0.176462i \(0.943535\pi\)
\(282\) 0 0
\(283\) 2.64575 0.157274 0.0786368 0.996903i \(-0.474943\pi\)
0.0786368 + 0.996903i \(0.474943\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 12.0000 0.705882
\(290\) 0 0
\(291\) 24.5000 20.7063i 1.43622 1.21382i
\(292\) 0 0
\(293\) 24.5967i 1.43696i −0.695549 0.718479i \(-0.744839\pi\)
0.695549 0.718479i \(-0.255161\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 26.4575 15.6525i 1.53522 0.908249i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 34.3948 1.96301 0.981507 0.191429i \(-0.0613121\pi\)
0.981507 + 0.191429i \(0.0613121\pi\)
\(308\) 0 0
\(309\) −3.50000 + 2.95804i −0.199108 + 0.168277i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 29.1033 1.64501 0.822507 0.568755i \(-0.192575\pi\)
0.822507 + 0.568755i \(0.192575\pi\)
\(314\) 0 0
\(315\) 17.5000 + 2.95804i 0.986013 + 0.166667i
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 35.0000 1.95962
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 13.2288 0.733799
\(326\) 0 0
\(327\) −14.5516 + 12.2984i −0.804707 + 0.680102i
\(328\) 0 0
\(329\) 29.5804i 1.63082i
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −18.5203 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) −7.00000 11.8322i −0.373632 0.631554i
\(352\) 0 0
\(353\) 29.0689i 1.54718i 0.633686 + 0.773590i \(0.281541\pi\)
−0.633686 + 0.773590i \(0.718459\pi\)
\(354\) 0 0
\(355\) 26.4575 1.40422
\(356\) 0 0
\(357\) 6.61438 + 7.82624i 0.350070 + 0.414208i
\(358\) 0 0
\(359\) 11.8322i 0.624477i 0.950004 + 0.312239i \(0.101079\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 31.7490 26.8328i 1.66639 1.40836i
\(364\) 0 0
\(365\) 23.6643i 1.23865i
\(366\) 0 0
\(367\) 18.5203 0.966750 0.483375 0.875413i \(-0.339411\pi\)
0.483375 + 0.875413i \(0.339411\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −12.5000 14.7902i −0.645497 0.763763i
\(376\) 0 0
\(377\) 15.6525i 0.806144i
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 35.7771i 1.82812i −0.405575 0.914062i \(-0.632929\pi\)
0.405575 0.914062i \(-0.367071\pi\)
\(384\) 0 0
\(385\) 35.0000 1.78377
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.91608i 0.299957i 0.988689 + 0.149979i \(0.0479205\pi\)
−0.988689 + 0.149979i \(0.952080\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2.23607i 0.112509i
\(396\) 0 0
\(397\) −34.3948 −1.72622 −0.863112 0.505013i \(-0.831488\pi\)
−0.863112 + 0.505013i \(0.831488\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 29.5804i 1.47717i −0.674158 0.738587i \(-0.735493\pi\)
0.674158 0.738587i \(-0.264507\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −6.61438 + 19.0066i −0.328671 + 0.944444i
\(406\) 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
\(413\) 0 0
\(414\) 0 0
\(415\) −20.0000 −0.981761
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −37.0000 −1.80327 −0.901635 0.432498i \(-0.857632\pi\)
−0.901635 + 0.432498i \(0.857632\pi\)
\(422\) 0 0
\(423\) 33.0719 + 5.59017i 1.60801 + 0.271803i
\(424\) 0 0
\(425\) 11.1803i 0.542326i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −17.5000 20.7063i −0.844908 0.999709i
\(430\) 0 0
\(431\) 41.4126i 1.99477i −0.0722525 0.997386i \(-0.523019\pi\)
0.0722525 0.997386i \(-0.476981\pi\)
\(432\) 0 0
\(433\) −10.5830 −0.508587 −0.254293 0.967127i \(-0.581843\pi\)
−0.254293 + 0.967127i \(0.581843\pi\)
\(434\) 0 0
\(435\) −17.5000 + 14.7902i −0.839061 + 0.709136i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 3.50000 20.7063i 0.166667 0.986013i
