Properties

Label 1152.2.f.a.575.2
Level $1152$
Weight $2$
Character 1152.575
Analytic conductor $9.199$
Analytic rank $0$
Dimension $4$
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] = [1152,2,Mod(575,1152)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1152.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1152 = 2^{7} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1152.f (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(9.19876631285\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 575.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1152.575
Dual form 1152.2.f.a.575.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{5} +2.82843i q^{7} +O(q^{10})\) \(q-1.41421 q^{5} +2.82843i q^{7} -4.00000i q^{11} +2.00000i q^{13} +1.41421i q^{17} -5.65685 q^{19} -4.00000 q^{23} -3.00000 q^{25} +7.07107 q^{29} +8.48528i q^{31} -4.00000i q^{35} -8.00000i q^{37} -4.24264i q^{41} -11.3137 q^{43} -12.0000 q^{47} -1.00000 q^{49} -12.7279 q^{53} +5.65685i q^{55} +8.00000i q^{61} -2.82843i q^{65} -5.65685 q^{67} -4.00000 q^{71} +8.00000 q^{73} +11.3137 q^{77} +2.82843i q^{79} -12.0000i q^{83} -2.00000i q^{85} +15.5563i q^{89} -5.65685 q^{91} +8.00000 q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{23} - 12 q^{25} - 48 q^{47} - 4 q^{49} - 16 q^{71} + 32 q^{73} + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(641\) \(901\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(5\) −1.41421 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) 2.82843i 1.06904i 0.845154 + 0.534522i \(0.179509\pi\)
−0.845154 + 0.534522i \(0.820491\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 4.00000i − 1.20605i −0.797724 0.603023i \(-0.793963\pi\)
0.797724 0.603023i \(-0.206037\pi\)
\(12\) 0 0
\(13\) 2.00000i 0.554700i 0.960769 + 0.277350i \(0.0894562\pi\)
−0.960769 + 0.277350i \(0.910544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421i 0.342997i 0.985184 + 0.171499i \(0.0548609\pi\)
−0.985184 + 0.171499i \(0.945139\pi\)
\(18\) 0 0
\(19\) −5.65685 −1.29777 −0.648886 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.07107 1.31306 0.656532 0.754298i \(-0.272023\pi\)
0.656532 + 0.754298i \(0.272023\pi\)
\(30\) 0 0
\(31\) 8.48528i 1.52400i 0.647576 + 0.762001i \(0.275783\pi\)
−0.647576 + 0.762001i \(0.724217\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 4.00000i − 0.676123i
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 0.657596i \(-0.771573\pi\)
0.753371 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 4.24264i − 0.662589i −0.943527 0.331295i \(-0.892515\pi\)
0.943527 0.331295i \(-0.107485\pi\)
\(42\) 0 0
\(43\) −11.3137 −1.72532 −0.862662 0.505781i \(-0.831205\pi\)
−0.862662 + 0.505781i \(0.831205\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −12.7279 −1.74831 −0.874157 0.485643i \(-0.838586\pi\)
−0.874157 + 0.485643i \(0.838586\pi\)
\(54\) 0 0
\(55\) 5.65685i 0.762770i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 8.00000i 1.02430i 0.858898 + 0.512148i \(0.171150\pi\)
−0.858898 + 0.512148i \(0.828850\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 2.82843i − 0.350823i
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −4.00000 −0.474713 −0.237356 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 8.00000 0.936329 0.468165 0.883641i \(-0.344915\pi\)
0.468165 + 0.883641i \(0.344915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 11.3137 1.28932
\(78\) 0 0
\(79\) 2.82843i 0.318223i 0.987261 + 0.159111i \(0.0508629\pi\)
−0.987261 + 0.159111i \(0.949137\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 12.0000i − 1.31717i −0.752506 0.658586i \(-0.771155\pi\)
