Properties

Label 9216.2.a.bs.1.8
Level $9216$
Weight $2$
Character 9216.1
Self dual yes
Analytic conductor $73.590$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9216,2,Mod(1,9216)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9216, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9216.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9216 = 2^{10} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9216.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: \(73.5901305028\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.3288334336.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} - 8x^{5} + 24x^{4} + 8x^{3} - 16x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 4608)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(3.49930\) of defining polynomial
Character \(\chi\) \(=\) 9216.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.97127 q^{5} +2.27411 q^{7} +O(q^{10})\) \(q+2.97127 q^{5} +2.27411 q^{7} -3.69552 q^{11} +3.41421 q^{13} +1.74053 q^{17} +7.76429 q^{19} -2.16478 q^{23} +3.82843 q^{25} -5.43275 q^{29} +8.70626 q^{31} +6.75699 q^{35} +0.585786 q^{37} +10.1445 q^{41} +1.33214 q^{43} -12.6173 q^{47} -1.82843 q^{49} +5.43275 q^{53} -10.9804 q^{55} +4.32957 q^{59} +7.41421 q^{61} +10.1445 q^{65} -6.43215 q^{67} -6.12293 q^{71} +10.4853 q^{73} -8.40401 q^{77} -2.27411 q^{79} -9.81845 q^{83} +5.17157 q^{85} +11.8851 q^{89} +7.76429 q^{91} +23.0698 q^{95} -15.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{13} + 8 q^{25} + 16 q^{37} + 8 q^{49} + 48 q^{61} + 16 q^{73} + 64 q^{85} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 2.97127 1.32879 0.664395 0.747381i \(-0.268689\pi\)
0.664395 + 0.747381i \(0.268689\pi\)
\(6\) 0 0
\(7\) 2.27411 0.859533 0.429766 0.902940i \(-0.358596\pi\)
0.429766 + 0.902940i \(0.358596\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −3.69552 −1.11424 −0.557120 0.830432i \(-0.688094\pi\)
−0.557120 + 0.830432i \(0.688094\pi\)
\(12\) 0 0
\(13\) 3.41421 0.946932 0.473466 0.880812i \(-0.343003\pi\)
0.473466 + 0.880812i \(0.343003\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.74053 0.422140 0.211070 0.977471i \(-0.432305\pi\)
0.211070 + 0.977471i \(0.432305\pi\)
\(18\) 0 0
\(19\) 7.76429 1.78125 0.890626 0.454737i \(-0.150267\pi\)
0.890626 + 0.454737i \(0.150267\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.16478 −0.451389 −0.225694 0.974198i \(-0.572465\pi\)
−0.225694 + 0.974198i \(0.572465\pi\)
\(24\) 0 0
\(25\) 3.82843 0.765685
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.43275 −1.00884 −0.504418 0.863460i \(-0.668293\pi\)
−0.504418 + 0.863460i \(0.668293\pi\)
\(30\) 0 0
\(31\) 8.70626 1.56369 0.781845 0.623472i \(-0.214279\pi\)
0.781845 + 0.623472i \(0.214279\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.75699 1.14214
\(36\) 0 0
\(37\) 0.585786 0.0963027 0.0481513 0.998840i \(-0.484667\pi\)
0.0481513 + 0.998840i \(0.484667\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.1445 1.58431 0.792155 0.610319i \(-0.208959\pi\)
0.792155 + 0.610319i \(0.208959\pi\)
\(42\) 0 0
\(43\) 1.33214 0.203150 0.101575 0.994828i \(-0.467612\pi\)
0.101575 + 0.994828i \(0.467612\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.6173 −1.84042 −0.920210 0.391424i \(-0.871982\pi\)
−0.920210 + 0.391424i \(0.871982\pi\)
\(48\) 0 0
\(49\) −1.82843 −0.261204
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.43275 0.746245 0.373122 0.927782i \(-0.378287\pi\)
0.373122 + 0.927782i \(0.378287\pi\)
\(54\) 0 0
\(55\) −10.9804 −1.48059
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.32957 0.563662 0.281831 0.959464i \(-0.409058\pi\)
0.281831 + 0.959464i \(0.409058\pi\)
\(60\) 0 0
\(61\) 7.41421 0.949293 0.474646 0.880177i \(-0.342576\pi\)
0.474646 + 0.880177i \(0.342576\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 10.1445 1.25828
\(66\) 0 0
\(67\) −6.43215 −0.785812 −0.392906 0.919579i \(-0.628530\pi\)
−0.392906 + 0.919579i \(0.628530\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.12293 −0.726659 −0.363329 0.931661i \(-0.618360\pi\)
−0.363329 + 0.931661i \(0.618360\pi\)
\(72\) 0 0
\(73\) 10.4853 1.22721 0.613605 0.789613i \(-0.289719\pi\)
