Properties

Label 4032.2.h.g.575.4
Level $4032$
Weight $2$
Character 4032.575
Analytic conductor $32.196$
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] = [4032,2,Mod(575,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.h (of order \(2\), degree \(1\), not 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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.5473632256.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 49x^{4} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 2016)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 575.4
Root \(1.81129 - 1.81129i\) of defining polynomial
Character \(\chi\) \(=\) 4032.575
Dual form 4032.2.h.g.575.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.20837i q^{5} +1.00000i q^{7} +O(q^{10})\) \(q-2.20837i q^{5} +1.00000i q^{7} -0.794156 q^{11} +5.12311 q^{13} -0.620058i q^{17} -4.00000i q^{19} +0.794156 q^{23} +0.123106 q^{25} +3.00252i q^{29} +6.24621i q^{31} +2.20837 q^{35} +11.1231 q^{37} -3.44849i q^{41} -4.00000i q^{43} -12.9020 q^{47} -1.00000 q^{49} +3.00252i q^{53} +1.75379i q^{55} -1.58831 q^{59} +2.00000 q^{61} -11.3137i q^{65} -5.12311i q^{67} +8.03932 q^{71} +1.12311 q^{73} -0.794156i q^{77} -11.3693i q^{79} +14.4903 q^{83} -1.36932 q^{85} +2.20837i q^{89} +5.12311i q^{91} -8.83348 q^{95} +1.12311 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{13} - 32 q^{25} + 56 q^{37} - 8 q^{49} + 16 q^{61} - 24 q^{73} + 88 q^{85} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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
\(5\) − 2.20837i − 0.987613i −0.869572 0.493806i \(-0.835605\pi\)
0.869572 0.493806i \(-0.164395\pi\)
\(6\) 0 0
\(7\) 1.00000i 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.794156 −0.239447 −0.119723 0.992807i \(-0.538201\pi\)
−0.119723 + 0.992807i \(0.538201\pi\)
\(12\) 0 0
\(13\) 5.12311 1.42089 0.710447 0.703751i \(-0.248493\pi\)
0.710447 + 0.703751i \(0.248493\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 0.620058i − 0.150386i −0.997169 0.0751931i \(-0.976043\pi\)
0.997169 0.0751931i \(-0.0239573\pi\)
\(18\) 0 0
\(19\) − 4.00000i − 0.917663i −0.888523 0.458831i \(-0.848268\pi\)
0.888523 0.458831i \(-0.151732\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.794156 0.165593 0.0827964 0.996566i \(-0.473615\pi\)
0.0827964 + 0.996566i \(0.473615\pi\)
\(24\) 0 0
\(25\) 0.123106 0.0246211
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00252i 0.557555i 0.960356 + 0.278777i \(0.0899292\pi\)
−0.960356 + 0.278777i \(0.910071\pi\)
\(30\) 0 0
\(31\) 6.24621i 1.12185i 0.827866 + 0.560926i \(0.189555\pi\)
−0.827866 + 0.560926i \(0.810445\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.20837 0.373283
\(36\) 0 0
\(37\) 11.1231 1.82863 0.914314 0.405007i \(-0.132731\pi\)
0.914314 + 0.405007i \(0.132731\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 3.44849i − 0.538563i −0.963062 0.269281i \(-0.913214\pi\)
0.963062 0.269281i \(-0.0867862\pi\)
\(42\) 0 0
\(43\) − 4.00000i − 0.609994i −0.952353 0.304997i \(-0.901344\pi\)
0.952353 0.304997i \(-0.0986555\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −12.9020 −1.88195 −0.940976 0.338472i \(-0.890090\pi\)
−0.940976 + 0.338472i \(0.890090\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.00252i 0.412428i 0.978507 + 0.206214i \(0.0661144\pi\)
−0.978507 + 0.206214i \(0.933886\pi\)
\(54\) 0 0
\(55\) 1.75379i 0.236481i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.58831 −0.206781 −0.103390 0.994641i \(-0.532969\pi\)
−0.103390 + 0.994641i \(0.532969\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 11.3137i − 1.40329i
\(66\) 0 0
\(67\) − 5.12311i − 0.625887i −0.949772 0.312943i \(-0.898685\pi\)
0.949772 0.312943i \(-0.101315\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.03932 0.954092 0.477046 0.878878i \(-0.341708\pi\)
0.477046 + 0.878878i \(0.341708\pi\)
\(72\) 0 0
\(73\) 1.12311 0.131450 0.0657248 0.997838i \(-0.479064\pi\)
