Properties

Label 8624.2.a.ce.1.1
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
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] = [8624,2,Mod(1,8624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8624.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.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: \(68.8629867032\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 77)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.23607 q^{3} +2.00000 q^{5} -1.47214 q^{9} +O(q^{10})\) \(q-1.23607 q^{3} +2.00000 q^{5} -1.47214 q^{9} +1.00000 q^{11} -3.23607 q^{13} -2.47214 q^{15} +3.23607 q^{17} +6.47214 q^{19} -2.47214 q^{23} -1.00000 q^{25} +5.52786 q^{27} +8.47214 q^{29} -2.76393 q^{31} -1.23607 q^{33} -8.47214 q^{37} +4.00000 q^{39} +11.2361 q^{41} -8.00000 q^{43} -2.94427 q^{45} +2.76393 q^{47} -4.00000 q^{51} -0.472136 q^{53} +2.00000 q^{55} -8.00000 q^{57} -1.23607 q^{59} +7.23607 q^{61} -6.47214 q^{65} -14.4721 q^{67} +3.05573 q^{69} +10.4721 q^{71} +0.763932 q^{73} +1.23607 q^{75} +8.94427 q^{79} -2.41641 q^{81} -11.4164 q^{83} +6.47214 q^{85} -10.4721 q^{87} -2.00000 q^{89} +3.41641 q^{93} +12.9443 q^{95} -17.4164 q^{97} -1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 4 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 4 q^{5} + 6 q^{9} + 2 q^{11} - 2 q^{13} + 4 q^{15} + 2 q^{17} + 4 q^{19} + 4 q^{23} - 2 q^{25} + 20 q^{27} + 8 q^{29} - 10 q^{31} + 2 q^{33} - 8 q^{37} + 8 q^{39} + 18 q^{41} - 16 q^{43} + 12 q^{45} + 10 q^{47} - 8 q^{51} + 8 q^{53} + 4 q^{55} - 16 q^{57} + 2 q^{59} + 10 q^{61} - 4 q^{65} - 20 q^{67} + 24 q^{69} + 12 q^{71} + 6 q^{73} - 2 q^{75} + 22 q^{81} + 4 q^{83} + 4 q^{85} - 12 q^{87} - 4 q^{89} - 20 q^{93} + 8 q^{95} - 8 q^{97} + 6 q^{99}+O(q^{100}) \) 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
\(3\) −1.23607 −0.713644 −0.356822 0.934172i \(-0.616140\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 0 0
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −1.47214 −0.490712
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −3.23607 −0.897524 −0.448762 0.893651i \(-0.648135\pi\)
−0.448762 + 0.893651i \(0.648135\pi\)
\(14\) 0 0
\(15\) −2.47214 −0.638303
\(16\) 0 0
\(17\) 3.23607 0.784862 0.392431 0.919781i \(-0.371634\pi\)
0.392431 + 0.919781i \(0.371634\pi\)
\(18\) 0 0
\(19\) 6.47214 1.48481 0.742405 0.669951i \(-0.233685\pi\)
0.742405 + 0.669951i \(0.233685\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.47214 −0.515476 −0.257738 0.966215i \(-0.582977\pi\)
−0.257738 + 0.966215i \(0.582977\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) 0 0
\(27\) 5.52786 1.06384
\(28\) 0 0
\(29\) 8.47214 1.57324 0.786618 0.617440i \(-0.211830\pi\)
0.786618 + 0.617440i \(0.211830\pi\)
\(30\) 0 0
\(31\) −2.76393 −0.496417 −0.248208 0.968707i \(-0.579842\pi\)
−0.248208 + 0.968707i \(0.579842\pi\)
\(32\) 0 0
\(33\) −1.23607 −0.215172
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.47214 −1.39281 −0.696405 0.717649i \(-0.745218\pi\)
−0.696405 + 0.717649i \(0.745218\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) 11.2361 1.75478 0.877390 0.479779i \(-0.159283\pi\)
0.877390 + 0.479779i \(0.159283\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) −2.94427 −0.438906
\(46\) 0 0
\(47\) 2.76393 0.403161 0.201580 0.979472i \(-0.435392\pi\)
0.201580 + 0.979472i \(0.435392\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −0.472136 −0.0648529 −0.0324264 0.999474i \(-0.510323\pi\)
−0.0324264 + 0.999474i \(0.510323\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) −8.00000 −1.05963
\(58\) 0 0
\(59\) −1.23607 −0.160922 −0.0804612 0.996758i \(-0.525639\pi\)
−0.0804612 + 0.996758i \(0.525639\pi\)
\(60\) 0 0
\(61\) 7.23607 0.926484 0.463242 0.886232i \(-0.346686\pi\)
0.463242 + 0.886232i \(0.346686\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −6.47214 −0.802770
\(66\) 0 0
\(67\) −14.4721 −1.76805 −0.884026 0.467437i \(-0.845177\pi\)
−0.884026 + 0.467437i \(0.845177\pi\)
\(68\) 0 0
\(69\) 3.05573 0.367866
\(70\) 0 0
\(71\) 10.4721 1.24281 0.621407 0.783488i \(-0.286561\pi\)
0.621407 + 0.783488i \(0.286561\pi\)
\(72\) 0 0