0.613605 + 0.789613i \(0.289719\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.40401 −0.957726
\(78\) 0 0
\(79\) −2.27411 −0.255857 −0.127929 0.991783i \(-0.540833\pi\)
−0.127929 + 0.991783i \(0.540833\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −9.81845 −1.07772 −0.538858 0.842397i \(-0.681144\pi\)
−0.538858 + 0.842397i \(0.681144\pi\)
\(84\) 0 0
\(85\) 5.17157 0.560936
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11.8851 1.25981 0.629907 0.776670i \(-0.283093\pi\)
0.629907 + 0.776670i \(0.283093\pi\)
\(90\) 0 0
\(91\) 7.76429 0.813919
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 23.0698 2.36691
\(96\) 0 0
\(97\) −15.3137 −1.55487 −0.777436 0.628962i \(-0.783480\pi\)
−0.777436 + 0.628962i \(0.783480\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.89422 0.785505 0.392752 0.919644i \(-0.371523\pi\)
0.392752 + 0.919644i \(0.371523\pi\)
\(102\) 0 0
\(103\) −8.70626 −0.857853 −0.428927 0.903339i \(-0.641108\pi\)
−0.428927 + 0.903339i \(0.641108\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.7206 −1.13307 −0.566537 0.824036i \(-0.691717\pi\)
−0.566537 + 0.824036i \(0.691717\pi\)
\(108\) 0 0
\(109\) −1.07107 −0.102590 −0.0512948 0.998684i \(-0.516335\pi\)
−0.0512948 + 0.998684i \(0.516335\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 16.8080 1.58117 0.790583 0.612355i \(-0.209778\pi\)
0.790583 + 0.612355i \(0.209778\pi\)
\(114\) 0 0
\(115\) −6.43215 −0.599801
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.95815 0.362843
\(120\) 0 0
\(121\) 2.65685 0.241532
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −3.48106 −0.311355
\(126\) 0 0
\(127\) −13.2545 −1.17614 −0.588072 0.808808i \(-0.700113\pi\)
−0.588072 + 0.808808i \(0.700113\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.5140 1.18072 0.590361 0.807140i \(-0.298986\pi\)
0.590361 + 0.807140i \(0.298986\pi\)
\(132\) 0 0
\(133\) 17.6569 1.53104
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −15.0675 −1.28730 −0.643652 0.765319i \(-0.722581\pi\)
−0.643652 + 0.765319i \(0.722581\pi\)
\(138\) 0 0
\(139\) 12.8643 1.09114 0.545568 0.838067i \(-0.316314\pi\)
0.545568 + 0.838067i \(0.316314\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −12.6173 −1.05511
\(144\) 0 0
\(145\) −16.1421 −1.34053
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 8.91380 0.730247 0.365124 0.930959i \(-0.381027\pi\)
0.365124 + 0.930959i \(0.381027\pi\)
\(150\) 0 0
\(151\) 13.2545 1.07863 0.539317 0.842103i \(-0.318682\pi\)
0.539317 + 0.842103i \(0.318682\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 25.8686 2.07782
\(156\) 0 0
\(157\) 5.75736 0.459487 0.229744 0.973251i \(-0.426211\pi\)
0.229744 + 0.973251i \(0.426211\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.92296 −0.387983
\(162\) 0 0
\(163\) 20.6286 1.61576 0.807878 0.589349i \(-0.200616\pi\)
0.807878 + 0.589349i \(0.200616\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.2764 1.64642 0.823210 0.567738i \(-0.192181\pi\)
0.823210 + 0.567738i \(0.192181\pi\)
\(168\) 0 0
\(169\) −1.34315 −0.103319
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −11.3753 −0.864846 −0.432423 0.901671i \(-0.642341\pi\)
−0.432423 + 0.901671i \(0.642341\pi\)
\(174\) 0 0
\(175\) 8.70626 0.658132
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −23.9665 −1.79134 −0.895669 0.444721i \(-0.853303\pi\)
−0.895669 + 0.444721i \(0.853303\pi\)
\(180\) 0 0
\(181\) 26.7279 1.98667 0.993335 0.115260i \(-0.0367700\pi\)
0.993335 + 0.115260i \(0.0367700\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.74053 0.127966
\(186\) 0 0
\(187\) −6.43215 −0.470366
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 16.9469 1.22623 0.613116 0.789993i \(-0.289916\pi\)
0.613116 + 0.789993i \(0.289916\pi\)
\(192\) 0 0
\(193\) −4.48528 −0.322858 −0.161429 0.986884i \(-0.551610\pi\)
−0.161429 + 0.986884i \(0.551610\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.3753 −0.810455 −0.405228 0.914216i \(-0.632808\pi\)
−0.405228 + 0.914216i \(0.632808\pi\)
\(198\) 0 0
\(199\) −0.390175 −0.0276588 −0.0138294 0.999904i \(-0.504402\pi\)
−0.0138294 + 0.999904i \(0.504402\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −12.3547 −0.867127
