Properties

Label 8112.2.a.cp.1.2
Level $8112$
Weight $2$
Character 8112.1
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
Dimension $3$
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] = [8112,2,Mod(1,8112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8112, 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("8112.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8112 = 2^{4} \cdot 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8112.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: \(64.7746461197\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 507)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.445042\) of defining polynomial
Character \(\chi\) \(=\) 8112.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.35690 q^{5} -2.24698 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +3.35690 q^{5} -2.24698 q^{7} +1.00000 q^{9} -4.93900 q^{11} +3.35690 q^{15} +0.911854 q^{17} +3.80194 q^{19} -2.24698 q^{21} -2.02715 q^{23} +6.26875 q^{25} +1.00000 q^{27} -3.93900 q^{29} +8.82908 q^{31} -4.93900 q^{33} -7.54288 q^{35} +8.80194 q^{37} +6.93900 q^{41} +2.28621 q^{43} +3.35690 q^{45} -3.80194 q^{47} -1.95108 q^{49} +0.911854 q^{51} +0.542877 q^{53} -16.5797 q^{55} +3.80194 q^{57} +4.71379 q^{59} +3.67994 q^{61} -2.24698 q^{63} +1.52111 q^{67} -2.02715 q^{69} -2.37867 q^{71} +7.41119 q^{73} +6.26875 q^{75} +11.0978 q^{77} +3.74094 q^{79} +1.00000 q^{81} -2.30798 q^{83} +3.06100 q^{85} -3.93900 q^{87} -10.0586 q^{89} +8.82908 q^{93} +12.7627 q^{95} -16.1293 q^{97} -4.93900 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 6 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 6 q^{5} - 2 q^{7} + 3 q^{9} - 5 q^{11} + 6 q^{15} - q^{17} + 7 q^{19} - 2 q^{21} + 11 q^{25} + 3 q^{27} - 2 q^{29} + 16 q^{31} - 5 q^{33} - 4 q^{35} + 22 q^{37} + 11 q^{41} + 15 q^{43} + 6 q^{45} - 7 q^{47} - 15 q^{49} - q^{51} - 17 q^{53} - 3 q^{55} + 7 q^{57} + 6 q^{59} - 13 q^{61} - 2 q^{63} - 11 q^{67} + 6 q^{73} + 11 q^{75} + 15 q^{77} - 3 q^{79} + 3 q^{81} - 12 q^{83} + 19 q^{85} - 2 q^{87} + q^{89} + 16 q^{93} + 21 q^{95} - 5 q^{97} - 5 q^{99}+O(q^{100}) \) 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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 3.35690 1.50125 0.750625 0.660729i \(-0.229753\pi\)
0.750625 + 0.660729i \(0.229753\pi\)
\(6\) 0 0
\(7\) −2.24698 −0.849278 −0.424639 0.905363i \(-0.639599\pi\)
−0.424639 + 0.905363i \(0.639599\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.93900 −1.48916 −0.744582 0.667531i \(-0.767351\pi\)
−0.744582 + 0.667531i \(0.767351\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 3.35690 0.866747
\(16\) 0 0
\(17\) 0.911854 0.221157 0.110579 0.993867i \(-0.464730\pi\)
0.110579 + 0.993867i \(0.464730\pi\)
\(18\) 0 0
\(19\) 3.80194 0.872224 0.436112 0.899892i \(-0.356355\pi\)
0.436112 + 0.899892i \(0.356355\pi\)
\(20\) 0 0
\(21\) −2.24698 −0.490331
\(22\) 0 0
\(23\) −2.02715 −0.422689 −0.211345 0.977412i \(-0.567784\pi\)
−0.211345 + 0.977412i \(0.567784\pi\)
\(24\) 0 0
\(25\) 6.26875 1.25375
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.93900 −0.731454 −0.365727 0.930722i \(-0.619180\pi\)
−0.365727 + 0.930722i \(0.619180\pi\)
\(30\) 0 0
\(31\) 8.82908 1.58575 0.792875 0.609384i \(-0.208583\pi\)
0.792875 + 0.609384i \(0.208583\pi\)
\(32\) 0 0
\(33\) −4.93900 −0.859770
\(34\) 0 0
\(35\) −7.54288 −1.27498
\(36\) 0 0
\(37\) 8.80194 1.44703 0.723515 0.690309i \(-0.242525\pi\)
0.723515 + 0.690309i \(0.242525\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.93900 1.08369 0.541845 0.840478i \(-0.317726\pi\)
0.541845 + 0.840478i \(0.317726\pi\)
\(42\) 0 0
\(43\) 2.28621 0.348643 0.174322 0.984689i \(-0.444227\pi\)
0.174322 + 0.984689i \(0.444227\pi\)
\(44\) 0 0
\(45\) 3.35690 0.500416
\(46\) 0 0
\(47\) −3.80194 −0.554570 −0.277285 0.960788i \(-0.589435\pi\)
−0.277285 + 0.960788i \(0.589435\pi\)
\(48\) 0 0
\(49\) −1.95108 −0.278726
\(50\) 0 0
\(51\) 0.911854 0.127685
\(52\) 0 0
\(53\) 0.542877 0.0745698 0.0372849 0.999305i \(-0.488129\pi\)
0.0372849 + 0.999305i \(0.488129\pi\)
\(54\) 0 0
\(55\) −16.5797 −2.23561
\(56\) 0 0
\(57\) 3.80194 0.503579
\(58\) 0 0
\(59\) 4.71379 0.613683 0.306842 0.951761i \(-0.400728\pi\)
0.306842 + 0.951761i \(0.400728\pi\)
\(60\) 0 0
\(61\) 3.67994 0.471168 0.235584 0.971854i \(-0.424300\pi\)