0.752506 0.658586i \(-0.228845\pi\)
\(84\) 0 0
\(85\) − 2.00000i − 0.216930i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.5563i 1.64897i 0.565884 + 0.824485i \(0.308535\pi\)
−0.565884 + 0.824485i \(0.691465\pi\)
\(90\) 0 0
\(91\) −5.65685 −0.592999
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −9.89949 −0.985037 −0.492518 0.870302i \(-0.663924\pi\)
−0.492518 + 0.870302i \(0.663924\pi\)
\(102\) 0 0
\(103\) 8.48528i 0.836080i 0.908429 + 0.418040i \(0.137283\pi\)
−0.908429 + 0.418040i \(0.862717\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 14.0000i 1.34096i 0.741929 + 0.670478i \(0.233911\pi\)
−0.741929 + 0.670478i \(0.766089\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.41421i 0.133038i 0.997785 + 0.0665190i \(0.0211893\pi\)
−0.997785 + 0.0665190i \(0.978811\pi\)
\(114\) 0 0
\(115\) 5.65685 0.527504
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) − 2.82843i − 0.250982i −0.992095 0.125491i \(-0.959949\pi\)
0.992095 0.125491i \(-0.0400507\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.00000i 0.698963i 0.936943 + 0.349482i \(0.113642\pi\)
−0.936943 + 0.349482i \(0.886358\pi\)
\(132\) 0 0
\(133\) − 16.0000i − 1.38738i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 18.3848i − 1.57072i −0.619041 0.785359i \(-0.712479\pi\)
0.619041 0.785359i \(-0.287521\pi\)
\(138\) 0 0
\(139\) 16.9706 1.43942 0.719712 0.694273i \(-0.244274\pi\)
0.719712 + 0.694273i \(0.244274\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.00000 0.668994
\(144\) 0 0
\(145\) −10.0000 −0.830455
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.89949 −0.810998 −0.405499 0.914095i \(-0.632902\pi\)
−0.405499 + 0.914095i \(0.632902\pi\)
\(150\) 0 0
\(151\) − 8.48528i − 0.690522i −0.938507 0.345261i \(-0.887790\pi\)
0.938507 0.345261i \(-0.112210\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 12.0000i − 0.963863i
\(156\) 0 0
\(157\) − 8.00000i − 0.638470i −0.947676 0.319235i \(-0.896574\pi\)
0.947676 0.319235i \(-0.103426\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 11.3137i − 0.891645i
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 16.0000 1.23812 0.619059 0.785345i \(-0.287514\pi\)
0.619059 + 0.785345i \(0.287514\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 7.07107 0.537603 0.268802 0.963196i \(-0.413372\pi\)
0.268802 + 0.963196i \(0.413372\pi\)
\(174\) 0 0
\(175\) − 8.48528i − 0.641427i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 24.0000i 1.79384i 0.442189 + 0.896922i \(0.354202\pi\)
−0.442189 + 0.896922i \(0.645798\pi\)
\(180\) 0 0
\(181\) − 6.00000i − 0.445976i −0.974821 0.222988i \(-0.928419\pi\)
0.974821 0.222988i \(-0.0715812\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 11.3137i 0.831800i
\(186\) 0 0
\(187\) 5.65685 0.413670
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.0000 −1.15772 −0.578860 0.815427i \(-0.696502\pi\)
−0.578860 + 0.815427i \(0.696502\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.41421 0.100759 0.0503793 0.998730i \(-0.483957\pi\)
0.0503793 + 0.998730i \(0.483957\pi\)
\(198\) 0 0
\(199\) − 25.4558i − 1.80452i −0.431196 0.902258i \(-0.641908\pi\)
0.431196 0.902258i \(-0.358092\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 20.0000i 1.40372i
\(204\) 0 0
\(205\) 6.00000i 0.419058i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 22.6274i 1.56517i
\(210\) 0 0
\(211\) 5.65685 0.389434 0.194717 0.980859i \(-0.437621\pi\)
0.194717 + 0.980859i \(0.437621\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 16.0000 1.09119
\(216\) 0 0
\(217\) −24.0000 −1.62923
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.82843 −0.190261