0.0657248 + 0.997838i \(0.479064\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.794156i − 0.0905024i
\(78\) 0 0
\(79\) − 11.3693i − 1.27915i −0.768729 0.639574i \(-0.779111\pi\)
0.768729 0.639574i \(-0.220889\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.4903 1.59052 0.795260 0.606268i \(-0.207334\pi\)
0.795260 + 0.606268i \(0.207334\pi\)
\(84\) 0 0
\(85\) −1.36932 −0.148523
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.20837i 0.234087i 0.993127 + 0.117043i \(0.0373416\pi\)
−0.993127 + 0.117043i \(0.962658\pi\)
\(90\) 0 0
\(91\) 5.12311i 0.537047i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.83348 −0.906296
\(96\) 0 0
\(97\) 1.12311 0.114034 0.0570170 0.998373i \(-0.481841\pi\)
0.0570170 + 0.998373i \(0.481841\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.03680i 0.501180i 0.968093 + 0.250590i \(0.0806246\pi\)
−0.968093 + 0.250590i \(0.919375\pi\)
\(102\) 0 0
\(103\) 12.0000i 1.18240i 0.806527 + 0.591198i \(0.201345\pi\)
−0.806527 + 0.591198i \(0.798655\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.38247 0.230322 0.115161 0.993347i \(-0.463262\pi\)
0.115161 + 0.993347i \(0.463262\pi\)
\(108\) 0 0
\(109\) 12.0000 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 20.3213i − 1.91167i −0.293911 0.955833i \(-0.594957\pi\)
0.293911 0.955833i \(-0.405043\pi\)
\(114\) 0 0
\(115\) − 1.75379i − 0.163542i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.620058 0.0568406
\(120\) 0 0
\(121\) −10.3693 −0.942665
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 11.3137i − 1.01193i
\(126\) 0 0
\(127\) 19.3693i 1.71875i 0.511347 + 0.859374i \(0.329147\pi\)
−0.511347 + 0.859374i \(0.670853\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.65685 0.494242 0.247121 0.968985i \(-0.420516\pi\)
0.247121 + 0.968985i \(0.420516\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 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
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.06854 −0.340229
\(144\) 0 0
\(145\) 6.63068 0.550648
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 13.0761i − 1.07124i −0.844460 0.535619i \(-0.820078\pi\)
0.844460 0.535619i \(-0.179922\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 13.7939 1.10796
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.794156i 0.0625882i
\(162\) 0 0
\(163\) − 17.6155i − 1.37975i −0.723926 0.689877i \(-0.757664\pi\)
0.723926 0.689877i \(-0.242336\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.24517 0.560648 0.280324 0.959905i \(-0.409558\pi\)
0.280324 + 0.959905i \(0.409558\pi\)
\(168\) 0 0
\(169\) 13.2462 1.01894
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 20.7672i − 1.57890i −0.613812 0.789452i \(-0.710365\pi\)
0.613812 0.789452i \(-0.289635\pi\)
\(174\) 0 0
\(175\) 0.123106i 0.00930591i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −20.9413 −1.56523 −0.782615 0.622506i \(-0.786114\pi\)
−0.782615 + 0.622506i \(0.786114\pi\)
\(180\) 0 0
\(181\) 5.12311 0.380797 0.190399 0.981707i \(-0.439022\pi\)
0.190399 + 0.981707i \(0.439022\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 24.5639i − 1.80598i
\(186\) 0 0
\(187\) 0.492423i 0.0360095i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.86270 −0.351853 −0.175926 0.984403i \(-0.556292\pi\)
−0.175926 + 0.984403i \(0.556292\pi\)
\(192\) 0 0
\(193\) −7.12311 −0.512732 −0.256366 0.966580i \(-0.582525\pi\)
−0.256366 + 0.966580i \(0.582525\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 14.3162i − 1.01999i −0.860178 0.509995i \(-0.829647\pi\)
0.860178 0.509995i \(-0.170353\pi\)
\(198\) 0 0
\(199\) − 5.75379i − 0.407875i −0.978984 0.203938i \(-0.934626\pi\)
0.978984 0.203938i \(-0.0653740\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −3.00252 −0.210736
\(204\) 0 0
\(205\) −7.61553 −0.531892
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.17662i 0.219732i
\(210\) 0 0
\(211\) − 16.4924i − 1.13539i −0.823241 0.567693i \(-0.807836\pi\)
0.823241 0.567693i \(-0.192164\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −8.83348 −0.602438
