Properties

Label 576.3.m.c.559.7
Level $576$
Weight $3$
Character 576.559
Analytic conductor $15.695$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [576,3,Mod(271,576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(576, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("576.271");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 576 = 2^{6} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 576.m (of order \(4\), degree \(2\), 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: \(15.6948632272\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 6 x^{14} - 4 x^{13} + 10 x^{12} + 56 x^{11} + 88 x^{10} - 128 x^{9} - 496 x^{8} - 512 x^{7} + 1408 x^{6} + 3584 x^{5} + 2560 x^{4} - 4096 x^{3} - 24576 x^{2} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{28} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 559.7
Root \(1.84258 + 0.777752i\) of defining polynomial
Character \(\chi\) \(=\) 576.559
Dual form 576.3.m.c.271.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.78830 - 4.78830i) q^{5} +10.3302 q^{7} +O(q^{10})\) \(q+(4.78830 - 4.78830i) q^{5} +10.3302 q^{7} +(-0.526169 - 0.526169i) q^{11} +(17.2840 + 17.2840i) q^{13} -4.71650 q^{17} +(2.53604 - 2.53604i) q^{19} -12.5864 q^{23} -20.8557i q^{25} +(2.19683 + 2.19683i) q^{29} -28.0521i q^{31} +(49.4644 - 49.4644i) q^{35} +(-32.1128 + 32.1128i) q^{37} +23.1145i q^{41} +(-4.79441 - 4.79441i) q^{43} -39.0095i q^{47} +57.7141 q^{49} +(27.9768 - 27.9768i) q^{53} -5.03891 q^{55} +(79.8538 + 79.8538i) q^{59} +(-36.7762 - 36.7762i) q^{61} +165.522 q^{65} +(10.9869 - 10.9869i) q^{67} +52.6605 q^{71} -67.8061i q^{73} +(-5.43545 - 5.43545i) q^{77} -56.4602i q^{79} +(-58.3697 + 58.3697i) q^{83} +(-22.5840 + 22.5840i) q^{85} -131.566i q^{89} +(178.548 + 178.548i) q^{91} -24.2866i q^{95} +60.9413 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{11} + 32 q^{19} - 128 q^{23} - 32 q^{29} + 96 q^{35} - 96 q^{37} - 160 q^{43} + 112 q^{49} + 160 q^{53} + 256 q^{55} - 128 q^{59} - 32 q^{61} + 32 q^{65} - 320 q^{67} + 512 q^{71} - 224 q^{77} - 160 q^{83} + 160 q^{85} + 480 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/576\mathbb{Z}\right)^\times\).

\(n\) \(65\) \(127\) \(325\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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
\(5\) 4.78830 4.78830i 0.957661 0.957661i −0.0414785 0.999139i \(-0.513207\pi\)
0.999139 + 0.0414785i \(0.0132068\pi\)
\(6\) 0 0
\(7\) 10.3302 1.47575 0.737875 0.674937i \(-0.235829\pi\)
0.737875 + 0.674937i \(0.235829\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.526169 0.526169i −0.0478335 0.0478335i 0.682785 0.730619i \(-0.260768\pi\)
−0.730619 + 0.682785i \(0.760768\pi\)
\(12\) 0 0
\(13\) 17.2840 + 17.2840i 1.32953 + 1.32953i 0.905774 + 0.423761i \(0.139290\pi\)
0.423761 + 0.905774i \(0.360710\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.71650 −0.277441 −0.138721 0.990332i \(-0.544299\pi\)
−0.138721 + 0.990332i \(0.544299\pi\)
\(18\) 0 0
\(19\) 2.53604 2.53604i 0.133476 0.133476i −0.637213 0.770688i \(-0.719913\pi\)
0.770688 + 0.637213i \(0.219913\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −12.5864 −0.547236 −0.273618 0.961838i \(-0.588220\pi\)
−0.273618 + 0.961838i \(0.588220\pi\)
\(24\) 0 0
\(25\) 20.8557i 0.834229i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.19683 + 2.19683i 0.0757526 + 0.0757526i 0.743968 0.668215i \(-0.232942\pi\)
−0.668215 + 0.743968i \(0.732942\pi\)
\(30\) 0 0
\(31\) 28.0521i 0.904908i −0.891788 0.452454i \(-0.850549\pi\)
0.891788 0.452454i \(-0.149451\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 49.4644 49.4644i 1.41327 1.41327i
\(36\) 0 0
\(37\) −32.1128 + 32.1128i −0.867914 + 0.867914i −0.992241 0.124327i \(-0.960323\pi\)
0.124327 + 0.992241i \(0.460323\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 23.1145i 0.563768i 0.959449 + 0.281884i \(0.0909593\pi\)
−0.959449 + 0.281884i \(0.909041\pi\)
\(42\) 0 0
\(43\) −4.79441 4.79441i −0.111498 0.111498i 0.649157 0.760655i \(-0.275122\pi\)