\(222\) 0 0
\(223\) − 14.1421i − 0.947027i −0.880786 0.473514i \(-0.842985\pi\)
0.880786 0.473514i \(-0.157015\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 4.00000i − 0.265489i −0.991150 0.132745i \(-0.957621\pi\)
0.991150 0.132745i \(-0.0423790\pi\)
\(228\) 0 0
\(229\) 22.0000i 1.45380i 0.686743 + 0.726900i \(0.259040\pi\)
−0.686743 + 0.726900i \(0.740960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 15.5563i − 1.01913i −0.860432 0.509565i \(-0.829806\pi\)
0.860432 0.509565i \(-0.170194\pi\)
\(234\) 0 0
\(235\) 16.9706 1.10704
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1.41421 0.0903508
\(246\) 0 0
\(247\) − 11.3137i − 0.719874i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.0000i 1.26239i 0.775625 + 0.631194i \(0.217435\pi\)
−0.775625 + 0.631194i \(0.782565\pi\)
\(252\) 0 0
\(253\) 16.0000i 1.00591i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 1.41421i − 0.0882162i −0.999027 0.0441081i \(-0.985955\pi\)
0.999027 0.0441081i \(-0.0140446\pi\)
\(258\) 0 0
\(259\) 22.6274 1.40600
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 0 0
\(265\) 18.0000 1.10573
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.07107 −0.431131 −0.215565 0.976489i \(-0.569159\pi\)
−0.215565 + 0.976489i \(0.569159\pi\)
\(270\) 0 0
\(271\) − 2.82843i − 0.171815i −0.996303 0.0859074i \(-0.972621\pi\)
0.996303 0.0859074i \(-0.0273789\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.0000i 0.723627i
\(276\) 0 0
\(277\) − 6.00000i − 0.360505i −0.983620 0.180253i \(-0.942309\pi\)
0.983620 0.180253i \(-0.0576915\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 26.8701i 1.60293i 0.598040 + 0.801467i \(0.295947\pi\)
−0.598040 + 0.801467i \(0.704053\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.41421 −0.0826192 −0.0413096 0.999146i \(-0.513153\pi\)
−0.0413096 + 0.999146i \(0.513153\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 8.00000i − 0.462652i
\(300\) 0 0
\(301\) − 32.0000i − 1.84445i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 11.3137i − 0.647821i
\(306\) 0 0
\(307\) 16.9706 0.968561 0.484281 0.874913i \(-0.339081\pi\)
0.484281 + 0.874913i \(0.339081\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −14.0000 −0.791327 −0.395663 0.918396i \(-0.629485\pi\)
−0.395663 + 0.918396i \(0.629485\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −18.3848 −1.03259 −0.516296 0.856410i \(-0.672690\pi\)
−0.516296 + 0.856410i \(0.672690\pi\)
\(318\) 0 0
\(319\) − 28.2843i − 1.58362i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 8.00000i − 0.445132i
\(324\) 0 0
\(325\) − 6.00000i − 0.332820i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 33.9411i − 1.87123i
\(330\) 0 0
\(331\) −5.65685 −0.310929 −0.155464 0.987841i \(-0.549687\pi\)
−0.155464 + 0.987841i \(0.549687\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 8.00000 0.437087
\(336\) 0 0
\(337\) −24.0000 −1.30736 −0.653682 0.756770i \(-0.726776\pi\)
−0.653682 + 0.756770i \(0.726776\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 33.9411 1.83801
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 20.0000i 1.07366i 0.843692 + 0.536828i \(0.180378\pi\)
−0.843692 + 0.536828i \(0.819622\pi\)
\(348\) 0 0
\(349\) − 8.00000i − 0.428230i −0.976808 0.214115i \(-0.931313\pi\)
0.976808 0.214115i \(-0.0686868\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 9.89949i − 0.526897i −0.964673 0.263448i \(-0.915140\pi\)
0.964673 0.263448i \(-0.0848599\pi\)
\(354\) 0 0
\(355\) 5.65685 0.300235
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.0000 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(360\) 0 0