\(216\) 0 0
\(217\) −6.24621 −0.424020
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 3.17662i − 0.213683i
\(222\) 0 0
\(223\) 2.24621i 0.150417i 0.997168 + 0.0752087i \(0.0239623\pi\)
−0.997168 + 0.0752087i \(0.976038\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −29.8726 −1.98271 −0.991356 0.131196i \(-0.958118\pi\)
−0.991356 + 0.131196i \(0.958118\pi\)
\(228\) 0 0
\(229\) −10.8769 −0.718765 −0.359383 0.933190i \(-0.617013\pi\)
−0.359383 + 0.933190i \(0.617013\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.83095i 0.381998i 0.981590 + 0.190999i \(0.0611728\pi\)
−0.981590 + 0.190999i \(0.938827\pi\)
\(234\) 0 0
\(235\) 28.4924i 1.85864i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.03932 0.520020 0.260010 0.965606i \(-0.416274\pi\)
0.260010 + 0.965606i \(0.416274\pi\)
\(240\) 0 0
\(241\) −9.12311 −0.587671 −0.293835 0.955856i \(-0.594932\pi\)
−0.293835 + 0.955856i \(0.594932\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.20837i 0.141088i
\(246\) 0 0
\(247\) − 20.4924i − 1.30390i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.9020 0.814368 0.407184 0.913346i \(-0.366511\pi\)
0.407184 + 0.913346i \(0.366511\pi\)
\(252\) 0 0
\(253\) −0.630683 −0.0396507
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.62511i 0.413263i 0.978419 + 0.206631i \(0.0662501\pi\)
−0.978419 + 0.206631i \(0.933750\pi\)
\(258\) 0 0
\(259\) 11.1231i 0.691156i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 26.5982 1.64011 0.820057 0.572281i \(-0.193941\pi\)
0.820057 + 0.572281i \(0.193941\pi\)
\(264\) 0 0
\(265\) 6.63068 0.407320
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 10.3455i − 0.630774i −0.948963 0.315387i \(-0.897866\pi\)
0.948963 0.315387i \(-0.102134\pi\)
\(270\) 0 0
\(271\) 4.00000i 0.242983i 0.992592 + 0.121491i \(0.0387677\pi\)
−0.992592 + 0.121491i \(0.961232\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −0.0977650 −0.00589545
\(276\) 0 0
\(277\) 9.12311 0.548154 0.274077 0.961708i \(-0.411628\pi\)
0.274077 + 0.961708i \(0.411628\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 15.9045i 0.948786i 0.880313 + 0.474393i \(0.157332\pi\)
−0.880313 + 0.474393i \(0.842668\pi\)
\(282\) 0 0
\(283\) 12.0000i 0.713326i 0.934233 + 0.356663i \(0.116086\pi\)
−0.934233 + 0.356663i \(0.883914\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.44849 0.203558
\(288\) 0 0
\(289\) 16.6155 0.977384
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 5.03680i − 0.294253i −0.989118 0.147126i \(-0.952998\pi\)
0.989118 0.147126i \(-0.0470024\pi\)
\(294\) 0 0
\(295\) 3.50758i 0.204219i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.06854 0.235290
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 4.41674i − 0.252902i
\(306\) 0 0
\(307\) − 14.2462i − 0.813074i −0.913634 0.406537i \(-0.866736\pi\)
0.913634 0.406537i \(-0.133264\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −16.0786 −0.911736 −0.455868 0.890047i \(-0.650671\pi\)
−0.455868 + 0.890047i \(0.650671\pi\)
\(312\) 0 0
\(313\) 20.2462 1.14438 0.572192 0.820120i \(-0.306093\pi\)
0.572192 + 0.820120i \(0.306093\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 24.3898i 1.36987i 0.728605 + 0.684935i \(0.240169\pi\)
−0.728605 + 0.684935i \(0.759831\pi\)
\(318\) 0 0
\(319\) − 2.38447i − 0.133505i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.48023 −0.138004
\(324\) 0 0
\(325\) 0.630683 0.0349840
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 12.9020i − 0.711311i
\(330\) 0 0
\(331\) − 0.492423i − 0.0270660i −0.999908 0.0135330i \(-0.995692\pi\)
0.999908 0.0135330i \(-0.00430782\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −11.3137 −0.618134
\(336\) 0 0
\(337\) 13.7538 0.749217 0.374608 0.927183i \(-0.377777\pi\)
0.374608 + 0.927183i \(0.377777\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 4.96046i − 0.268624i
\(342\) 0 0
\(343\) − 1.00000i − 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 25.0099 1.34260 0.671300 0.741185i \(-0.265736\pi\)