0.235584 + 0.971854i \(0.424300\pi\)
\(62\) 0 0
\(63\) −2.24698 −0.283093
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.52111 0.185833 0.0929164 0.995674i \(-0.470381\pi\)
0.0929164 + 0.995674i \(0.470381\pi\)
\(68\) 0 0
\(69\) −2.02715 −0.244040
\(70\) 0 0
\(71\) −2.37867 −0.282296 −0.141148 0.989989i \(-0.545079\pi\)
−0.141148 + 0.989989i \(0.545079\pi\)
\(72\) 0 0
\(73\) 7.41119 0.867414 0.433707 0.901054i \(-0.357205\pi\)
0.433707 + 0.901054i \(0.357205\pi\)
\(74\) 0 0
\(75\) 6.26875 0.723853
\(76\) 0 0
\(77\) 11.0978 1.26472
\(78\) 0 0
\(79\) 3.74094 0.420888 0.210444 0.977606i \(-0.432509\pi\)
0.210444 + 0.977606i \(0.432509\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −2.30798 −0.253334 −0.126667 0.991945i \(-0.540428\pi\)
−0.126667 + 0.991945i \(0.540428\pi\)
\(84\) 0 0
\(85\) 3.06100 0.332012
\(86\) 0 0
\(87\) −3.93900 −0.422305
\(88\) 0 0
\(89\) −10.0586 −1.06621 −0.533105 0.846049i \(-0.678975\pi\)
−0.533105 + 0.846049i \(0.678975\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 8.82908 0.915533
\(94\) 0 0
\(95\) 12.7627 1.30943
\(96\) 0 0
\(97\) −16.1293 −1.63768 −0.818841 0.574021i \(-0.805383\pi\)
−0.818841 + 0.574021i \(0.805383\pi\)
\(98\) 0 0
\(99\) −4.93900 −0.496388
\(100\) 0 0
\(101\) −9.94869 −0.989932 −0.494966 0.868912i \(-0.664819\pi\)
−0.494966 + 0.868912i \(0.664819\pi\)
\(102\) 0 0
\(103\) 10.9879 1.08267 0.541336 0.840806i \(-0.317919\pi\)
0.541336 + 0.840806i \(0.317919\pi\)
\(104\) 0 0
\(105\) −7.54288 −0.736109
\(106\) 0 0
\(107\) 9.87263 0.954423 0.477211 0.878789i \(-0.341648\pi\)
0.477211 + 0.878789i \(0.341648\pi\)
\(108\) 0 0
\(109\) 20.2446 1.93908 0.969540 0.244934i \(-0.0787662\pi\)
0.969540 + 0.244934i \(0.0787662\pi\)
\(110\) 0 0
\(111\) 8.80194 0.835443
\(112\) 0 0
\(113\) 9.69202 0.911749 0.455874 0.890044i \(-0.349327\pi\)
0.455874 + 0.890044i \(0.349327\pi\)
\(114\) 0 0
\(115\) −6.80492 −0.634562
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.04892 −0.187824
\(120\) 0 0
\(121\) 13.3937 1.21761
\(122\) 0 0
\(123\) 6.93900 0.625669
\(124\) 0 0
\(125\) 4.25906 0.380942
\(126\) 0 0
\(127\) −13.8116 −1.22558 −0.612792 0.790244i \(-0.709954\pi\)
−0.612792 + 0.790244i \(0.709954\pi\)
\(128\) 0 0
\(129\) 2.28621 0.201289
\(130\) 0 0
\(131\) −2.99462 −0.261641 −0.130821 0.991406i \(-0.541761\pi\)
−0.130821 + 0.991406i \(0.541761\pi\)
\(132\) 0 0
\(133\) −8.54288 −0.740761
\(134\) 0 0
\(135\) 3.35690 0.288916
\(136\) 0 0
\(137\) 23.0194 1.96668 0.983339 0.181781i \(-0.0581862\pi\)
0.983339 + 0.181781i \(0.0581862\pi\)
\(138\) 0 0
\(139\) −0.982542 −0.0833381 −0.0416690 0.999131i \(-0.513268\pi\)
−0.0416690 + 0.999131i \(0.513268\pi\)
\(140\) 0 0
\(141\) −3.80194 −0.320181
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −13.2228 −1.09810
\(146\) 0 0
\(147\) −1.95108 −0.160923
\(148\) 0 0
\(149\) 10.2591 0.840455 0.420228 0.907419i \(-0.361950\pi\)
0.420228 + 0.907419i \(0.361950\pi\)
\(150\) 0 0
\(151\) 20.1685 1.64129 0.820646 0.571438i \(-0.193614\pi\)
0.820646 + 0.571438i \(0.193614\pi\)
\(152\) 0 0
\(153\) 0.911854 0.0737190
\(154\) 0 0
\(155\) 29.6383 2.38061
\(156\) 0 0
\(157\) 10.4383 0.833070 0.416535 0.909120i \(-0.363244\pi\)
0.416535 + 0.909120i \(0.363244\pi\)
\(158\) 0 0
\(159\) 0.542877 0.0430529
\(160\) 0 0
\(161\) 4.55496 0.358981
\(162\) 0 0
\(163\) 11.0465 0.865231 0.432615 0.901579i \(-0.357591\pi\)
0.432615 + 0.901579i \(0.357591\pi\)
\(164\) 0 0
\(165\) −16.5797 −1.29073
\(166\) 0 0
\(167\) −8.10992 −0.627564 −0.313782 0.949495i \(-0.601596\pi\)
−0.313782 + 0.949495i \(0.601596\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 3.80194 0.290741
\(172\) 0 0
\(173\) −18.0562 −1.37279 −0.686394 0.727230i \(-0.740808\pi\)
−0.686394 + 0.727230i \(0.740808\pi\)
\(174\) 0 0
\(175\) −14.0858 −1.06478
\(176\) 0 0
\(177\) 4.71379 0.354310
\(178\) 0 0
\(179\) 19.8702 1.48517 0.742585 0.669751i \(-0.233599\pi\)
0.742585 + 0.669751i \(0.233599\pi\)
\(180\) 0 0
\(181\) −10.0828 −0.749446 −0.374723 0.927137i \(-0.622262\pi\)
−0.374723 + 0.927137i \(0.622262\pi\)
\(182\) 0 0
\(183\) 3.67994 0.272029
\(184\) 0 0
\(185\) 29.5472 2.17235
\(186\) 0 0
\(187\) −4.50365 −0.329339
\(188\) 0 0
\(189\) −2.24698 −0.163444
\(190\) 0 0
\(191\) 6.58748 0.476653 0.238327 0.971185i \(-0.423401\pi\)
