Properties

Label 8112.2.a.bs.1.2
Level $8112$
Weight $2$
Character 8112.1
Self dual yes
Analytic conductor $64.775$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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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: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 156)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +3.46410 q^{5} +1.00000 q^{9} +3.46410 q^{11} +3.46410 q^{15} -6.00000 q^{17} +6.92820 q^{19} +7.00000 q^{25} +1.00000 q^{27} -6.00000 q^{29} +6.92820 q^{31} +3.46410 q^{33} -3.46410 q^{41} +8.00000 q^{43} +3.46410 q^{45} +3.46410 q^{47} -7.00000 q^{49} -6.00000 q^{51} +6.00000 q^{53} +12.0000 q^{55} +6.92820 q^{57} -3.46410 q^{59} +10.0000 q^{61} -13.8564 q^{67} -10.3923 q^{71} +6.92820 q^{73} +7.00000 q^{75} +8.00000 q^{79} +1.00000 q^{81} +3.46410 q^{83} -20.7846 q^{85} -6.00000 q^{87} +17.3205 q^{89} +6.92820 q^{93} +24.0000 q^{95} +6.92820 q^{97} +3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 2 q^{9} - 12 q^{17} + 14 q^{25} + 2 q^{27} - 12 q^{29} + 16 q^{43} - 14 q^{49} - 12 q^{51} + 12 q^{53} + 24 q^{55} + 20 q^{61} + 14 q^{75} + 16 q^{79} + 2 q^{81} - 12 q^{87} + 48 q^{95}+O(q^{100}) \) 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
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) 3.46410 0.894427
\(16\) 0 0
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) 6.92820 1.58944 0.794719 0.606977i \(-0.207618\pi\)
0.794719 + 0.606977i \(0.207618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 6.92820 1.24434 0.622171 0.782881i \(-0.286251\pi\)
0.622171 + 0.782881i \(0.286251\pi\)
\(32\) 0 0
\(33\) 3.46410 0.603023
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 3.46410 0.516398
\(46\) 0 0
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 12.0000 1.61808
\(56\) 0 0
\(57\) 6.92820 0.917663
\(58\) 0 0
\(59\) −3.46410 −0.450988 −0.225494 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) 10.0000 1.28037 0.640184 0.768221i \(-0.278858\pi\)
0.640184 + 0.768221i \(0.278858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −13.8564 −1.69283 −0.846415 0.532524i \(-0.821244\pi\)
−0.846415 + 0.532524i \(0.821244\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.3923 −1.23334 −0.616670 0.787222i \(-0.711519\pi\)
−0.616670 + 0.787222i \(0.711519\pi\)
\(72\) 0 0
\(73\) 6.92820 0.810885 0.405442 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) 0 0
\(75\) 7.00000 0.808290
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.46410 0.380235 0.190117 0.981761i \(-0.439113\pi\)
0.190117 + 0.981761i \(0.439113\pi\)
\(84\) 0 0
\(85\) −20.7846 −2.25441
\(86\) 0 0
\(87\) −6.00000 −0.643268
\(88\) 0 0
\(89\) 17.3205 1.83597 0.917985 0.396615i \(-0.129815\pi\)
0.917985 + 0.396615i \(0.129815\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.92820 0.718421
\(94\) 0 0
\(95\) 24.0000 2.46235
\(96\) 0 0
\(97\) 6.92820 0.703452 0.351726 0.936103i \(-0.385595\pi\)
0.351726 + 0.936103i \(0.385595\pi\)
\(98\) 0 0
\(99\) 3.46410 0.348155
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) −6.92820 −0.663602 −0.331801 0.943349i \(-0.607656\pi\)
−0.331801 + 0.943349i \(0.607656\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −3.46410 −0.312348
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 8.00000 0.704361
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 3.46410 0.298142
\(136\) 0 0
\(137\) 10.3923 0.887875 0.443937 0.896058i \(-0.353581\pi\)
0.443937 + 0.896058i \(0.353581\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 3.46410 0.291730
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −20.7846 −1.72607
\(146\) 0 0
\(147\) −7.00000 −0.577350
\(148\) 0 0