0.671300 + 0.741185i \(0.265736\pi\)
\(348\) 0 0
\(349\) −24.2462 −1.29787 −0.648935 0.760844i \(-0.724785\pi\)
−0.648935 + 0.760844i \(0.724785\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 1.86017i − 0.0990071i −0.998774 0.0495035i \(-0.984236\pi\)
0.998774 0.0495035i \(-0.0157639\pi\)
\(354\) 0 0
\(355\) − 17.7538i − 0.942273i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.0784 1.53470 0.767350 0.641228i \(-0.221575\pi\)
0.767350 + 0.641228i \(0.221575\pi\)
\(360\) 0 0
\(361\) 3.00000 0.157895
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 2.48023i − 0.129821i
\(366\) 0 0
\(367\) − 36.4924i − 1.90489i −0.304712 0.952444i \(-0.598560\pi\)
0.304712 0.952444i \(-0.401440\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −3.00252 −0.155883
\(372\) 0 0
\(373\) 24.7386 1.28092 0.640459 0.767992i \(-0.278744\pi\)
0.640459 + 0.767992i \(0.278744\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 15.3823i 0.792226i
\(378\) 0 0
\(379\) 28.0000i 1.43826i 0.694874 + 0.719132i \(0.255460\pi\)
−0.694874 + 0.719132i \(0.744540\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −28.2843 −1.44526 −0.722629 0.691236i \(-0.757067\pi\)
−0.722629 + 0.691236i \(0.757067\pi\)
\(384\) 0 0
\(385\) −1.75379 −0.0893814
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 1.06602i − 0.0540493i −0.999635 0.0270246i \(-0.991397\pi\)
0.999635 0.0270246i \(-0.00860326\pi\)
\(390\) 0 0
\(391\) − 0.492423i − 0.0249029i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −25.1076 −1.26330
\(396\) 0 0
\(397\) −14.0000 −0.702640 −0.351320 0.936255i \(-0.614267\pi\)
−0.351320 + 0.936255i \(0.614267\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 4.59084i − 0.229255i −0.993409 0.114628i \(-0.963432\pi\)
0.993409 0.114628i \(-0.0365675\pi\)
\(402\) 0 0
\(403\) 32.0000i 1.59403i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.83348 −0.437859
\(408\) 0 0
\(409\) 27.3693 1.35333 0.676663 0.736293i \(-0.263425\pi\)
0.676663 + 0.736293i \(0.263425\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 1.58831i − 0.0781557i
\(414\) 0 0
\(415\) − 32.0000i − 1.57082i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −21.7355 −1.06185 −0.530924 0.847419i \(-0.678155\pi\)
−0.530924 + 0.847419i \(0.678155\pi\)
\(420\) 0 0
\(421\) −33.1231 −1.61432 −0.807161 0.590332i \(-0.798997\pi\)
−0.807161 + 0.590332i \(0.798997\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 0.0763326i − 0.00370268i
\(426\) 0 0
\(427\) 2.00000i 0.0967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 34.7353 1.67314 0.836570 0.547860i \(-0.184557\pi\)
0.836570 + 0.547860i \(0.184557\pi\)
\(432\) 0 0
\(433\) 28.7386 1.38109 0.690545 0.723289i \(-0.257371\pi\)
0.690545 + 0.723289i \(0.257371\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 3.17662i − 0.151958i
\(438\) 0 0
\(439\) 32.9848i 1.57428i 0.616774 + 0.787140i \(0.288439\pi\)
−0.616774 + 0.787140i \(0.711561\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −6.45101 −0.306497 −0.153248 0.988188i \(-0.548973\pi\)
−0.153248 + 0.988188i \(0.548973\pi\)
\(444\) 0 0
\(445\) 4.87689 0.231187
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 28.8066i 1.35947i 0.733460 + 0.679733i \(0.237904\pi\)
−0.733460 + 0.679733i \(0.762096\pi\)
\(450\) 0 0
\(451\) 2.73863i 0.128957i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 11.3137 0.530395
\(456\) 0 0
\(457\) −24.0000 −1.12267 −0.561336 0.827588i \(-0.689713\pi\)
−0.561336 + 0.827588i \(0.689713\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 6.27691i 0.292345i 0.989259 + 0.146172i \(0.0466954\pi\)
−0.989259 + 0.146172i \(0.953305\pi\)
\(462\) 0 0
\(463\) − 23.8617i − 1.10895i −0.832201 0.554475i \(-0.812919\pi\)
0.832201 0.554475i \(-0.187081\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16.0786 −0.744031 −0.372015 0.928227i \(-0.621333\pi\)
−0.372015 + 0.928227i \(0.621333\pi\)
\(468\) 0 0
\(469\) 5.12311 0.236563
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.17662i 0.146061i
