Properties

Label 9680.2.a.bk.1.1
Level $9680$
Weight $2$
Character 9680.1
Self dual yes
Analytic conductor $77.295$
Analytic rank $1$
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] = [9680,2,Mod(1,9680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9680, 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("9680.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9680 = 2^{4} \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9680.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: \(77.2951891566\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 4840)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9680.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.41421 q^{3} +1.00000 q^{5} +2.41421 q^{7} +2.82843 q^{9} +O(q^{10})\) \(q-2.41421 q^{3} +1.00000 q^{5} +2.41421 q^{7} +2.82843 q^{9} -2.00000 q^{13} -2.41421 q^{15} -5.65685 q^{17} +4.82843 q^{19} -5.82843 q^{21} -3.65685 q^{23} +1.00000 q^{25} +0.414214 q^{27} -2.00000 q^{29} +5.65685 q^{31} +2.41421 q^{35} -4.00000 q^{37} +4.82843 q^{39} +9.48528 q^{41} +3.58579 q^{43} +2.82843 q^{45} -7.58579 q^{47} -1.17157 q^{49} +13.6569 q^{51} -7.65685 q^{53} -11.6569 q^{57} +11.3137 q^{59} -1.00000 q^{61} +6.82843 q^{63} -2.00000 q^{65} +6.41421 q^{67} +8.82843 q^{69} +4.00000 q^{71} -4.00000 q^{73} -2.41421 q^{75} -14.4853 q^{79} -9.48528 q^{81} -13.3137 q^{83} -5.65685 q^{85} +4.82843 q^{87} +2.65685 q^{89} -4.82843 q^{91} -13.6569 q^{93} +4.82843 q^{95} -17.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} - 4 q^{13} - 2 q^{15} + 4 q^{19} - 6 q^{21} + 4 q^{23} + 2 q^{25} - 2 q^{27} - 4 q^{29} + 2 q^{35} - 8 q^{37} + 4 q^{39} + 2 q^{41} + 10 q^{43} - 18 q^{47} - 8 q^{49} + 16 q^{51} - 4 q^{53} - 12 q^{57} - 2 q^{61} + 8 q^{63} - 4 q^{65} + 10 q^{67} + 12 q^{69} + 8 q^{71} - 8 q^{73} - 2 q^{75} - 12 q^{79} - 2 q^{81} - 4 q^{83} + 4 q^{87} - 6 q^{89} - 4 q^{91} - 16 q^{93} + 4 q^{95} - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −2.41421 −1.39385 −0.696923 0.717146i \(-0.745448\pi\)
−0.696923 + 0.717146i \(0.745448\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.41421 0.912487 0.456243 0.889855i \(-0.349195\pi\)
0.456243 + 0.889855i \(0.349195\pi\)
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) 0 0
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −2.41421 −0.623347
\(16\) 0 0
\(17\) −5.65685 −1.37199 −0.685994 0.727607i \(-0.740633\pi\)
−0.685994 + 0.727607i \(0.740633\pi\)
\(18\) 0 0
\(19\) 4.82843 1.10772 0.553859 0.832611i \(-0.313155\pi\)
0.553859 + 0.832611i \(0.313155\pi\)
\(20\) 0 0
\(21\) −5.82843 −1.27187
\(22\) 0 0
\(23\) −3.65685 −0.762507 −0.381253 0.924471i \(-0.624507\pi\)
−0.381253 + 0.924471i \(0.624507\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0.414214 0.0797154
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) 5.65685 1.01600 0.508001 0.861357i \(-0.330385\pi\)
0.508001 + 0.861357i \(0.330385\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.41421 0.408077
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) 4.82843 0.773167
\(40\) 0 0
\(41\) 9.48528 1.48135 0.740676 0.671862i \(-0.234505\pi\)
0.740676 + 0.671862i \(0.234505\pi\)
\(42\) 0 0
\(43\) 3.58579 0.546827 0.273414 0.961897i \(-0.411847\pi\)
0.273414 + 0.961897i \(0.411847\pi\)
\(44\) 0 0
\(45\) 2.82843 0.421637
\(46\) 0 0
\(47\) −7.58579 −1.10650 −0.553250 0.833015i \(-0.686613\pi\)
−0.553250 + 0.833015i \(0.686613\pi\)
\(48\) 0 0
\(49\) −1.17157 −0.167368
\(50\) 0 0
\(51\) 13.6569 1.91234
\(52\) 0 0
\(53\) −7.65685 −1.05175 −0.525875 0.850562i \(-0.676262\pi\)
−0.525875 + 0.850562i \(0.676262\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −11.6569 −1.54399
\(58\) 0 0
\(59\) 11.3137 1.47292 0.736460 0.676481i \(-0.236496\pi\)
0.736460 + 0.676481i \(0.236496\pi\)
\(60\) 0 0
\(61\) −1.00000 −0.128037 −0.0640184 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 6.82843 0.860301
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 6.41421 0.783621 0.391810 0.920046i \(-0.371849\pi\)
0.391810 + 0.920046i \(0.371849\pi\)
\(68\) 0 0