\(149\) −17.3205 −1.41895 −0.709476 0.704730i \(-0.751068\pi\)
−0.709476 + 0.704730i \(0.751068\pi\)
\(150\) 0 0
\(151\) 6.92820 0.563809 0.281905 0.959442i \(-0.409034\pi\)
0.281905 + 0.959442i \(0.409034\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) 0 0
\(155\) 24.0000 1.92773
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 6.00000 0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −20.7846 −1.62798 −0.813988 0.580881i \(-0.802708\pi\)
−0.813988 + 0.580881i \(0.802708\pi\)
\(164\) 0 0
\(165\) 12.0000 0.934199
\(166\) 0 0
\(167\) 17.3205 1.34030 0.670151 0.742225i \(-0.266230\pi\)
0.670151 + 0.742225i \(0.266230\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 6.92820 0.529813
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −3.46410 −0.260378
\(178\) 0 0
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 22.0000 1.63525 0.817624 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −20.7846 −1.51992
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −13.8564 −0.997406 −0.498703 0.866773i \(-0.666190\pi\)
−0.498703 + 0.866773i \(0.666190\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 17.3205 1.23404 0.617018 0.786949i \(-0.288341\pi\)
0.617018 + 0.786949i \(0.288341\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −13.8564 −0.977356
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 24.0000 1.66011
\(210\) 0 0
\(211\) −4.00000 −0.275371 −0.137686 0.990476i \(-0.543966\pi\)
−0.137686 + 0.990476i \(0.543966\pi\)
\(212\) 0 0
\(213\) −10.3923 −0.712069
\(214\) 0 0
\(215\) 27.7128 1.89000
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 6.92820 0.468165
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 6.92820 0.463947 0.231973 0.972722i \(-0.425482\pi\)
0.231973 + 0.972722i \(0.425482\pi\)
\(224\) 0 0
\(225\) 7.00000 0.466667
\(226\) 0 0
\(227\) 10.3923 0.689761 0.344881 0.938647i \(-0.387919\pi\)
0.344881 + 0.938647i \(0.387919\pi\)
\(228\) 0 0
\(229\) −6.92820 −0.457829 −0.228914 0.973447i \(-0.573518\pi\)
−0.228914 + 0.973447i \(0.573518\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −10.3923 −0.672222 −0.336111 0.941822i \(-0.609112\pi\)
−0.336111 + 0.941822i \(0.609112\pi\)
\(240\) 0 0
\(241\) 20.7846 1.33885 0.669427 0.742878i \(-0.266540\pi\)
0.669427 + 0.742878i \(0.266540\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −24.2487 −1.54919
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 3.46410 0.219529
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −20.7846 −1.30158
\(256\) 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
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 20.7846 1.27679
\(266\) 0 0
\(267\) 17.3205 1.06000
\(268\) 0 0
\(269\) −30.0000 −1.82913 −0.914566 0.404436i \(-0.867468\pi\)
−0.914566 + 0.404436i \(0.867468\pi\)
\(270\) 0 0
\(271\) −27.7128 −1.68343 −0.841717 0.539919i \(-0.818455\pi\)
−0.841717 + 0.539919i \(0.818455\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 24.2487 1.46225
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) 6.92820 0.414781
\(280\) 0 0
\(281\) −17.3205 −1.03325 −0.516627 0.856210i \(-0.672813\pi\)
−0.516627 + 0.856210i \(0.672813\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 24.0000 1.42164
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 6.92820 0.406138
\(292\) 0 0
\(293\) −3.46410 −0.202375 −0.101187 0.994867i \(-0.532264\pi\)
−0.101187 + 0.994867i \(0.532264\pi\)
\(294\) 0 0
\(295\) −12.0000 −0.698667
\(296\) 0 0
\(297\) 3.46410 0.201008
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) 34.6410 1.98354
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 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
\(317\) −24.2487 −1.36194 −0.680972 0.732310i \(-0.738442\pi\)
