Properties

Label 2520.2.a.y.1.2
Level $2520$
Weight $2$
Character 2520.1
Self dual yes
Analytic conductor $20.122$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2520,2,Mod(1,2520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2520, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2520.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2520 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2520.a (trivial)

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: \(20.1223013094\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2520.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.00000 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +1.00000 q^{7} +5.12311 q^{11} +2.00000 q^{13} +5.12311 q^{17} +7.12311 q^{19} -5.12311 q^{23} +1.00000 q^{25} -3.12311 q^{29} -10.2462 q^{31} +1.00000 q^{35} -1.12311 q^{37} -8.24621 q^{41} +7.12311 q^{43} -7.12311 q^{47} +1.00000 q^{49} +7.12311 q^{53} +5.12311 q^{55} -4.00000 q^{59} +2.87689 q^{61} +2.00000 q^{65} +7.12311 q^{67} -6.00000 q^{71} +12.2462 q^{73} +5.12311 q^{77} +8.00000 q^{79} +5.12311 q^{85} +12.2462 q^{89} +2.00000 q^{91} +7.12311 q^{95} +6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 2 q^{7} + 2 q^{11} + 4 q^{13} + 2 q^{17} + 6 q^{19} - 2 q^{23} + 2 q^{25} + 2 q^{29} - 4 q^{31} + 2 q^{35} + 6 q^{37} + 6 q^{43} - 6 q^{47} + 2 q^{49} + 6 q^{53} + 2 q^{55} - 8 q^{59} + 14 q^{61} + 4 q^{65} + 6 q^{67} - 12 q^{71} + 8 q^{73} + 2 q^{77} + 16 q^{79} + 2 q^{85} + 8 q^{89} + 4 q^{91} + 6 q^{95} + 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.12311 1.54467 0.772337 0.635213i \(-0.219088\pi\)
0.772337 + 0.635213i \(0.219088\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.12311 1.24254 0.621268 0.783598i \(-0.286618\pi\)
0.621268 + 0.783598i \(0.286618\pi\)
\(18\) 0 0
\(19\) 7.12311 1.63415 0.817076 0.576530i \(-0.195593\pi\)
0.817076 + 0.576530i \(0.195593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −5.12311 −1.06824 −0.534121 0.845408i \(-0.679357\pi\)
−0.534121 + 0.845408i \(0.679357\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.12311 −0.579946 −0.289973 0.957035i \(-0.593646\pi\)
−0.289973 + 0.957035i \(0.593646\pi\)
\(30\) 0 0
\(31\) −10.2462 −1.84027 −0.920137 0.391597i \(-0.871923\pi\)
−0.920137 + 0.391597i \(0.871923\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) −1.12311 −0.184637 −0.0923187 0.995730i \(-0.529428\pi\)
−0.0923187 + 0.995730i \(0.529428\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −8.24621 −1.28784 −0.643921 0.765092i \(-0.722693\pi\)
−0.643921 + 0.765092i \(0.722693\pi\)
\(42\) 0 0
\(43\) 7.12311 1.08626 0.543132 0.839648i \(-0.317238\pi\)
0.543132 + 0.839648i \(0.317238\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.12311 −1.03901 −0.519506 0.854467i \(-0.673884\pi\)
−0.519506 + 0.854467i \(0.673884\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 7.12311 0.978434 0.489217 0.872162i \(-0.337283\pi\)
0.489217 + 0.872162i \(0.337283\pi\)
\(54\) 0 0
\(55\) 5.12311 0.690799
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.00000 −0.520756 −0.260378 0.965507i \(-0.583847\pi\)
−0.260378 + 0.965507i \(0.583847\pi\)
\(60\) 0 0
\(61\) 2.87689 0.368349 0.184174 0.982894i \(-0.441039\pi\)
0.184174 + 0.982894i \(0.441039\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) 7.12311 0.870226 0.435113 0.900376i \(-0.356708\pi\)
0.435113 + 0.900376i \(0.356708\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 12.2462 1.43331 0.716655 0.697428i \(-0.245672\pi\)
0.716655 + 0.697428i \(0.245672\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.12311 0.583832