−0.760655 + 0.649157i \(0.775122\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 39.0095i 0.829989i −0.909824 0.414994i \(-0.863784\pi\)
0.909824 0.414994i \(-0.136216\pi\)
\(48\) 0 0
\(49\) 57.7141 1.17784
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 27.9768 27.9768i 0.527864 0.527864i −0.392071 0.919935i \(-0.628241\pi\)
0.919935 + 0.392071i \(0.128241\pi\)
\(54\) 0 0
\(55\) −5.03891 −0.0916166
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 79.8538 + 79.8538i 1.35345 + 1.35345i 0.881764 + 0.471691i \(0.156356\pi\)
0.471691 + 0.881764i \(0.343644\pi\)
\(60\) 0 0
\(61\) −36.7762 36.7762i −0.602888 0.602888i 0.338190 0.941078i \(-0.390185\pi\)
−0.941078 + 0.338190i \(0.890185\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 165.522 2.54649
\(66\) 0 0
\(67\) 10.9869 10.9869i 0.163984 0.163984i −0.620345 0.784329i \(-0.713008\pi\)
0.784329 + 0.620345i \(0.213008\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 52.6605 0.741697 0.370849 0.928693i \(-0.379067\pi\)
0.370849 + 0.928693i \(0.379067\pi\)
\(72\) 0 0
\(73\) 67.8061i 0.928850i −0.885612 0.464425i \(-0.846261\pi\)
0.885612 0.464425i \(-0.153739\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.43545 5.43545i −0.0705903 0.0705903i
\(78\) 0 0
\(79\) 56.4602i 0.714686i −0.933973 0.357343i \(-0.883683\pi\)
0.933973 0.357343i \(-0.116317\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −58.3697 + 58.3697i −0.703249 + 0.703249i −0.965107 0.261857i \(-0.915665\pi\)
0.261857 + 0.965107i \(0.415665\pi\)
\(84\) 0 0
\(85\) −22.5840 + 22.5840i −0.265694 + 0.265694i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 131.566i 1.47827i −0.673558 0.739135i \(-0.735235\pi\)
0.673558 0.739135i \(-0.264765\pi\)
\(90\) 0 0
\(91\) 178.548 + 178.548i 1.96206 + 1.96206i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 24.2866i 0.255649i
\(96\) 0 0
\(97\) 60.9413 0.628261 0.314131 0.949380i \(-0.398287\pi\)
0.314131 + 0.949380i \(0.398287\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −109.986 + 109.986i −1.08897 + 1.08897i −0.0933326 + 0.995635i \(0.529752\pi\)
−0.995635 + 0.0933326i \(0.970248\pi\)
\(102\) 0 0
\(103\) −173.295 −1.68248 −0.841239 0.540663i \(-0.818174\pi\)
−0.841239 + 0.540663i \(0.818174\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −25.4747 25.4747i −0.238081 0.238081i 0.577974 0.816055i \(-0.303844\pi\)
−0.816055 + 0.577974i \(0.803844\pi\)
\(108\) 0 0
\(109\) 33.0605 + 33.0605i 0.303307 + 0.303307i 0.842306 0.538999i \(-0.181197\pi\)
−0.538999 + 0.842306i \(0.681197\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −140.159 −1.24034 −0.620171 0.784466i \(-0.712937\pi\)
−0.620171 + 0.784466i \(0.712937\pi\)
\(114\) 0 0
\(115\) −60.2677 + 60.2677i −0.524067 + 0.524067i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −48.7226 −0.409434
\(120\) 0 0
\(121\) 120.446i 0.995424i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 19.8441 + 19.8441i 0.158752 + 0.158752i
\(126\) 0 0
\(127\) 40.8458i 0.321620i 0.986985 + 0.160810i \(0.0514107\pi\)
−0.986985 + 0.160810i \(0.948589\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −75.0168 + 75.0168i −0.572647 + 0.572647i −0.932867 0.360220i \(-0.882702\pi\)
0.360220 + 0.932867i \(0.382702\pi\)
\(132\) 0 0
\(133\) 26.1979 26.1979i 0.196977 0.196977i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 134.028i 0.978308i −0.872197 0.489154i \(-0.837306\pi\)
0.872197 0.489154i \(-0.162694\pi\)
\(138\) 0 0
\(139\) −22.8798 22.8798i −0.164603 0.164603i 0.619999 0.784602i \(-0.287133\pi\)
−0.784602 + 0.619999i \(0.787133\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 18.1885i 0.127193i
\(144\) 0 0
\(145\) 21.0381 0.145091
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.32124 9.32124i 0.0625587 0.0625587i −0.675135 0.737694i \(-0.735915\pi\)
0.737694 + 0.675135i \(0.235915\pi\)
\(150\) 0 0
\(151\) 50.5403 0.334704 0.167352 0.985897i \(-0.446478\pi\)
0.167352 + 0.985897i \(0.446478\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −134.322 134.322i −0.866595 0.866595i