−0.680972 + 0.732310i \(0.738442\pi\)
\(318\) 0 0
\(319\) −20.7846 −1.16371
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −41.5692 −2.31297
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −6.92820 −0.383131
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −13.8564 −0.761617 −0.380808 0.924654i \(-0.624354\pi\)
−0.380808 + 0.924654i \(0.624354\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −48.0000 −2.62252
\(336\) 0 0
\(337\) −10.0000 −0.544735 −0.272367 0.962193i \(-0.587807\pi\)
−0.272367 + 0.962193i \(0.587807\pi\)
\(338\) 0 0
\(339\) −18.0000 −0.977626
\(340\) 0 0
\(341\) 24.0000 1.29967
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000 0.644194 0.322097 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10.3923 0.553127 0.276563 0.960996i \(-0.410804\pi\)
0.276563 + 0.960996i \(0.410804\pi\)
\(354\) 0 0
\(355\) −36.0000 −1.91068
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −24.2487 −1.27980 −0.639899 0.768459i \(-0.721024\pi\)
−0.639899 + 0.768459i \(0.721024\pi\)
\(360\) 0 0
\(361\) 29.0000 1.52632
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 24.0000 1.25622
\(366\) 0 0
\(367\) 28.0000 1.46159 0.730794 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) −3.46410 −0.180334
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.00000 −0.103556 −0.0517780 0.998659i \(-0.516489\pi\)
−0.0517780 + 0.998659i \(0.516489\pi\)
\(374\) 0 0
\(375\) 6.92820 0.357771
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −20.7846 −1.06763 −0.533817 0.845600i \(-0.679243\pi\)
−0.533817 + 0.845600i \(0.679243\pi\)
\(380\) 0 0
\(381\) 16.0000 0.819705
\(382\) 0 0
\(383\) 24.2487 1.23905 0.619526 0.784976i \(-0.287325\pi\)
0.619526 + 0.784976i \(0.287325\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 8.00000 0.406663
\(388\) 0 0
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 27.7128 1.39438
\(396\) 0 0
\(397\) −13.8564 −0.695433 −0.347717 0.937600i \(-0.613043\pi\)
−0.347717 + 0.937600i \(0.613043\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.3923 −0.518967 −0.259483 0.965748i \(-0.583552\pi\)
−0.259483 + 0.965748i \(0.583552\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 3.46410 0.172133
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 20.7846 1.02773 0.513866 0.857870i \(-0.328213\pi\)
0.513866 + 0.857870i \(0.328213\pi\)
\(410\) 0 0
\(411\) 10.3923 0.512615
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 12.0000 0.589057
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −20.7846 −1.01298 −0.506490 0.862246i \(-0.669057\pi\)
−0.506490 + 0.862246i \(0.669057\pi\)
\(422\) 0 0
\(423\) 3.46410 0.168430
\(424\) 0 0
\(425\) −42.0000 −2.03730
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −31.1769 −1.50174 −0.750870 0.660451i \(-0.770365\pi\)
−0.750870 + 0.660451i \(0.770365\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) −20.7846 −0.996546
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 20.0000 0.954548 0.477274 0.878755i \(-0.341625\pi\)
0.477274 + 0.878755i \(0.341625\pi\)
\(440\) 0 0
\(441\) −7.00000 −0.333333
\(442\) 0 0
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 60.0000 2.84427
\(446\) 0 0
\(447\) −17.3205 −0.819232
\(448\) 0 0
\(449\) 3.46410 0.163481 0.0817405 0.996654i \(-0.473952\pi\)
0.0817405 + 0.996654i \(0.473952\pi\)
\(450\) 0 0
\(451\) −12.0000 −0.565058
\(452\) 0 0
\(453\) 6.92820 0.325515
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 6.92820 0.324088 0.162044 0.986784i \(-0.448191\pi\)
0.162044 + 0.986784i \(0.448191\pi\)
\(458\) 0 0
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) −10.3923 −0.484018 −0.242009 0.970274i \(-0.577806\pi\)