\(69\) 8.82843 1.06282
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 0 0
\(75\) −2.41421 −0.278769
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −14.4853 −1.62972 −0.814861 0.579657i \(-0.803187\pi\)
−0.814861 + 0.579657i \(0.803187\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) −13.3137 −1.46137 −0.730685 0.682715i \(-0.760799\pi\)
−0.730685 + 0.682715i \(0.760799\pi\)
\(84\) 0 0
\(85\) −5.65685 −0.613572
\(86\) 0 0
\(87\) 4.82843 0.517662
\(88\) 0 0
\(89\) 2.65685 0.281626 0.140813 0.990036i \(-0.455028\pi\)
0.140813 + 0.990036i \(0.455028\pi\)
\(90\) 0 0
\(91\) −4.82843 −0.506157
\(92\) 0 0
\(93\) −13.6569 −1.41615
\(94\) 0 0
\(95\) 4.82843 0.495386
\(96\) 0 0
\(97\) −17.3137 −1.75794 −0.878970 0.476876i \(-0.841769\pi\)
−0.878970 + 0.476876i \(0.841769\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −10.1716 −1.01211 −0.506055 0.862501i \(-0.668897\pi\)
−0.506055 + 0.862501i \(0.668897\pi\)
\(102\) 0 0
\(103\) −13.3137 −1.31184 −0.655919 0.754831i \(-0.727719\pi\)
−0.655919 + 0.754831i \(0.727719\pi\)
\(104\) 0 0
\(105\) −5.82843 −0.568796
\(106\) 0 0
\(107\) 15.2426 1.47356 0.736781 0.676132i \(-0.236345\pi\)
0.736781 + 0.676132i \(0.236345\pi\)
\(108\) 0 0
\(109\) −16.6569 −1.59544 −0.797719 0.603030i \(-0.793960\pi\)
−0.797719 + 0.603030i \(0.793960\pi\)
\(110\) 0 0
\(111\) 9.65685 0.916588
\(112\) 0 0
\(113\) 17.3137 1.62874 0.814368 0.580348i \(-0.197084\pi\)
0.814368 + 0.580348i \(0.197084\pi\)
\(114\) 0 0
\(115\) −3.65685 −0.341003
\(116\) 0 0
\(117\) −5.65685 −0.522976
\(118\) 0 0
\(119\) −13.6569 −1.25192
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −22.8995 −2.06478
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −4.41421 −0.391698 −0.195849 0.980634i \(-0.562746\pi\)
−0.195849 + 0.980634i \(0.562746\pi\)
\(128\) 0 0
\(129\) −8.65685 −0.762194
\(130\) 0 0
\(131\) −2.48528 −0.217140 −0.108570 0.994089i \(-0.534627\pi\)
−0.108570 + 0.994089i \(0.534627\pi\)
\(132\) 0 0
\(133\) 11.6569 1.01078
\(134\) 0 0
\(135\) 0.414214 0.0356498
\(136\) 0 0
\(137\) 13.3137 1.13747 0.568733 0.822522i \(-0.307434\pi\)
0.568733 + 0.822522i \(0.307434\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) 18.3137 1.54229
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) 2.82843 0.233285
\(148\) 0 0
\(149\) 17.0000 1.39269 0.696347 0.717705i \(-0.254807\pi\)
0.696347 + 0.717705i \(0.254807\pi\)
\(150\) 0 0
\(151\) 5.65685 0.460348 0.230174 0.973149i \(-0.426070\pi\)
0.230174 + 0.973149i \(0.426070\pi\)
\(152\) 0 0
\(153\) −16.0000 −1.29352
\(154\) 0 0
\(155\) 5.65685 0.454369
\(156\) 0 0
\(157\) 17.6569 1.40917 0.704585 0.709619i \(-0.251133\pi\)
0.704585 + 0.709619i \(0.251133\pi\)
\(158\) 0 0
\(159\) 18.4853 1.46598
\(160\) 0 0
\(161\) −8.82843 −0.695778
\(162\) 0 0
\(163\) −7.58579 −0.594165 −0.297082 0.954852i \(-0.596014\pi\)
−0.297082 + 0.954852i \(0.596014\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.41421 0.496347 0.248173 0.968716i \(-0.420170\pi\)
0.248173 + 0.968716i \(0.420170\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 13.6569 1.04437
\(172\) 0 0
\(173\) 3.65685 0.278025 0.139013 0.990291i \(-0.455607\pi\)
0.139013 + 0.990291i \(0.455607\pi\)
\(174\) 0 0
\(175\) 2.41421 0.182497
\(176\) 0 0
\(177\) −27.3137 −2.05302
\(178\) 0 0
\(179\) 17.7990 1.33036 0.665179 0.746684i \(-0.268355\pi\)
0.665179 + 0.746684i \(0.268355\pi\)
\(180\) 0 0
\(181\) −25.8284 −1.91981 −0.959906 0.280322i \(-0.909559\pi\)
−0.959906 + 0.280322i \(0.909559\pi\)
\(182\) 0 0
\(183\) 2.41421 0.178464
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −16.8284 −1.21766 −0.608831 0.793300i \(-0.708361\pi\)
−0.608831 + 0.793300i \(0.708361\pi\)
\(192\) 0 0
\(193\) −1.31371 −0.0945628 −0.0472814 0.998882i \(-0.515056\pi\)
−0.0472814 + 0.998882i \(0.515056\pi\)
\(194\) 0 0
\(195\) 4.82843 0.345771
\(196\) 0 0
\(197\) 6.97056 0.496632 0.248316 0.968679i \(-0.420123\pi\)
0.248316 + 0.968679i \(0.420123\pi\)
\(198\) 0 0
\(199\) 7.17157 0.508379 0.254190 0.967154i \(-0.418191\pi\)
0.254190 + 0.967154i \(0.418191\pi\)