\(361\) 13.0000 0.684211
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11.3137 −0.592187
\(366\) 0 0
\(367\) 8.48528i 0.442928i 0.975169 + 0.221464i \(0.0710835\pi\)
−0.975169 + 0.221464i \(0.928916\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 36.0000i − 1.86903i
\(372\) 0 0
\(373\) 24.0000i 1.24267i 0.783544 + 0.621336i \(0.213410\pi\)
−0.783544 + 0.621336i \(0.786590\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.1421i 0.728357i
\(378\) 0 0
\(379\) 16.9706 0.871719 0.435860 0.900015i \(-0.356444\pi\)
0.435860 + 0.900015i \(0.356444\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) −16.0000 −0.815436
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 12.7279 0.645331 0.322666 0.946513i \(-0.395421\pi\)
0.322666 + 0.946513i \(0.395421\pi\)
\(390\) 0 0
\(391\) − 5.65685i − 0.286079i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 4.00000i − 0.201262i
\(396\) 0 0
\(397\) − 8.00000i − 0.401508i −0.979642 0.200754i \(-0.935661\pi\)
0.979642 0.200754i \(-0.0643393\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 24.0416i 1.20058i 0.799782 + 0.600291i \(0.204949\pi\)
−0.799782 + 0.600291i \(0.795051\pi\)
\(402\) 0 0
\(403\) −16.9706 −0.845364
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −32.0000 −1.58618
\(408\) 0 0
\(409\) −16.0000 −0.791149 −0.395575 0.918434i \(-0.629455\pi\)
−0.395575 + 0.918434i \(0.629455\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 16.9706i 0.833052i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 36.0000i 1.75872i 0.476162 + 0.879358i \(0.342028\pi\)
−0.476162 + 0.879358i \(0.657972\pi\)
\(420\) 0 0
\(421\) 26.0000i 1.26716i 0.773676 + 0.633581i \(0.218416\pi\)
−0.773676 + 0.633581i \(0.781584\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 4.24264i − 0.205798i
\(426\) 0 0
\(427\) −22.6274 −1.09502
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.0000 −0.578020 −0.289010 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 22.6274 1.08242
\(438\) 0 0
\(439\) − 8.48528i − 0.404980i −0.979284 0.202490i \(-0.935097\pi\)
0.979284 0.202490i \(-0.0649034\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 36.0000i − 1.71041i −0.518289 0.855206i \(-0.673431\pi\)
0.518289 0.855206i \(-0.326569\pi\)
\(444\) 0 0
\(445\) − 22.0000i − 1.04290i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 12.7279i − 0.600668i −0.953834 0.300334i \(-0.902902\pi\)
0.953834 0.300334i \(-0.0970981\pi\)
\(450\) 0 0
\(451\) −16.9706 −0.799113
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 8.00000 0.375046
\(456\) 0 0
\(457\) 32.0000 1.49690 0.748448 0.663193i \(-0.230799\pi\)
0.748448 + 0.663193i \(0.230799\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −29.6985 −1.38320 −0.691598 0.722282i \(-0.743093\pi\)
−0.691598 + 0.722282i \(0.743093\pi\)
\(462\) 0 0
\(463\) 25.4558i 1.18303i 0.806293 + 0.591517i \(0.201471\pi\)
−0.806293 + 0.591517i \(0.798529\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.0000i 0.555294i 0.960683 + 0.277647i \(0.0895545\pi\)
−0.960683 + 0.277647i \(0.910445\pi\)
\(468\) 0 0
\(469\) − 16.0000i − 0.738811i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 45.2548i 2.08082i
\(474\) 0 0
\(475\) 16.9706 0.778663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20.0000 −0.913823 −0.456912 0.889512i \(-0.651044\pi\)
−0.456912 + 0.889512i \(0.651044\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8.48528i 0.384505i 0.981346 + 0.192252i \(0.0615792\pi\)
−0.981346 + 0.192252i \(0.938421\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 8.00000i − 0.361035i −0.983572 0.180517i \(-0.942223\pi\)
0.983572 0.180517i \(-0.0577772\pi\)