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −26.4575 31.3050i −1.25140 1.48067i
\(448\) 0 0
\(449\) 41.4126i 1.95438i 0.212368 + 0.977190i \(0.431882\pi\)
−0.212368 + 0.977190i \(0.568118\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 22.4889 19.0066i 1.05662 0.893007i
\(454\) 0 0
\(455\) 15.6525i 0.733799i
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) −10.0000 + 5.91608i −0.466760 + 0.276139i
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 42.4853i 1.96598i −0.183646 0.982992i \(-0.558790\pi\)
0.183646 0.982992i \(-0.441210\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −28.0000 + 23.6643i −1.29017 + 1.09039i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 41.4126i 1.88045i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 29.5804i 1.33494i 0.744635 + 0.667472i \(0.232624\pi\)
−0.744635 + 0.667472i \(0.767376\pi\)
\(492\) 0 0
\(493\) −13.2288 −0.595793
\(494\) 0 0
\(495\) −6.61438 + 39.1312i −0.297294 + 1.75882i
\(496\) 0 0
\(497\) 31.3050i 1.40422i
\(498\) 0 0
\(499\) −41.0000 −1.83541 −0.917706 0.397260i \(-0.869961\pi\)
−0.917706 + 0.397260i \(0.869961\pi\)
\(500\) 0 0
\(501\) −27.5000 32.5384i −1.22861 1.45371i
\(502\) 0 0
\(503\) 38.0132i 1.69492i 0.530857 + 0.847461i \(0.321870\pi\)
−0.530857 + 0.847461i \(0.678130\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7.93725 6.70820i 0.352506 0.297922i
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 28.0000 1.23865
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.91608i 0.260694i
\(516\) 0 0
\(517\) 66.1438 2.90900
\(518\) 0 0
\(519\) 12.5000 + 14.7902i 0.548689 + 0.649218i
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −37.0405 −1.61967 −0.809834 0.586659i \(-0.800443\pi\)
−0.809834 + 0.586659i \(0.800443\pi\)
\(524\) 0 0
\(525\) −17.5000 + 14.7902i −0.763763 + 0.645497i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −13.2288 15.6525i −0.570863 0.675454i
\(538\) 0 0
\(539\) 41.4126i 1.78377i
\(540\) 0 0
\(541\) −43.0000 −1.84871 −0.924357 0.381528i \(-0.875398\pi\)
−0.924357 + 0.381528i \(0.875398\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 24.5967i 1.05361i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −2.64575 −0.112509
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −17.5000 + 14.7902i −0.738851 + 0.624443i
\(562\) 0 0
\(563\) 44.7214i 1.88478i 0.334515 + 0.942390i \(0.391427\pi\)
−0.334515 + 0.942390i \(0.608573\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 22.4889 + 7.82624i 0.944444 + 0.328671i
\(568\) 0 0
\(569\) 47.3286i 1.98412i −0.125767 0.992060i \(-0.540139\pi\)
0.125767 0.992060i \(-0.459861\pi\)
\(570\) 0 0
\(571\) −32.0000 −1.33916 −0.669579 0.742741i \(-0.733526\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) 6.61438 + 7.82624i 0.276320 + 0.326946i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −34.3948 −1.43187 −0.715936 0.698165i \(-0.754000\pi\)
−0.715936 + 0.698165i \(0.754000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 23.6643i 0.981761i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 17.5000 + 2.95804i 0.723536 + 0.122300i
\(586\) 0 0
\(587\) 8.94427i 0.369170i −0.982817 0.184585i \(-0.940906\pi\)
0.982817 0.184585i \(-0.0590940\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 42.4853i 1.74466i 0.488916 + 0.872331i \(0.337392\pi\)
−0.488916 + 0.872331i \(0.662608\pi\)
\(594\) 0 0
\(595\) −13.2288 −0.542326
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 29.5804i 1.20862i 0.796748 + 0.604311i \(0.206552\pi\)