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 5.12311 0.555679
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.2462 1.29810 0.649048 0.760748i \(-0.275167\pi\)
0.649048 + 0.760748i \(0.275167\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.12311 0.730815
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −2.24621 −0.221326 −0.110663 0.993858i \(-0.535297\pi\)
−0.110663 + 0.993858i \(0.535297\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17.1231 −1.65535 −0.827677 0.561205i \(-0.810338\pi\)
−0.827677 + 0.561205i \(0.810338\pi\)
\(108\) 0 0
\(109\) −4.24621 −0.406713 −0.203357 0.979105i \(-0.565185\pi\)
−0.203357 + 0.979105i \(0.565185\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 19.1231 1.79895 0.899475 0.436972i \(-0.143949\pi\)
0.899475 + 0.436972i \(0.143949\pi\)
\(114\) 0 0
\(115\) −5.12311 −0.477732
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.12311 0.469634
\(120\) 0 0
\(121\) 15.2462 1.38602
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −20.4924 −1.81841 −0.909204 0.416350i \(-0.863309\pi\)
−0.909204 + 0.416350i \(0.863309\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −1.75379 −0.153229 −0.0766146 0.997061i \(-0.524411\pi\)
−0.0766146 + 0.997061i \(0.524411\pi\)
\(132\) 0 0
\(133\) 7.12311 0.617652
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.3693 −1.14222 −0.571109 0.820874i \(-0.693487\pi\)
−0.571109 + 0.820874i \(0.693487\pi\)
\(138\) 0 0
\(139\) −9.36932 −0.794695 −0.397348 0.917668i \(-0.630069\pi\)
−0.397348 + 0.917668i \(0.630069\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.2462 0.856831
\(144\) 0 0
\(145\) −3.12311 −0.259360
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.876894 0.0718380 0.0359190 0.999355i \(-0.488564\pi\)
0.0359190 + 0.999355i \(0.488564\pi\)
\(150\) 0 0
\(151\) 20.4924 1.66765 0.833825 0.552029i \(-0.186146\pi\)
0.833825 + 0.552029i \(0.186146\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −10.2462 −0.822995
\(156\) 0 0
\(157\) 18.4924 1.47586 0.737928 0.674879i \(-0.235804\pi\)
0.737928 + 0.674879i \(0.235804\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.12311 −0.403757
\(162\) 0 0
\(163\) 15.1231 1.18453 0.592267 0.805742i \(-0.298233\pi\)
0.592267 + 0.805742i \(0.298233\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −0.876894 −0.0678561 −0.0339281 0.999424i \(-0.510802\pi\)
−0.0339281 + 0.999424i \(0.510802\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.24621 −0.322833 −0.161417 0.986886i \(-0.551606\pi\)
−0.161417 + 0.986886i \(0.551606\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −19.3693 −1.44773 −0.723865 0.689941i \(-0.757636\pi\)
−0.723865 + 0.689941i \(0.757636\pi\)
\(180\) 0 0
\(181\) 21.1231 1.57007 0.785034 0.619453i \(-0.212645\pi\)
0.785034 + 0.619453i \(0.212645\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.12311 −0.0825724
\(186\) 0 0
\(187\) 26.2462 1.91931
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −14.0000 −1.01300 −0.506502 0.862239i \(-0.669062\pi\)
−0.506502 + 0.862239i \(0.669062\pi\)
\(192\) 0 0
\(193\) −22.4924 −1.61904 −0.809520 0.587092i \(-0.800273\pi\)
−0.809520 + 0.587092i \(0.800273\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.12311 0.507500 0.253750 0.967270i \(-0.418336\pi\)
0.253750 + 0.967270i \(0.418336\pi\)
\(198\) 0 0
\(199\) 14.2462 1.00989 0.504944 0.863152i \(-0.331513\pi\)
0.504944 + 0.863152i \(0.331513\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.12311 −0.219199
\(204\) 0 0
\(205\) −8.24621 −0.575940
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 36.4924 2.52423
\(210\) 0 0
\(211\) −5.75379 −0.396107 −0.198054 0.980191i \(-0.563462\pi\)