0.238327 + 0.971185i \(0.423401\pi\)
\(192\) 0 0
\(193\) 10.8672 0.782242 0.391121 0.920339i \(-0.372087\pi\)
0.391121 + 0.920339i \(0.372087\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.24160 0.658437 0.329218 0.944254i \(-0.393215\pi\)
0.329218 + 0.944254i \(0.393215\pi\)
\(198\) 0 0
\(199\) 1.56465 0.110915 0.0554574 0.998461i \(-0.482338\pi\)
0.0554574 + 0.998461i \(0.482338\pi\)
\(200\) 0 0
\(201\) 1.52111 0.107291
\(202\) 0 0
\(203\) 8.85086 0.621208
\(204\) 0 0
\(205\) 23.2935 1.62689
\(206\) 0 0
\(207\) −2.02715 −0.140896
\(208\) 0 0
\(209\) −18.7778 −1.29889
\(210\) 0 0
\(211\) −23.2446 −1.60022 −0.800112 0.599851i \(-0.795226\pi\)
−0.800112 + 0.599851i \(0.795226\pi\)
\(212\) 0 0
\(213\) −2.37867 −0.162984
\(214\) 0 0
\(215\) 7.67456 0.523401
\(216\) 0 0
\(217\) −19.8388 −1.34674
\(218\) 0 0
\(219\) 7.41119 0.500802
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −10.7385 −0.719106 −0.359553 0.933125i \(-0.617071\pi\)
−0.359553 + 0.933125i \(0.617071\pi\)
\(224\) 0 0
\(225\) 6.26875 0.417917
\(226\) 0 0
\(227\) 6.97584 0.463003 0.231501 0.972835i \(-0.425636\pi\)
0.231501 + 0.972835i \(0.425636\pi\)
\(228\) 0 0
\(229\) −16.8049 −1.11050 −0.555250 0.831683i \(-0.687378\pi\)
−0.555250 + 0.831683i \(0.687378\pi\)
\(230\) 0 0
\(231\) 11.0978 0.730184
\(232\) 0 0
\(233\) 8.69202 0.569433 0.284717 0.958612i \(-0.408100\pi\)
0.284717 + 0.958612i \(0.408100\pi\)
\(234\) 0 0
\(235\) −12.7627 −0.832547
\(236\) 0 0
\(237\) 3.74094 0.243000
\(238\) 0 0
\(239\) 22.9191 1.48252 0.741258 0.671220i \(-0.234229\pi\)
0.741258 + 0.671220i \(0.234229\pi\)
\(240\) 0 0
\(241\) 21.9801 1.41587 0.707933 0.706280i \(-0.249628\pi\)
0.707933 + 0.706280i \(0.249628\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −6.54958 −0.418437
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −2.30798 −0.146262
\(250\) 0 0
\(251\) −26.6437 −1.68174 −0.840868 0.541241i \(-0.817955\pi\)
−0.840868 + 0.541241i \(0.817955\pi\)
\(252\) 0 0
\(253\) 10.0121 0.629454
\(254\) 0 0
\(255\) 3.06100 0.191687
\(256\) 0 0
\(257\) 23.1444 1.44371 0.721853 0.692047i \(-0.243291\pi\)
0.721853 + 0.692047i \(0.243291\pi\)
\(258\) 0 0
\(259\) −19.7778 −1.22893
\(260\) 0 0
\(261\) −3.93900 −0.243818
\(262\) 0 0
\(263\) 18.5284 1.14251 0.571255 0.820773i \(-0.306457\pi\)
0.571255 + 0.820773i \(0.306457\pi\)
\(264\) 0 0
\(265\) 1.82238 0.111948
\(266\) 0 0
\(267\) −10.0586 −0.615577
\(268\) 0 0
\(269\) 9.75840 0.594980 0.297490 0.954725i \(-0.403851\pi\)
0.297490 + 0.954725i \(0.403851\pi\)
\(270\) 0 0
\(271\) 22.8019 1.38512 0.692560 0.721361i \(-0.256483\pi\)
0.692560 + 0.721361i \(0.256483\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −30.9614 −1.86704
\(276\) 0 0
\(277\) −23.9705 −1.44025 −0.720123 0.693847i \(-0.755914\pi\)
−0.720123 + 0.693847i \(0.755914\pi\)
\(278\) 0 0
\(279\) 8.82908 0.528583
\(280\) 0 0
\(281\) −4.12498 −0.246076 −0.123038 0.992402i \(-0.539264\pi\)
−0.123038 + 0.992402i \(0.539264\pi\)
\(282\) 0 0
\(283\) 15.6558 0.930639 0.465320 0.885143i \(-0.345939\pi\)
0.465320 + 0.885143i \(0.345939\pi\)
\(284\) 0 0
\(285\) 12.7627 0.755998
\(286\) 0 0
\(287\) −15.5918 −0.920354
\(288\) 0 0
\(289\) −16.1685 −0.951090
\(290\) 0 0
\(291\) −16.1293 −0.945516
\(292\) 0 0
\(293\) 22.5948 1.32000 0.660001 0.751265i \(-0.270556\pi\)
0.660001 + 0.751265i \(0.270556\pi\)
\(294\) 0 0
\(295\) 15.8237 0.921292
\(296\) 0 0
\(297\) −4.93900 −0.286590
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −5.13706 −0.296095
\(302\) 0 0
\(303\) −9.94869 −0.571537
\(304\) 0 0
\(305\) 12.3532 0.707341
\(306\) 0 0
\(307\) 6.55496 0.374111 0.187056 0.982349i \(-0.440106\pi\)
0.187056 + 0.982349i \(0.440106\pi\)
\(308\) 0 0
\(309\) 10.9879 0.625081
\(310\) 0 0
\(311\) 12.0392 0.682682 0.341341 0.939940i \(-0.389119\pi\)
0.341341 + 0.939940i \(0.389119\pi\)
\(312\) 0 0
\(313\) −33.8950 −1.91586 −0.957929 0.287005i \(-0.907340\pi\)
−0.957929 + 0.287005i \(0.907340\pi\)
\(314\) 0 0
\(315\) −7.54288 −0.424993
\(316\) 0 0
\(317\) −4.49827 −0.252648 −0.126324 0.991989i \(-0.540318\pi\)
−0.126324 + 0.991989i \(0.540318\pi\)
\(318\) 0 0
\(319\) 19.4547 1.08926
\(320\) 0 0
\(321\) 9.87263 0.551036
\(322\) 0 0
\(323\) 3.46681 0.192899
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 20.2446 1.11953
\(328\) 0 0
\(329\) 8.54288 0.470984