−0.796748 + 0.604311i \(0.793448\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 53.6656i 2.18182i
\(606\) 0 0
\(607\) −44.9778 −1.82559 −0.912796 0.408416i \(-0.866081\pi\)
−0.912796 + 0.408416i \(0.866081\pi\)
\(608\) 0 0
\(609\) 17.5000 + 20.7063i 0.709136 + 0.839061i
\(610\) 0 0
\(611\) 29.5804i 1.19669i
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −47.0000 −1.87104 −0.935520 0.353273i \(-0.885069\pi\)
−0.935520 + 0.353273i \(0.885069\pi\)
\(632\) 0 0
\(633\) 30.4261 25.7148i 1.20933 1.02207i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −18.5203 −0.733799
\(638\) 0 0
\(639\) 35.0000 + 5.91608i 1.38458 + 0.234036i
\(640\) 0 0
\(641\) 47.3286i 1.86937i −0.355479 0.934684i \(-0.615682\pi\)
0.355479 0.934684i \(-0.384318\pi\)
\(642\) 0 0
\(643\) 50.2693 1.98243 0.991213 0.132273i \(-0.0422275\pi\)
0.991213 + 0.132273i \(0.0422275\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 17.8885i 0.703271i 0.936137 + 0.351636i \(0.114374\pi\)
−0.936137 + 0.351636i \(0.885626\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −5.29150 + 31.3050i −0.206441 + 1.22132i
\(658\) 0 0
\(659\) 5.91608i 0.230458i −0.993339 0.115229i \(-0.963240\pi\)
0.993339 0.115229i \(-0.0367601\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 6.61438 + 7.82624i 0.256881 + 0.303946i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 38.5000 32.5384i 1.48850 1.25801i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −13.2288 22.3607i −0.509175 0.860663i
\(676\) 0 0
\(677\) 51.4296i 1.97660i −0.152527 0.988299i \(-0.548741\pi\)
0.152527 0.988299i \(-0.451259\pi\)
\(678\) 0 0
\(679\) 49.0000 1.88045
\(680\) 0 0
\(681\) 32.5000 + 38.4545i 1.24540 + 1.47358i
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 46.3006 + 7.82624i 1.75882 + 0.297294i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 41.4126i 1.56413i 0.623196 + 0.782065i \(0.285834\pi\)
−0.623196 + 0.782065i \(0.714166\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) −33.0719 + 27.9508i −1.24556 + 1.05269i
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.00000 −0.0375558 −0.0187779 0.999824i \(-0.505978\pi\)
−0.0187779 + 0.999824i \(0.505978\pi\)
\(710\) 0 0
\(711\) 0.500000 2.95804i 0.0187515 0.110935i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 35.0000 1.30893
\(716\) 0 0
\(717\) −33.0719 39.1312i −1.23509 1.46138i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −7.00000 −0.260694
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 29.5804i 1.09859i
\(726\) 0 0
\(727\) −5.29150 −0.196251 −0.0981255 0.995174i \(-0.531285\pi\)
−0.0981255 + 0.995174i \(0.531285\pi\)
\(728\) 0 0
\(729\) −13.0000 + 23.6643i −0.481481 + 0.876456i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −50.2693 −1.85674 −0.928369 0.371660i \(-0.878789\pi\)
−0.928369 + 0.371660i \(0.878789\pi\)
\(734\) 0 0
\(735\) 17.5000 + 20.7063i 0.645497 + 0.763763i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −11.0000 −0.404642 −0.202321 0.979319i \(-0.564848\pi\)
−0.202321 + 0.979319i \(0.564848\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 52.9150 1.93866
\(746\) 0 0
\(747\) −26.4575 4.47214i −0.968030 0.163627i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 13.0000 0.474377 0.237188 0.971464i \(-0.423774\pi\)
0.237188 + 0.971464i \(0.423774\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 38.0132i 1.38344i
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −29.1033 −1.05361
\(764\) 0 0
\(765\) 2.50000 14.7902i 0.0903877 0.534741i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 5.00000 + 5.91608i 0.180071 + 0.213062i