\(204\) 0 0
\(205\) 30.1421 2.10522
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −28.6931 −1.98474
\(210\) 0 0
\(211\) −18.1929 −1.25245 −0.626225 0.779643i \(-0.715401\pi\)
−0.626225 + 0.779643i \(0.715401\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.95815 0.269944
\(216\) 0 0
\(217\) 19.7990 1.34404
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.94253 0.399738
\(222\) 0 0
\(223\) −10.5902 −0.709172 −0.354586 0.935023i \(-0.615378\pi\)
−0.354586 + 0.935023i \(0.615378\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.42742 0.161113 0.0805567 0.996750i \(-0.474330\pi\)
0.0805567 + 0.996750i \(0.474330\pi\)
\(228\) 0 0
\(229\) 10.9289 0.722204 0.361102 0.932526i \(-0.382401\pi\)
0.361102 + 0.932526i \(0.382401\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 21.7310 1.42364 0.711822 0.702360i \(-0.247870\pi\)
0.711822 + 0.702360i \(0.247870\pi\)
\(234\) 0 0
\(235\) −37.4893 −2.44553
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −18.7402 −1.21220 −0.606102 0.795387i \(-0.707268\pi\)
−0.606102 + 0.795387i \(0.707268\pi\)
\(240\) 0 0
\(241\) −18.1421 −1.16864 −0.584319 0.811524i \(-0.698638\pi\)
−0.584319 + 0.811524i \(0.698638\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.43275 −0.347085
\(246\) 0 0
\(247\) 26.5090 1.68672
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.4776 1.16630 0.583148 0.812366i \(-0.301821\pi\)
0.583148 + 0.812366i \(0.301821\pi\)
\(252\) 0 0
\(253\) 8.00000 0.502956
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.92296 0.307086 0.153543 0.988142i \(-0.450932\pi\)
0.153543 + 0.988142i \(0.450932\pi\)
\(258\) 0 0
\(259\) 1.33214 0.0827753
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −16.5754 −1.02209 −0.511043 0.859555i \(-0.670741\pi\)
−0.511043 + 0.859555i \(0.670741\pi\)
\(264\) 0 0
\(265\) 16.1421 0.991604
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 25.7218 1.56829 0.784144 0.620579i \(-0.213103\pi\)
0.784144 + 0.620579i \(0.213103\pi\)
\(270\) 0 0
\(271\) 11.3705 0.690711 0.345356 0.938472i \(-0.387758\pi\)
0.345356 + 0.938472i \(0.387758\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −14.1480 −0.853158
\(276\) 0 0
\(277\) −8.38478 −0.503792 −0.251896 0.967754i \(-0.581054\pi\)
−0.251896 + 0.967754i \(0.581054\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −18.8472 −1.12433 −0.562164 0.827026i \(-0.690031\pi\)
−0.562164 + 0.827026i \(0.690031\pi\)
\(282\) 0 0
\(283\) −24.6250 −1.46381 −0.731903 0.681409i \(-0.761368\pi\)
−0.731903 + 0.681409i \(0.761368\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 23.0698 1.36177
\(288\) 0 0
\(289\) −13.9706 −0.821798
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4.41317 −0.257820 −0.128910 0.991656i \(-0.541148\pi\)
−0.128910 + 0.991656i \(0.541148\pi\)
\(294\) 0 0
\(295\) 12.8643 0.748989
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −7.39104 −0.427435
\(300\) 0 0
\(301\) 3.02944 0.174614
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 22.0296 1.26141
\(306\) 0 0
\(307\) −6.43215 −0.367102 −0.183551 0.983010i \(-0.558759\pi\)
−0.183551 + 0.983010i \(0.558759\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.32957 −0.245507 −0.122754 0.992437i \(-0.539173\pi\)
−0.122754 + 0.992437i \(0.539173\pi\)
\(312\) 0 0
\(313\) 18.1421 1.02545 0.512727 0.858552i \(-0.328635\pi\)
0.512727 + 0.858552i \(0.328635\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.93338 0.557914 0.278957 0.960304i \(-0.410011\pi\)
0.278957 + 0.960304i \(0.410011\pi\)
\(318\) 0 0
\(319\) 20.0768 1.12409
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 13.5140 0.751937
\(324\) 0 0
\(325\) 13.0711 0.725052
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −28.6931 −1.58190
\(330\) 0 0
\(331\) 12.8643 0.707086 0.353543 0.935418i \(-0.384977\pi\)
0.353543 + 0.935418i \(0.384977\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −19.1116 −1.04418
\(336\) 0 0
\(337\) −2.48528 −0.135382 −0.0676910 0.997706i \(-0.521563\pi\)
−0.0676910 + 0.997706i \(0.521563\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −32.1741 −1.74233
\(342\) 0 0
\(343\) −20.0768 −1.08405