\(73\) 0.763932 0.0894115 0.0447057 0.999000i \(-0.485765\pi\)
0.0447057 + 0.999000i \(0.485765\pi\)
\(74\) 0 0
\(75\) 1.23607 0.142729
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.94427 1.00631 0.503155 0.864196i \(-0.332173\pi\)
0.503155 + 0.864196i \(0.332173\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 0 0
\(83\) −11.4164 −1.25311 −0.626557 0.779376i \(-0.715536\pi\)
−0.626557 + 0.779376i \(0.715536\pi\)
\(84\) 0 0
\(85\) 6.47214 0.702002
\(86\) 0 0
\(87\) −10.4721 −1.12273
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 3.41641 0.354265
\(94\) 0 0
\(95\) 12.9443 1.32805
\(96\) 0 0
\(97\) −17.4164 −1.76837 −0.884184 0.467139i \(-0.845285\pi\)
−0.884184 + 0.467139i \(0.845285\pi\)
\(98\) 0 0
\(99\) −1.47214 −0.147955
\(100\) 0 0
\(101\) 4.76393 0.474029 0.237014 0.971506i \(-0.423831\pi\)
0.237014 + 0.971506i \(0.423831\pi\)
\(102\) 0 0
\(103\) 7.70820 0.759512 0.379756 0.925087i \(-0.376008\pi\)
0.379756 + 0.925087i \(0.376008\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.00000 0.386695 0.193347 0.981130i \(-0.438066\pi\)
0.193347 + 0.981130i \(0.438066\pi\)
\(108\) 0 0
\(109\) −4.47214 −0.428353 −0.214176 0.976795i \(-0.568707\pi\)
−0.214176 + 0.976795i \(0.568707\pi\)
\(110\) 0 0
\(111\) 10.4721 0.993971
\(112\) 0 0
\(113\) 2.00000 0.188144 0.0940721 0.995565i \(-0.470012\pi\)
0.0940721 + 0.995565i \(0.470012\pi\)
\(114\) 0 0
\(115\) −4.94427 −0.461056
\(116\) 0 0
\(117\) 4.76393 0.440426
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −13.8885 −1.25229
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −3.05573 −0.271152 −0.135576 0.990767i \(-0.543288\pi\)
−0.135576 + 0.990767i \(0.543288\pi\)
\(128\) 0 0
\(129\) 9.88854 0.870638
\(130\) 0 0
\(131\) 21.8885 1.91241 0.956205 0.292696i \(-0.0945525\pi\)
0.956205 + 0.292696i \(0.0945525\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 11.0557 0.951526
\(136\) 0 0
\(137\) 16.4721 1.40731 0.703655 0.710542i \(-0.251550\pi\)
0.703655 + 0.710542i \(0.251550\pi\)
\(138\) 0 0
\(139\) −1.52786 −0.129592 −0.0647959 0.997899i \(-0.520640\pi\)
−0.0647959 + 0.997899i \(0.520640\pi\)
\(140\) 0 0
\(141\) −3.41641 −0.287713
\(142\) 0 0
\(143\) −3.23607 −0.270614
\(144\) 0 0
\(145\) 16.9443 1.40715
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 14.0000 1.14692 0.573462 0.819232i \(-0.305600\pi\)
0.573462 + 0.819232i \(0.305600\pi\)
\(150\) 0 0
\(151\) −8.94427 −0.727875 −0.363937 0.931423i \(-0.618568\pi\)
−0.363937 + 0.931423i \(0.618568\pi\)
\(152\) 0 0
\(153\) −4.76393 −0.385141
\(154\) 0 0
\(155\) −5.52786 −0.444009
\(156\) 0 0
\(157\) −10.9443 −0.873448 −0.436724 0.899596i \(-0.643861\pi\)
−0.436724 + 0.899596i \(0.643861\pi\)
\(158\) 0 0
\(159\) 0.583592 0.0462819
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.41641 0.267594 0.133797 0.991009i \(-0.457283\pi\)
0.133797 + 0.991009i \(0.457283\pi\)
\(164\) 0 0
\(165\) −2.47214 −0.192456
\(166\) 0 0
\(167\) 4.94427 0.382599 0.191300 0.981532i \(-0.438730\pi\)
0.191300 + 0.981532i \(0.438730\pi\)
\(168\) 0 0
\(169\) −2.52786 −0.194451
\(170\) 0 0
\(171\) −9.52786 −0.728614
\(172\) 0 0
\(173\) 12.7639 0.970424 0.485212 0.874397i \(-0.338742\pi\)
0.485212 + 0.874397i \(0.338742\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.52786 0.114841
\(178\) 0 0
\(179\) 8.94427 0.668526 0.334263 0.942480i \(-0.391513\pi\)
0.334263 + 0.942480i \(0.391513\pi\)
\(180\) 0 0
\(181\) 25.4164 1.88919 0.944593 0.328243i \(-0.106456\pi\)
0.944593 + 0.328243i \(0.106456\pi\)
\(182\) 0 0
\(183\) −8.94427 −0.661180
\(184\) 0 0
\(185\) −16.9443 −1.24577
\(186\) 0 0
\(187\) 3.23607 0.236645
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.05573 0.221105 0.110552 0.993870i \(-0.464738\pi\)
0.110552 + 0.993870i \(0.464738\pi\)
\(192\) 0 0
\(193\) 11.8885 0.855756 0.427878 0.903836i \(-0.359261\pi\)
0.427878 + 0.903836i \(0.359261\pi\)
\(194\) 0 0
\(195\) 8.00000 0.572892
\(196\) 0 0
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −2.18034 −0.154560 −0.0772801 0.997009i \(-0.524624\pi\)
−0.0772801 + 0.997009i \(0.524624\pi\)
\(200\) 0 0
\(201\) 17.8885 1.26176
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 22.4721 1.56952