−0.242009 + 0.970274i \(0.577806\pi\)
\(462\) 0 0
\(463\) 13.8564 0.643962 0.321981 0.946746i \(-0.395651\pi\)
0.321981 + 0.946746i \(0.395651\pi\)
\(464\) 0 0
\(465\) 24.0000 1.11297
\(466\) 0 0
\(467\) 36.0000 1.66588 0.832941 0.553362i \(-0.186655\pi\)
0.832941 + 0.553362i \(0.186655\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 2.00000 0.0921551
\(472\) 0 0
\(473\) 27.7128 1.27424
\(474\) 0 0
\(475\) 48.4974 2.22521
\(476\) 0 0
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) −3.46410 −0.158279 −0.0791394 0.996864i \(-0.525217\pi\)
−0.0791394 + 0.996864i \(0.525217\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 24.0000 1.08978
\(486\) 0 0
\(487\) −6.92820 −0.313947 −0.156973 0.987603i \(-0.550174\pi\)
−0.156973 + 0.987603i \(0.550174\pi\)
\(488\) 0 0
\(489\) −20.7846 −0.939913
\(490\) 0 0
\(491\) −36.0000 −1.62466 −0.812329 0.583200i \(-0.801800\pi\)
−0.812329 + 0.583200i \(0.801800\pi\)
\(492\) 0 0
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 12.0000 0.539360
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 17.3205 0.773823
\(502\) 0 0
\(503\) −24.0000 −1.07011 −0.535054 0.844818i \(-0.679709\pi\)
−0.535054 + 0.844818i \(0.679709\pi\)
\(504\) 0 0
\(505\) −20.7846 −0.924903
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −24.2487 −1.07481 −0.537403 0.843326i \(-0.680594\pi\)
−0.537403 + 0.843326i \(0.680594\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 6.92820 0.305888
\(514\) 0 0
\(515\) −13.8564 −0.610586
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) −6.00000 −0.263371
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −41.5692 −1.81078
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −3.46410 −0.150329
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 41.5692 1.79719
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) −24.2487 −1.04447
\(540\) 0 0
\(541\) 20.7846 0.893600 0.446800 0.894634i \(-0.352564\pi\)
0.446800 + 0.894634i \(0.352564\pi\)
\(542\) 0 0
\(543\) 22.0000 0.944110
\(544\) 0 0
\(545\) −24.0000 −1.02805
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) 10.0000 0.426790
\(550\) 0 0
\(551\) −41.5692 −1.77091
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 24.2487 1.02745 0.513725 0.857955i \(-0.328265\pi\)
0.513725 + 0.857955i \(0.328265\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −20.7846 −0.877527
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 0 0
\(565\) −62.3538 −2.62325
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −18.0000 −0.754599 −0.377300 0.926091i \(-0.623147\pi\)
−0.377300 + 0.926091i \(0.623147\pi\)
\(570\) 0 0
\(571\) 28.0000 1.17176 0.585882 0.810397i \(-0.300748\pi\)
0.585882 + 0.810397i \(0.300748\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −13.8564 −0.576850 −0.288425 0.957503i \(-0.593132\pi\)
−0.288425 + 0.957503i \(0.593132\pi\)
\(578\) 0 0
\(579\) −13.8564 −0.575853
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 20.7846 0.860811
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 31.1769 1.28681 0.643404 0.765526i \(-0.277521\pi\)
0.643404 + 0.765526i \(0.277521\pi\)
\(588\) 0 0
\(589\) 48.0000 1.97781
\(590\) 0 0
\(591\) 17.3205 0.712470
\(592\) 0 0
\(593\) −24.2487 −0.995775 −0.497888 0.867242i \(-0.665891\pi\)
−0.497888 + 0.867242i \(0.665891\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 0 0
\(603\) −13.8564 −0.564276
\(604\) 0 0
\(605\) 3.46410 0.140836
\(606\) 0 0
\(607\) 28.0000 1.13648 0.568242 0.822861i \(-0.307624\pi\)
0.568242 + 0.822861i \(0.307624\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 13.8564 0.559655 0.279827 0.960050i \(-0.409723\pi\)
0.279827 + 0.960050i \(0.409723\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −31.1769 −1.25514 −0.627568 0.778562i \(-0.715949\pi\)