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −23.3324 −1.25255 −0.626275 0.779602i \(-0.715421\pi\)
−0.626275 + 0.779602i \(0.715421\pi\)
\(348\) 0 0
\(349\) 30.2426 1.61885 0.809426 0.587222i \(-0.199779\pi\)
0.809426 + 0.587222i \(0.199779\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −35.6552 −1.89773 −0.948867 0.315675i \(-0.897769\pi\)
−0.948867 + 0.315675i \(0.897769\pi\)
\(354\) 0 0
\(355\) −18.1929 −0.965577
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.4525 −0.551662 −0.275831 0.961206i \(-0.588953\pi\)
−0.275831 + 0.961206i \(0.588953\pi\)
\(360\) 0 0
\(361\) 41.2843 2.17286
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 31.1546 1.63070
\(366\) 0 0
\(367\) −26.1188 −1.36339 −0.681695 0.731637i \(-0.738757\pi\)
−0.681695 + 0.731637i \(0.738757\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 12.3547 0.641422
\(372\) 0 0
\(373\) 30.7279 1.59103 0.795516 0.605933i \(-0.207200\pi\)
0.795516 + 0.605933i \(0.207200\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −18.5486 −0.955299
\(378\) 0 0
\(379\) 14.1964 0.729222 0.364611 0.931160i \(-0.381202\pi\)
0.364611 + 0.931160i \(0.381202\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 3.95815 0.202252 0.101126 0.994874i \(-0.467755\pi\)
0.101126 + 0.994874i \(0.467755\pi\)
\(384\) 0 0
\(385\) −24.9706 −1.27262
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 17.3178 0.878048 0.439024 0.898475i \(-0.355324\pi\)
0.439024 + 0.898475i \(0.355324\pi\)
\(390\) 0 0
\(391\) −3.76787 −0.190549
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −6.75699 −0.339981
\(396\) 0 0
\(397\) 20.3848 1.02308 0.511541 0.859259i \(-0.329075\pi\)
0.511541 + 0.859259i \(0.329075\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −35.3566 −1.76562 −0.882812 0.469727i \(-0.844352\pi\)
−0.882812 + 0.469727i \(0.844352\pi\)
\(402\) 0 0
\(403\) 29.7250 1.48071
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.16478 −0.107304
\(408\) 0 0
\(409\) 9.65685 0.477501 0.238750 0.971081i \(-0.423262\pi\)
0.238750 + 0.971081i \(0.423262\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 9.84591 0.484486
\(414\) 0 0
\(415\) −29.1732 −1.43206
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 33.2597 1.62484 0.812420 0.583072i \(-0.198149\pi\)
0.812420 + 0.583072i \(0.198149\pi\)
\(420\) 0 0
\(421\) −8.38478 −0.408649 −0.204324 0.978903i \(-0.565500\pi\)
−0.204324 + 0.978903i \(0.565500\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.66348 0.323226
\(426\) 0 0
\(427\) 16.8607 0.815948
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −6.49435 −0.312822 −0.156411 0.987692i \(-0.549992\pi\)
−0.156411 + 0.987692i \(0.549992\pi\)
\(432\) 0 0
\(433\) −4.97056 −0.238870 −0.119435 0.992842i \(-0.538108\pi\)
−0.119435 + 0.992842i \(0.538108\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −16.8080 −0.804037
\(438\) 0 0
\(439\) −9.48661 −0.452771 −0.226386 0.974038i \(-0.572691\pi\)
−0.226386 + 0.974038i \(0.572691\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 9.81845 0.466489 0.233244 0.972418i \(-0.425066\pi\)
0.233244 + 0.972418i \(0.425066\pi\)
\(444\) 0 0
\(445\) 35.3137 1.67403
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.66348 0.314469 0.157235 0.987561i \(-0.449742\pi\)
0.157235 + 0.987561i \(0.449742\pi\)
\(450\) 0 0
\(451\) −37.4893 −1.76530
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 23.0698 1.08153
\(456\) 0 0
\(457\) 7.31371 0.342121 0.171060 0.985261i \(-0.445281\pi\)
0.171060 + 0.985261i \(0.445281\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −24.2799 −1.13083 −0.565414 0.824807i \(-0.691284\pi\)
−0.565414 + 0.824807i \(0.691284\pi\)
\(462\) 0 0
\(463\) −31.4474 −1.46148 −0.730741 0.682655i \(-0.760825\pi\)
−0.730741 + 0.682655i \(0.760825\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.2095 0.796360 0.398180 0.917307i \(-0.369642\pi\)
0.398180 + 0.917307i \(0.369642\pi\)
\(468\) 0 0
\(469\) −14.6274 −0.675431
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.92296 −0.226358
\(474\) 0 0
\(475\) 29.7250 1.36388
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −31.3575 −1.43276 −0.716381 0.697710i \(-0.754203\pi\)