\(200\) 0 0
\(201\) −15.4853 −1.09225
\(202\) 0 0
\(203\) −4.82843 −0.338889
\(204\) 0 0
\(205\) 9.48528 0.662481
\(206\) 0 0
\(207\) −10.3431 −0.718898
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −8.00000 −0.550743 −0.275371 0.961338i \(-0.588801\pi\)
−0.275371 + 0.961338i \(0.588801\pi\)
\(212\) 0 0
\(213\) −9.65685 −0.661677
\(214\) 0 0
\(215\) 3.58579 0.244549
\(216\) 0 0
\(217\) 13.6569 0.927088
\(218\) 0 0
\(219\) 9.65685 0.652550
\(220\) 0 0
\(221\) 11.3137 0.761042
\(222\) 0 0
\(223\) −7.92893 −0.530961 −0.265480 0.964116i \(-0.585531\pi\)
−0.265480 + 0.964116i \(0.585531\pi\)
\(224\) 0 0
\(225\) 2.82843 0.188562
\(226\) 0 0
\(227\) 26.0711 1.73040 0.865199 0.501429i \(-0.167192\pi\)
0.865199 + 0.501429i \(0.167192\pi\)
\(228\) 0 0
\(229\) 11.4853 0.758969 0.379484 0.925198i \(-0.376101\pi\)
0.379484 + 0.925198i \(0.376101\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.34315 −0.284529 −0.142264 0.989829i \(-0.545438\pi\)
−0.142264 + 0.989829i \(0.545438\pi\)
\(234\) 0 0
\(235\) −7.58579 −0.494842
\(236\) 0 0
\(237\) 34.9706 2.27158
\(238\) 0 0
\(239\) −24.1421 −1.56162 −0.780812 0.624765i \(-0.785195\pi\)
−0.780812 + 0.624765i \(0.785195\pi\)
\(240\) 0 0
\(241\) −16.1716 −1.04170 −0.520851 0.853647i \(-0.674385\pi\)
−0.520851 + 0.853647i \(0.674385\pi\)
\(242\) 0 0
\(243\) 21.6569 1.38929
\(244\) 0 0
\(245\) −1.17157 −0.0748490
\(246\) 0 0
\(247\) −9.65685 −0.614451
\(248\) 0 0
\(249\) 32.1421 2.03693
\(250\) 0 0
\(251\) −0.686292 −0.0433183 −0.0216592 0.999765i \(-0.506895\pi\)
−0.0216592 + 0.999765i \(0.506895\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 13.6569 0.855225
\(256\) 0 0
\(257\) 12.9706 0.809081 0.404541 0.914520i \(-0.367431\pi\)
0.404541 + 0.914520i \(0.367431\pi\)
\(258\) 0 0
\(259\) −9.65685 −0.600048
\(260\) 0 0
\(261\) −5.65685 −0.350150
\(262\) 0 0
\(263\) 24.6274 1.51859 0.759296 0.650746i \(-0.225544\pi\)
0.759296 + 0.650746i \(0.225544\pi\)
\(264\) 0 0
\(265\) −7.65685 −0.470357
\(266\) 0 0
\(267\) −6.41421 −0.392543
\(268\) 0 0
\(269\) −16.3137 −0.994664 −0.497332 0.867560i \(-0.665687\pi\)
−0.497332 + 0.867560i \(0.665687\pi\)
\(270\) 0 0
\(271\) −11.3137 −0.687259 −0.343629 0.939105i \(-0.611656\pi\)
−0.343629 + 0.939105i \(0.611656\pi\)
\(272\) 0 0
\(273\) 11.6569 0.705505
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −28.6274 −1.72005 −0.860027 0.510248i \(-0.829554\pi\)
−0.860027 + 0.510248i \(0.829554\pi\)
\(278\) 0 0
\(279\) 16.0000 0.957895
\(280\) 0 0
\(281\) 5.31371 0.316989 0.158495 0.987360i \(-0.449336\pi\)
0.158495 + 0.987360i \(0.449336\pi\)
\(282\) 0 0
\(283\) −20.2132 −1.20155 −0.600775 0.799418i \(-0.705141\pi\)
−0.600775 + 0.799418i \(0.705141\pi\)
\(284\) 0 0
\(285\) −11.6569 −0.690492
\(286\) 0 0
\(287\) 22.8995 1.35171
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 41.7990 2.45030
\(292\) 0 0
\(293\) 19.3137 1.12832 0.564159 0.825666i \(-0.309200\pi\)
0.564159 + 0.825666i \(0.309200\pi\)
\(294\) 0 0
\(295\) 11.3137 0.658710
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.31371 0.422963
\(300\) 0 0
\(301\) 8.65685 0.498973
\(302\) 0 0
\(303\) 24.5563 1.41073
\(304\) 0 0
\(305\) −1.00000 −0.0572598
\(306\) 0 0
\(307\) −25.3137 −1.44473 −0.722365 0.691512i \(-0.756945\pi\)
−0.722365 + 0.691512i \(0.756945\pi\)
\(308\) 0 0
\(309\) 32.1421 1.82850
\(310\) 0 0
\(311\) −12.8284 −0.727433 −0.363717 0.931510i \(-0.618492\pi\)
−0.363717 + 0.931510i \(0.618492\pi\)
\(312\) 0 0
\(313\) −21.3137 −1.20472 −0.602361 0.798224i \(-0.705773\pi\)
−0.602361 + 0.798224i \(0.705773\pi\)
\(314\) 0 0
\(315\) 6.82843 0.384738
\(316\) 0 0
\(317\) −2.68629 −0.150877 −0.0754386 0.997150i \(-0.524036\pi\)
−0.0754386 + 0.997150i \(0.524036\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −36.7990 −2.05392
\(322\) 0 0
\(323\) −27.3137 −1.51978
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 40.2132 2.22380
\(328\) 0 0
\(329\) −18.3137 −1.00967
\(330\) 0 0
\(331\) −4.82843 −0.265394 −0.132697 0.991157i \(-0.542364\pi\)
−0.132697 + 0.991157i \(0.542364\pi\)