\(474\) 0 0
\(475\) − 0.492423i − 0.0225939i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −33.0492 −1.51006 −0.755028 0.655692i \(-0.772377\pi\)
−0.755028 + 0.655692i \(0.772377\pi\)
\(480\) 0 0
\(481\) 56.9848 2.59829
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 2.48023i − 0.112622i
\(486\) 0 0
\(487\) 8.00000i 0.362515i 0.983436 + 0.181257i \(0.0580167\pi\)
−0.983436 + 0.181257i \(0.941983\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 26.5982 1.20036 0.600180 0.799865i \(-0.295096\pi\)
0.600180 + 0.799865i \(0.295096\pi\)
\(492\) 0 0
\(493\) 1.86174 0.0838485
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 8.03932i 0.360613i
\(498\) 0 0
\(499\) 28.9848i 1.29754i 0.760985 + 0.648770i \(0.224716\pi\)
−0.760985 + 0.648770i \(0.775284\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 33.9411 1.51336 0.756680 0.653785i \(-0.226820\pi\)
0.756680 + 0.653785i \(0.226820\pi\)
\(504\) 0 0
\(505\) 11.1231 0.494972
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 34.9094i − 1.54733i −0.633594 0.773666i \(-0.718421\pi\)
0.633594 0.773666i \(-0.281579\pi\)
\(510\) 0 0
\(511\) 1.12311i 0.0496833i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 26.5004 1.16775
\(516\) 0 0
\(517\) 10.2462 0.450628
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 34.0175i 1.49033i 0.666879 + 0.745166i \(0.267630\pi\)
−0.666879 + 0.745166i \(0.732370\pi\)
\(522\) 0 0
\(523\) − 28.4924i − 1.24589i −0.782267 0.622943i \(-0.785937\pi\)
0.782267 0.622943i \(-0.214063\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.87301 0.168711
\(528\) 0 0
\(529\) −22.3693 −0.972579
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 17.6670i − 0.765241i
\(534\) 0 0
\(535\) − 5.26137i − 0.227469i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.794156 0.0342067
\(540\) 0 0
\(541\) −6.87689 −0.295661 −0.147830 0.989013i \(-0.547229\pi\)
−0.147830 + 0.989013i \(0.547229\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 26.5004i − 1.13515i
\(546\) 0 0
\(547\) − 10.8769i − 0.465062i −0.972589 0.232531i \(-0.925299\pi\)
0.972589 0.232531i \(-0.0747008\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 12.0101 0.511647
\(552\) 0 0
\(553\) 11.3693 0.483473
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 18.3848i 0.778988i 0.921029 + 0.389494i \(0.127350\pi\)
−0.921029 + 0.389494i \(0.872650\pi\)
\(558\) 0 0
\(559\) − 20.4924i − 0.866737i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −41.8827 −1.76514 −0.882572 0.470177i \(-0.844190\pi\)
−0.882572 + 0.470177i \(0.844190\pi\)
\(564\) 0 0
\(565\) −44.8769 −1.88799
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 22.4533i − 0.941292i −0.882322 0.470646i \(-0.844021\pi\)
0.882322 0.470646i \(-0.155979\pi\)
\(570\) 0 0
\(571\) 34.8769i 1.45955i 0.683686 + 0.729776i \(0.260376\pi\)
−0.683686 + 0.729776i \(0.739624\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0.0977650 0.00407708
\(576\) 0 0
\(577\) −38.9848 −1.62296 −0.811480 0.584380i \(-0.801338\pi\)
−0.811480 + 0.584380i \(0.801338\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 14.4903i 0.601160i
\(582\) 0 0
\(583\) − 2.38447i − 0.0987547i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.2056 0.503780 0.251890 0.967756i \(-0.418948\pi\)
0.251890 + 0.967756i \(0.418948\pi\)
\(588\) 0 0
\(589\) 24.9848 1.02948
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 42.1545i 1.73108i 0.500840 + 0.865540i \(0.333024\pi\)
−0.500840 + 0.865540i \(0.666976\pi\)
\(594\) 0 0
\(595\) − 1.36932i − 0.0561365i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 25.9018 1.05832 0.529160 0.848522i \(-0.322507\pi\)
0.529160 + 0.848522i \(0.322507\pi\)
\(600\) 0 0
\(601\) −34.9848 −1.42706 −0.713531 0.700624i \(-0.752905\pi\)
−0.713531 + 0.700624i \(0.752905\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 22.8993i 0.930988i
\(606\) 0 0
\(607\) 28.4924i 1.15647i 0.815870 + 0.578236i \(0.196259\pi\)