−0.716381 + 0.697710i \(0.754203\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −45.5011 −2.06610
\(486\) 0 0
\(487\) 28.7831 1.30429 0.652143 0.758096i \(-0.273870\pi\)
0.652143 + 0.758096i \(0.273870\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7.39104 −0.333553 −0.166776 0.985995i \(-0.553336\pi\)
−0.166776 + 0.985995i \(0.553336\pi\)
\(492\) 0 0
\(493\) −9.45584 −0.425870
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −13.9242 −0.624587
\(498\) 0 0
\(499\) 6.43215 0.287943 0.143971 0.989582i \(-0.454013\pi\)
0.143971 + 0.989582i \(0.454013\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10.4525 −0.466054 −0.233027 0.972470i \(-0.574863\pi\)
−0.233027 + 0.972470i \(0.574863\pi\)
\(504\) 0 0
\(505\) 23.4558 1.04377
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −6.45232 −0.285994 −0.142997 0.989723i \(-0.545674\pi\)
−0.142997 + 0.989723i \(0.545674\pi\)
\(510\) 0 0
\(511\) 23.8447 1.05483
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −25.8686 −1.13991
\(516\) 0 0
\(517\) 46.6274 2.05067
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −23.4715 −1.02831 −0.514153 0.857699i \(-0.671894\pi\)
−0.514153 + 0.857699i \(0.671894\pi\)
\(522\) 0 0
\(523\) −5.10001 −0.223008 −0.111504 0.993764i \(-0.535567\pi\)
−0.111504 + 0.993764i \(0.535567\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 15.1535 0.660096
\(528\) 0 0
\(529\) −18.3137 −0.796248
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 34.6356 1.50024
\(534\) 0 0
\(535\) −34.8250 −1.50562
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.75699 0.291044
\(540\) 0 0
\(541\) −18.7279 −0.805176 −0.402588 0.915381i \(-0.631889\pi\)
−0.402588 + 0.915381i \(0.631889\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.18243 −0.136320
\(546\) 0 0
\(547\) 3.99643 0.170875 0.0854374 0.996344i \(-0.472771\pi\)
0.0854374 + 0.996344i \(0.472771\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −42.1814 −1.79699
\(552\) 0 0
\(553\) −5.17157 −0.219918
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.97127 −0.125897 −0.0629483 0.998017i \(-0.520050\pi\)
−0.0629483 + 0.998017i \(0.520050\pi\)
\(558\) 0 0
\(559\) 4.54822 0.192369
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −13.6228 −0.574131 −0.287065 0.957911i \(-0.592680\pi\)
−0.287065 + 0.957911i \(0.592680\pi\)
\(564\) 0 0
\(565\) 49.9411 2.10104
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.6256 −0.571215 −0.285607 0.958347i \(-0.592195\pi\)
−0.285607 + 0.958347i \(0.592195\pi\)
\(570\) 0 0
\(571\) −24.6250 −1.03053 −0.515263 0.857032i \(-0.672306\pi\)
−0.515263 + 0.857032i \(0.672306\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −8.28772 −0.345622
\(576\) 0 0
\(577\) −25.1127 −1.04546 −0.522728 0.852500i \(-0.675086\pi\)
−0.522728 + 0.852500i \(0.675086\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −22.3282 −0.926331
\(582\) 0 0
\(583\) −20.0768 −0.831496
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −6.12293 −0.252721 −0.126360 0.991984i \(-0.540330\pi\)
−0.126360 + 0.991984i \(0.540330\pi\)
\(588\) 0 0
\(589\) 67.5980 2.78533
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 16.8080 0.690223 0.345111 0.938562i \(-0.387841\pi\)
0.345111 + 0.938562i \(0.387841\pi\)
\(594\) 0 0
\(595\) 11.7607 0.482143
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −18.7402 −0.765705 −0.382852 0.923810i \(-0.625058\pi\)
−0.382852 + 0.923810i \(0.625058\pi\)
\(600\) 0 0
\(601\) −15.1716 −0.618861 −0.309431 0.950922i \(-0.600138\pi\)
−0.309431 + 0.950922i \(0.600138\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.89422 0.320946
\(606\) 0 0
\(607\) 39.7634 1.61395 0.806974 0.590587i \(-0.201104\pi\)
0.806974 + 0.590587i \(0.201104\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −43.0781 −1.74275
\(612\) 0 0
\(613\) 0.100505 0.00405936 0.00202968 0.999998i \(-0.499354\pi\)
0.00202968 + 0.999998i \(0.499354\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 18.1929 0.731233 0.365617 0.930766i \(-0.380858\pi\)
0.365617 + 0.930766i \(0.380858\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 27.0279 1.08285
\(624\) 0 0
\(625\) −29.4853 −1.17941