\(332\) 0 0
\(333\) −11.3137 −0.619987
\(334\) 0 0
\(335\) 6.41421 0.350446
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) −41.7990 −2.27021
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −19.7279 −1.06521
\(344\) 0 0
\(345\) 8.82843 0.475307
\(346\) 0 0
\(347\) 1.38478 0.0743387 0.0371693 0.999309i \(-0.488166\pi\)
0.0371693 + 0.999309i \(0.488166\pi\)
\(348\) 0 0
\(349\) 5.31371 0.284436 0.142218 0.989835i \(-0.454577\pi\)
0.142218 + 0.989835i \(0.454577\pi\)
\(350\) 0 0
\(351\) −0.828427 −0.0442182
\(352\) 0 0
\(353\) −34.6274 −1.84303 −0.921516 0.388341i \(-0.873048\pi\)
−0.921516 + 0.388341i \(0.873048\pi\)
\(354\) 0 0
\(355\) 4.00000 0.212298
\(356\) 0 0
\(357\) 32.9706 1.74499
\(358\) 0 0
\(359\) −3.17157 −0.167389 −0.0836946 0.996491i \(-0.526672\pi\)
−0.0836946 + 0.996491i \(0.526672\pi\)
\(360\) 0 0
\(361\) 4.31371 0.227037
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −5.24264 −0.273664 −0.136832 0.990594i \(-0.543692\pi\)
−0.136832 + 0.990594i \(0.543692\pi\)
\(368\) 0 0
\(369\) 26.8284 1.39663
\(370\) 0 0
\(371\) −18.4853 −0.959708
\(372\) 0 0
\(373\) 20.9706 1.08581 0.542907 0.839793i \(-0.317323\pi\)
0.542907 + 0.839793i \(0.317323\pi\)
\(374\) 0 0
\(375\) −2.41421 −0.124669
\(376\) 0 0
\(377\) 4.00000 0.206010
\(378\) 0 0
\(379\) 20.1421 1.03463 0.517316 0.855794i \(-0.326931\pi\)
0.517316 + 0.855794i \(0.326931\pi\)
\(380\) 0 0
\(381\) 10.6569 0.545967
\(382\) 0 0
\(383\) −30.9706 −1.58252 −0.791261 0.611479i \(-0.790575\pi\)
−0.791261 + 0.611479i \(0.790575\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.1421 0.515554
\(388\) 0 0
\(389\) 26.3137 1.33416 0.667079 0.744987i \(-0.267544\pi\)
0.667079 + 0.744987i \(0.267544\pi\)
\(390\) 0 0
\(391\) 20.6863 1.04615
\(392\) 0 0
\(393\) 6.00000 0.302660
\(394\) 0 0
\(395\) −14.4853 −0.728834
\(396\) 0 0
\(397\) −24.0000 −1.20453 −0.602263 0.798298i \(-0.705734\pi\)
−0.602263 + 0.798298i \(0.705734\pi\)
\(398\) 0 0
\(399\) −28.1421 −1.40887
\(400\) 0 0
\(401\) 6.31371 0.315292 0.157646 0.987496i \(-0.449610\pi\)
0.157646 + 0.987496i \(0.449610\pi\)
\(402\) 0 0
\(403\) −11.3137 −0.563576
\(404\) 0 0
\(405\) −9.48528 −0.471327
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 39.7696 1.96648 0.983239 0.182322i \(-0.0583613\pi\)
0.983239 + 0.182322i \(0.0583613\pi\)
\(410\) 0 0
\(411\) −32.1421 −1.58545
\(412\) 0 0
\(413\) 27.3137 1.34402
\(414\) 0 0
\(415\) −13.3137 −0.653544
\(416\) 0 0
\(417\) −28.9706 −1.41869
\(418\) 0 0
\(419\) 17.5147 0.855650 0.427825 0.903862i \(-0.359280\pi\)
0.427825 + 0.903862i \(0.359280\pi\)
\(420\) 0 0
\(421\) −7.68629 −0.374607 −0.187303 0.982302i \(-0.559975\pi\)
−0.187303 + 0.982302i \(0.559975\pi\)
\(422\) 0 0
\(423\) −21.4558 −1.04322
\(424\) 0 0
\(425\) −5.65685 −0.274398
\(426\) 0 0
\(427\) −2.41421 −0.116832
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 17.5147 0.843654 0.421827 0.906676i \(-0.361389\pi\)
0.421827 + 0.906676i \(0.361389\pi\)
\(432\) 0 0
\(433\) −28.9706 −1.39224 −0.696118 0.717927i \(-0.745091\pi\)
−0.696118 + 0.717927i \(0.745091\pi\)
\(434\) 0 0
\(435\) 4.82843 0.231505
\(436\) 0 0
\(437\) −17.6569 −0.844642
\(438\) 0 0
\(439\) −24.2843 −1.15903 −0.579513 0.814963i \(-0.696757\pi\)
−0.579513 + 0.814963i \(0.696757\pi\)
\(440\) 0 0
\(441\) −3.31371 −0.157796
\(442\) 0 0
\(443\) −9.24264 −0.439131 −0.219566 0.975598i \(-0.570464\pi\)
−0.219566 + 0.975598i \(0.570464\pi\)
\(444\) 0 0
\(445\) 2.65685 0.125947
\(446\) 0 0
\(447\) −41.0416 −1.94120
\(448\) 0 0
\(449\) −40.4558 −1.90923 −0.954615 0.297844i \(-0.903733\pi\)
−0.954615 + 0.297844i \(0.903733\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −13.6569 −0.641655
\(454\) 0 0
\(455\) −4.82843 −0.226360
\(456\) 0 0
\(457\) −3.31371 −0.155009 −0.0775044 0.996992i \(-0.524695\pi\)
−0.0775044 + 0.996992i \(0.524695\pi\)
\(458\) 0 0
\(459\) −2.34315 −0.109369
\(460\) 0 0
\(461\) 9.14214 0.425792 0.212896 0.977075i \(-0.431710\pi\)
0.212896 + 0.977075i \(0.431710\pi\)
\(462\) 0 0
\(463\) −15.2426 −0.708386 −0.354193 0.935172i \(-0.615244\pi\)