−0.198054 + 0.980191i \(0.563462\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 7.12311 0.485792
\(216\) 0 0
\(217\) −10.2462 −0.695558
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 10.2462 0.689235
\(222\) 0 0
\(223\) 8.00000 0.535720 0.267860 0.963458i \(-0.413684\pi\)
0.267860 + 0.963458i \(0.413684\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.00000 −0.265489 −0.132745 0.991150i \(-0.542379\pi\)
−0.132745 + 0.991150i \(0.542379\pi\)
\(228\) 0 0
\(229\) 19.3693 1.27996 0.639980 0.768391i \(-0.278943\pi\)
0.639980 + 0.768391i \(0.278943\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 16.8769 1.10564 0.552821 0.833300i \(-0.313551\pi\)
0.552821 + 0.833300i \(0.313551\pi\)
\(234\) 0 0
\(235\) −7.12311 −0.464660
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −10.0000 −0.646846 −0.323423 0.946254i \(-0.604834\pi\)
−0.323423 + 0.946254i \(0.604834\pi\)
\(240\) 0 0
\(241\) −28.7386 −1.85122 −0.925609 0.378481i \(-0.876447\pi\)
−0.925609 + 0.378481i \(0.876447\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) 14.2462 0.906465
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −20.0000 −1.26239 −0.631194 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) −26.2462 −1.65009
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 17.1231 1.06811 0.534055 0.845450i \(-0.320668\pi\)
0.534055 + 0.845450i \(0.320668\pi\)
\(258\) 0 0
\(259\) −1.12311 −0.0697864
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.36932 −0.207761 −0.103880 0.994590i \(-0.533126\pi\)
−0.103880 + 0.994590i \(0.533126\pi\)
\(264\) 0 0
\(265\) 7.12311 0.437569
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −24.2462 −1.47832 −0.739159 0.673531i \(-0.764777\pi\)
−0.739159 + 0.673531i \(0.764777\pi\)
\(270\) 0 0
\(271\) 10.2462 0.622413 0.311207 0.950342i \(-0.399267\pi\)
0.311207 + 0.950342i \(0.399267\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.12311 0.308935
\(276\) 0 0
\(277\) −13.1231 −0.788491 −0.394245 0.919005i \(-0.628994\pi\)
−0.394245 + 0.919005i \(0.628994\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 10.2462 0.611238 0.305619 0.952154i \(-0.401137\pi\)
0.305619 + 0.952154i \(0.401137\pi\)
\(282\) 0 0
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −8.24621 −0.486758
\(288\) 0 0
\(289\) 9.24621 0.543895
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) −4.00000 −0.232889
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −10.2462 −0.592554
\(300\) 0 0
\(301\) 7.12311 0.410569
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.87689 0.164730
\(306\) 0 0
\(307\) 12.0000 0.684876 0.342438 0.939540i \(-0.388747\pi\)
0.342438 + 0.939540i \(0.388747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −24.4924 −1.38884 −0.694419 0.719571i \(-0.744339\pi\)
−0.694419 + 0.719571i \(0.744339\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.36932 0.301571 0.150785 0.988567i \(-0.451820\pi\)
0.150785 + 0.988567i \(0.451820\pi\)
\(318\) 0 0
\(319\) −16.0000 −0.895828
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 36.4924 2.03049
\(324\) 0 0
\(325\) 2.00000 0.110940
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −7.12311 −0.392710
\(330\) 0 0
\(331\) −30.7386 −1.68955 −0.844774 0.535123i \(-0.820265\pi\)
−0.844774 + 0.535123i \(0.820265\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 7.12311 0.389177
\(336\) 0 0
\(337\) −34.0000 −1.85210 −0.926049 0.377403i \(-0.876817\pi\)
−0.926049 + 0.377403i \(0.876817\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −52.4924 −2.84262
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 13.6155 0.730920 0.365460 0.930827i \(-0.380912\pi\)