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.01958 0.0406532
\(630\) 0 0
\(631\) 3.05446 0.121596 0.0607981 0.998150i \(-0.480635\pi\)
0.0607981 + 0.998150i \(0.480635\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −39.3826 −1.56285
\(636\) 0 0
\(637\) −6.24264 −0.247342
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.74053 0.0687467 0.0343734 0.999409i \(-0.489056\pi\)
0.0343734 + 0.999409i \(0.489056\pi\)
\(642\) 0 0
\(643\) 23.2929 0.918582 0.459291 0.888286i \(-0.348104\pi\)
0.459291 + 0.888286i \(0.348104\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 35.3157 1.38840 0.694201 0.719781i \(-0.255758\pi\)
0.694201 + 0.719781i \(0.255758\pi\)
\(648\) 0 0
\(649\) −16.0000 −0.628055
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 13.4144 0.524947 0.262474 0.964939i \(-0.415462\pi\)
0.262474 + 0.964939i \(0.415462\pi\)
\(654\) 0 0
\(655\) 40.1536 1.56893
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7.39104 0.287914 0.143957 0.989584i \(-0.454017\pi\)
0.143957 + 0.989584i \(0.454017\pi\)
\(660\) 0 0
\(661\) 3.61522 0.140616 0.0703080 0.997525i \(-0.477602\pi\)
0.0703080 + 0.997525i \(0.477602\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 52.4632 2.03444
\(666\) 0 0
\(667\) 11.7607 0.455377
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −27.3994 −1.05774
\(672\) 0 0
\(673\) 17.6569 0.680622 0.340311 0.940313i \(-0.389468\pi\)
0.340311 + 0.940313i \(0.389468\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.5298 1.63455 0.817277 0.576244i \(-0.195482\pi\)
0.817277 + 0.576244i \(0.195482\pi\)
\(678\) 0 0
\(679\) −34.8250 −1.33646
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 40.6507 1.55546 0.777728 0.628601i \(-0.216372\pi\)
0.777728 + 0.628601i \(0.216372\pi\)
\(684\) 0 0
\(685\) −44.7696 −1.71056
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 18.5486 0.706644
\(690\) 0 0
\(691\) 7.76429 0.295368 0.147684 0.989035i \(-0.452818\pi\)
0.147684 + 0.989035i \(0.452818\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 38.2233 1.44989
\(696\) 0 0
\(697\) 17.6569 0.668801
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 32.6839 1.23446 0.617228 0.786785i \(-0.288256\pi\)
0.617228 + 0.786785i \(0.288256\pi\)
\(702\) 0 0
\(703\) 4.54822 0.171539
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 17.9523 0.675167
\(708\) 0 0
\(709\) 18.7279 0.703342 0.351671 0.936124i \(-0.385614\pi\)
0.351671 + 0.936124i \(0.385614\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −18.8472 −0.705832
\(714\) 0 0
\(715\) −37.4893 −1.40202
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 14.7821 0.551278 0.275639 0.961261i \(-0.411111\pi\)
0.275639 + 0.961261i \(0.411111\pi\)
\(720\) 0 0
\(721\) −19.7990 −0.737353
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −20.7989 −0.772451
\(726\) 0 0
\(727\) 30.6670 1.13738 0.568688 0.822553i \(-0.307451\pi\)
0.568688 + 0.822553i \(0.307451\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.31863 0.0857577
\(732\) 0 0
\(733\) −1.55635 −0.0574851 −0.0287425 0.999587i \(-0.509150\pi\)
−0.0287425 + 0.999587i \(0.509150\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 23.7701 0.875584
\(738\) 0 0
\(739\) 24.6250 0.905846 0.452923 0.891550i \(-0.350381\pi\)
0.452923 + 0.891550i \(0.350381\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 18.7402 0.687512 0.343756 0.939059i \(-0.388301\pi\)
0.343756 + 0.939059i \(0.388301\pi\)
\(744\) 0 0
\(745\) 26.4853 0.970346
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −26.6539 −0.973914
\(750\) 0 0
\(751\) 40.8670 1.49126 0.745629 0.666361i \(-0.232149\pi\)
0.745629 + 0.666361i \(0.232149\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 39.3826 1.43328
\(756\) 0 0
\(757\) −1.95837 −0.0711781 −0.0355891 0.999367i \(-0.511331\pi\)
−0.0355891 + 0.999367i \(0.511331\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 23.4715 0.850842 0.425421 0.904996i \(-0.360126\pi\)
0.425421 + 0.904996i \(0.360126\pi\)
\(762\) 0 0
\(763\) −2.43573 −0.0881792
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.7821 0.533750
\(768\) 0 0
\(769\) −30.8284 −1.11170 −0.555851 0.831282i \(-0.687607\pi\)