\(206\) 0 0
\(207\) 3.63932 0.252950
\(208\) 0 0
\(209\) 6.47214 0.447687
\(210\) 0 0
\(211\) 13.8885 0.956127 0.478063 0.878325i \(-0.341339\pi\)
0.478063 + 0.878325i \(0.341339\pi\)
\(212\) 0 0
\(213\) −12.9443 −0.886927
\(214\) 0 0
\(215\) −16.0000 −1.09119
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −0.944272 −0.0638080
\(220\) 0 0
\(221\) −10.4721 −0.704432
\(222\) 0 0
\(223\) −10.1803 −0.681726 −0.340863 0.940113i \(-0.610719\pi\)
−0.340863 + 0.940113i \(0.610719\pi\)
\(224\) 0 0
\(225\) 1.47214 0.0981424
\(226\) 0 0
\(227\) 5.88854 0.390836 0.195418 0.980720i \(-0.437394\pi\)
0.195418 + 0.980720i \(0.437394\pi\)
\(228\) 0 0
\(229\) 4.47214 0.295527 0.147764 0.989023i \(-0.452793\pi\)
0.147764 + 0.989023i \(0.452793\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.41641 0.616889 0.308445 0.951242i \(-0.400192\pi\)
0.308445 + 0.951242i \(0.400192\pi\)
\(234\) 0 0
\(235\) 5.52786 0.360598
\(236\) 0 0
\(237\) −11.0557 −0.718147
\(238\) 0 0
\(239\) 9.88854 0.639637 0.319818 0.947479i \(-0.396378\pi\)
0.319818 + 0.947479i \(0.396378\pi\)
\(240\) 0 0
\(241\) 13.1246 0.845431 0.422715 0.906263i \(-0.361077\pi\)
0.422715 + 0.906263i \(0.361077\pi\)
\(242\) 0 0
\(243\) −13.5967 −0.872232
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −20.9443 −1.33265
\(248\) 0 0
\(249\) 14.1115 0.894277
\(250\) 0 0
\(251\) 4.29180 0.270896 0.135448 0.990784i \(-0.456753\pi\)
0.135448 + 0.990784i \(0.456753\pi\)
\(252\) 0 0
\(253\) −2.47214 −0.155422
\(254\) 0 0
\(255\) −8.00000 −0.500979
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −12.4721 −0.772006
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −0.944272 −0.0580062
\(266\) 0 0
\(267\) 2.47214 0.151292
\(268\) 0 0
\(269\) −13.4164 −0.818013 −0.409006 0.912532i \(-0.634125\pi\)
−0.409006 + 0.912532i \(0.634125\pi\)
\(270\) 0 0
\(271\) −10.4721 −0.636137 −0.318068 0.948068i \(-0.603034\pi\)
−0.318068 + 0.948068i \(0.603034\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) −19.8885 −1.19499 −0.597493 0.801874i \(-0.703837\pi\)
−0.597493 + 0.801874i \(0.703837\pi\)
\(278\) 0 0
\(279\) 4.06888 0.243598
\(280\) 0 0
\(281\) −3.52786 −0.210455 −0.105227 0.994448i \(-0.533557\pi\)
−0.105227 + 0.994448i \(0.533557\pi\)
\(282\) 0 0
\(283\) 29.8885 1.77669 0.888345 0.459177i \(-0.151856\pi\)
0.888345 + 0.459177i \(0.151856\pi\)
\(284\) 0 0
\(285\) −16.0000 −0.947758
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6.52786 −0.383992
\(290\) 0 0
\(291\) 21.5279 1.26199
\(292\) 0 0
\(293\) 25.1246 1.46780 0.733898 0.679260i \(-0.237699\pi\)
0.733898 + 0.679260i \(0.237699\pi\)
\(294\) 0 0
\(295\) −2.47214 −0.143933
\(296\) 0 0
\(297\) 5.52786 0.320759
\(298\) 0 0
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −5.88854 −0.338288
\(304\) 0 0
\(305\) 14.4721 0.828672
\(306\) 0 0
\(307\) −8.94427 −0.510477 −0.255238 0.966878i \(-0.582154\pi\)
−0.255238 + 0.966878i \(0.582154\pi\)
\(308\) 0 0
\(309\) −9.52786 −0.542021
\(310\) 0 0
\(311\) −8.29180 −0.470185 −0.235092 0.971973i \(-0.575539\pi\)
−0.235092 + 0.971973i \(0.575539\pi\)
\(312\) 0 0
\(313\) −14.9443 −0.844700 −0.422350 0.906433i \(-0.638795\pi\)
−0.422350 + 0.906433i \(0.638795\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 14.0000 0.786318 0.393159 0.919470i \(-0.371382\pi\)
0.393159 + 0.919470i \(0.371382\pi\)
\(318\) 0 0
\(319\) 8.47214 0.474349
\(320\) 0 0
\(321\) −4.94427 −0.275962
\(322\) 0 0
\(323\) 20.9443 1.16537
\(324\) 0 0
\(325\) 3.23607 0.179505
\(326\) 0 0
\(327\) 5.52786 0.305692
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.8885 0.763383 0.381692 0.924290i \(-0.375342\pi\)
0.381692 + 0.924290i \(0.375342\pi\)
\(332\) 0 0
\(333\) 12.4721 0.683469
\(334\) 0 0
\(335\) −28.9443 −1.58139
\(336\) 0 0
\(337\) −11.5279 −0.627963 −0.313981 0.949429i \(-0.601663\pi\)
−0.313981 + 0.949429i \(0.601663\pi\)
\(338\) 0 0
\(339\) −2.47214 −0.134268
\(340\) 0 0
\(341\) −2.76393 −0.149675
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 6.11146 0.329030
\(346\) 0 0
\(347\) −20.9443 −1.12435 −0.562174 0.827019i \(-0.690035\pi\)
−0.562174 + 0.827019i \(0.690035\pi\)