−0.354193 + 0.935172i \(0.615244\pi\)
\(464\) 0 0
\(465\) −13.6569 −0.633321
\(466\) 0 0
\(467\) −42.5563 −1.96927 −0.984636 0.174617i \(-0.944131\pi\)
−0.984636 + 0.174617i \(0.944131\pi\)
\(468\) 0 0
\(469\) 15.4853 0.715044
\(470\) 0 0
\(471\) −42.6274 −1.96417
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.82843 0.221543
\(476\) 0 0
\(477\) −21.6569 −0.991599
\(478\) 0 0
\(479\) −1.79899 −0.0821979 −0.0410990 0.999155i \(-0.513086\pi\)
−0.0410990 + 0.999155i \(0.513086\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 21.3137 0.969807
\(484\) 0 0
\(485\) −17.3137 −0.786175
\(486\) 0 0
\(487\) −18.6863 −0.846757 −0.423378 0.905953i \(-0.639156\pi\)
−0.423378 + 0.905953i \(0.639156\pi\)
\(488\) 0 0
\(489\) 18.3137 0.828175
\(490\) 0 0
\(491\) 17.5147 0.790428 0.395214 0.918589i \(-0.370670\pi\)
0.395214 + 0.918589i \(0.370670\pi\)
\(492\) 0 0
\(493\) 11.3137 0.509544
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 9.65685 0.433169
\(498\) 0 0
\(499\) 37.7990 1.69212 0.846058 0.533092i \(-0.178970\pi\)
0.846058 + 0.533092i \(0.178970\pi\)
\(500\) 0 0
\(501\) −15.4853 −0.691831
\(502\) 0 0
\(503\) 9.38478 0.418446 0.209223 0.977868i \(-0.432906\pi\)
0.209223 + 0.977868i \(0.432906\pi\)
\(504\) 0 0
\(505\) −10.1716 −0.452629
\(506\) 0 0
\(507\) 21.7279 0.964971
\(508\) 0 0
\(509\) 11.4853 0.509076 0.254538 0.967063i \(-0.418077\pi\)
0.254538 + 0.967063i \(0.418077\pi\)
\(510\) 0 0
\(511\) −9.65685 −0.427194
\(512\) 0 0
\(513\) 2.00000 0.0883022
\(514\) 0 0
\(515\) −13.3137 −0.586672
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −8.82843 −0.387525
\(520\) 0 0
\(521\) 6.02944 0.264154 0.132077 0.991239i \(-0.457835\pi\)
0.132077 + 0.991239i \(0.457835\pi\)
\(522\) 0 0
\(523\) −13.3137 −0.582168 −0.291084 0.956698i \(-0.594016\pi\)
−0.291084 + 0.956698i \(0.594016\pi\)
\(524\) 0 0
\(525\) −5.82843 −0.254373
\(526\) 0 0
\(527\) −32.0000 −1.39394
\(528\) 0 0
\(529\) −9.62742 −0.418583
\(530\) 0 0
\(531\) 32.0000 1.38868
\(532\) 0 0
\(533\) −18.9706 −0.821706
\(534\) 0 0
\(535\) 15.2426 0.658997
\(536\) 0 0
\(537\) −42.9706 −1.85432
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 5.14214 0.221078 0.110539 0.993872i \(-0.464742\pi\)
0.110539 + 0.993872i \(0.464742\pi\)
\(542\) 0 0
\(543\) 62.3553 2.67592
\(544\) 0 0
\(545\) −16.6569 −0.713501
\(546\) 0 0
\(547\) 26.0000 1.11168 0.555840 0.831289i \(-0.312397\pi\)
0.555840 + 0.831289i \(0.312397\pi\)
\(548\) 0 0
\(549\) −2.82843 −0.120714
\(550\) 0 0
\(551\) −9.65685 −0.411396
\(552\) 0 0
\(553\) −34.9706 −1.48710
\(554\) 0 0
\(555\) 9.65685 0.409911
\(556\) 0 0
\(557\) −22.6274 −0.958754 −0.479377 0.877609i \(-0.659137\pi\)
−0.479377 + 0.877609i \(0.659137\pi\)
\(558\) 0 0
\(559\) −7.17157 −0.303325
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.89949 0.122199 0.0610996 0.998132i \(-0.480539\pi\)
0.0610996 + 0.998132i \(0.480539\pi\)
\(564\) 0 0
\(565\) 17.3137 0.728393
\(566\) 0 0
\(567\) −22.8995 −0.961688
\(568\) 0 0
\(569\) 16.4558 0.689865 0.344932 0.938628i \(-0.387902\pi\)
0.344932 + 0.938628i \(0.387902\pi\)
\(570\) 0 0
\(571\) 23.4558 0.981597 0.490798 0.871273i \(-0.336705\pi\)
0.490798 + 0.871273i \(0.336705\pi\)
\(572\) 0 0
\(573\) 40.6274 1.69723
\(574\) 0 0
\(575\) −3.65685 −0.152501
\(576\) 0 0
\(577\) −33.9411 −1.41299 −0.706494 0.707719i \(-0.749724\pi\)
−0.706494 + 0.707719i \(0.749724\pi\)
\(578\) 0 0
\(579\) 3.17157 0.131806
\(580\) 0 0
\(581\) −32.1421 −1.33348
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −5.65685 −0.233882
\(586\) 0 0
\(587\) −36.6985 −1.51471 −0.757354 0.653004i \(-0.773508\pi\)
−0.757354 + 0.653004i \(0.773508\pi\)
\(588\) 0 0
\(589\) 27.3137 1.12544
\(590\) 0 0
\(591\) −16.8284 −0.692229
\(592\) 0 0
\(593\) 23.6569 0.971471 0.485735 0.874106i \(-0.338552\pi\)
0.485735 + 0.874106i \(0.338552\pi\)
\(594\) 0 0
\(595\) −13.6569 −0.559876
\(596\) 0 0
\(597\) −17.3137 −0.708603
\(598\) 0 0
\(599\) −15.8579 −0.647935 −0.323967 0.946068i \(-0.605017\pi\)
−0.323967 + 0.946068i \(0.605017\pi\)