−0.555851 + 0.831282i \(0.687607\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 21.8184 0.784755 0.392377 0.919804i \(-0.371653\pi\)
0.392377 + 0.919804i \(0.371653\pi\)
\(774\) 0 0
\(775\) 33.3313 1.19730
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 78.7652 2.82206
\(780\) 0 0
\(781\) 22.6274 0.809673
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 17.1067 0.610563
\(786\) 0 0
\(787\) −14.1964 −0.506049 −0.253024 0.967460i \(-0.581425\pi\)
−0.253024 + 0.967460i \(0.581425\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 38.2233 1.35906
\(792\) 0 0
\(793\) 25.3137 0.898916
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −18.7597 −0.664503 −0.332252 0.943191i \(-0.607808\pi\)
−0.332252 + 0.943191i \(0.607808\pi\)
\(798\) 0 0
\(799\) −21.9607 −0.776915
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −38.7485 −1.36741
\(804\) 0 0
\(805\) −14.6274 −0.515549
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 37.3957 1.31476 0.657382 0.753558i \(-0.271664\pi\)
0.657382 + 0.753558i \(0.271664\pi\)
\(810\) 0 0
\(811\) 24.3965 0.856676 0.428338 0.903619i \(-0.359099\pi\)
0.428338 + 0.903619i \(0.359099\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 61.2931 2.14700
\(816\) 0 0
\(817\) 10.3431 0.361861
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 30.6448 1.06951 0.534755 0.845007i \(-0.320404\pi\)
0.534755 + 0.845007i \(0.320404\pi\)
\(822\) 0 0
\(823\) −2.27411 −0.0792705 −0.0396352 0.999214i \(-0.512620\pi\)
−0.0396352 + 0.999214i \(0.512620\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −25.2346 −0.877492 −0.438746 0.898611i \(-0.644577\pi\)
−0.438746 + 0.898611i \(0.644577\pi\)
\(828\) 0 0
\(829\) 21.5563 0.748683 0.374341 0.927291i \(-0.377869\pi\)
0.374341 + 0.927291i \(0.377869\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.18243 −0.110265
\(834\) 0 0
\(835\) 63.2179 2.18775
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −12.9887 −0.448420 −0.224210 0.974541i \(-0.571980\pi\)
−0.224210 + 0.974541i \(0.571980\pi\)
\(840\) 0 0
\(841\) 0.514719 0.0177489
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −3.99084 −0.137289
\(846\) 0 0
\(847\) 6.04198 0.207605
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1.26810 −0.0434700
\(852\) 0 0
\(853\) −14.5269 −0.497392 −0.248696 0.968582i \(-0.580002\pi\)
−0.248696 + 0.968582i \(0.580002\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.5486 −0.633606 −0.316803 0.948491i \(-0.602609\pi\)
−0.316803 + 0.948491i \(0.602609\pi\)
\(858\) 0 0
\(859\) −20.6286 −0.703839 −0.351919 0.936030i \(-0.614471\pi\)
−0.351919 + 0.936030i \(0.614471\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −33.5223 −1.14111 −0.570556 0.821259i \(-0.693272\pi\)
−0.570556 + 0.821259i \(0.693272\pi\)
\(864\) 0 0
\(865\) −33.7990 −1.14920
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 8.40401 0.285087
\(870\) 0 0
\(871\) −21.9607 −0.744111
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −7.91630 −0.267620
\(876\) 0 0
\(877\) −45.8406 −1.54793 −0.773964 0.633230i \(-0.781729\pi\)
−0.773964 + 0.633230i \(0.781729\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −14.7689 −0.497576 −0.248788 0.968558i \(-0.580032\pi\)
−0.248788 + 0.968558i \(0.580032\pi\)
\(882\) 0 0
\(883\) −16.8607 −0.567409 −0.283704 0.958912i \(-0.591563\pi\)
−0.283704 + 0.958912i \(0.591563\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −43.6034 −1.46406 −0.732029 0.681273i \(-0.761427\pi\)
−0.732029 + 0.681273i \(0.761427\pi\)
\(888\) 0 0
\(889\) −30.1421 −1.01093
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −97.9643 −3.27825
\(894\) 0 0
\(895\) −71.2108 −2.38031
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −47.2989 −1.57751
\(900\) 0 0
\(901\) 9.45584 0.315020
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 79.4158 2.63987
\(906\) 0 0
\(907\) 10.4286 0.346275 0.173138 0.984898i \(-0.444609\pi\)
0.173138 + 0.984898i \(0.444609\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 26.6565 0.883170 0.441585 0.897219i \(-0.354416\pi\)
0.441585 + 0.897219i \(0.354416\pi\)
\(912\) 0 0
\(913\) 36.2843 1.20083
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 30.7322 1.01487
\(918\) 0 0