0.365460 + 0.930827i \(0.380912\pi\)
\(348\) 0 0
\(349\) 17.1231 0.916579 0.458289 0.888803i \(-0.348462\pi\)
0.458289 + 0.888803i \(0.348462\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.8769 0.791817 0.395909 0.918290i \(-0.370430\pi\)
0.395909 + 0.918290i \(0.370430\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.7386 1.09454 0.547272 0.836955i \(-0.315666\pi\)
0.547272 + 0.836955i \(0.315666\pi\)
\(360\) 0 0
\(361\) 31.7386 1.67045
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.2462 0.640996
\(366\) 0 0
\(367\) 12.4924 0.652099 0.326050 0.945353i \(-0.394282\pi\)
0.326050 + 0.945353i \(0.394282\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.12311 0.369813
\(372\) 0 0
\(373\) −27.3693 −1.41713 −0.708565 0.705646i \(-0.750657\pi\)
−0.708565 + 0.705646i \(0.750657\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.24621 −0.321696
\(378\) 0 0
\(379\) −5.75379 −0.295552 −0.147776 0.989021i \(-0.547212\pi\)
−0.147776 + 0.989021i \(0.547212\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 19.1231 0.977145 0.488573 0.872523i \(-0.337518\pi\)
0.488573 + 0.872523i \(0.337518\pi\)
\(384\) 0 0
\(385\) 5.12311 0.261098
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.87689 −0.247268 −0.123634 0.992328i \(-0.539455\pi\)
−0.123634 + 0.992328i \(0.539455\pi\)
\(390\) 0 0
\(391\) −26.2462 −1.32733
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.00000 0.402524
\(396\) 0 0
\(397\) −16.2462 −0.815374 −0.407687 0.913122i \(-0.633665\pi\)
−0.407687 + 0.913122i \(0.633665\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.2462 1.31067 0.655337 0.755337i \(-0.272527\pi\)
0.655337 + 0.755337i \(0.272527\pi\)
\(402\) 0 0
\(403\) −20.4924 −1.02080
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.75379 −0.285205
\(408\) 0 0
\(409\) −40.2462 −1.99005 −0.995024 0.0996402i \(-0.968231\pi\)
−0.995024 + 0.0996402i \(0.968231\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −4.00000 −0.196827
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −32.4924 −1.58736 −0.793679 0.608336i \(-0.791837\pi\)
−0.793679 + 0.608336i \(0.791837\pi\)
\(420\) 0 0
\(421\) 10.4924 0.511369 0.255685 0.966760i \(-0.417699\pi\)
0.255685 + 0.966760i \(0.417699\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.12311 0.248507
\(426\) 0 0
\(427\) 2.87689 0.139223
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −26.4924 −1.27610 −0.638048 0.769997i \(-0.720258\pi\)
−0.638048 + 0.769997i \(0.720258\pi\)
\(432\) 0 0
\(433\) 24.2462 1.16520 0.582599 0.812760i \(-0.302036\pi\)
0.582599 + 0.812760i \(0.302036\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −36.4924 −1.74567
\(438\) 0 0
\(439\) −4.49242 −0.214412 −0.107206 0.994237i \(-0.534190\pi\)
−0.107206 + 0.994237i \(0.534190\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 21.6155 1.02698 0.513492 0.858094i \(-0.328351\pi\)
0.513492 + 0.858094i \(0.328351\pi\)
\(444\) 0 0
\(445\) 12.2462 0.580526
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −26.7386 −1.26187 −0.630937 0.775834i \(-0.717329\pi\)
−0.630937 + 0.775834i \(0.717329\pi\)
\(450\) 0 0
\(451\) −42.2462 −1.98930
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 24.7386 1.15722 0.578612 0.815603i \(-0.303594\pi\)
0.578612 + 0.815603i \(0.303594\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.49242 0.302382 0.151191 0.988505i \(-0.451689\pi\)
0.151191 + 0.988505i \(0.451689\pi\)
\(462\) 0 0
\(463\) −25.7538 −1.19688 −0.598440 0.801168i \(-0.704213\pi\)
−0.598440 + 0.801168i \(0.704213\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 13.7538 0.636449 0.318225 0.948015i \(-0.396913\pi\)