\(330\) 0 0
\(331\) 11.2131 0.616329 0.308165 0.951333i \(-0.400285\pi\)
0.308165 + 0.951333i \(0.400285\pi\)
\(332\) 0 0
\(333\) 8.80194 0.482343
\(334\) 0 0
\(335\) 5.10620 0.278981
\(336\) 0 0
\(337\) 7.04892 0.383979 0.191989 0.981397i \(-0.438506\pi\)
0.191989 + 0.981397i \(0.438506\pi\)
\(338\) 0 0
\(339\) 9.69202 0.526398
\(340\) 0 0
\(341\) −43.6069 −2.36144
\(342\) 0 0
\(343\) 20.1129 1.08599
\(344\) 0 0
\(345\) −6.80492 −0.366365
\(346\) 0 0
\(347\) 2.70410 0.145164 0.0725819 0.997362i \(-0.476876\pi\)
0.0725819 + 0.997362i \(0.476876\pi\)
\(348\) 0 0
\(349\) −0.415502 −0.0222413 −0.0111207 0.999938i \(-0.503540\pi\)
−0.0111207 + 0.999938i \(0.503540\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −32.1672 −1.71209 −0.856044 0.516904i \(-0.827084\pi\)
−0.856044 + 0.516904i \(0.827084\pi\)
\(354\) 0 0
\(355\) −7.98493 −0.423796
\(356\) 0 0
\(357\) −2.04892 −0.108440
\(358\) 0 0
\(359\) −22.3521 −1.17970 −0.589850 0.807513i \(-0.700813\pi\)
−0.589850 + 0.807513i \(0.700813\pi\)
\(360\) 0 0
\(361\) −4.54527 −0.239225
\(362\) 0 0
\(363\) 13.3937 0.702989
\(364\) 0 0
\(365\) 24.8786 1.30221
\(366\) 0 0
\(367\) 2.30260 0.120195 0.0600974 0.998193i \(-0.480859\pi\)
0.0600974 + 0.998193i \(0.480859\pi\)
\(368\) 0 0
\(369\) 6.93900 0.361230
\(370\) 0 0
\(371\) −1.21983 −0.0633305
\(372\) 0 0
\(373\) −19.2760 −0.998076 −0.499038 0.866580i \(-0.666313\pi\)
−0.499038 + 0.866580i \(0.666313\pi\)
\(374\) 0 0
\(375\) 4.25906 0.219937
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 7.33944 0.377002 0.188501 0.982073i \(-0.439637\pi\)
0.188501 + 0.982073i \(0.439637\pi\)
\(380\) 0 0
\(381\) −13.8116 −0.707591
\(382\) 0 0
\(383\) −19.0901 −0.975457 −0.487728 0.872995i \(-0.662174\pi\)
−0.487728 + 0.872995i \(0.662174\pi\)
\(384\) 0 0
\(385\) 37.2543 1.89865
\(386\) 0 0
\(387\) 2.28621 0.116214
\(388\) 0 0
\(389\) −23.9879 −1.21624 −0.608118 0.793847i \(-0.708075\pi\)
−0.608118 + 0.793847i \(0.708075\pi\)
\(390\) 0 0
\(391\) −1.84846 −0.0934807
\(392\) 0 0
\(393\) −2.99462 −0.151059
\(394\) 0 0
\(395\) 12.5579 0.631859
\(396\) 0 0
\(397\) −4.41789 −0.221728 −0.110864 0.993836i \(-0.535362\pi\)
−0.110864 + 0.993836i \(0.535362\pi\)
\(398\) 0 0
\(399\) −8.54288 −0.427679
\(400\) 0 0
\(401\) −20.4088 −1.01917 −0.509583 0.860421i \(-0.670200\pi\)
−0.509583 + 0.860421i \(0.670200\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.35690 0.166805
\(406\) 0 0
\(407\) −43.4728 −2.15487
\(408\) 0 0
\(409\) −22.1491 −1.09520 −0.547602 0.836739i \(-0.684459\pi\)
−0.547602 + 0.836739i \(0.684459\pi\)
\(410\) 0 0
\(411\) 23.0194 1.13546
\(412\) 0 0
\(413\) −10.5918 −0.521188
\(414\) 0 0
\(415\) −7.74764 −0.380317
\(416\) 0 0
\(417\) −0.982542 −0.0481153
\(418\) 0 0
\(419\) −14.7560 −0.720878 −0.360439 0.932783i \(-0.617373\pi\)
−0.360439 + 0.932783i \(0.617373\pi\)
\(420\) 0 0
\(421\) 8.47219 0.412909 0.206455 0.978456i \(-0.433807\pi\)
0.206455 + 0.978456i \(0.433807\pi\)
\(422\) 0 0
\(423\) −3.80194 −0.184857
\(424\) 0 0
\(425\) 5.71618 0.277276
\(426\) 0 0
\(427\) −8.26875 −0.400153
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.39181 −0.115210 −0.0576048 0.998339i \(-0.518346\pi\)
−0.0576048 + 0.998339i \(0.518346\pi\)
\(432\) 0 0
\(433\) −10.4286 −0.501169 −0.250584 0.968095i \(-0.580623\pi\)
−0.250584 + 0.968095i \(0.580623\pi\)
\(434\) 0 0
\(435\) −13.2228 −0.633986
\(436\) 0 0
\(437\) −7.70709 −0.368680
\(438\) 0 0
\(439\) 32.5502 1.55353 0.776767 0.629787i \(-0.216858\pi\)
0.776767 + 0.629787i \(0.216858\pi\)
\(440\) 0 0
\(441\) −1.95108 −0.0929087
\(442\) 0 0
\(443\) −9.58211 −0.455260 −0.227630 0.973748i \(-0.573098\pi\)
−0.227630 + 0.973748i \(0.573098\pi\)
\(444\) 0 0
\(445\) −33.7657 −1.60065
\(446\) 0 0
\(447\) 10.2591 0.485237
\(448\) 0 0
\(449\) −28.3937 −1.33998 −0.669992 0.742369i \(-0.733702\pi\)
−0.669992 + 0.742369i \(0.733702\pi\)
\(450\) 0 0
\(451\) −34.2717 −1.61379
\(452\) 0 0
\(453\) 20.1685 0.947600
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 4.99569 0.233688 0.116844 0.993150i \(-0.462722\pi\)
0.116844 + 0.993150i \(0.462722\pi\)
\(458\) 0 0
\(459\) 0.911854 0.0425617
\(460\) 0 0
\(461\) 1.35258 0.0629961 0.0314981 0.999504i \(-0.489972\pi\)
0.0314981 + 0.999504i \(0.489972\pi\)
\(462\) 0 0
\(463\) 3.36467 0.156369 0.0781846 0.996939i \(-0.475088\pi\)