\(919\) 59.8402 1.97395 0.986974 0.160881i \(-0.0514335\pi\)
0.986974 + 0.160881i \(0.0514335\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −20.9050 −0.688097
\(924\) 0 0
\(925\) 2.24264 0.0737376
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.18243 0.104412 0.0522060 0.998636i \(-0.483375\pi\)
0.0522060 + 0.998636i \(0.483375\pi\)
\(930\) 0 0
\(931\) −14.1964 −0.465270
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −19.1116 −0.625018
\(936\) 0 0
\(937\) 15.9411 0.520774 0.260387 0.965504i \(-0.416150\pi\)
0.260387 + 0.965504i \(0.416150\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −31.2420 −1.01846 −0.509231 0.860630i \(-0.670070\pi\)
−0.509231 + 0.860630i \(0.670070\pi\)
\(942\) 0 0
\(943\) −21.9607 −0.715140
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 30.0894 0.977774 0.488887 0.872347i \(-0.337403\pi\)
0.488887 + 0.872347i \(0.337403\pi\)
\(948\) 0 0
\(949\) 35.7990 1.16208
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0.298627 0.00967349 0.00483674 0.999988i \(-0.498460\pi\)
0.00483674 + 0.999988i \(0.498460\pi\)
\(954\) 0 0
\(955\) 50.3536 1.62941
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −34.2651 −1.10648
\(960\) 0 0
\(961\) 44.7990 1.44513
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −13.3270 −0.429010
\(966\) 0 0
\(967\) −34.1116 −1.09696 −0.548478 0.836165i \(-0.684793\pi\)
−0.548478 + 0.836165i \(0.684793\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −14.6733 −0.470888 −0.235444 0.971888i \(-0.575654\pi\)
−0.235444 + 0.971888i \(0.575654\pi\)
\(972\) 0 0
\(973\) 29.2548 0.937867
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 17.1067 0.547290 0.273645 0.961831i \(-0.411771\pi\)
0.273645 + 0.961831i \(0.411771\pi\)
\(978\) 0 0
\(979\) −43.9215 −1.40374
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 17.3183 0.552367 0.276184 0.961105i \(-0.410930\pi\)
0.276184 + 0.961105i \(0.410930\pi\)
\(984\) 0 0
\(985\) −33.7990 −1.07693
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.88380 −0.0916995
\(990\) 0 0
\(991\) 42.4277 1.34776 0.673881 0.738840i \(-0.264626\pi\)
0.673881 + 0.738840i \(0.264626\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.15932 −0.0367528
\(996\) 0 0
\(997\) −38.0416 −1.20479 −0.602395 0.798198i \(-0.705787\pi\)
−0.602395 + 0.798198i \(0.705787\pi\)
\(998\) 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 9216.2.a.bs.1.8 8
3.2 odd 2 inner 9216.2.a.bs.1.2 8
4.3 odd 2 inner 9216.2.a.bs.1.7 8
8.3 odd 2 9216.2.a.br.1.1 8
8.5 even 2 9216.2.a.br.1.2 8
12.11 even 2 inner 9216.2.a.bs.1.1 8
24.5 odd 2 9216.2.a.br.1.8 8
24.11 even 2 9216.2.a.br.1.7 8
32.3 odd 8 4608.2.k.bk.1153.2 yes 16
32.5 even 8 4608.2.k.bl.3457.8 yes 16
32.11 odd 8 4608.2.k.bk.3457.1 yes 16
32.13 even 8 4608.2.k.bl.1153.7 yes 16
32.19 odd 8 4608.2.k.bl.1153.8 yes 16
32.21 even 8 4608.2.k.bk.3457.2 yes 16
32.27 odd 8 4608.2.k.bl.3457.7 yes 16
32.29 even 8 4608.2.k.bk.1153.1 16
96.5 odd 8 4608.2.k.bl.3457.2 yes 16
96.11 even 8 4608.2.k.bk.3457.7 yes 16
96.29 odd 8 4608.2.k.bk.1153.7 yes 16
96.35 even 8 4608.2.k.bk.1153.8 yes 16
96.53 odd 8 4608.2.k.bk.3457.8 yes 16
96.59 even 8 4608.2.k.bl.3457.1 yes 16
96.77 odd 8 4608.2.k.bl.1153.1 yes 16
96.83 even 8 4608.2.k.bl.1153.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.k.bk.1153.1 16 32.29 even 8
4608.2.k.bk.1153.2 yes 16 32.3 odd 8
4608.2.k.bk.1153.7 yes 16 96.29 odd 8
4608.2.k.bk.1153.8 yes 16 96.35 even 8
4608.2.k.bk.3457.1 yes 16 32.11 odd 8
4608.2.k.bk.3457.2 yes 16 32.21 even 8
4608.2.k.bk.3457.7 yes 16 96.11 even 8
4608.2.k.bk.3457.8 yes 16 96.53 odd 8
4608.2.k.bl.1153.1 yes 16 96.77 odd 8
4608.2.k.bl.1153.2 yes 16 96.83 even 8
4608.2.k.bl.1153.7 yes 16 32.13 even 8
4608.2.k.bl.1153.8 yes 16 32.19 odd 8
4608.2.k.bl.3457.1 yes 16 96.59 even 8
4608.2.k.bl.3457.2 yes 16 96.5 odd 8
4608.2.k.bl.3457.7 yes 16 32.27 odd 8
4608.2.k.bl.3457.8 yes 16 32.5 even 8
9216.2.a.br.1.1 8 8.3 odd 2
9216.2.a.br.1.2 8 8.5 even 2
9216.2.a.br.1.7 8 24.11 even 2
9216.2.a.br.1.8 8 24.5 odd 2
9216.2.a.bs.1.1 8 12.11 even 2 inner
9216.2.a.bs.1.2 8 3.2 odd 2 inner
9216.2.a.bs.1.7 8 4.3 odd 2 inner
9216.2.a.bs.1.8 8 1.1 even 1 trivial