0.318225 + 0.948015i \(0.396913\pi\)
\(468\) 0 0
\(469\) 7.12311 0.328914
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 36.4924 1.67792
\(474\) 0 0
\(475\) 7.12311 0.326831
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −2.24621 −0.102418
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.00000 0.272446
\(486\) 0 0
\(487\) −20.4924 −0.928600 −0.464300 0.885678i \(-0.653694\pi\)
−0.464300 + 0.885678i \(0.653694\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −35.3693 −1.59619 −0.798097 0.602528i \(-0.794160\pi\)
−0.798097 + 0.602528i \(0.794160\pi\)
\(492\) 0 0
\(493\) −16.0000 −0.720604
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) 2.24621 0.100554 0.0502771 0.998735i \(-0.483990\pi\)
0.0502771 + 0.998735i \(0.483990\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −7.61553 −0.339560 −0.169780 0.985482i \(-0.554306\pi\)
−0.169780 + 0.985482i \(0.554306\pi\)
\(504\) 0 0
\(505\) −6.00000 −0.266996
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −10.0000 −0.443242 −0.221621 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 12.2462 0.541740
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.24621 −0.0989799
\(516\) 0 0
\(517\) −36.4924 −1.60493
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 34.9848 1.53271 0.766357 0.642415i \(-0.222067\pi\)
0.766357 + 0.642415i \(0.222067\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −52.4924 −2.28661
\(528\) 0 0
\(529\) 3.24621 0.141140
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −16.4924 −0.714366
\(534\) 0 0
\(535\) −17.1231 −0.740296
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.12311 0.220668
\(540\) 0 0
\(541\) −12.7386 −0.547677 −0.273838 0.961776i \(-0.588293\pi\)
−0.273838 + 0.961776i \(0.588293\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −4.24621 −0.181888
\(546\) 0 0
\(547\) −21.3693 −0.913686 −0.456843 0.889547i \(-0.651020\pi\)
−0.456843 + 0.889547i \(0.651020\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −22.2462 −0.947720
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −31.1231 −1.31873 −0.659364 0.751824i \(-0.729174\pi\)
−0.659364 + 0.751824i \(0.729174\pi\)
\(558\) 0 0
\(559\) 14.2462 0.602551
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.50758 0.316407 0.158203 0.987407i \(-0.449430\pi\)
0.158203 + 0.987407i \(0.449430\pi\)
\(564\) 0 0
\(565\) 19.1231 0.804515
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 13.7538 0.576589 0.288294 0.957542i \(-0.406912\pi\)
0.288294 + 0.957542i \(0.406912\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.12311 −0.213648
\(576\) 0 0
\(577\) −34.4924 −1.43594 −0.717969 0.696075i \(-0.754928\pi\)
−0.717969 + 0.696075i \(0.754928\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 36.4924 1.51136
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0.492423 0.0203245 0.0101622 0.999948i \(-0.496765\pi\)
0.0101622 + 0.999948i \(0.496765\pi\)
\(588\) 0 0
\(589\) −72.9848 −3.00729
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −7.36932 −0.302622 −0.151311 0.988486i \(-0.548349\pi\)
−0.151311 + 0.988486i \(0.548349\pi\)
\(594\) 0 0
\(595\) 5.12311 0.210027
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 34.9848 1.42944 0.714721 0.699410i \(-0.246554\pi\)
0.714721 + 0.699410i \(0.246554\pi\)
\(600\) 0 0
\(601\) 26.0000 1.06056 0.530281 0.847822i \(-0.322086\pi\)
0.530281 + 0.847822i \(0.322086\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 15.2462 0.619847
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −14.2462 −0.576340
\(612\) 0 0
\(613\) 26.1080 1.05449 0.527245 0.849713i \(-0.323225\pi\)