0.0781846 + 0.996939i \(0.475088\pi\)
\(464\) 0 0
\(465\) 29.6383 1.37444
\(466\) 0 0
\(467\) −6.91079 −0.319793 −0.159897 0.987134i \(-0.551116\pi\)
−0.159897 + 0.987134i \(0.551116\pi\)
\(468\) 0 0
\(469\) −3.41789 −0.157824
\(470\) 0 0
\(471\) 10.4383 0.480973
\(472\) 0 0
\(473\) −11.2916 −0.519188
\(474\) 0 0
\(475\) 23.8334 1.09355
\(476\) 0 0
\(477\) 0.542877 0.0248566
\(478\) 0 0
\(479\) −3.51573 −0.160638 −0.0803189 0.996769i \(-0.525594\pi\)
−0.0803189 + 0.996769i \(0.525594\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 4.55496 0.207258
\(484\) 0 0
\(485\) −54.1444 −2.45857
\(486\) 0 0
\(487\) 12.8479 0.582193 0.291096 0.956694i \(-0.405980\pi\)
0.291096 + 0.956694i \(0.405980\pi\)
\(488\) 0 0
\(489\) 11.0465 0.499541
\(490\) 0 0
\(491\) −28.6708 −1.29390 −0.646948 0.762534i \(-0.723955\pi\)
−0.646948 + 0.762534i \(0.723955\pi\)
\(492\) 0 0
\(493\) −3.59179 −0.161766
\(494\) 0 0
\(495\) −16.5797 −0.745203
\(496\) 0 0
\(497\) 5.34481 0.239748
\(498\) 0 0
\(499\) 33.5555 1.50215 0.751076 0.660215i \(-0.229535\pi\)
0.751076 + 0.660215i \(0.229535\pi\)
\(500\) 0 0
\(501\) −8.10992 −0.362324
\(502\) 0 0
\(503\) −21.5633 −0.961461 −0.480730 0.876868i \(-0.659628\pi\)
−0.480730 + 0.876868i \(0.659628\pi\)
\(504\) 0 0
\(505\) −33.3967 −1.48613
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −8.89008 −0.394046 −0.197023 0.980399i \(-0.563127\pi\)
−0.197023 + 0.980399i \(0.563127\pi\)
\(510\) 0 0
\(511\) −16.6528 −0.736676
\(512\) 0 0
\(513\) 3.80194 0.167860
\(514\) 0 0
\(515\) 36.8853 1.62536
\(516\) 0 0
\(517\) 18.7778 0.825846
\(518\) 0 0
\(519\) −18.0562 −0.792580
\(520\) 0 0
\(521\) 19.3478 0.847642 0.423821 0.905746i \(-0.360688\pi\)
0.423821 + 0.905746i \(0.360688\pi\)
\(522\) 0 0
\(523\) 12.5948 0.550731 0.275366 0.961340i \(-0.411201\pi\)
0.275366 + 0.961340i \(0.411201\pi\)
\(524\) 0 0
\(525\) −14.0858 −0.614753
\(526\) 0 0
\(527\) 8.05084 0.350700
\(528\) 0 0
\(529\) −18.8907 −0.821334
\(530\) 0 0
\(531\) 4.71379 0.204561
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 33.1414 1.43283
\(536\) 0 0
\(537\) 19.8702 0.857464
\(538\) 0 0
\(539\) 9.63640 0.415069
\(540\) 0 0
\(541\) 29.0019 1.24689 0.623445 0.781867i \(-0.285733\pi\)
0.623445 + 0.781867i \(0.285733\pi\)
\(542\) 0 0
\(543\) −10.0828 −0.432693
\(544\) 0 0
\(545\) 67.9590 2.91104
\(546\) 0 0
\(547\) −27.7006 −1.18439 −0.592197 0.805793i \(-0.701739\pi\)
−0.592197 + 0.805793i \(0.701739\pi\)
\(548\) 0 0
\(549\) 3.67994 0.157056
\(550\) 0 0
\(551\) −14.9758 −0.637992
\(552\) 0 0
\(553\) −8.40581 −0.357452
\(554\) 0 0
\(555\) 29.5472 1.25421
\(556\) 0 0
\(557\) 6.80971 0.288537 0.144268 0.989539i \(-0.453917\pi\)
0.144268 + 0.989539i \(0.453917\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −4.50365 −0.190144
\(562\) 0 0
\(563\) 19.2524 0.811390 0.405695 0.914008i \(-0.367030\pi\)
0.405695 + 0.914008i \(0.367030\pi\)
\(564\) 0 0
\(565\) 32.5351 1.36876
\(566\) 0 0
\(567\) −2.24698 −0.0943643
\(568\) 0 0
\(569\) 7.84846 0.329025 0.164512 0.986375i \(-0.447395\pi\)
0.164512 + 0.986375i \(0.447395\pi\)
\(570\) 0 0
\(571\) −29.8568 −1.24947 −0.624735 0.780837i \(-0.714793\pi\)
−0.624735 + 0.780837i \(0.714793\pi\)
\(572\) 0 0
\(573\) 6.58748 0.275196
\(574\) 0 0
\(575\) −12.7077 −0.529947
\(576\) 0 0
\(577\) 8.97823 0.373769 0.186884 0.982382i \(-0.440161\pi\)
0.186884 + 0.982382i \(0.440161\pi\)
\(578\) 0 0
\(579\) 10.8672 0.451627
\(580\) 0 0
\(581\) 5.18598 0.215151
\(582\) 0 0
\(583\) −2.68127 −0.111047
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −23.8538 −0.984553 −0.492277 0.870439i \(-0.663835\pi\)
−0.492277 + 0.870439i \(0.663835\pi\)
\(588\) 0 0
\(589\) 33.5676 1.38313
\(590\) 0 0
\(591\) 9.24160 0.380149
\(592\) 0 0
\(593\) 11.9866 0.492230 0.246115 0.969241i \(-0.420846\pi\)
0.246115 + 0.969241i \(0.420846\pi\)
\(594\) 0 0
\(595\) −6.87800 −0.281971
\(596\) 0 0
\(597\) 1.56465 0.0640367
\(598\) 0 0
\(599\) −29.1142 −1.18958 −0.594788 0.803883i \(-0.702764\pi\)
−0.594788 + 0.803883i \(0.702764\pi\)
\(600\) 0 0
\(601\) 37.1366 1.51483 0.757417 0.652932i \(-0.226461\pi\)
0.757417 + 0.652932i \(0.226461\pi\)
\(602\) 0 0
\(603\) 1.52111 0.0619442
\(604\) 0 0
\(605\) 44.9614 1.82794
\(606\) 0 0
\(607\) 14.2325 0.577680 0.288840 0.957377i \(-0.406731\pi\)
0.288840 + 0.957377i \(0.406731\pi\)