\(492\) 0 0
\(493\) 10.0000i 0.450377i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 11.3137i − 0.507489i
\(498\) 0 0
\(499\) −22.6274 −1.01294 −0.506471 0.862257i \(-0.669050\pi\)
−0.506471 + 0.862257i \(0.669050\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12.0000 −0.535054 −0.267527 0.963550i \(-0.586206\pi\)
−0.267527 + 0.963550i \(0.586206\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −38.1838 −1.69247 −0.846233 0.532813i \(-0.821135\pi\)
−0.846233 + 0.532813i \(0.821135\pi\)
\(510\) 0 0
\(511\) 22.6274i 1.00098i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 12.0000i − 0.528783i
\(516\) 0 0
\(517\) 48.0000i 2.11104i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 29.6985i − 1.30111i −0.759457 0.650557i \(-0.774535\pi\)
0.759457 0.650557i \(-0.225465\pi\)
\(522\) 0 0
\(523\) −39.5980 −1.73150 −0.865749 0.500478i \(-0.833158\pi\)
−0.865749 + 0.500478i \(0.833158\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −12.0000 −0.522728
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.48528 0.367538
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.00000i 0.172292i
\(540\) 0 0
\(541\) − 2.00000i − 0.0859867i −0.999075 0.0429934i \(-0.986311\pi\)
0.999075 0.0429934i \(-0.0136894\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 19.7990i − 0.848096i
\(546\) 0 0
\(547\) −33.9411 −1.45122 −0.725609 0.688107i \(-0.758442\pi\)
−0.725609 + 0.688107i \(0.758442\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −40.0000 −1.70406
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −18.3848 −0.778988 −0.389494 0.921029i \(-0.627350\pi\)
−0.389494 + 0.921029i \(0.627350\pi\)
\(558\) 0 0
\(559\) − 22.6274i − 0.957038i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 12.0000i − 0.505740i −0.967500 0.252870i \(-0.918626\pi\)
0.967500 0.252870i \(-0.0813744\pi\)
\(564\) 0 0
\(565\) − 2.00000i − 0.0841406i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 7.07107i − 0.296435i −0.988955 0.148217i \(-0.952646\pi\)
0.988955 0.148217i \(-0.0473535\pi\)
\(570\) 0 0
\(571\) −33.9411 −1.42039 −0.710196 0.704004i \(-0.751394\pi\)
−0.710196 + 0.704004i \(0.751394\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) 10.0000 0.416305 0.208153 0.978096i \(-0.433255\pi\)
0.208153 + 0.978096i \(0.433255\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 33.9411 1.40812
\(582\) 0 0
\(583\) 50.9117i 2.10855i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 32.0000i − 1.32078i −0.750922 0.660391i \(-0.770391\pi\)
0.750922 0.660391i \(-0.229609\pi\)
\(588\) 0 0
\(589\) − 48.0000i − 1.97781i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 43.8406i 1.80032i 0.435561 + 0.900159i \(0.356550\pi\)
−0.435561 + 0.900159i \(0.643450\pi\)
\(594\) 0 0
\(595\) 5.65685 0.231908
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −46.0000 −1.87638 −0.938190 0.346122i \(-0.887498\pi\)
−0.938190 + 0.346122i \(0.887498\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.07107 0.287480
\(606\) 0 0
\(607\) 19.7990i 0.803616i 0.915724 + 0.401808i \(0.131618\pi\)
−0.915724 + 0.401808i \(0.868382\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 24.0000i − 0.970936i
\(612\) 0 0
\(613\) − 24.0000i − 0.969351i −0.874694 0.484675i \(-0.838938\pi\)
0.874694 0.484675i \(-0.161062\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 7.07107i − 0.284670i −0.989819 0.142335i \(-0.954539\pi\)
0.989819 0.142335i \(-0.0454611\pi\)
\(618\) 0 0
\(619\) 33.9411 1.36421 0.682105 0.731255i \(-0.261065\pi\)
0.682105 + 0.731255i \(0.261065\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −44.0000 −1.76282