0.527245 + 0.849713i \(0.323225\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −11.6155 −0.467624 −0.233812 0.972282i \(-0.575120\pi\)
−0.233812 + 0.972282i \(0.575120\pi\)
\(618\) 0 0
\(619\) 17.3693 0.698132 0.349066 0.937098i \(-0.386499\pi\)
0.349066 + 0.937098i \(0.386499\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 12.2462 0.490634
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5.75379 −0.229419
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −20.4924 −0.813217
\(636\) 0 0
\(637\) 2.00000 0.0792429
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −24.4924 −0.967393 −0.483696 0.875236i \(-0.660706\pi\)
−0.483696 + 0.875236i \(0.660706\pi\)
\(642\) 0 0
\(643\) 14.7386 0.581235 0.290617 0.956839i \(-0.406139\pi\)
0.290617 + 0.956839i \(0.406139\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 29.3693 1.15463 0.577313 0.816523i \(-0.304101\pi\)
0.577313 + 0.816523i \(0.304101\pi\)
\(648\) 0 0
\(649\) −20.4924 −0.804398
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 27.1231 1.06141 0.530705 0.847557i \(-0.321927\pi\)
0.530705 + 0.847557i \(0.321927\pi\)
\(654\) 0 0
\(655\) −1.75379 −0.0685262
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 41.6155 1.62111 0.810555 0.585662i \(-0.199165\pi\)
0.810555 + 0.585662i \(0.199165\pi\)
\(660\) 0 0
\(661\) −13.1231 −0.510430 −0.255215 0.966884i \(-0.582146\pi\)
−0.255215 + 0.966884i \(0.582146\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 7.12311 0.276222
\(666\) 0 0
\(667\) 16.0000 0.619522
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14.7386 0.568979
\(672\) 0 0
\(673\) 10.4924 0.404453 0.202227 0.979339i \(-0.435182\pi\)
0.202227 + 0.979339i \(0.435182\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 12.7386 0.489585 0.244793 0.969575i \(-0.421280\pi\)
0.244793 + 0.969575i \(0.421280\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 25.1231 0.961309 0.480654 0.876910i \(-0.340399\pi\)
0.480654 + 0.876910i \(0.340399\pi\)
\(684\) 0 0
\(685\) −13.3693 −0.510815
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 14.2462 0.542737
\(690\) 0 0
\(691\) 4.87689 0.185526 0.0927629 0.995688i \(-0.470430\pi\)
0.0927629 + 0.995688i \(0.470430\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9.36932 −0.355398
\(696\) 0 0
\(697\) −42.2462 −1.60019
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −22.6307 −0.854749 −0.427375 0.904075i \(-0.640561\pi\)
−0.427375 + 0.904075i \(0.640561\pi\)
\(702\) 0 0
\(703\) −8.00000 −0.301726
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 52.4924 1.96586
\(714\) 0 0
\(715\) 10.2462 0.383187
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 38.2462 1.42634 0.713171 0.700990i \(-0.247258\pi\)
0.713171 + 0.700990i \(0.247258\pi\)
\(720\) 0 0
\(721\) −2.24621 −0.0836533
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.12311 −0.115989
\(726\) 0 0
\(727\) 13.7538 0.510100 0.255050 0.966928i \(-0.417908\pi\)
0.255050 + 0.966928i \(0.417908\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 36.4924 1.34972
\(732\) 0 0
\(733\) 34.9848 1.29219 0.646097 0.763255i \(-0.276400\pi\)
0.646097 + 0.763255i \(0.276400\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 36.4924 1.34422
\(738\) 0 0
\(739\) −18.2462 −0.671198 −0.335599 0.942005i \(-0.608939\pi\)
−0.335599 + 0.942005i \(0.608939\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −15.3693 −0.563846 −0.281923 0.959437i \(-0.590972\pi\)
−0.281923 + 0.959437i \(0.590972\pi\)
\(744\) 0 0
\(745\) 0.876894 0.0321269
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −17.1231 −0.625665
\(750\) 0 0
\(751\) −24.0000 −0.875772 −0.437886 0.899030i \(-0.644273\pi\)