\(608\) 0 0
\(609\) 8.85086 0.358655
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 30.3139 1.22437 0.612184 0.790715i \(-0.290291\pi\)
0.612184 + 0.790715i \(0.290291\pi\)
\(614\) 0 0
\(615\) 23.2935 0.939285
\(616\) 0 0
\(617\) −24.9638 −1.00500 −0.502501 0.864576i \(-0.667587\pi\)
−0.502501 + 0.864576i \(0.667587\pi\)
\(618\) 0 0
\(619\) −35.6122 −1.43138 −0.715688 0.698420i \(-0.753887\pi\)
−0.715688 + 0.698420i \(0.753887\pi\)
\(620\) 0 0
\(621\) −2.02715 −0.0813466
\(622\) 0 0
\(623\) 22.6015 0.905509
\(624\) 0 0
\(625\) −17.0465 −0.681861
\(626\) 0 0
\(627\) −18.7778 −0.749912
\(628\) 0 0
\(629\) 8.02608 0.320021
\(630\) 0 0
\(631\) −23.8829 −0.950763 −0.475382 0.879780i \(-0.657690\pi\)
−0.475382 + 0.879780i \(0.657690\pi\)
\(632\) 0 0
\(633\) −23.2446 −0.923889
\(634\) 0 0
\(635\) −46.3642 −1.83991
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2.37867 −0.0940986
\(640\) 0 0
\(641\) −24.6577 −0.973920 −0.486960 0.873424i \(-0.661894\pi\)
−0.486960 + 0.873424i \(0.661894\pi\)
\(642\) 0 0
\(643\) −47.8165 −1.88570 −0.942850 0.333218i \(-0.891866\pi\)
−0.942850 + 0.333218i \(0.891866\pi\)
\(644\) 0 0
\(645\) 7.67456 0.302186
\(646\) 0 0
\(647\) 5.38942 0.211880 0.105940 0.994373i \(-0.466215\pi\)
0.105940 + 0.994373i \(0.466215\pi\)
\(648\) 0 0
\(649\) −23.2814 −0.913876
\(650\) 0 0
\(651\) −19.8388 −0.777543
\(652\) 0 0
\(653\) −5.03790 −0.197148 −0.0985741 0.995130i \(-0.531428\pi\)
−0.0985741 + 0.995130i \(0.531428\pi\)
\(654\) 0 0
\(655\) −10.0526 −0.392789
\(656\) 0 0
\(657\) 7.41119 0.289138
\(658\) 0 0
\(659\) 43.9812 1.71326 0.856632 0.515927i \(-0.172553\pi\)
0.856632 + 0.515927i \(0.172553\pi\)
\(660\) 0 0
\(661\) −38.2194 −1.48656 −0.743280 0.668980i \(-0.766731\pi\)
−0.743280 + 0.668980i \(0.766731\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −28.6775 −1.11207
\(666\) 0 0
\(667\) 7.98493 0.309178
\(668\) 0 0
\(669\) −10.7385 −0.415176
\(670\) 0 0
\(671\) −18.1752 −0.701647
\(672\) 0 0
\(673\) −6.66487 −0.256912 −0.128456 0.991715i \(-0.541002\pi\)
−0.128456 + 0.991715i \(0.541002\pi\)
\(674\) 0 0
\(675\) 6.26875 0.241284
\(676\) 0 0
\(677\) 4.80194 0.184553 0.0922767 0.995733i \(-0.470586\pi\)
0.0922767 + 0.995733i \(0.470586\pi\)
\(678\) 0 0
\(679\) 36.2422 1.39085
\(680\) 0 0
\(681\) 6.97584 0.267315
\(682\) 0 0
\(683\) −11.2591 −0.430816 −0.215408 0.976524i \(-0.569108\pi\)
−0.215408 + 0.976524i \(0.569108\pi\)
\(684\) 0 0
\(685\) 77.2737 2.95247
\(686\) 0 0
\(687\) −16.8049 −0.641148
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 24.7144 0.940179 0.470090 0.882619i \(-0.344222\pi\)
0.470090 + 0.882619i \(0.344222\pi\)
\(692\) 0 0
\(693\) 11.0978 0.421572
\(694\) 0 0
\(695\) −3.29829 −0.125111
\(696\) 0 0
\(697\) 6.32736 0.239666
\(698\) 0 0
\(699\) 8.69202 0.328762
\(700\) 0 0
\(701\) 25.8920 0.977927 0.488964 0.872304i \(-0.337375\pi\)
0.488964 + 0.872304i \(0.337375\pi\)
\(702\) 0 0
\(703\) 33.4644 1.26213
\(704\) 0 0
\(705\) −12.7627 −0.480671
\(706\) 0 0
\(707\) 22.3545 0.840728
\(708\) 0 0
\(709\) 0.0851621 0.00319833 0.00159916 0.999999i \(-0.499491\pi\)
0.00159916 + 0.999999i \(0.499491\pi\)
\(710\) 0 0
\(711\) 3.74094 0.140296
\(712\) 0 0
\(713\) −17.8979 −0.670280
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 22.9191 0.855931
\(718\) 0 0
\(719\) −27.0508 −1.00883 −0.504413 0.863463i \(-0.668291\pi\)
−0.504413 + 0.863463i \(0.668291\pi\)
\(720\) 0 0
\(721\) −24.6896 −0.919490
\(722\) 0 0
\(723\) 21.9801 0.817451
\(724\) 0 0
\(725\) −24.6926 −0.917061
\(726\) 0 0
\(727\) 47.1584 1.74901 0.874503 0.485019i \(-0.161187\pi\)
0.874503 + 0.485019i \(0.161187\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.08469 0.0771050
\(732\) 0 0
\(733\) −9.68186 −0.357608 −0.178804 0.983885i \(-0.557223\pi\)
−0.178804 + 0.983885i \(0.557223\pi\)
\(734\) 0 0
\(735\) −6.54958 −0.241585
\(736\) 0 0
\(737\) −7.51275 −0.276736
\(738\) 0 0
\(739\) −36.3139 −1.33583 −0.667915 0.744238i \(-0.732813\pi\)
−0.667915 + 0.744238i \(0.732813\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 41.7536 1.53179 0.765896 0.642965i \(-0.222296\pi\)
0.765896 + 0.642965i \(0.222296\pi\)
\(744\) 0 0
\(745\) 34.4386 1.26173
\(746\) 0 0
\(747\) −2.30798 −0.0844445
\(748\) 0 0
\(749\) −22.1836 −0.810571
\(750\) 0 0