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 11.3137 0.451107
\(630\) 0 0
\(631\) 42.4264i 1.68897i 0.535580 + 0.844484i \(0.320093\pi\)
−0.535580 + 0.844484i \(0.679907\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.00000i 0.158735i
\(636\) 0 0
\(637\) − 2.00000i − 0.0792429i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.41421i 0.0558581i 0.999610 + 0.0279290i \(0.00889125\pi\)
−0.999610 + 0.0279290i \(0.991109\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.00000 0.157256 0.0786281 0.996904i \(-0.474946\pi\)
0.0786281 + 0.996904i \(0.474946\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 29.6985 1.16219 0.581096 0.813835i \(-0.302624\pi\)
0.581096 + 0.813835i \(0.302624\pi\)
\(654\) 0 0
\(655\) − 11.3137i − 0.442063i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 16.0000i − 0.623272i −0.950202 0.311636i \(-0.899123\pi\)
0.950202 0.311636i \(-0.100877\pi\)
\(660\) 0 0
\(661\) 40.0000i 1.55582i 0.628376 + 0.777910i \(0.283720\pi\)
−0.628376 + 0.777910i \(0.716280\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 22.6274i 0.877454i
\(666\) 0 0
\(667\) −28.2843 −1.09517
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 32.0000 1.23535
\(672\) 0 0
\(673\) −2.00000 −0.0770943 −0.0385472 0.999257i \(-0.512273\pi\)
−0.0385472 + 0.999257i \(0.512273\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 21.2132 0.815290 0.407645 0.913141i \(-0.366350\pi\)
0.407645 + 0.913141i \(0.366350\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 12.0000i 0.459167i 0.973289 + 0.229584i \(0.0737364\pi\)
−0.973289 + 0.229584i \(0.926264\pi\)
\(684\) 0 0
\(685\) 26.0000i 0.993409i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 25.4558i − 0.969790i
\(690\) 0 0
\(691\) −22.6274 −0.860788 −0.430394 0.902641i \(-0.641625\pi\)
−0.430394 + 0.902641i \(0.641625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −24.0000 −0.910372
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −26.8701 −1.01487 −0.507434 0.861691i \(-0.669406\pi\)
−0.507434 + 0.861691i \(0.669406\pi\)
\(702\) 0 0
\(703\) 45.2548i 1.70682i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 28.0000i − 1.05305i
\(708\) 0 0
\(709\) − 26.0000i − 0.976450i −0.872718 0.488225i \(-0.837644\pi\)
0.872718 0.488225i \(-0.162356\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 33.9411i − 1.27111i
\(714\) 0 0
\(715\) −11.3137 −0.423109
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.00000 0.149175 0.0745874 0.997214i \(-0.476236\pi\)
0.0745874 + 0.997214i \(0.476236\pi\)
\(720\) 0 0
\(721\) −24.0000 −0.893807
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −21.2132 −0.787839
\(726\) 0 0
\(727\) − 14.1421i − 0.524503i −0.965000 0.262251i \(-0.915535\pi\)
0.965000 0.262251i \(-0.0844650\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 16.0000i − 0.591781i
\(732\) 0 0
\(733\) − 30.0000i − 1.10808i −0.832492 0.554038i \(-0.813086\pi\)
0.832492 0.554038i \(-0.186914\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 22.6274i 0.833492i
\(738\) 0 0
\(739\) 33.9411 1.24854 0.624272 0.781207i \(-0.285396\pi\)
0.624272 + 0.781207i \(0.285396\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 14.0000 0.512920
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 19.7990i 0.722475i 0.932474 + 0.361238i \(0.117646\pi\)
−0.932474 + 0.361238i \(0.882354\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.0000i 0.436725i
\(756\) 0 0
\(757\) 6.00000i 0.218074i 0.994038 + 0.109037i \(0.0347767\pi\)
−0.994038 + 0.109037i \(0.965223\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 26.8701i − 0.974039i −0.873391 0.487019i \(-0.838084\pi\)