−0.627568 + 0.778562i \(0.715949\pi\)
\(618\) 0 0
\(619\) −27.7128 −1.11387 −0.556936 0.830555i \(-0.688023\pi\)
−0.556936 + 0.830555i \(0.688023\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 24.0000 0.958468
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 20.7846 0.827422 0.413711 0.910408i \(-0.364232\pi\)
0.413711 + 0.910408i \(0.364232\pi\)
\(632\) 0 0
\(633\) −4.00000 −0.158986
\(634\) 0 0
\(635\) 55.4256 2.19950
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −10.3923 −0.411113
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) 41.5692 1.63933 0.819665 0.572843i \(-0.194160\pi\)
0.819665 + 0.572843i \(0.194160\pi\)
\(644\) 0 0
\(645\) 27.7128 1.09119
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.00000 −0.234798 −0.117399 0.993085i \(-0.537456\pi\)
−0.117399 + 0.993085i \(0.537456\pi\)
\(654\) 0 0
\(655\) 41.5692 1.62424
\(656\) 0 0
\(657\) 6.92820 0.270295
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 13.8564 0.538952 0.269476 0.963007i \(-0.413150\pi\)
0.269476 + 0.963007i \(0.413150\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 6.92820 0.267860
\(670\) 0 0
\(671\) 34.6410 1.33730
\(672\) 0 0
\(673\) −14.0000 −0.539660 −0.269830 0.962908i \(-0.586968\pi\)
−0.269830 + 0.962908i \(0.586968\pi\)
\(674\) 0 0
\(675\) 7.00000 0.269430
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 10.3923 0.398234
\(682\) 0 0
\(683\) 3.46410 0.132550 0.0662751 0.997801i \(-0.478889\pi\)
0.0662751 + 0.997801i \(0.478889\pi\)
\(684\) 0 0
\(685\) 36.0000 1.37549
\(686\) 0 0
\(687\) −6.92820 −0.264327
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −27.7128 −1.05425 −0.527123 0.849789i \(-0.676729\pi\)
−0.527123 + 0.849789i \(0.676729\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −27.7128 −1.05121
\(696\) 0 0
\(697\) 20.7846 0.787273
\(698\) 0 0
\(699\) −18.0000 −0.680823
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 12.0000 0.451946
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 6.92820 0.260194 0.130097 0.991501i \(-0.458471\pi\)
0.130097 + 0.991501i \(0.458471\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −10.3923 −0.388108
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 20.7846 0.772988
\(724\) 0 0
\(725\) −42.0000 −1.55984
\(726\) 0 0
\(727\) 16.0000 0.593407 0.296704 0.954970i \(-0.404113\pi\)
0.296704 + 0.954970i \(0.404113\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −48.0000 −1.77534
\(732\) 0 0
\(733\) 6.92820 0.255899 0.127950 0.991781i \(-0.459160\pi\)
0.127950 + 0.991781i \(0.459160\pi\)
\(734\) 0 0
\(735\) −24.2487 −0.894427
\(736\) 0 0
\(737\) −48.0000 −1.76810
\(738\) 0 0
\(739\) 27.7128 1.01943 0.509716 0.860343i \(-0.329750\pi\)
0.509716 + 0.860343i \(0.329750\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −10.3923 −0.381257 −0.190628 0.981662i \(-0.561053\pi\)
−0.190628 + 0.981662i \(0.561053\pi\)
\(744\) 0 0
\(745\) −60.0000 −2.19823
\(746\) 0 0
\(747\) 3.46410 0.126745
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −28.0000 −1.02173 −0.510867 0.859660i \(-0.670676\pi\)
−0.510867 + 0.859660i \(0.670676\pi\)
\(752\) 0 0
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −31.1769 −1.13016 −0.565081 0.825035i \(-0.691155\pi\)
−0.565081 + 0.825035i \(0.691155\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −20.7846 −0.751469
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 27.7128 0.999350 0.499675 0.866213i \(-0.333453\pi\)
0.499675 + 0.866213i \(0.333453\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) 51.9615 1.86893 0.934463 0.356060i \(-0.115880\pi\)
0.934463 + 0.356060i \(0.115880\pi\)
\(774\) 0 0
\(775\) 48.4974 1.74208