−0.437886 + 0.899030i \(0.644273\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 20.4924 0.745796
\(756\) 0 0
\(757\) −20.6307 −0.749835 −0.374917 0.927058i \(-0.622329\pi\)
−0.374917 + 0.927058i \(0.622329\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −34.0000 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) 0 0
\(763\) −4.24621 −0.153723
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.00000 −0.288863
\(768\) 0 0
\(769\) −28.2462 −1.01858 −0.509292 0.860594i \(-0.670093\pi\)
−0.509292 + 0.860594i \(0.670093\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.50758 −0.0542238 −0.0271119 0.999632i \(-0.508631\pi\)
−0.0271119 + 0.999632i \(0.508631\pi\)
\(774\) 0 0
\(775\) −10.2462 −0.368055
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −58.7386 −2.10453
\(780\) 0 0
\(781\) −30.7386 −1.09991
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 18.4924 0.660023
\(786\) 0 0
\(787\) −9.75379 −0.347685 −0.173843 0.984773i \(-0.555618\pi\)
−0.173843 + 0.984773i \(0.555618\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 19.1231 0.679939
\(792\) 0 0
\(793\) 5.75379 0.204323
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −53.2311 −1.88554 −0.942770 0.333443i \(-0.891790\pi\)
−0.942770 + 0.333443i \(0.891790\pi\)
\(798\) 0 0
\(799\) −36.4924 −1.29101
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 62.7386 2.21400
\(804\) 0 0
\(805\) −5.12311 −0.180566
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −16.0000 −0.562530 −0.281265 0.959630i \(-0.590754\pi\)
−0.281265 + 0.959630i \(0.590754\pi\)
\(810\) 0 0
\(811\) 31.6155 1.11017 0.555086 0.831793i \(-0.312685\pi\)
0.555086 + 0.831793i \(0.312685\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 15.1231 0.529739
\(816\) 0 0
\(817\) 50.7386 1.77512
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −46.3542 −1.61777 −0.808886 0.587966i \(-0.799929\pi\)
−0.808886 + 0.587966i \(0.799929\pi\)
\(822\) 0 0
\(823\) −22.2462 −0.775454 −0.387727 0.921774i \(-0.626740\pi\)
−0.387727 + 0.921774i \(0.626740\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.38447 0.0829162 0.0414581 0.999140i \(-0.486800\pi\)
0.0414581 + 0.999140i \(0.486800\pi\)
\(828\) 0 0
\(829\) 7.86174 0.273049 0.136525 0.990637i \(-0.456407\pi\)
0.136525 + 0.990637i \(0.456407\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.12311 0.177505
\(834\) 0 0
\(835\) −0.876894 −0.0303462
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −35.2311 −1.21631 −0.608156 0.793818i \(-0.708090\pi\)
−0.608156 + 0.793818i \(0.708090\pi\)
\(840\) 0 0
\(841\) −19.2462 −0.663662
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 15.2462 0.523866
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 5.75379 0.197237
\(852\) 0 0
\(853\) −41.2311 −1.41172 −0.705862 0.708349i \(-0.749440\pi\)
−0.705862 + 0.708349i \(0.749440\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −35.3693 −1.20819 −0.604096 0.796911i \(-0.706466\pi\)
−0.604096 + 0.796911i \(0.706466\pi\)
\(858\) 0 0
\(859\) 23.1231 0.788950 0.394475 0.918907i \(-0.370926\pi\)
0.394475 + 0.918907i \(0.370926\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −31.8617 −1.08459 −0.542293 0.840189i \(-0.682444\pi\)
−0.542293 + 0.840189i \(0.682444\pi\)
\(864\) 0 0
\(865\) −4.24621 −0.144376
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 40.9848 1.39032
\(870\) 0 0
\(871\) 14.2462 0.482714
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −13.1231 −0.443136 −0.221568 0.975145i \(-0.571117\pi\)
−0.221568 + 0.975145i \(0.571117\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −40.7386 −1.37252 −0.686260 0.727357i \(-0.740749\pi\)