\(751\) 22.1927 0.809823 0.404911 0.914356i \(-0.367302\pi\)
0.404911 + 0.914356i \(0.367302\pi\)
\(752\) 0 0
\(753\) −26.6437 −0.970950
\(754\) 0 0
\(755\) 67.7036 2.46399
\(756\) 0 0
\(757\) 9.23729 0.335735 0.167868 0.985810i \(-0.446312\pi\)
0.167868 + 0.985810i \(0.446312\pi\)
\(758\) 0 0
\(759\) 10.0121 0.363416
\(760\) 0 0
\(761\) −7.50173 −0.271937 −0.135969 0.990713i \(-0.543415\pi\)
−0.135969 + 0.990713i \(0.543415\pi\)
\(762\) 0 0
\(763\) −45.4892 −1.64682
\(764\) 0 0
\(765\) 3.06100 0.110671
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 5.14005 0.185355 0.0926774 0.995696i \(-0.470457\pi\)
0.0926774 + 0.995696i \(0.470457\pi\)
\(770\) 0 0
\(771\) 23.1444 0.833524
\(772\) 0 0
\(773\) −8.50173 −0.305786 −0.152893 0.988243i \(-0.548859\pi\)
−0.152893 + 0.988243i \(0.548859\pi\)
\(774\) 0 0
\(775\) 55.3473 1.98813
\(776\) 0 0
\(777\) −19.7778 −0.709524
\(778\) 0 0
\(779\) 26.3817 0.945221
\(780\) 0 0
\(781\) 11.7482 0.420385
\(782\) 0 0
\(783\) −3.93900 −0.140768
\(784\) 0 0
\(785\) 35.0404 1.25065
\(786\) 0 0
\(787\) 7.78554 0.277525 0.138762 0.990326i \(-0.455688\pi\)
0.138762 + 0.990326i \(0.455688\pi\)
\(788\) 0 0
\(789\) 18.5284 0.659629
\(790\) 0 0
\(791\) −21.7778 −0.774329
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 1.82238 0.0646332
\(796\) 0 0
\(797\) 21.3840 0.757462 0.378731 0.925507i \(-0.376361\pi\)
0.378731 + 0.925507i \(0.376361\pi\)
\(798\) 0 0
\(799\) −3.46681 −0.122647
\(800\) 0 0
\(801\) −10.0586 −0.355403
\(802\) 0 0
\(803\) −36.6039 −1.29172
\(804\) 0 0
\(805\) 15.2905 0.538920
\(806\) 0 0
\(807\) 9.75840 0.343512
\(808\) 0 0
\(809\) 2.28621 0.0803788 0.0401894 0.999192i \(-0.487204\pi\)
0.0401894 + 0.999192i \(0.487204\pi\)
\(810\) 0 0
\(811\) −17.8079 −0.625320 −0.312660 0.949865i \(-0.601220\pi\)
−0.312660 + 0.949865i \(0.601220\pi\)
\(812\) 0 0
\(813\) 22.8019 0.799699
\(814\) 0 0
\(815\) 37.0820 1.29893
\(816\) 0 0
\(817\) 8.69202 0.304095
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.9054 0.485302 0.242651 0.970114i \(-0.421983\pi\)
0.242651 + 0.970114i \(0.421983\pi\)
\(822\) 0 0
\(823\) 7.24831 0.252660 0.126330 0.991988i \(-0.459680\pi\)
0.126330 + 0.991988i \(0.459680\pi\)
\(824\) 0 0
\(825\) −30.9614 −1.07794
\(826\) 0 0
\(827\) −54.1191 −1.88191 −0.940953 0.338536i \(-0.890068\pi\)
−0.940953 + 0.338536i \(0.890068\pi\)
\(828\) 0 0
\(829\) −5.79178 −0.201157 −0.100578 0.994929i \(-0.532069\pi\)
−0.100578 + 0.994929i \(0.532069\pi\)
\(830\) 0 0
\(831\) −23.9705 −0.831526
\(832\) 0 0
\(833\) −1.77910 −0.0616422
\(834\) 0 0
\(835\) −27.2241 −0.942130
\(836\) 0 0
\(837\) 8.82908 0.305178
\(838\) 0 0
\(839\) 5.29350 0.182752 0.0913760 0.995816i \(-0.470873\pi\)
0.0913760 + 0.995816i \(0.470873\pi\)
\(840\) 0 0
\(841\) −13.4843 −0.464975
\(842\) 0 0
\(843\) −4.12498 −0.142072
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −30.0954 −1.03409
\(848\) 0 0
\(849\) 15.6558 0.537305
\(850\) 0 0
\(851\) −17.8428 −0.611644
\(852\) 0 0
\(853\) −13.5961 −0.465522 −0.232761 0.972534i \(-0.574776\pi\)
−0.232761 + 0.972534i \(0.574776\pi\)
\(854\) 0 0
\(855\) 12.7627 0.436475
\(856\) 0 0
\(857\) −23.8323 −0.814097 −0.407048 0.913407i \(-0.633442\pi\)
−0.407048 + 0.913407i \(0.633442\pi\)
\(858\) 0 0
\(859\) 26.9861 0.920754 0.460377 0.887723i \(-0.347714\pi\)
0.460377 + 0.887723i \(0.347714\pi\)
\(860\) 0 0
\(861\) −15.5918 −0.531367
\(862\) 0 0
\(863\) 27.0291 0.920080 0.460040 0.887898i \(-0.347835\pi\)
0.460040 + 0.887898i \(0.347835\pi\)
\(864\) 0 0
\(865\) −60.6128 −2.06090
\(866\) 0 0
\(867\) −16.1685 −0.549112
\(868\) 0 0
\(869\) −18.4765 −0.626772
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −16.1293 −0.545894
\(874\) 0 0
\(875\) −9.57002 −0.323526
\(876\) 0 0
\(877\) 40.8780 1.38035 0.690176 0.723642i \(-0.257533\pi\)
0.690176 + 0.723642i \(0.257533\pi\)
\(878\) 0 0
\(879\) 22.5948 0.762103
\(880\) 0 0
\(881\) −22.4964 −0.757921 −0.378961 0.925413i \(-0.623718\pi\)
−0.378961 + 0.925413i \(0.623718\pi\)
\(882\) 0 0
\(883\) 4.16315 0.140101 0.0700505 0.997543i \(-0.477684\pi\)
0.0700505 + 0.997543i \(0.477684\pi\)
\(884\) 0 0
\(885\) 15.8237 0.531908
\(886\) 0 0
\(887\) 26.3002 0.883075 0.441537 0.897243i \(-0.354433\pi\)
0.441537 + 0.897243i \(0.354433\pi\)
\(888\) 0 0
\(889\) 31.0344 1.04086
\(890\) 0 0
\(891\) −4.93900 −0.165463
\(892\) 0 0