\(772\) 0 0
\(773\) 2.23607i 0.0804258i 0.999191 + 0.0402129i \(0.0128036\pi\)
−0.999191 + 0.0402129i \(0.987196\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 70.0000 2.50480
\(782\) 0 0
\(783\) −26.4575 + 15.6525i −0.945514 + 0.559374i
\(784\) 0 0
\(785\) 47.3286i 1.68923i
\(786\) 0 0
\(787\) −44.9778 −1.60328 −0.801642 0.597804i \(-0.796040\pi\)
−0.801642 + 0.597804i \(0.796040\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 55.9017i 1.98014i 0.140576 + 0.990070i \(0.455105\pi\)
−0.140576 + 0.990070i \(0.544895\pi\)
\(798\) 0 0
\(799\) −25.0000 −0.884436
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 62.6099i 2.20946i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 41.4126i 1.45599i 0.685583 + 0.727994i \(0.259547\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) 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
\(819\) 3.50000 20.7063i 0.122300 0.723536i
\(820\) 0 0
\(821\) 5.91608i 0.206473i 0.994657 + 0.103236i \(0.0329198\pi\)
−0.994657 + 0.103236i \(0.967080\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) −33.0719 39.1312i −1.15142 1.36237i
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 15.6525i 0.542326i
\(834\) 0 0
\(835\) 55.0000 1.90335
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −6.00000 −0.206897
\(842\) 0 0
\(843\) −6.61438 7.82624i −0.227811 0.269550i
\(844\) 0 0
\(845\) 13.4164i 0.461538i
\(846\) 0 0
\(847\) 63.4980 2.18182
\(848\) 0 0
\(849\) −3.50000 + 2.95804i −0.120120 + 0.101520i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −42.3320 −1.44942 −0.724710 0.689054i \(-0.758026\pi\)
−0.724710 + 0.689054i \(0.758026\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 49.1935i 1.68042i 0.542263 + 0.840209i \(0.317568\pi\)
−0.542263 + 0.840209i \(0.682432\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −25.0000 −0.850026
\(866\) 0 0
\(867\) −15.8745 + 13.4164i −0.539127 + 0.455645i
\(868\) 0 0
\(869\) 5.91608i 0.200689i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −9.26013 + 54.7837i −0.313408 + 1.85415i
\(874\) 0 0
\(875\) 29.5804i 1.00000i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 27.5000 + 32.5384i 0.927552 + 1.09749i
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 35.7771i 1.20128i −0.799521 0.600639i \(-0.794913\pi\)
0.799521 0.600639i \(-0.205087\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −17.5000 + 50.2867i −0.586272 + 1.68467i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 26.4575 0.884377
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 59.1608i 1.96008i −0.198789 0.980042i \(-0.563701\pi\)
0.198789 0.980042i \(-0.436299\pi\)
\(912\) 0 0
\(913\) −52.9150 −1.75123
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −29.0000 −0.956622 −0.478311 0.878191i \(-0.658751\pi\)
−0.478311 + 0.878191i \(0.658751\pi\)
\(920\) 0 0
\(921\) −45.5000 + 38.4545i −1.49928 + 1.26712i
\(922\) 0 0
\(923\) 31.3050i 1.03042i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.32288 7.82624i 0.0434489 0.257047i
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 29.5804i 0.967382i
\(936\) 0 0
\(937\) 60.8523 1.98796 0.993979 0.109574i \(-0.0349486\pi\)
0.993979 + 0.109574i \(0.0349486\pi\)
\(938\) 0 0
\(939\) −38.5000 + 32.5384i −1.25640 + 1.06185i
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −26.4575 + 15.6525i −0.860663 + 0.509175i
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 28.0000 0.908918
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) −13.2288 −0.428073