\(348\) 0 0
\(349\) 7.23607 0.387338 0.193669 0.981067i \(-0.437961\pi\)
0.193669 + 0.981067i \(0.437961\pi\)
\(350\) 0 0
\(351\) −17.8885 −0.954820
\(352\) 0 0
\(353\) −19.8885 −1.05856 −0.529280 0.848447i \(-0.677538\pi\)
−0.529280 + 0.848447i \(0.677538\pi\)
\(354\) 0 0
\(355\) 20.9443 1.11161
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 24.9443 1.31651 0.658254 0.752796i \(-0.271295\pi\)
0.658254 + 0.752796i \(0.271295\pi\)
\(360\) 0 0
\(361\) 22.8885 1.20466
\(362\) 0 0
\(363\) −1.23607 −0.0648767
\(364\) 0 0
\(365\) 1.52786 0.0799721
\(366\) 0 0
\(367\) −23.1246 −1.20709 −0.603547 0.797327i \(-0.706247\pi\)
−0.603547 + 0.797327i \(0.706247\pi\)
\(368\) 0 0
\(369\) −16.5410 −0.861091
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) 0 0
\(375\) 14.8328 0.765963
\(376\) 0 0
\(377\) −27.4164 −1.41202
\(378\) 0 0
\(379\) −37.3050 −1.91623 −0.958113 0.286389i \(-0.907545\pi\)
−0.958113 + 0.286389i \(0.907545\pi\)
\(380\) 0 0
\(381\) 3.77709 0.193506
\(382\) 0 0
\(383\) −4.65248 −0.237730 −0.118865 0.992910i \(-0.537926\pi\)
−0.118865 + 0.992910i \(0.537926\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 11.7771 0.598663
\(388\) 0 0
\(389\) 15.8885 0.805581 0.402791 0.915292i \(-0.368040\pi\)
0.402791 + 0.915292i \(0.368040\pi\)
\(390\) 0 0
\(391\) −8.00000 −0.404577
\(392\) 0 0
\(393\) −27.0557 −1.36478
\(394\) 0 0
\(395\) 17.8885 0.900070
\(396\) 0 0
\(397\) 35.8885 1.80119 0.900597 0.434655i \(-0.143130\pi\)
0.900597 + 0.434655i \(0.143130\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 22.9443 1.14578 0.572891 0.819631i \(-0.305822\pi\)
0.572891 + 0.819631i \(0.305822\pi\)
\(402\) 0 0
\(403\) 8.94427 0.445546
\(404\) 0 0
\(405\) −4.83282 −0.240145
\(406\) 0 0
\(407\) −8.47214 −0.419948
\(408\) 0 0
\(409\) −9.12461 −0.451183 −0.225592 0.974222i \(-0.572431\pi\)
−0.225592 + 0.974222i \(0.572431\pi\)
\(410\) 0 0
\(411\) −20.3607 −1.00432
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −22.8328 −1.12082
\(416\) 0 0
\(417\) 1.88854 0.0924824
\(418\) 0 0
\(419\) 24.6525 1.20435 0.602176 0.798363i \(-0.294300\pi\)
0.602176 + 0.798363i \(0.294300\pi\)
\(420\) 0 0
\(421\) 22.3607 1.08979 0.544896 0.838503i \(-0.316569\pi\)
0.544896 + 0.838503i \(0.316569\pi\)
\(422\) 0 0
\(423\) −4.06888 −0.197836
\(424\) 0 0
\(425\) −3.23607 −0.156972
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 8.47214 0.407145 0.203572 0.979060i \(-0.434745\pi\)
0.203572 + 0.979060i \(0.434745\pi\)
\(434\) 0 0
\(435\) −20.9443 −1.00420
\(436\) 0 0
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) 10.4721 0.499808 0.249904 0.968271i \(-0.419601\pi\)
0.249904 + 0.968271i \(0.419601\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.9443 1.18514 0.592569 0.805520i \(-0.298114\pi\)
0.592569 + 0.805520i \(0.298114\pi\)
\(444\) 0 0
\(445\) −4.00000 −0.189618
\(446\) 0 0
\(447\) −17.3050 −0.818496
\(448\) 0 0
\(449\) −28.4721 −1.34368 −0.671842 0.740695i \(-0.734496\pi\)
−0.671842 + 0.740695i \(0.734496\pi\)
\(450\) 0 0
\(451\) 11.2361 0.529086
\(452\) 0 0
\(453\) 11.0557 0.519443
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −28.8328 −1.34874 −0.674371 0.738393i \(-0.735585\pi\)
−0.674371 + 0.738393i \(0.735585\pi\)
\(458\) 0 0
\(459\) 17.8885 0.834966
\(460\) 0 0
\(461\) 12.1803 0.567295 0.283647 0.958929i \(-0.408455\pi\)
0.283647 + 0.958929i \(0.408455\pi\)
\(462\) 0 0
\(463\) 5.52786 0.256902 0.128451 0.991716i \(-0.459000\pi\)
0.128451 + 0.991716i \(0.459000\pi\)
\(464\) 0 0
\(465\) 6.83282 0.316864
\(466\) 0 0
\(467\) 24.0689 1.11378 0.556888 0.830588i \(-0.311995\pi\)
0.556888 + 0.830588i \(0.311995\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 13.5279 0.623331
\(472\) 0 0
\(473\) −8.00000 −0.367840
\(474\) 0 0
\(475\) −6.47214 −0.296962
\(476\) 0 0
\(477\) 0.695048 0.0318241
\(478\) 0 0
\(479\) 13.5279 0.618104 0.309052 0.951045i \(-0.399988\pi\)
0.309052 + 0.951045i \(0.399988\pi\)
\(480\) 0 0
\(481\) 27.4164 1.25008
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −34.8328 −1.58168
\(486\) 0 0
\(487\) 36.3607 1.64766 0.823830 0.566837i \(-0.191833\pi\)
0.823830 + 0.566837i \(0.191833\pi\)