−0.815870 + 0.578236i \(0.803741\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −66.0984 −2.67406
\(612\) 0 0
\(613\) 5.50758 0.222449 0.111224 0.993795i \(-0.464523\pi\)
0.111224 + 0.993795i \(0.464523\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 33.5715i 1.35154i 0.737114 + 0.675769i \(0.236188\pi\)
−0.737114 + 0.675769i \(0.763812\pi\)
\(618\) 0 0
\(619\) − 22.7386i − 0.913943i −0.889481 0.456971i \(-0.848934\pi\)
0.889481 0.456971i \(-0.151066\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −2.20837 −0.0884764
\(624\) 0 0
\(625\) −24.3693 −0.974773
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 6.89697i − 0.275000i
\(630\) 0 0
\(631\) 43.3693i 1.72651i 0.504772 + 0.863253i \(0.331577\pi\)
−0.504772 + 0.863253i \(0.668423\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 42.7746 1.69746
\(636\) 0 0
\(637\) −5.12311 −0.202985
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.1447i 0.677173i 0.940935 + 0.338587i \(0.109949\pi\)
−0.940935 + 0.338587i \(0.890051\pi\)
\(642\) 0 0
\(643\) 12.0000i 0.473234i 0.971603 + 0.236617i \(0.0760386\pi\)
−0.971603 + 0.236617i \(0.923961\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −18.5589 −0.729625 −0.364812 0.931081i \(-0.618867\pi\)
−0.364812 + 0.931081i \(0.618867\pi\)
\(648\) 0 0
\(649\) 1.26137 0.0495130
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 21.2132i 0.830137i 0.909790 + 0.415068i \(0.136242\pi\)
−0.909790 + 0.415068i \(0.863758\pi\)
\(654\) 0 0
\(655\) − 12.4924i − 0.488119i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 37.9119 1.47684 0.738419 0.674342i \(-0.235573\pi\)
0.738419 + 0.674342i \(0.235573\pi\)
\(660\) 0 0
\(661\) −8.73863 −0.339893 −0.169947 0.985453i \(-0.554360\pi\)
−0.169947 + 0.985453i \(0.554360\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 8.83348i − 0.342548i
\(666\) 0 0
\(667\) 2.38447i 0.0923271i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.58831 −0.0613161
\(672\) 0 0
\(673\) −44.1080 −1.70024 −0.850118 0.526592i \(-0.823470\pi\)
−0.850118 + 0.526592i \(0.823470\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 44.6348i 1.71545i 0.514106 + 0.857727i \(0.328124\pi\)
−0.514106 + 0.857727i \(0.671876\pi\)
\(678\) 0 0
\(679\) 1.12311i 0.0431008i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22.5297 −0.862073 −0.431037 0.902334i \(-0.641852\pi\)
−0.431037 + 0.902334i \(0.641852\pi\)
\(684\) 0 0
\(685\) −40.6004 −1.55126
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 15.3823i 0.586017i
\(690\) 0 0
\(691\) − 13.7538i − 0.523219i −0.965174 0.261609i \(-0.915747\pi\)
0.965174 0.261609i \(-0.0842532\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −2.13826 −0.0809924
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 8.31118i − 0.313909i −0.987606 0.156955i \(-0.949832\pi\)
0.987606 0.156955i \(-0.0501676\pi\)
\(702\) 0 0
\(703\) − 44.4924i − 1.67806i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −5.03680 −0.189428
\(708\) 0 0
\(709\) 4.00000 0.150223 0.0751116 0.997175i \(-0.476069\pi\)
0.0751116 + 0.997175i \(0.476069\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.96046i 0.185771i
\(714\) 0 0
\(715\) 8.98485i 0.336014i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0.696391 0.0259710 0.0129855 0.999916i \(-0.495866\pi\)
0.0129855 + 0.999916i \(0.495866\pi\)
\(720\) 0 0
\(721\) −12.0000 −0.446903
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.369628i 0.0137276i
\(726\) 0 0
\(727\) 44.9848i 1.66840i 0.551465 + 0.834198i \(0.314069\pi\)
−0.551465 + 0.834198i \(0.685931\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −2.48023 −0.0917347
\(732\) 0 0
\(733\) −48.3542 −1.78600 −0.893001 0.450055i \(-0.851404\pi\)
−0.893001 + 0.450055i \(0.851404\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.06854i 0.149867i
\(738\) 0 0
\(739\) 9.61553i 0.353713i 0.984237 + 0.176856i \(0.0565928\pi\)
−0.984237 + 0.176856i \(0.943407\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 14.5881 0.535185 0.267593 0.963532i \(-0.413772\pi\)