\(600\) 0 0
\(601\) −39.9411 −1.62923 −0.814616 0.580000i \(-0.803052\pi\)
−0.814616 + 0.580000i \(0.803052\pi\)
\(602\) 0 0
\(603\) 18.1421 0.738805
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 33.5980 1.36370 0.681850 0.731492i \(-0.261176\pi\)
0.681850 + 0.731492i \(0.261176\pi\)
\(608\) 0 0
\(609\) 11.6569 0.472360
\(610\) 0 0
\(611\) 15.1716 0.613776
\(612\) 0 0
\(613\) −14.9706 −0.604655 −0.302328 0.953204i \(-0.597764\pi\)
−0.302328 + 0.953204i \(0.597764\pi\)
\(614\) 0 0
\(615\) −22.8995 −0.923397
\(616\) 0 0
\(617\) −40.0000 −1.61034 −0.805170 0.593045i \(-0.797926\pi\)
−0.805170 + 0.593045i \(0.797926\pi\)
\(618\) 0 0
\(619\) −7.31371 −0.293963 −0.146981 0.989139i \(-0.546956\pi\)
−0.146981 + 0.989139i \(0.546956\pi\)
\(620\) 0 0
\(621\) −1.51472 −0.0607836
\(622\) 0 0
\(623\) 6.41421 0.256980
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 22.6274 0.902214
\(630\) 0 0
\(631\) −34.3431 −1.36718 −0.683590 0.729867i \(-0.739582\pi\)
−0.683590 + 0.729867i \(0.739582\pi\)
\(632\) 0 0
\(633\) 19.3137 0.767651
\(634\) 0 0
\(635\) −4.41421 −0.175173
\(636\) 0 0
\(637\) 2.34315 0.0928388
\(638\) 0 0
\(639\) 11.3137 0.447563
\(640\) 0 0
\(641\) −48.6274 −1.92067 −0.960334 0.278853i \(-0.910046\pi\)
−0.960334 + 0.278853i \(0.910046\pi\)
\(642\) 0 0
\(643\) −0.272078 −0.0107297 −0.00536485 0.999986i \(-0.501708\pi\)
−0.00536485 + 0.999986i \(0.501708\pi\)
\(644\) 0 0
\(645\) −8.65685 −0.340863
\(646\) 0 0
\(647\) 29.8701 1.17431 0.587157 0.809473i \(-0.300247\pi\)
0.587157 + 0.809473i \(0.300247\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −32.9706 −1.29222
\(652\) 0 0
\(653\) −7.65685 −0.299636 −0.149818 0.988714i \(-0.547869\pi\)
−0.149818 + 0.988714i \(0.547869\pi\)
\(654\) 0 0
\(655\) −2.48528 −0.0971080
\(656\) 0 0
\(657\) −11.3137 −0.441390
\(658\) 0 0
\(659\) −23.1716 −0.902636 −0.451318 0.892363i \(-0.649046\pi\)
−0.451318 + 0.892363i \(0.649046\pi\)
\(660\) 0 0
\(661\) 35.3431 1.37469 0.687345 0.726332i \(-0.258776\pi\)
0.687345 + 0.726332i \(0.258776\pi\)
\(662\) 0 0
\(663\) −27.3137 −1.06078
\(664\) 0 0
\(665\) 11.6569 0.452033
\(666\) 0 0
\(667\) 7.31371 0.283188
\(668\) 0 0
\(669\) 19.1421 0.740078
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −16.9706 −0.654167 −0.327084 0.944995i \(-0.606066\pi\)
−0.327084 + 0.944995i \(0.606066\pi\)
\(674\) 0 0
\(675\) 0.414214 0.0159431
\(676\) 0 0
\(677\) 8.68629 0.333841 0.166921 0.985970i \(-0.446618\pi\)
0.166921 + 0.985970i \(0.446618\pi\)
\(678\) 0 0
\(679\) −41.7990 −1.60410
\(680\) 0 0
\(681\) −62.9411 −2.41191
\(682\) 0 0
\(683\) −19.8701 −0.760307 −0.380153 0.924923i \(-0.624129\pi\)
−0.380153 + 0.924923i \(0.624129\pi\)
\(684\) 0 0
\(685\) 13.3137 0.508691
\(686\) 0 0
\(687\) −27.7279 −1.05789
\(688\) 0 0
\(689\) 15.3137 0.583406
\(690\) 0 0
\(691\) 14.4853 0.551046 0.275523 0.961294i \(-0.411149\pi\)
0.275523 + 0.961294i \(0.411149\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) −53.6569 −2.03240
\(698\) 0 0
\(699\) 10.4853 0.396590
\(700\) 0 0
\(701\) −39.9411 −1.50856 −0.754278 0.656555i \(-0.772013\pi\)
−0.754278 + 0.656555i \(0.772013\pi\)
\(702\) 0 0
\(703\) −19.3137 −0.728430
\(704\) 0 0
\(705\) 18.3137 0.689734
\(706\) 0 0
\(707\) −24.5563 −0.923537
\(708\) 0 0
\(709\) −35.1421 −1.31979 −0.659895 0.751358i \(-0.729399\pi\)
−0.659895 + 0.751358i \(0.729399\pi\)
\(710\) 0 0
\(711\) −40.9706 −1.53652
\(712\) 0 0
\(713\) −20.6863 −0.774708
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 58.2843 2.17667
\(718\) 0 0
\(719\) 31.1716 1.16250 0.581252 0.813724i \(-0.302563\pi\)
0.581252 + 0.813724i \(0.302563\pi\)
\(720\) 0 0
\(721\) −32.1421 −1.19704
\(722\) 0 0
\(723\) 39.0416 1.45197
\(724\) 0 0
\(725\) −2.00000 −0.0742781
\(726\) 0 0
\(727\) −52.0711 −1.93121 −0.965605 0.260015i \(-0.916272\pi\)
−0.965605 + 0.260015i \(0.916272\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) −20.2843 −0.750241
\(732\) 0 0
\(733\) −40.2843 −1.48793 −0.743967 0.668217i \(-0.767058\pi\)
−0.743967 + 0.668217i \(0.767058\pi\)
\(734\) 0 0