\(488\) 0 0
\(489\) −4.22291 −0.190967
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 27.4164 1.23477
\(494\) 0 0
\(495\) −2.94427 −0.132335
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.52786 −0.0683966 −0.0341983 0.999415i \(-0.510888\pi\)
−0.0341983 + 0.999415i \(0.510888\pi\)
\(500\) 0 0
\(501\) −6.11146 −0.273040
\(502\) 0 0
\(503\) 23.4164 1.04409 0.522043 0.852919i \(-0.325170\pi\)
0.522043 + 0.852919i \(0.325170\pi\)
\(504\) 0 0
\(505\) 9.52786 0.423984
\(506\) 0 0
\(507\) 3.12461 0.138769
\(508\) 0 0
\(509\) −40.4721 −1.79390 −0.896948 0.442136i \(-0.854221\pi\)
−0.896948 + 0.442136i \(0.854221\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 35.7771 1.57960
\(514\) 0 0
\(515\) 15.4164 0.679328
\(516\) 0 0
\(517\) 2.76393 0.121558
\(518\) 0 0
\(519\) −15.7771 −0.692537
\(520\) 0 0
\(521\) −30.3607 −1.33013 −0.665063 0.746787i \(-0.731595\pi\)
−0.665063 + 0.746787i \(0.731595\pi\)
\(522\) 0 0
\(523\) 44.0000 1.92399 0.961993 0.273075i \(-0.0880406\pi\)
0.961993 + 0.273075i \(0.0880406\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.94427 −0.389619
\(528\) 0 0
\(529\) −16.8885 −0.734285
\(530\) 0 0
\(531\) 1.81966 0.0789665
\(532\) 0 0
\(533\) −36.3607 −1.57496
\(534\) 0 0
\(535\) 8.00000 0.345870
\(536\) 0 0
\(537\) −11.0557 −0.477090
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 20.8328 0.895673 0.447836 0.894116i \(-0.352195\pi\)
0.447836 + 0.894116i \(0.352195\pi\)
\(542\) 0 0
\(543\) −31.4164 −1.34821
\(544\) 0 0
\(545\) −8.94427 −0.383131
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) −10.6525 −0.454637
\(550\) 0 0
\(551\) 54.8328 2.33596
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 20.9443 0.889035
\(556\) 0 0
\(557\) −38.9443 −1.65012 −0.825061 0.565044i \(-0.808859\pi\)
−0.825061 + 0.565044i \(0.808859\pi\)
\(558\) 0 0
\(559\) 25.8885 1.09497
\(560\) 0 0
\(561\) −4.00000 −0.168880
\(562\) 0 0
\(563\) 12.5836 0.530335 0.265168 0.964202i \(-0.414573\pi\)
0.265168 + 0.964202i \(0.414573\pi\)
\(564\) 0 0
\(565\) 4.00000 0.168281
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7.52786 0.315584 0.157792 0.987472i \(-0.449562\pi\)
0.157792 + 0.987472i \(0.449562\pi\)
\(570\) 0 0
\(571\) 15.0557 0.630063 0.315031 0.949081i \(-0.397985\pi\)
0.315031 + 0.949081i \(0.397985\pi\)
\(572\) 0 0
\(573\) −3.77709 −0.157790
\(574\) 0 0
\(575\) 2.47214 0.103095
\(576\) 0 0
\(577\) 19.5279 0.812956 0.406478 0.913661i \(-0.366757\pi\)
0.406478 + 0.913661i \(0.366757\pi\)
\(578\) 0 0
\(579\) −14.6950 −0.610705
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.472136 −0.0195539
\(584\) 0 0
\(585\) 9.52786 0.393929
\(586\) 0 0
\(587\) 27.1246 1.11955 0.559776 0.828644i \(-0.310887\pi\)
0.559776 + 0.828644i \(0.310887\pi\)
\(588\) 0 0
\(589\) −17.8885 −0.737085
\(590\) 0 0
\(591\) 2.47214 0.101690
\(592\) 0 0
\(593\) 45.7082 1.87701 0.938505 0.345264i \(-0.112211\pi\)
0.938505 + 0.345264i \(0.112211\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.69505 0.110301
\(598\) 0 0
\(599\) 23.4164 0.956768 0.478384 0.878151i \(-0.341223\pi\)
0.478384 + 0.878151i \(0.341223\pi\)
\(600\) 0 0
\(601\) 37.1246 1.51434 0.757172 0.653215i \(-0.226580\pi\)
0.757172 + 0.653215i \(0.226580\pi\)
\(602\) 0 0
\(603\) 21.3050 0.867605
\(604\) 0 0
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) 12.9443 0.525392 0.262696 0.964879i \(-0.415388\pi\)
0.262696 + 0.964879i \(0.415388\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −8.94427 −0.361847
\(612\) 0 0
\(613\) 15.3050 0.618161 0.309081 0.951036i \(-0.399979\pi\)
0.309081 + 0.951036i \(0.399979\pi\)
\(614\) 0 0
\(615\) −27.7771 −1.12008
\(616\) 0 0
\(617\) 6.58359 0.265045 0.132523 0.991180i \(-0.457692\pi\)
0.132523 + 0.991180i \(0.457692\pi\)
\(618\) 0 0
\(619\) 11.1246 0.447136 0.223568 0.974688i \(-0.428230\pi\)
0.223568 + 0.974688i \(0.428230\pi\)
\(620\) 0 0
\(621\) −13.6656 −0.548383
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) −27.4164 −1.09316
\(630\) 0 0
\(631\) 24.0000 0.955425 0.477712 0.878516i \(-0.341466\pi\)
0.477712 + 0.878516i \(0.341466\pi\)
\(632\) 0 0
\(633\) −17.1672 −0.682334
\(634\) 0 0