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) −36.0000 −1.28818
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) 0 0
\(785\) 6.92820 0.247278
\(786\) 0 0
\(787\) −27.7128 −0.987855 −0.493928 0.869503i \(-0.664439\pi\)
−0.493928 + 0.869503i \(0.664439\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 20.7846 0.737154
\(796\) 0 0
\(797\) 30.0000 1.06265 0.531327 0.847167i \(-0.321693\pi\)
0.531327 + 0.847167i \(0.321693\pi\)
\(798\) 0 0
\(799\) −20.7846 −0.735307
\(800\) 0 0
\(801\) 17.3205 0.611990
\(802\) 0 0
\(803\) 24.0000 0.846942
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −30.0000 −1.05605
\(808\) 0 0
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) −6.92820 −0.243282 −0.121641 0.992574i \(-0.538816\pi\)
−0.121641 + 0.992574i \(0.538816\pi\)
\(812\) 0 0
\(813\) −27.7128 −0.971931
\(814\) 0 0
\(815\) −72.0000 −2.52205
\(816\) 0 0
\(817\) 55.4256 1.93910
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −3.46410 −0.120898 −0.0604490 0.998171i \(-0.519253\pi\)
−0.0604490 + 0.998171i \(0.519253\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 24.2487 0.844232
\(826\) 0 0
\(827\) 17.3205 0.602293 0.301147 0.953578i \(-0.402631\pi\)
0.301147 + 0.953578i \(0.402631\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 0 0
\(833\) 42.0000 1.45521
\(834\) 0 0
\(835\) 60.0000 2.07639
\(836\) 0 0
\(837\) 6.92820 0.239474
\(838\) 0 0
\(839\) 10.3923 0.358782 0.179391 0.983778i \(-0.442587\pi\)
0.179391 + 0.983778i \(0.442587\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −17.3205 −0.596550
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −20.0000 −0.686398
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −13.8564 −0.474434 −0.237217 0.971457i \(-0.576235\pi\)
−0.237217 + 0.971457i \(0.576235\pi\)
\(854\) 0 0
\(855\) 24.0000 0.820783
\(856\) 0 0
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) −28.0000 −0.955348 −0.477674 0.878537i \(-0.658520\pi\)
−0.477674 + 0.878537i \(0.658520\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 38.1051 1.29711 0.648557 0.761166i \(-0.275373\pi\)
0.648557 + 0.761166i \(0.275373\pi\)
\(864\) 0 0
\(865\) −20.7846 −0.706698
\(866\) 0 0
\(867\) 19.0000 0.645274
\(868\) 0 0
\(869\) 27.7128 0.940093
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 6.92820 0.234484
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 41.5692 1.40369 0.701846 0.712328i \(-0.252359\pi\)
0.701846 + 0.712328i \(0.252359\pi\)
\(878\) 0 0
\(879\) −3.46410 −0.116841
\(880\) 0 0
\(881\) 6.00000 0.202145 0.101073 0.994879i \(-0.467773\pi\)
0.101073 + 0.994879i \(0.467773\pi\)
\(882\) 0 0
\(883\) −20.0000 −0.673054 −0.336527 0.941674i \(-0.609252\pi\)
−0.336527 + 0.941674i \(0.609252\pi\)
\(884\) 0 0
\(885\) −12.0000 −0.403376
\(886\) 0 0
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 3.46410 0.116052
\(892\) 0 0
\(893\) 24.0000 0.803129
\(894\) 0 0
\(895\) −41.5692 −1.38951
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −41.5692 −1.38641
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 76.2102 2.53331
\(906\) 0 0
\(907\) 28.0000 0.929725 0.464862 0.885383i \(-0.346104\pi\)
0.464862 + 0.885383i \(0.346104\pi\)
\(908\) 0 0
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 34.6410 1.14520
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −4.00000 −0.131377
\(928\) 0 0
\(929\) −31.1769 −1.02288 −0.511441 0.859319i \(-0.670888\pi\)
−0.511441 + 0.859319i \(0.670888\pi\)
\(930\) 0 0
\(931\) −48.4974 −1.58944
\(932\) 0 0
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) −72.0000 −2.35465
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 10.3923 0.338779 0.169390 0.985549i \(-0.445820\pi\)