0.267593 + 0.963532i \(0.413772\pi\)
\(744\) 0 0
\(745\) −28.8769 −1.05797
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.38247i 0.0870534i
\(750\) 0 0
\(751\) − 36.4924i − 1.33163i −0.746118 0.665814i \(-0.768085\pi\)
0.746118 0.665814i \(-0.231915\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −16.4924 −0.599427 −0.299714 0.954029i \(-0.596891\pi\)
−0.299714 + 0.954029i \(0.596891\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 27.3160i 0.990205i 0.868835 + 0.495102i \(0.164869\pi\)
−0.868835 + 0.495102i \(0.835131\pi\)
\(762\) 0 0
\(763\) 12.0000i 0.434429i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.13709 −0.293813
\(768\) 0 0
\(769\) 23.7538 0.856584 0.428292 0.903641i \(-0.359116\pi\)
0.428292 + 0.903641i \(0.359116\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 8.21342i − 0.295416i −0.989031 0.147708i \(-0.952810\pi\)
0.989031 0.147708i \(-0.0471896\pi\)
\(774\) 0 0
\(775\) 0.768944i 0.0276213i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −13.7939 −0.494219
\(780\) 0 0
\(781\) −6.38447 −0.228454
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 30.9172i − 1.10348i
\(786\) 0 0
\(787\) − 46.7386i − 1.66605i −0.553234 0.833026i \(-0.686606\pi\)
0.553234 0.833026i \(-0.313394\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 20.3213 0.722542
\(792\) 0 0
\(793\) 10.2462 0.363854
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.5271i 0.691686i 0.938292 + 0.345843i \(0.112407\pi\)
−0.938292 + 0.345843i \(0.887593\pi\)
\(798\) 0 0
\(799\) 8.00000i 0.283020i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −0.891921 −0.0314752
\(804\) 0 0
\(805\) 1.75379 0.0618129
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 16.2527i − 0.571416i −0.958317 0.285708i \(-0.907771\pi\)
0.958317 0.285708i \(-0.0922287\pi\)
\(810\) 0 0
\(811\) − 5.75379i − 0.202043i −0.994884 0.101021i \(-0.967789\pi\)
0.994884 0.101021i \(-0.0322111\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −38.9016 −1.36266
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22.8015i 0.795778i 0.917433 + 0.397889i \(0.130257\pi\)
−0.917433 + 0.397889i \(0.869743\pi\)
\(822\) 0 0
\(823\) 4.63068i 0.161415i 0.996738 + 0.0807077i \(0.0257180\pi\)
−0.996738 + 0.0807077i \(0.974282\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 54.8825 1.90845 0.954225 0.299089i \(-0.0966828\pi\)
0.954225 + 0.299089i \(0.0966828\pi\)
\(828\) 0 0
\(829\) 17.6155 0.611813 0.305906 0.952062i \(-0.401041\pi\)
0.305906 + 0.952062i \(0.401041\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.620058i 0.0214837i
\(834\) 0 0
\(835\) − 16.0000i − 0.553703i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −9.72540 −0.335758 −0.167879 0.985808i \(-0.553692\pi\)
−0.167879 + 0.985808i \(0.553692\pi\)
\(840\) 0 0
\(841\) 19.9848 0.689133
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 29.2525i − 1.00632i
\(846\) 0 0
\(847\) − 10.3693i − 0.356294i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.83348 0.302808
\(852\) 0 0
\(853\) −15.7538 −0.539399 −0.269700 0.962944i \(-0.586924\pi\)
−0.269700 + 0.962944i \(0.586924\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 37.3896i − 1.27720i −0.769537 0.638602i \(-0.779513\pi\)
0.769537 0.638602i \(-0.220487\pi\)
\(858\) 0 0
\(859\) − 31.2311i − 1.06559i −0.846244 0.532795i \(-0.821142\pi\)
0.846244 0.532795i \(-0.178858\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −37.0200 −1.26017 −0.630087 0.776524i \(-0.716981\pi\)
−0.630087 + 0.776524i \(0.716981\pi\)
\(864\) 0 0
\(865\) −45.8617 −1.55935
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 9.02901i 0.306288i
\(870\) 0 0
\(871\) − 26.2462i − 0.889319i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 11.3137 0.382473
\(876\) 0 0
\(877\) 23.7538 0.802108 0.401054 0.916054i \(-0.368644\pi\)
0.401054 + 0.916054i \(0.368644\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 37.1941i 1.25310i 0.779381 + 0.626550i \(0.215534\pi\)
−0.779381 + 0.626550i \(0.784466\pi\)