0.873391 0.487019i \(-0.161916\pi\)
\(762\) 0 0
\(763\) −39.5980 −1.43354
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −35.3553 −1.27164 −0.635822 0.771836i \(-0.719339\pi\)
−0.635822 + 0.771836i \(0.719339\pi\)
\(774\) 0 0
\(775\) − 25.4558i − 0.914401i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24.0000i 0.859889i
\(780\) 0 0
\(781\) 16.0000i 0.572525i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 11.3137i 0.403804i
\(786\) 0 0
\(787\) 28.2843 1.00823 0.504113 0.863638i \(-0.331820\pi\)
0.504113 + 0.863638i \(0.331820\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) −16.0000 −0.568177
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −15.5563 −0.551034 −0.275517 0.961296i \(-0.588849\pi\)
−0.275517 + 0.961296i \(0.588849\pi\)
\(798\) 0 0
\(799\) − 16.9706i − 0.600375i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 32.0000i − 1.12926i
\(804\) 0 0
\(805\) 16.0000i 0.563926i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.3848i 0.646374i 0.946335 + 0.323187i \(0.104754\pi\)
−0.946335 + 0.323187i \(0.895246\pi\)
\(810\) 0 0
\(811\) 16.9706 0.595917 0.297959 0.954579i \(-0.403694\pi\)
0.297959 + 0.954579i \(0.403694\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 64.0000 2.23908
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.41421 0.0493564 0.0246782 0.999695i \(-0.492144\pi\)
0.0246782 + 0.999695i \(0.492144\pi\)
\(822\) 0 0
\(823\) − 8.48528i − 0.295778i −0.989004 0.147889i \(-0.952752\pi\)
0.989004 0.147889i \(-0.0472479\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 8.00000i − 0.278187i −0.990279 0.139094i \(-0.955581\pi\)
0.990279 0.139094i \(-0.0444189\pi\)
\(828\) 0 0
\(829\) 30.0000i 1.04194i 0.853574 + 0.520972i \(0.174430\pi\)
−0.853574 + 0.520972i \(0.825570\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 1.41421i − 0.0489996i
\(834\) 0 0
\(835\) −22.6274 −0.783054
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 52.0000 1.79524 0.897620 0.440771i \(-0.145295\pi\)
0.897620 + 0.440771i \(0.145295\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −12.7279 −0.437854
\(846\) 0 0
\(847\) − 14.1421i − 0.485930i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 32.0000i 1.09695i
\(852\) 0 0
\(853\) − 24.0000i − 0.821744i −0.911693 0.410872i \(-0.865224\pi\)
0.911693 0.410872i \(-0.134776\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 52.3259i 1.78742i 0.448646 + 0.893709i \(0.351906\pi\)
−0.448646 + 0.893709i \(0.648094\pi\)
\(858\) 0 0
\(859\) 33.9411 1.15806 0.579028 0.815308i \(-0.303432\pi\)
0.579028 + 0.815308i \(0.303432\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −32.0000 −1.08929 −0.544646 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.3137 0.383791
\(870\) 0 0
\(871\) − 11.3137i − 0.383350i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 32.0000i 1.08180i
\(876\) 0 0
\(877\) − 8.00000i − 0.270141i −0.990836 0.135070i \(-0.956874\pi\)
0.990836 0.135070i \(-0.0431261\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 12.7279i − 0.428815i −0.976744 0.214407i \(-0.931218\pi\)
0.976744 0.214407i \(-0.0687820\pi\)
\(882\) 0 0
\(883\) −33.9411 −1.14221 −0.571105 0.820877i \(-0.693485\pi\)
−0.571105 + 0.820877i \(0.693485\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −8.00000 −0.268614 −0.134307 0.990940i \(-0.542881\pi\)
−0.134307 + 0.990940i \(0.542881\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 67.8823 2.27159
\(894\) 0 0
\(895\) − 33.9411i − 1.13453i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 60.0000i 2.00111i
\(900\) 0 0
\(901\) − 18.0000i − 0.599667i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8.48528i 0.282060i