0.169390 + 0.985549i \(0.445820\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −45.0333 −1.46339 −0.731693 0.681634i \(-0.761270\pi\)
−0.731693 + 0.681634i \(0.761270\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −24.2487 −0.786318
\(952\) 0 0
\(953\) 6.00000 0.194359 0.0971795 0.995267i \(-0.469018\pi\)
0.0971795 + 0.995267i \(0.469018\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −20.7846 −0.671871
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 0 0
\(965\) −48.0000 −1.54517
\(966\) 0 0
\(967\) 34.6410 1.11398 0.556990 0.830519i \(-0.311956\pi\)
0.556990 + 0.830519i \(0.311956\pi\)
\(968\) 0 0
\(969\) −41.5692 −1.33540
\(970\) 0 0
\(971\) −60.0000 −1.92549 −0.962746 0.270408i \(-0.912841\pi\)
−0.962746 + 0.270408i \(0.912841\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −24.2487 −0.775785 −0.387893 0.921705i \(-0.626797\pi\)
−0.387893 + 0.921705i \(0.626797\pi\)
\(978\) 0 0
\(979\) 60.0000 1.91761
\(980\) 0 0
\(981\) −6.92820 −0.221201
\(982\) 0 0
\(983\) −31.1769 −0.994389 −0.497195 0.867639i \(-0.665636\pi\)
−0.497195 + 0.867639i \(0.665636\pi\)
\(984\) 0 0
\(985\) 60.0000 1.91176
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −20.0000 −0.635321 −0.317660 0.948205i \(-0.602897\pi\)
−0.317660 + 0.948205i \(0.602897\pi\)
\(992\) 0 0
\(993\) −13.8564 −0.439720
\(994\) 0 0
\(995\) −27.7128 −0.878555
\(996\) 0 0
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) 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 8112.2.a.bs.1.2 2
4.3 odd 2 2028.2.a.g.1.2 2
12.11 even 2 6084.2.a.v.1.1 2
13.5 odd 4 624.2.c.f.337.1 2
13.8 odd 4 624.2.c.f.337.2 2
13.12 even 2 inner 8112.2.a.bs.1.1 2
39.5 even 4 1872.2.c.c.1585.2 2
39.8 even 4 1872.2.c.c.1585.1 2
52.3 odd 6 2028.2.i.i.529.2 4
52.7 even 12 2028.2.q.c.361.1 2
52.11 even 12 2028.2.q.b.1837.1 2
52.15 even 12 2028.2.q.c.1837.1 2
52.19 even 12 2028.2.q.b.361.1 2
52.23 odd 6 2028.2.i.i.529.1 4
52.31 even 4 156.2.b.a.25.1 2
52.35 odd 6 2028.2.i.i.2005.2 4
52.43 odd 6 2028.2.i.i.2005.1 4
52.47 even 4 156.2.b.a.25.2 yes 2
52.51 odd 2 2028.2.a.g.1.1 2
104.5 odd 4 2496.2.c.e.961.2 2
104.21 odd 4 2496.2.c.e.961.1 2
104.83 even 4 2496.2.c.l.961.2 2
104.99 even 4 2496.2.c.l.961.1 2
156.47 odd 4 468.2.b.a.181.1 2
156.83 odd 4 468.2.b.a.181.2 2
156.155 even 2 6084.2.a.v.1.2 2
260.47 odd 4 3900.2.j.h.649.4 4
260.83 odd 4 3900.2.j.h.649.1 4
260.99 even 4 3900.2.c.c.3301.2 2
260.187 odd 4 3900.2.j.h.649.3 4
260.203 odd 4 3900.2.j.h.649.2 4
260.239 even 4 3900.2.c.c.3301.1 2
364.83 odd 4 7644.2.e.g.4705.2 2
364.307 odd 4 7644.2.e.g.4705.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
156.2.b.a.25.1 2 52.31 even 4
156.2.b.a.25.2 yes 2 52.47 even 4
468.2.b.a.181.1 2 156.47 odd 4
468.2.b.a.181.2 2 156.83 odd 4
624.2.c.f.337.1 2 13.5 odd 4
624.2.c.f.337.2 2 13.8 odd 4
1872.2.c.c.1585.1 2 39.8 even 4
1872.2.c.c.1585.2 2 39.5 even 4
2028.2.a.g.1.1 2 52.51 odd 2
2028.2.a.g.1.2 2 4.3 odd 2
2028.2.i.i.529.1 4 52.23 odd 6
2028.2.i.i.529.2 4 52.3 odd 6
2028.2.i.i.2005.1 4 52.43 odd 6
2028.2.i.i.2005.2 4 52.35 odd 6
2028.2.q.b.361.1 2 52.19 even 12
2028.2.q.b.1837.1 2 52.11 even 12
2028.2.q.c.361.1 2 52.7 even 12
2028.2.q.c.1837.1 2 52.15 even 12
2496.2.c.e.961.1 2 104.21 odd 4
2496.2.c.e.961.2 2 104.5 odd 4
2496.2.c.l.961.1 2 104.99 even 4
2496.2.c.l.961.2 2 104.83 even 4
3900.2.c.c.3301.1 2 260.239 even 4
3900.2.c.c.3301.2 2 260.99 even 4
3900.2.j.h.649.1 4 260.83 odd 4
3900.2.j.h.649.2 4 260.203 odd 4
3900.2.j.h.649.3 4 260.187 odd 4
3900.2.j.h.649.4 4 260.47 odd 4
6084.2.a.v.1.1 2 12.11 even 2
6084.2.a.v.1.2 2 156.155 even 2
7644.2.e.g.4705.1 2 364.307 odd 4
7644.2.e.g.4705.2 2 364.83 odd 4
8112.2.a.bs.1.1 2 13.12 even 2 inner
8112.2.a.bs.1.2 2 1.1 even 1 trivial