−0.686260 + 0.727357i \(0.740749\pi\)
\(882\) 0 0
\(883\) −49.8617 −1.67798 −0.838991 0.544146i \(-0.816854\pi\)
−0.838991 + 0.544146i \(0.816854\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 47.1231 1.58224 0.791120 0.611662i \(-0.209499\pi\)
0.791120 + 0.611662i \(0.209499\pi\)
\(888\) 0 0
\(889\) −20.4924 −0.687294
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −50.7386 −1.69790
\(894\) 0 0
\(895\) −19.3693 −0.647445
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 32.0000 1.06726
\(900\) 0 0
\(901\) 36.4924 1.21574
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 21.1231 0.702156
\(906\) 0 0
\(907\) 18.6307 0.618622 0.309311 0.950961i \(-0.399902\pi\)
0.309311 + 0.950961i \(0.399902\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −46.0000 −1.52405 −0.762024 0.647549i \(-0.775794\pi\)
−0.762024 + 0.647549i \(0.775794\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.75379 −0.0579152
\(918\) 0 0
\(919\) 19.5076 0.643496 0.321748 0.946825i \(-0.395730\pi\)
0.321748 + 0.946825i \(0.395730\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 0 0
\(925\) −1.12311 −0.0369275
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −14.4924 −0.475481 −0.237740 0.971329i \(-0.576407\pi\)
−0.237740 + 0.971329i \(0.576407\pi\)
\(930\) 0 0
\(931\) 7.12311 0.233450
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 26.2462 0.858343
\(936\) 0 0
\(937\) 16.2462 0.530741 0.265370 0.964147i \(-0.414506\pi\)
0.265370 + 0.964147i \(0.414506\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −52.7386 −1.71923 −0.859615 0.510942i \(-0.829297\pi\)
−0.859615 + 0.510942i \(0.829297\pi\)
\(942\) 0 0
\(943\) 42.2462 1.37573
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −53.6155 −1.74227 −0.871135 0.491043i \(-0.836616\pi\)
−0.871135 + 0.491043i \(0.836616\pi\)
\(948\) 0 0
\(949\) 24.4924 0.795058
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 11.1231 0.360313 0.180156 0.983638i \(-0.442340\pi\)
0.180156 + 0.983638i \(0.442340\pi\)
\(954\) 0 0
\(955\) −14.0000 −0.453029
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −13.3693 −0.431718
\(960\) 0 0
\(961\) 73.9848 2.38661
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −22.4924 −0.724057
\(966\) 0 0
\(967\) 53.4773 1.71971 0.859856 0.510536i \(-0.170553\pi\)
0.859856 + 0.510536i \(0.170553\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.73863 0.0878869 0.0439435 0.999034i \(-0.486008\pi\)
0.0439435 + 0.999034i \(0.486008\pi\)
\(972\) 0 0
\(973\) −9.36932 −0.300367
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 9.86174 0.315505 0.157752 0.987479i \(-0.449575\pi\)
0.157752 + 0.987479i \(0.449575\pi\)
\(978\) 0 0
\(979\) 62.7386 2.00514
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 35.1231 1.12025 0.560127 0.828407i \(-0.310753\pi\)
0.560127 + 0.828407i \(0.310753\pi\)
\(984\) 0 0
\(985\) 7.12311 0.226961
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −36.4924 −1.16039
\(990\) 0 0
\(991\) −48.9848 −1.55605 −0.778027 0.628230i \(-0.783780\pi\)
−0.778027 + 0.628230i \(0.783780\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 14.2462 0.451635
\(996\) 0 0
\(997\) 12.2462 0.387841 0.193921 0.981017i \(-0.437880\pi\)
0.193921 + 0.981017i \(0.437880\pi\)
\(998\) 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 2520.2.a.y.1.2 yes 2
3.2 odd 2 2520.2.a.u.1.1 2
4.3 odd 2 5040.2.a.bv.1.1 2
12.11 even 2 5040.2.a.bs.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2520.2.a.u.1.1 2 3.2 odd 2
2520.2.a.y.1.2 yes 2 1.1 even 1 trivial
5040.2.a.bs.1.2 2 12.11 even 2
5040.2.a.bv.1.1 2 4.3 odd 2