\(956\) 0 0
\(957\) −46.3006 + 39.1312i −1.49669 + 1.26493i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −17.5000 + 14.7902i −0.560449 + 0.473665i
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 5.50000 32.5384i 0.175601 1.03887i
\(982\) 0 0
\(983\) 29.0689i 0.927153i −0.886057 0.463577i \(-0.846566\pi\)
0.886057 0.463577i \(-0.153434\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 33.0719 + 39.1312i 1.05269 + 1.24556i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 52.0000 1.65183 0.825917 0.563791i \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 0 0
\(993\) 10.5830 8.94427i 0.335842 0.283838i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −18.5203 −0.586542 −0.293271 0.956029i \(-0.594744\pi\)
−0.293271 + 0.956029i \(0.594744\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1680.2.k.b.209.2 4
3.2 odd 2 inner 1680.2.k.b.209.1 4
4.3 odd 2 105.2.g.b.104.3 yes 4
5.4 even 2 inner 1680.2.k.b.209.3 4
7.6 odd 2 inner 1680.2.k.b.209.3 4
12.11 even 2 105.2.g.b.104.4 yes 4
15.14 odd 2 inner 1680.2.k.b.209.4 4
20.3 even 4 525.2.b.f.251.4 4
20.7 even 4 525.2.b.f.251.1 4
20.19 odd 2 105.2.g.b.104.2 yes 4
21.20 even 2 inner 1680.2.k.b.209.4 4
28.3 even 6 735.2.p.b.509.4 8
28.11 odd 6 735.2.p.b.509.1 8
28.19 even 6 735.2.p.b.374.2 8
28.23 odd 6 735.2.p.b.374.3 8
28.27 even 2 105.2.g.b.104.2 yes 4
35.34 odd 2 CM 1680.2.k.b.209.2 4
60.23 odd 4 525.2.b.f.251.2 4
60.47 odd 4 525.2.b.f.251.3 4
60.59 even 2 105.2.g.b.104.1 4
84.11 even 6 735.2.p.b.509.3 8
84.23 even 6 735.2.p.b.374.1 8
84.47 odd 6 735.2.p.b.374.4 8
84.59 odd 6 735.2.p.b.509.2 8
84.83 odd 2 105.2.g.b.104.1 4
105.104 even 2 inner 1680.2.k.b.209.1 4
140.19 even 6 735.2.p.b.374.3 8
140.27 odd 4 525.2.b.f.251.4 4
140.39 odd 6 735.2.p.b.509.4 8
140.59 even 6 735.2.p.b.509.1 8
140.79 odd 6 735.2.p.b.374.2 8
140.83 odd 4 525.2.b.f.251.1 4
140.139 even 2 105.2.g.b.104.3 yes 4
420.59 odd 6 735.2.p.b.509.3 8
420.83 even 4 525.2.b.f.251.3 4
420.167 even 4 525.2.b.f.251.2 4
420.179 even 6 735.2.p.b.509.2 8
420.299 odd 6 735.2.p.b.374.1 8
420.359 even 6 735.2.p.b.374.4 8
420.419 odd 2 105.2.g.b.104.4 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.2.g.b.104.1 4 60.59 even 2
105.2.g.b.104.1 4 84.83 odd 2
105.2.g.b.104.2 yes 4 20.19 odd 2
105.2.g.b.104.2 yes 4 28.27 even 2
105.2.g.b.104.3 yes 4 4.3 odd 2
105.2.g.b.104.3 yes 4 140.139 even 2
105.2.g.b.104.4 yes 4 12.11 even 2
105.2.g.b.104.4 yes 4 420.419 odd 2
525.2.b.f.251.1 4 20.7 even 4
525.2.b.f.251.1 4 140.83 odd 4
525.2.b.f.251.2 4 60.23 odd 4
525.2.b.f.251.2 4 420.167 even 4
525.2.b.f.251.3 4 60.47 odd 4
525.2.b.f.251.3 4 420.83 even 4
525.2.b.f.251.4 4 20.3 even 4
525.2.b.f.251.4 4 140.27 odd 4
735.2.p.b.374.1 8 84.23 even 6
735.2.p.b.374.1 8 420.299 odd 6
735.2.p.b.374.2 8 28.19 even 6
735.2.p.b.374.2 8 140.79 odd 6
735.2.p.b.374.3 8 28.23 odd 6
735.2.p.b.374.3 8 140.19 even 6
735.2.p.b.374.4 8 84.47 odd 6
735.2.p.b.374.4 8 420.359 even 6
735.2.p.b.509.1 8 28.11 odd 6
735.2.p.b.509.1 8 140.59 even 6
735.2.p.b.509.2 8 84.59 odd 6
735.2.p.b.509.2 8 420.179 even 6
735.2.p.b.509.3 8 84.11 even 6
735.2.p.b.509.3 8 420.59 odd 6
735.2.p.b.509.4 8 28.3 even 6
735.2.p.b.509.4 8 140.39 odd 6
1680.2.k.b.209.1 4 3.2 odd 2 inner
1680.2.k.b.209.1 4 105.104 even 2 inner
1680.2.k.b.209.2 4 1.1 even 1 trivial
1680.2.k.b.209.2 4 35.34 odd 2 CM
1680.2.k.b.209.3 4 5.4 even 2 inner
1680.2.k.b.209.3 4 7.6 odd 2 inner
1680.2.k.b.209.4 4 15.14 odd 2 inner
1680.2.k.b.209.4 4 21.20 even 2 inner