\(635\) −6.11146 −0.242526
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −15.4164 −0.609864
\(640\) 0 0
\(641\) −15.5279 −0.613314 −0.306657 0.951820i \(-0.599210\pi\)
−0.306657 + 0.951820i \(0.599210\pi\)
\(642\) 0 0
\(643\) 11.1246 0.438712 0.219356 0.975645i \(-0.429604\pi\)
0.219356 + 0.975645i \(0.429604\pi\)
\(644\) 0 0
\(645\) 19.7771 0.778722
\(646\) 0 0
\(647\) 36.0689 1.41801 0.709007 0.705201i \(-0.249143\pi\)
0.709007 + 0.705201i \(0.249143\pi\)
\(648\) 0 0
\(649\) −1.23607 −0.0485199
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 25.0557 0.980506 0.490253 0.871580i \(-0.336904\pi\)
0.490253 + 0.871580i \(0.336904\pi\)
\(654\) 0 0
\(655\) 43.7771 1.71051
\(656\) 0 0
\(657\) −1.12461 −0.0438753
\(658\) 0 0
\(659\) 17.8885 0.696839 0.348419 0.937339i \(-0.386719\pi\)
0.348419 + 0.937339i \(0.386719\pi\)
\(660\) 0 0
\(661\) 40.8328 1.58821 0.794106 0.607779i \(-0.207939\pi\)
0.794106 + 0.607779i \(0.207939\pi\)
\(662\) 0 0
\(663\) 12.9443 0.502714
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −20.9443 −0.810965
\(668\) 0 0
\(669\) 12.5836 0.486510
\(670\) 0 0
\(671\) 7.23607 0.279345
\(672\) 0 0
\(673\) −21.4164 −0.825542 −0.412771 0.910835i \(-0.635439\pi\)
−0.412771 + 0.910835i \(0.635439\pi\)
\(674\) 0 0
\(675\) −5.52786 −0.212768
\(676\) 0 0
\(677\) 9.70820 0.373117 0.186558 0.982444i \(-0.440267\pi\)
0.186558 + 0.982444i \(0.440267\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −7.27864 −0.278918
\(682\) 0 0
\(683\) 5.88854 0.225319 0.112659 0.993634i \(-0.464063\pi\)
0.112659 + 0.993634i \(0.464063\pi\)
\(684\) 0 0
\(685\) 32.9443 1.25874
\(686\) 0 0
\(687\) −5.52786 −0.210901
\(688\) 0 0
\(689\) 1.52786 0.0582070
\(690\) 0 0
\(691\) 18.5410 0.705334 0.352667 0.935749i \(-0.385275\pi\)
0.352667 + 0.935749i \(0.385275\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −3.05573 −0.115910
\(696\) 0 0
\(697\) 36.3607 1.37726
\(698\) 0 0
\(699\) −11.6393 −0.440240
\(700\) 0 0
\(701\) −15.5279 −0.586479 −0.293240 0.956039i \(-0.594733\pi\)
−0.293240 + 0.956039i \(0.594733\pi\)
\(702\) 0 0
\(703\) −54.8328 −2.06806
\(704\) 0 0
\(705\) −6.83282 −0.257339
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −14.9443 −0.561244 −0.280622 0.959818i \(-0.590541\pi\)
−0.280622 + 0.959818i \(0.590541\pi\)
\(710\) 0 0
\(711\) −13.1672 −0.493808
\(712\) 0 0
\(713\) 6.83282 0.255891
\(714\) 0 0
\(715\) −6.47214 −0.242044
\(716\) 0 0
\(717\) −12.2229 −0.456473
\(718\) 0 0
\(719\) −51.4853 −1.92008 −0.960039 0.279867i \(-0.909710\pi\)
−0.960039 + 0.279867i \(0.909710\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −16.2229 −0.603337
\(724\) 0 0
\(725\) −8.47214 −0.314647
\(726\) 0 0
\(727\) 25.0132 0.927687 0.463843 0.885917i \(-0.346470\pi\)
0.463843 + 0.885917i \(0.346470\pi\)
\(728\) 0 0
\(729\) 24.0557 0.890953
\(730\) 0 0
\(731\) −25.8885 −0.957522
\(732\) 0 0
\(733\) −8.76393 −0.323703 −0.161852 0.986815i \(-0.551747\pi\)
−0.161852 + 0.986815i \(0.551747\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −14.4721 −0.533088
\(738\) 0 0
\(739\) −24.9443 −0.917590 −0.458795 0.888542i \(-0.651719\pi\)
−0.458795 + 0.888542i \(0.651719\pi\)
\(740\) 0 0
\(741\) 25.8885 0.951039
\(742\) 0 0
\(743\) −1.88854 −0.0692840 −0.0346420 0.999400i \(-0.511029\pi\)
−0.0346420 + 0.999400i \(0.511029\pi\)
\(744\) 0 0
\(745\) 28.0000 1.02584
\(746\) 0 0
\(747\) 16.8065 0.614918
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −29.5279 −1.07749 −0.538744 0.842470i \(-0.681101\pi\)
−0.538744 + 0.842470i \(0.681101\pi\)
\(752\) 0 0
\(753\) −5.30495 −0.193323
\(754\) 0 0
\(755\) −17.8885 −0.651031
\(756\) 0 0
\(757\) 15.8885 0.577479 0.288739 0.957408i \(-0.406764\pi\)
0.288739 + 0.957408i \(0.406764\pi\)
\(758\) 0 0
\(759\) 3.05573 0.110916
\(760\) 0 0
\(761\) 31.5967 1.14538 0.572691 0.819772i \(-0.305900\pi\)
0.572691 + 0.819772i \(0.305900\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −9.52786 −0.344481
\(766\) 0 0
\(767\) 4.00000 0.144432
\(768\) 0 0
\(769\) −18.2918 −0.659619 −0.329810 0.944047i \(-0.606985\pi\)
−0.329810 + 0.944047i \(0.606985\pi\)
\(770\) 0 0
\(771\) −7.41641 −0.267095
\(772\) 0 0