\(882\) 0 0
\(883\) 28.9848i 0.975418i 0.873006 + 0.487709i \(0.162167\pi\)
−0.873006 + 0.487709i \(0.837833\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −33.0492 −1.10968 −0.554842 0.831956i \(-0.687221\pi\)
−0.554842 + 0.831956i \(0.687221\pi\)
\(888\) 0 0
\(889\) −19.3693 −0.649626
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 51.6081i 1.72700i
\(894\) 0 0
\(895\) 46.2462i 1.54584i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −18.7544 −0.625494
\(900\) 0 0
\(901\) 1.86174 0.0620235
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 11.3137i − 0.376080i
\(906\) 0 0
\(907\) − 24.4924i − 0.813258i −0.913593 0.406629i \(-0.866704\pi\)
0.913593 0.406629i \(-0.133296\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −41.0885 −1.36132 −0.680662 0.732598i \(-0.738308\pi\)
−0.680662 + 0.732598i \(0.738308\pi\)
\(912\) 0 0
\(913\) −11.5076 −0.380845
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.65685i 0.186806i
\(918\) 0 0
\(919\) 32.9848i 1.08807i 0.839063 + 0.544035i \(0.183104\pi\)
−0.839063 + 0.544035i \(0.816896\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 41.1863 1.35566
\(924\) 0 0
\(925\) 1.36932 0.0450229
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 24.8358i − 0.814836i −0.913242 0.407418i \(-0.866429\pi\)
0.913242 0.407418i \(-0.133571\pi\)
\(930\) 0 0
\(931\) 4.00000i 0.131095i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.08745 0.0355634
\(936\) 0 0
\(937\) −11.2614 −0.367893 −0.183946 0.982936i \(-0.558887\pi\)
−0.183946 + 0.982936i \(0.558887\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30.4926i 0.994032i 0.867741 + 0.497016i \(0.165571\pi\)
−0.867741 + 0.497016i \(0.834429\pi\)
\(942\) 0 0
\(943\) − 2.73863i − 0.0891822i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −6.45101 −0.209630 −0.104815 0.994492i \(-0.533425\pi\)
−0.104815 + 0.994492i \(0.533425\pi\)
\(948\) 0 0
\(949\) 5.75379 0.186776
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 22.1051i 0.716055i 0.933711 + 0.358028i \(0.116551\pi\)
−0.933711 + 0.358028i \(0.883449\pi\)
\(954\) 0 0
\(955\) 10.7386i 0.347494i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 18.3848 0.593675
\(960\) 0 0
\(961\) −8.01515 −0.258553
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 15.7304i 0.506381i
\(966\) 0 0
\(967\) 33.1231i 1.06517i 0.846377 + 0.532584i \(0.178779\pi\)
−0.846377 + 0.532584i \(0.821221\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −20.1472 −0.646554 −0.323277 0.946304i \(-0.604785\pi\)
−0.323277 + 0.946304i \(0.604785\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 24.3898i − 0.780300i −0.920751 0.390150i \(-0.872423\pi\)
0.920751 0.390150i \(-0.127577\pi\)
\(978\) 0 0
\(979\) − 1.75379i − 0.0560513i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 23.3238 0.743914 0.371957 0.928250i \(-0.378687\pi\)
0.371957 + 0.928250i \(0.378687\pi\)
\(984\) 0 0
\(985\) −31.6155 −1.00735
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 3.17662i − 0.101011i
\(990\) 0 0
\(991\) − 24.0000i − 0.762385i −0.924496 0.381193i \(-0.875513\pi\)
0.924496 0.381193i \(-0.124487\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −12.7065 −0.402823
\(996\) 0 0
\(997\) −0.246211 −0.00779759 −0.00389879 0.999992i \(-0.501241\pi\)
−0.00389879 + 0.999992i \(0.501241\pi\)
\(998\) 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 4032.2.h.g.575.4 8
3.2 odd 2 inner 4032.2.h.g.575.6 8
4.3 odd 2 inner 4032.2.h.g.575.3 8
8.3 odd 2 2016.2.h.e.575.5 yes 8
8.5 even 2 2016.2.h.e.575.6 yes 8
12.11 even 2 inner 4032.2.h.g.575.5 8
24.5 odd 2 2016.2.h.e.575.4 yes 8
24.11 even 2 2016.2.h.e.575.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2016.2.h.e.575.3 8 24.11 even 2
2016.2.h.e.575.4 yes 8 24.5 odd 2
2016.2.h.e.575.5 yes 8 8.3 odd 2
2016.2.h.e.575.6 yes 8 8.5 even 2
4032.2.h.g.575.3 8 4.3 odd 2 inner
4032.2.h.g.575.4 8 1.1 even 1 trivial
4032.2.h.g.575.5 8 12.11 even 2 inner
4032.2.h.g.575.6 8 3.2 odd 2 inner