\(735\) 2.82843 0.104328
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −33.5147 −1.23286 −0.616429 0.787410i \(-0.711421\pi\)
−0.616429 + 0.787410i \(0.711421\pi\)
\(740\) 0 0
\(741\) 23.3137 0.856450
\(742\) 0 0
\(743\) 32.4142 1.18916 0.594581 0.804036i \(-0.297318\pi\)
0.594581 + 0.804036i \(0.297318\pi\)
\(744\) 0 0
\(745\) 17.0000 0.622832
\(746\) 0 0
\(747\) −37.6569 −1.37779
\(748\) 0 0
\(749\) 36.7990 1.34461
\(750\) 0 0
\(751\) −4.82843 −0.176192 −0.0880959 0.996112i \(-0.528078\pi\)
−0.0880959 + 0.996112i \(0.528078\pi\)
\(752\) 0 0
\(753\) 1.65685 0.0603791
\(754\) 0 0
\(755\) 5.65685 0.205874
\(756\) 0 0
\(757\) −30.9706 −1.12564 −0.562822 0.826578i \(-0.690284\pi\)
−0.562822 + 0.826578i \(0.690284\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.00000 −0.217500 −0.108750 0.994069i \(-0.534685\pi\)
−0.108750 + 0.994069i \(0.534685\pi\)
\(762\) 0 0
\(763\) −40.2132 −1.45582
\(764\) 0 0
\(765\) −16.0000 −0.578481
\(766\) 0 0
\(767\) −22.6274 −0.817029
\(768\) 0 0
\(769\) 18.0000 0.649097 0.324548 0.945869i \(-0.394788\pi\)
0.324548 + 0.945869i \(0.394788\pi\)
\(770\) 0 0
\(771\) −31.3137 −1.12774
\(772\) 0 0
\(773\) −28.9706 −1.04200 −0.520999 0.853557i \(-0.674441\pi\)
−0.520999 + 0.853557i \(0.674441\pi\)
\(774\) 0 0
\(775\) 5.65685 0.203200
\(776\) 0 0
\(777\) 23.3137 0.836375
\(778\) 0 0
\(779\) 45.7990 1.64092
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −0.828427 −0.0296056
\(784\) 0 0
\(785\) 17.6569 0.630200
\(786\) 0 0
\(787\) 0.213203 0.00759988 0.00379994 0.999993i \(-0.498790\pi\)
0.00379994 + 0.999993i \(0.498790\pi\)
\(788\) 0 0
\(789\) −59.4558 −2.11668
\(790\) 0 0
\(791\) 41.7990 1.48620
\(792\) 0 0
\(793\) 2.00000 0.0710221
\(794\) 0 0
\(795\) 18.4853 0.655605
\(796\) 0 0
\(797\) 34.3431 1.21650 0.608248 0.793747i \(-0.291872\pi\)
0.608248 + 0.793747i \(0.291872\pi\)
\(798\) 0 0
\(799\) 42.9117 1.51811
\(800\) 0 0
\(801\) 7.51472 0.265520
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −8.82843 −0.311161
\(806\) 0 0
\(807\) 39.3848 1.38641
\(808\) 0 0
\(809\) −24.6274 −0.865854 −0.432927 0.901429i \(-0.642519\pi\)
−0.432927 + 0.901429i \(0.642519\pi\)
\(810\) 0 0
\(811\) −5.65685 −0.198639 −0.0993195 0.995056i \(-0.531667\pi\)
−0.0993195 + 0.995056i \(0.531667\pi\)
\(812\) 0 0
\(813\) 27.3137 0.957934
\(814\) 0 0
\(815\) −7.58579 −0.265719
\(816\) 0 0
\(817\) 17.3137 0.605730
\(818\) 0 0
\(819\) −13.6569 −0.477209
\(820\) 0 0
\(821\) 3.97056 0.138574 0.0692868 0.997597i \(-0.477928\pi\)
0.0692868 + 0.997597i \(0.477928\pi\)
\(822\) 0 0
\(823\) 1.92893 0.0672383 0.0336192 0.999435i \(-0.489297\pi\)
0.0336192 + 0.999435i \(0.489297\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −32.3553 −1.12511 −0.562553 0.826761i \(-0.690181\pi\)
−0.562553 + 0.826761i \(0.690181\pi\)
\(828\) 0 0
\(829\) 32.9411 1.14409 0.572046 0.820221i \(-0.306150\pi\)
0.572046 + 0.820221i \(0.306150\pi\)
\(830\) 0 0
\(831\) 69.1127 2.39749
\(832\) 0 0
\(833\) 6.62742 0.229626
\(834\) 0 0
\(835\) 6.41421 0.221973
\(836\) 0 0
\(837\) 2.34315 0.0809910
\(838\) 0 0
\(839\) −41.7990 −1.44306 −0.721531 0.692382i \(-0.756561\pi\)
−0.721531 + 0.692382i \(0.756561\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −12.8284 −0.441834
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 48.7990 1.67478
\(850\) 0 0
\(851\) 14.6274 0.501421
\(852\) 0 0
\(853\) 29.6569 1.01543 0.507716 0.861525i \(-0.330490\pi\)
0.507716 + 0.861525i \(0.330490\pi\)
\(854\) 0 0
\(855\) 13.6569 0.467055
\(856\) 0 0
\(857\) 10.9706 0.374747 0.187374 0.982289i \(-0.440002\pi\)
0.187374 + 0.982289i \(0.440002\pi\)
\(858\) 0 0
\(859\) 7.85786 0.268107 0.134053 0.990974i \(-0.457201\pi\)
0.134053 + 0.990974i \(0.457201\pi\)
\(860\) 0 0
\(861\) −55.2843 −1.88408
\(862\) 0 0
\(863\) 7.87006 0.267900 0.133950 0.990988i \(-0.457234\pi\)
0.133950 + 0.990988i \(0.457234\pi\)
\(864\) 0 0
\(865\) 3.65685 0.124337
\(866\) 0 0
\(867\) −36.2132 −1.22986
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) −12.8284 −0.434675
\(872\) 0 0
\(873\) −48.9706 −1.65740