\(773\) 38.3607 1.37974 0.689869 0.723934i \(-0.257668\pi\)
0.689869 + 0.723934i \(0.257668\pi\)
\(774\) 0 0
\(775\) 2.76393 0.0992834
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 72.7214 2.60551
\(780\) 0 0
\(781\) 10.4721 0.374722
\(782\) 0 0
\(783\) 46.8328 1.67367
\(784\) 0 0
\(785\) −21.8885 −0.781236
\(786\) 0 0
\(787\) −43.4164 −1.54763 −0.773814 0.633413i \(-0.781653\pi\)
−0.773814 + 0.633413i \(0.781653\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −23.4164 −0.831541
\(794\) 0 0
\(795\) 1.16718 0.0413958
\(796\) 0 0
\(797\) 14.9443 0.529353 0.264677 0.964337i \(-0.414735\pi\)
0.264677 + 0.964337i \(0.414735\pi\)
\(798\) 0 0
\(799\) 8.94427 0.316426
\(800\) 0 0
\(801\) 2.94427 0.104031
\(802\) 0 0
\(803\) 0.763932 0.0269586
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 16.5836 0.583770
\(808\) 0 0
\(809\) 21.0557 0.740280 0.370140 0.928976i \(-0.379310\pi\)
0.370140 + 0.928976i \(0.379310\pi\)
\(810\) 0 0
\(811\) −34.8328 −1.22315 −0.611573 0.791188i \(-0.709463\pi\)
−0.611573 + 0.791188i \(0.709463\pi\)
\(812\) 0 0
\(813\) 12.9443 0.453975
\(814\) 0 0
\(815\) 6.83282 0.239343
\(816\) 0 0
\(817\) −51.7771 −1.81145
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 44.8328 1.56468 0.782338 0.622854i \(-0.214027\pi\)
0.782338 + 0.622854i \(0.214027\pi\)
\(822\) 0 0
\(823\) 14.1115 0.491894 0.245947 0.969283i \(-0.420901\pi\)
0.245947 + 0.969283i \(0.420901\pi\)
\(824\) 0 0
\(825\) 1.23607 0.0430344
\(826\) 0 0
\(827\) 12.9443 0.450116 0.225058 0.974345i \(-0.427743\pi\)
0.225058 + 0.974345i \(0.427743\pi\)
\(828\) 0 0
\(829\) −36.8328 −1.27926 −0.639628 0.768684i \(-0.720912\pi\)
−0.639628 + 0.768684i \(0.720912\pi\)
\(830\) 0 0
\(831\) 24.5836 0.852795
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 9.88854 0.342207
\(836\) 0 0
\(837\) −15.2786 −0.528107
\(838\) 0 0
\(839\) 44.0689 1.52143 0.760713 0.649088i \(-0.224849\pi\)
0.760713 + 0.649088i \(0.224849\pi\)
\(840\) 0 0
\(841\) 42.7771 1.47507
\(842\) 0 0
\(843\) 4.36068 0.150190
\(844\) 0 0
\(845\) −5.05573 −0.173922
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −36.9443 −1.26792
\(850\) 0 0
\(851\) 20.9443 0.717960
\(852\) 0 0
\(853\) 30.6525 1.04952 0.524760 0.851250i \(-0.324155\pi\)
0.524760 + 0.851250i \(0.324155\pi\)
\(854\) 0 0
\(855\) −19.0557 −0.651692
\(856\) 0 0
\(857\) −15.2361 −0.520454 −0.260227 0.965547i \(-0.583797\pi\)
−0.260227 + 0.965547i \(0.583797\pi\)
\(858\) 0 0
\(859\) −26.5410 −0.905568 −0.452784 0.891620i \(-0.649569\pi\)
−0.452784 + 0.891620i \(0.649569\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.05573 0.104018 0.0520091 0.998647i \(-0.483438\pi\)
0.0520091 + 0.998647i \(0.483438\pi\)
\(864\) 0 0
\(865\) 25.5279 0.867973
\(866\) 0 0
\(867\) 8.06888 0.274034
\(868\) 0 0
\(869\) 8.94427 0.303414
\(870\) 0 0
\(871\) 46.8328 1.58687
\(872\) 0 0
\(873\) 25.6393 0.867760
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 14.5836 0.492453 0.246226 0.969212i \(-0.420809\pi\)
0.246226 + 0.969212i \(0.420809\pi\)
\(878\) 0 0
\(879\) −31.0557 −1.04748
\(880\) 0 0
\(881\) −2.58359 −0.0870434 −0.0435217 0.999052i \(-0.513858\pi\)
−0.0435217 + 0.999052i \(0.513858\pi\)
\(882\) 0 0
\(883\) 8.94427 0.300999 0.150499 0.988610i \(-0.451912\pi\)
0.150499 + 0.988610i \(0.451912\pi\)
\(884\) 0 0
\(885\) 3.05573 0.102717
\(886\) 0 0
\(887\) −4.36068 −0.146417 −0.0732086 0.997317i \(-0.523324\pi\)
−0.0732086 + 0.997317i \(0.523324\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −2.41641 −0.0809527
\(892\) 0 0
\(893\) 17.8885 0.598617
\(894\) 0 0
\(895\) 17.8885 0.597948
\(896\) 0 0
\(897\) −9.88854 −0.330169
\(898\) 0 0
\(899\) −23.4164 −0.780981
\(900\) 0 0
\(901\) −1.52786 −0.0509005
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 50.8328 1.68974
\(906\) 0 0
\(907\) −22.4721 −0.746175 −0.373088 0.927796i \(-0.621701\pi\)
−0.373088 + 0.927796i \(0.621701\pi\)
\(908\) 0 0
\(909\) −7.01316 −0.232612
\(910\) 0 0
\(911\) −42.4721 −1.40716 −0.703582 0.710614i \(-0.748417\pi\)
−0.703582 + 0.710614i \(0.748417\pi\)
\(912\) 0 0
\(913\) −11.4164 −0.377828
\(914\) 0 0
\(915\) −17.8885 −0.591377