\(906\) 0 0
\(907\) 33.9411 1.12700 0.563498 0.826117i \(-0.309455\pi\)
0.563498 + 0.826117i \(0.309455\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −32.0000 −1.06021 −0.530104 0.847933i \(-0.677847\pi\)
−0.530104 + 0.847933i \(0.677847\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −22.6274 −0.747223
\(918\) 0 0
\(919\) − 31.1127i − 1.02631i −0.858295 0.513157i \(-0.828476\pi\)
0.858295 0.513157i \(-0.171524\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 8.00000i − 0.263323i
\(924\) 0 0
\(925\) 24.0000i 0.789115i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 32.5269i − 1.06717i −0.845745 0.533587i \(-0.820844\pi\)
0.845745 0.533587i \(-0.179156\pi\)
\(930\) 0 0
\(931\) 5.65685 0.185396
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −8.00000 −0.261628
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 15.5563 0.507122 0.253561 0.967319i \(-0.418398\pi\)
0.253561 + 0.967319i \(0.418398\pi\)
\(942\) 0 0
\(943\) 16.9706i 0.552638i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 16.0000i 0.519930i 0.965618 + 0.259965i \(0.0837111\pi\)
−0.965618 + 0.259965i \(0.916289\pi\)
\(948\) 0 0
\(949\) 16.0000i 0.519382i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 41.0122i − 1.32852i −0.747504 0.664258i \(-0.768748\pi\)
0.747504 0.664258i \(-0.231252\pi\)
\(954\) 0 0
\(955\) 22.6274 0.732206
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 52.0000 1.67917
\(960\) 0 0
\(961\) −41.0000 −1.32258
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −14.1421 −0.455251
\(966\) 0 0
\(967\) 31.1127i 1.00052i 0.865876 + 0.500258i \(0.166762\pi\)
−0.865876 + 0.500258i \(0.833238\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 20.0000i − 0.641831i −0.947108 0.320915i \(-0.896010\pi\)
0.947108 0.320915i \(-0.103990\pi\)
\(972\) 0 0
\(973\) 48.0000i 1.53881i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 21.2132i 0.678671i 0.940666 + 0.339335i \(0.110202\pi\)
−0.940666 + 0.339335i \(0.889798\pi\)
\(978\) 0 0
\(979\) 62.2254 1.98873
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 45.2548 1.43902
\(990\) 0 0
\(991\) − 48.0833i − 1.52742i −0.645562 0.763708i \(-0.723377\pi\)
0.645562 0.763708i \(-0.276623\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 36.0000i 1.14128i
\(996\) 0 0
\(997\) − 40.0000i − 1.26681i −0.773819 0.633406i \(-0.781656\pi\)
0.773819 0.633406i \(-0.218344\pi\)
\(998\) 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 1152.2.f.a.575.2 yes 4
3.2 odd 2 1152.2.f.d.575.4 yes 4
4.3 odd 2 1152.2.f.d.575.1 yes 4
8.3 odd 2 1152.2.f.d.575.3 yes 4
8.5 even 2 inner 1152.2.f.a.575.4 yes 4
12.11 even 2 inner 1152.2.f.a.575.3 yes 4
16.3 odd 4 2304.2.c.g.2303.2 2
16.5 even 4 2304.2.c.h.2303.1 2
16.11 odd 4 2304.2.c.b.2303.1 2
16.13 even 4 2304.2.c.a.2303.2 2
24.5 odd 2 1152.2.f.d.575.2 yes 4
24.11 even 2 inner 1152.2.f.a.575.1 4
48.5 odd 4 2304.2.c.b.2303.2 2
48.11 even 4 2304.2.c.h.2303.2 2
48.29 odd 4 2304.2.c.g.2303.1 2
48.35 even 4 2304.2.c.a.2303.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.f.a.575.1 4 24.11 even 2 inner
1152.2.f.a.575.2 yes 4 1.1 even 1 trivial
1152.2.f.a.575.3 yes 4 12.11 even 2 inner
1152.2.f.a.575.4 yes 4 8.5 even 2 inner
1152.2.f.d.575.1 yes 4 4.3 odd 2
1152.2.f.d.575.2 yes 4 24.5 odd 2
1152.2.f.d.575.3 yes 4 8.3 odd 2
1152.2.f.d.575.4 yes 4 3.2 odd 2
2304.2.c.a.2303.1 2 48.35 even 4
2304.2.c.a.2303.2 2 16.13 even 4
2304.2.c.b.2303.1 2 16.11 odd 4
2304.2.c.b.2303.2 2 48.5 odd 4
2304.2.c.g.2303.1 2 48.29 odd 4
2304.2.c.g.2303.2 2 16.3 odd 4
2304.2.c.h.2303.1 2 16.5 even 4
2304.2.c.h.2303.2 2 48.11 even 4