\(893\) −14.4547 −0.483709
\(894\) 0 0
\(895\) 66.7023 2.22961
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −34.7778 −1.15990
\(900\) 0 0
\(901\) 0.495024 0.0164916
\(902\) 0 0
\(903\) −5.13706 −0.170951
\(904\) 0 0
\(905\) −33.8468 −1.12511
\(906\) 0 0
\(907\) 57.9114 1.92292 0.961458 0.274952i \(-0.0886620\pi\)
0.961458 + 0.274952i \(0.0886620\pi\)
\(908\) 0 0
\(909\) −9.94869 −0.329977
\(910\) 0 0
\(911\) 0.286799 0.00950208 0.00475104 0.999989i \(-0.498488\pi\)
0.00475104 + 0.999989i \(0.498488\pi\)
\(912\) 0 0
\(913\) 11.3991 0.377255
\(914\) 0 0
\(915\) 12.3532 0.408383
\(916\) 0 0
\(917\) 6.72886 0.222206
\(918\) 0 0
\(919\) 31.1239 1.02668 0.513342 0.858184i \(-0.328407\pi\)
0.513342 + 0.858184i \(0.328407\pi\)
\(920\) 0 0
\(921\) 6.55496 0.215993
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 55.1771 1.81421
\(926\) 0 0
\(927\) 10.9879 0.360891
\(928\) 0 0
\(929\) 7.62671 0.250224 0.125112 0.992143i \(-0.460071\pi\)
0.125112 + 0.992143i \(0.460071\pi\)
\(930\) 0 0
\(931\) −7.41789 −0.243112
\(932\) 0 0
\(933\) 12.0392 0.394147
\(934\) 0 0
\(935\) −15.1183 −0.494421
\(936\) 0 0
\(937\) −5.67324 −0.185337 −0.0926683 0.995697i \(-0.529540\pi\)
−0.0926683 + 0.995697i \(0.529540\pi\)
\(938\) 0 0
\(939\) −33.8950 −1.10612
\(940\) 0 0
\(941\) 41.5394 1.35415 0.677073 0.735916i \(-0.263248\pi\)
0.677073 + 0.735916i \(0.263248\pi\)
\(942\) 0 0
\(943\) −14.0664 −0.458064
\(944\) 0 0
\(945\) −7.54288 −0.245370
\(946\) 0 0
\(947\) 47.3110 1.53740 0.768700 0.639610i \(-0.220904\pi\)
0.768700 + 0.639610i \(0.220904\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −4.49827 −0.145866
\(952\) 0 0
\(953\) −34.3435 −1.11249 −0.556247 0.831017i \(-0.687759\pi\)
−0.556247 + 0.831017i \(0.687759\pi\)
\(954\) 0 0
\(955\) 22.1135 0.715576
\(956\) 0 0
\(957\) 19.4547 0.628882
\(958\) 0 0
\(959\) −51.7241 −1.67026
\(960\) 0 0
\(961\) 46.9527 1.51460
\(962\) 0 0
\(963\) 9.87263 0.318141
\(964\) 0 0
\(965\) 36.4802 1.17434
\(966\) 0 0
\(967\) −48.5096 −1.55996 −0.779982 0.625802i \(-0.784772\pi\)
−0.779982 + 0.625802i \(0.784772\pi\)
\(968\) 0 0
\(969\) 3.46681 0.111370
\(970\) 0 0
\(971\) −41.4650 −1.33068 −0.665338 0.746542i \(-0.731712\pi\)
−0.665338 + 0.746542i \(0.731712\pi\)
\(972\) 0 0
\(973\) 2.20775 0.0707772
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −2.09677 −0.0670816 −0.0335408 0.999437i \(-0.510678\pi\)
−0.0335408 + 0.999437i \(0.510678\pi\)
\(978\) 0 0
\(979\) 49.6795 1.58776
\(980\) 0 0
\(981\) 20.2446 0.646360
\(982\) 0 0
\(983\) −25.2336 −0.804826 −0.402413 0.915458i \(-0.631828\pi\)
−0.402413 + 0.915458i \(0.631828\pi\)
\(984\) 0 0
\(985\) 31.0231 0.988478
\(986\) 0 0
\(987\) 8.54288 0.271923
\(988\) 0 0
\(989\) −4.63448 −0.147368
\(990\) 0 0
\(991\) −12.0489 −0.382746 −0.191373 0.981517i \(-0.561294\pi\)
−0.191373 + 0.981517i \(0.561294\pi\)
\(992\) 0 0
\(993\) 11.2131 0.355838
\(994\) 0 0
\(995\) 5.25236 0.166511
\(996\) 0 0
\(997\) −1.43403 −0.0454160 −0.0227080 0.999742i \(-0.507229\pi\)
−0.0227080 + 0.999742i \(0.507229\pi\)
\(998\) 0 0
\(999\) 8.80194 0.278481
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 8112.2.a.cp.1.2 3
4.3 odd 2 507.2.a.l.1.1 yes 3
12.11 even 2 1521.2.a.n.1.3 3
13.12 even 2 8112.2.a.cg.1.2 3
52.3 odd 6 507.2.e.i.22.3 6
52.7 even 12 507.2.j.i.361.3 12
52.11 even 12 507.2.j.i.316.4 12
52.15 even 12 507.2.j.i.316.3 12
52.19 even 12 507.2.j.i.361.4 12
52.23 odd 6 507.2.e.l.22.1 6
52.31 even 4 507.2.b.f.337.4 6
52.35 odd 6 507.2.e.i.484.3 6
52.43 odd 6 507.2.e.l.484.1 6
52.47 even 4 507.2.b.f.337.3 6
52.51 odd 2 507.2.a.i.1.3 3
156.47 odd 4 1521.2.b.k.1351.4 6
156.83 odd 4 1521.2.b.k.1351.3 6
156.155 even 2 1521.2.a.s.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
507.2.a.i.1.3 3 52.51 odd 2
507.2.a.l.1.1 yes 3 4.3 odd 2
507.2.b.f.337.3 6 52.47 even 4
507.2.b.f.337.4 6 52.31 even 4
507.2.e.i.22.3 6 52.3 odd 6
507.2.e.i.484.3 6 52.35 odd 6
507.2.e.l.22.1 6 52.23 odd 6
507.2.e.l.484.1 6 52.43 odd 6
507.2.j.i.316.3 12 52.15 even 12
507.2.j.i.316.4 12 52.11 even 12
507.2.j.i.361.3 12 52.7 even 12
507.2.j.i.361.4 12 52.19 even 12
1521.2.a.n.1.3 3 12.11 even 2
1521.2.a.s.1.1 3 156.155 even 2
1521.2.b.k.1351.3 6 156.83 odd 4
1521.2.b.k.1351.4 6 156.47 odd 4
8112.2.a.cg.1.2 3 13.12 even 2
8112.2.a.cp.1.2 3 1.1 even 1 trivial