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −41.8885 −1.38178 −0.690888 0.722962i \(-0.742780\pi\)
−0.690888 + 0.722962i \(0.742780\pi\)
\(920\) 0 0
\(921\) 11.0557 0.364299
\(922\) 0 0
\(923\) −33.8885 −1.11546
\(924\) 0 0
\(925\) 8.47214 0.278562
\(926\) 0 0
\(927\) −11.3475 −0.372702
\(928\) 0 0
\(929\) 52.2492 1.71424 0.857121 0.515116i \(-0.172251\pi\)
0.857121 + 0.515116i \(0.172251\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 10.2492 0.335545
\(934\) 0 0
\(935\) 6.47214 0.211661
\(936\) 0 0
\(937\) 10.6525 0.348001 0.174001 0.984746i \(-0.444331\pi\)
0.174001 + 0.984746i \(0.444331\pi\)
\(938\) 0 0
\(939\) 18.4721 0.602815
\(940\) 0 0
\(941\) −7.59675 −0.247647 −0.123823 0.992304i \(-0.539516\pi\)
−0.123823 + 0.992304i \(0.539516\pi\)
\(942\) 0 0
\(943\) −27.7771 −0.904546
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5.16718 0.167911 0.0839555 0.996470i \(-0.473245\pi\)
0.0839555 + 0.996470i \(0.473245\pi\)
\(948\) 0 0
\(949\) −2.47214 −0.0802489
\(950\) 0 0
\(951\) −17.3050 −0.561152
\(952\) 0 0
\(953\) 22.9443 0.743238 0.371619 0.928385i \(-0.378803\pi\)
0.371619 + 0.928385i \(0.378803\pi\)
\(954\) 0 0
\(955\) 6.11146 0.197762
\(956\) 0 0
\(957\) −10.4721 −0.338516
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −23.3607 −0.753570
\(962\) 0 0
\(963\) −5.88854 −0.189756
\(964\) 0 0
\(965\) 23.7771 0.765412
\(966\) 0 0
\(967\) 13.8885 0.446625 0.223313 0.974747i \(-0.428313\pi\)
0.223313 + 0.974747i \(0.428313\pi\)
\(968\) 0 0
\(969\) −25.8885 −0.831660
\(970\) 0 0
\(971\) 11.1246 0.357006 0.178503 0.983939i \(-0.442875\pi\)
0.178503 + 0.983939i \(0.442875\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) 22.9443 0.734052 0.367026 0.930211i \(-0.380376\pi\)
0.367026 + 0.930211i \(0.380376\pi\)
\(978\) 0 0
\(979\) −2.00000 −0.0639203
\(980\) 0 0
\(981\) 6.58359 0.210198
\(982\) 0 0
\(983\) 21.8197 0.695939 0.347970 0.937506i \(-0.386871\pi\)
0.347970 + 0.937506i \(0.386871\pi\)
\(984\) 0 0
\(985\) −4.00000 −0.127451
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 19.7771 0.628875
\(990\) 0 0
\(991\) −54.2492 −1.72328 −0.861642 0.507517i \(-0.830563\pi\)
−0.861642 + 0.507517i \(0.830563\pi\)
\(992\) 0 0
\(993\) −17.1672 −0.544784
\(994\) 0 0
\(995\) −4.36068 −0.138243
\(996\) 0 0
\(997\) −1.34752 −0.0426765 −0.0213383 0.999772i \(-0.506793\pi\)
−0.0213383 + 0.999772i \(0.506793\pi\)
\(998\) 0 0
\(999\) −46.8328 −1.48172
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 8624.2.a.ce.1.1 2
4.3 odd 2 539.2.a.f.1.2 2
7.6 odd 2 1232.2.a.m.1.2 2
12.11 even 2 4851.2.a.y.1.1 2
28.3 even 6 539.2.e.i.177.1 4
28.11 odd 6 539.2.e.j.177.1 4
28.19 even 6 539.2.e.i.67.1 4
28.23 odd 6 539.2.e.j.67.1 4
28.27 even 2 77.2.a.d.1.2 2
44.43 even 2 5929.2.a.m.1.1 2
56.13 odd 2 4928.2.a.bv.1.1 2
56.27 even 2 4928.2.a.bm.1.2 2
84.83 odd 2 693.2.a.h.1.1 2
140.27 odd 4 1925.2.b.h.1849.3 4
140.83 odd 4 1925.2.b.h.1849.2 4
140.139 even 2 1925.2.a.r.1.1 2
308.27 even 10 847.2.f.a.729.1 4
308.83 odd 10 847.2.f.m.729.1 4
308.139 odd 10 847.2.f.b.148.1 4
308.167 odd 10 847.2.f.m.323.1 4
308.195 odd 10 847.2.f.b.372.1 4
308.223 even 10 847.2.f.n.372.1 4
308.251 even 10 847.2.f.a.323.1 4
308.279 even 10 847.2.f.n.148.1 4
308.307 odd 2 847.2.a.f.1.1 2
924.923 even 2 7623.2.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.a.d.1.2 2 28.27 even 2
539.2.a.f.1.2 2 4.3 odd 2
539.2.e.i.67.1 4 28.19 even 6
539.2.e.i.177.1 4 28.3 even 6
539.2.e.j.67.1 4 28.23 odd 6
539.2.e.j.177.1 4 28.11 odd 6
693.2.a.h.1.1 2 84.83 odd 2
847.2.a.f.1.1 2 308.307 odd 2
847.2.f.a.323.1 4 308.251 even 10
847.2.f.a.729.1 4 308.27 even 10
847.2.f.b.148.1 4 308.139 odd 10
847.2.f.b.372.1 4 308.195 odd 10
847.2.f.m.323.1 4 308.167 odd 10
847.2.f.m.729.1 4 308.83 odd 10
847.2.f.n.148.1 4 308.279 even 10
847.2.f.n.372.1 4 308.223 even 10
1232.2.a.m.1.2 2 7.6 odd 2
1925.2.a.r.1.1 2 140.139 even 2
1925.2.b.h.1849.2 4 140.83 odd 4
1925.2.b.h.1849.3 4 140.27 odd 4
4851.2.a.y.1.1 2 12.11 even 2
4928.2.a.bm.1.2 2 56.27 even 2
4928.2.a.bv.1.1 2 56.13 odd 2
5929.2.a.m.1.1 2 44.43 even 2
7623.2.a.bl.1.2 2 924.923 even 2
8624.2.a.ce.1.1 2 1.1 even 1 trivial