\(874\) 0 0
\(875\) 2.41421 0.0816153
\(876\) 0 0
\(877\) 47.3137 1.59767 0.798835 0.601550i \(-0.205450\pi\)
0.798835 + 0.601550i \(0.205450\pi\)
\(878\) 0 0
\(879\) −46.6274 −1.57270
\(880\) 0 0
\(881\) 2.51472 0.0847230 0.0423615 0.999102i \(-0.486512\pi\)
0.0423615 + 0.999102i \(0.486512\pi\)
\(882\) 0 0
\(883\) 32.3431 1.08843 0.544217 0.838945i \(-0.316827\pi\)
0.544217 + 0.838945i \(0.316827\pi\)
\(884\) 0 0
\(885\) −27.3137 −0.918140
\(886\) 0 0
\(887\) −0.615224 −0.0206572 −0.0103286 0.999947i \(-0.503288\pi\)
−0.0103286 + 0.999947i \(0.503288\pi\)
\(888\) 0 0
\(889\) −10.6569 −0.357419
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −36.6274 −1.22569
\(894\) 0 0
\(895\) 17.7990 0.594955
\(896\) 0 0
\(897\) −17.6569 −0.589545
\(898\) 0 0
\(899\) −11.3137 −0.377333
\(900\) 0 0
\(901\) 43.3137 1.44299
\(902\) 0 0
\(903\) −20.8995 −0.695492
\(904\) 0 0
\(905\) −25.8284 −0.858566
\(906\) 0 0
\(907\) −40.6985 −1.35137 −0.675686 0.737190i \(-0.736152\pi\)
−0.675686 + 0.737190i \(0.736152\pi\)
\(908\) 0 0
\(909\) −28.7696 −0.954226
\(910\) 0 0
\(911\) 27.1716 0.900234 0.450117 0.892969i \(-0.351382\pi\)
0.450117 + 0.892969i \(0.351382\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 2.41421 0.0798114
\(916\) 0 0
\(917\) −6.00000 −0.198137
\(918\) 0 0
\(919\) 32.4264 1.06965 0.534824 0.844963i \(-0.320378\pi\)
0.534824 + 0.844963i \(0.320378\pi\)
\(920\) 0 0
\(921\) 61.1127 2.01373
\(922\) 0 0
\(923\) −8.00000 −0.263323
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) −37.6569 −1.23681
\(928\) 0 0
\(929\) 34.0000 1.11550 0.557752 0.830008i \(-0.311664\pi\)
0.557752 + 0.830008i \(0.311664\pi\)
\(930\) 0 0
\(931\) −5.65685 −0.185396
\(932\) 0 0
\(933\) 30.9706 1.01393
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 30.0000 0.980057 0.490029 0.871706i \(-0.336986\pi\)
0.490029 + 0.871706i \(0.336986\pi\)
\(938\) 0 0
\(939\) 51.4558 1.67920
\(940\) 0 0
\(941\) −51.8284 −1.68956 −0.844779 0.535115i \(-0.820268\pi\)
−0.844779 + 0.535115i \(0.820268\pi\)
\(942\) 0 0
\(943\) −34.6863 −1.12954
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −10.0000 −0.324956 −0.162478 0.986712i \(-0.551949\pi\)
−0.162478 + 0.986712i \(0.551949\pi\)
\(948\) 0 0
\(949\) 8.00000 0.259691
\(950\) 0 0
\(951\) 6.48528 0.210300
\(952\) 0 0
\(953\) −24.2843 −0.786645 −0.393322 0.919401i \(-0.628674\pi\)
−0.393322 + 0.919401i \(0.628674\pi\)
\(954\) 0 0
\(955\) −16.8284 −0.544555
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 32.1421 1.03792
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 43.1127 1.38929
\(964\) 0 0
\(965\) −1.31371 −0.0422898
\(966\) 0 0
\(967\) 41.5980 1.33770 0.668850 0.743397i \(-0.266787\pi\)
0.668850 + 0.743397i \(0.266787\pi\)
\(968\) 0 0
\(969\) 65.9411 2.11833
\(970\) 0 0
\(971\) 3.71573 0.119243 0.0596217 0.998221i \(-0.481011\pi\)
0.0596217 + 0.998221i \(0.481011\pi\)
\(972\) 0 0
\(973\) 28.9706 0.928754
\(974\) 0 0
\(975\) 4.82843 0.154633
\(976\) 0 0
\(977\) 2.62742 0.0840585 0.0420293 0.999116i \(-0.486618\pi\)
0.0420293 + 0.999116i \(0.486618\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −47.1127 −1.50419
\(982\) 0 0
\(983\) 5.92893 0.189104 0.0945518 0.995520i \(-0.469858\pi\)
0.0945518 + 0.995520i \(0.469858\pi\)
\(984\) 0 0
\(985\) 6.97056 0.222101
\(986\) 0 0
\(987\) 44.2132 1.40732
\(988\) 0 0
\(989\) −13.1127 −0.416960
\(990\) 0 0
\(991\) −58.4853 −1.85785 −0.928923 0.370273i \(-0.879264\pi\)
−0.928923 + 0.370273i \(0.879264\pi\)
\(992\) 0 0
\(993\) 11.6569 0.369919
\(994\) 0 0
\(995\) 7.17157 0.227354
\(996\) 0 0
\(997\) 29.3137 0.928374 0.464187 0.885737i \(-0.346346\pi\)
0.464187 + 0.885737i \(0.346346\pi\)
\(998\) 0 0
\(999\) −1.65685 −0.0524205
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 9680.2.a.bk.1.1 2
4.3 odd 2 4840.2.a.o.1.2 2
11.10 odd 2 9680.2.a.bj.1.1 2
44.43 even 2 4840.2.a.p.1.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4840.2.a.o.1.2 2 4.3 odd 2
4840.2.a.p.1.2 yes 2 44.43 even 2
9680.2.a.bj.1.1 2 11.10 odd 2
9680.2.a.bk.1.1 2 1.1 even 1 trivial