\(156\) 0 0
\(157\) −95.8844 95.8844i −0.610729 0.610729i 0.332407 0.943136i \(-0.392139\pi\)
−0.943136 + 0.332407i \(0.892139\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −130.021 −0.807584
\(162\) 0 0
\(163\) 140.885 140.885i 0.864324 0.864324i −0.127513 0.991837i \(-0.540699\pi\)
0.991837 + 0.127513i \(0.0406994\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −107.849 −0.645800 −0.322900 0.946433i \(-0.604658\pi\)
−0.322900 + 0.946433i \(0.604658\pi\)
\(168\) 0 0
\(169\) 428.470i 2.53533i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 53.8845 + 53.8845i 0.311471 + 0.311471i 0.845479 0.534008i \(-0.179315\pi\)
−0.534008 + 0.845479i \(0.679315\pi\)
\(174\) 0 0
\(175\) 215.445i 1.23111i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 104.178 104.178i 0.582002 0.582002i −0.353451 0.935453i \(-0.614992\pi\)
0.935453 + 0.353451i \(0.114992\pi\)
\(180\) 0 0
\(181\) −205.498 + 205.498i −1.13535 + 1.13535i −0.146073 + 0.989274i \(0.546664\pi\)
−0.989274 + 0.146073i \(0.953336\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 307.532i 1.66233i
\(186\) 0 0
\(187\) 2.48167 + 2.48167i 0.0132710 + 0.0132710i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 248.255i 1.29977i 0.760034 + 0.649883i \(0.225182\pi\)
−0.760034 + 0.649883i \(0.774818\pi\)
\(192\) 0 0
\(193\) −129.921 −0.673166 −0.336583 0.941654i \(-0.609271\pi\)
−0.336583 + 0.941654i \(0.609271\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −237.001 + 237.001i −1.20305 + 1.20305i −0.229816 + 0.973234i \(0.573812\pi\)
−0.973234 + 0.229816i \(0.926188\pi\)
\(198\) 0 0
\(199\) −246.508 −1.23873 −0.619366 0.785102i \(-0.712610\pi\)
−0.619366 + 0.785102i \(0.712610\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 22.6938 + 22.6938i 0.111792 + 0.111792i
\(204\) 0 0
\(205\) 110.679 + 110.679i 0.539898 + 0.539898i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.66877 −0.0127692
\(210\) 0 0
\(211\) 13.4139 13.4139i 0.0635728 0.0635728i −0.674606 0.738178i \(-0.735686\pi\)
0.738178 + 0.674606i \(0.235686\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −45.9142 −0.213554
\(216\) 0 0
\(217\) 289.786i 1.33542i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −81.5197 81.5197i −0.368867 0.368867i
\(222\) 0 0
\(223\) 295.580i 1.32547i 0.748854 + 0.662735i \(0.230604\pi\)
−0.748854 + 0.662735i \(0.769396\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −97.0742 + 97.0742i −0.427640 + 0.427640i −0.887824 0.460184i \(-0.847783\pi\)
0.460184 + 0.887824i \(0.347783\pi\)
\(228\) 0 0
\(229\) 34.2565 34.2565i 0.149592 0.149592i −0.628344 0.777936i \(-0.716267\pi\)
0.777936 + 0.628344i \(0.216267\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 62.8176i 0.269604i 0.990873 + 0.134802i \(0.0430398\pi\)
−0.990873 + 0.134802i \(0.956960\pi\)
\(234\) 0 0
\(235\) −186.789 186.789i −0.794848 0.794848i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 355.910i 1.48916i −0.667532 0.744581i \(-0.732649\pi\)
0.667532 0.744581i \(-0.267351\pi\)
\(240\) 0 0
\(241\) 66.2545 0.274915 0.137458 0.990508i \(-0.456107\pi\)
0.137458 + 0.990508i \(0.456107\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 276.352 276.352i 1.12797 1.12797i
\(246\) 0 0
\(247\) 87.6655 0.354921
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 325.395 + 325.395i 1.29640 + 1.29640i 0.930757 + 0.365638i \(0.119149\pi\)
0.365638 + 0.930757i \(0.380851\pi\)
\(252\) 0 0
\(253\) 6.62259 + 6.62259i 0.0261762 + 0.0261762i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 312.011 1.21405 0.607026 0.794682i \(-0.292362\pi\)
0.607026 + 0.794682i \(0.292362\pi\)
\(258\) 0 0
\(259\) −331.733 + 331.733i −1.28082 + 1.28082i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −168.163 −0.639403 −0.319702 0.947518i \(-0.603583\pi\)
−0.319702 + 0.947518i \(0.603583\pi\)
\(264\) 0 0
\(265\) 267.923i 1.01103i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −212.116 212.116i −0.788535 0.788535i 0.192719 0.981254i \(-0.438269\pi\)
−0.981254 + 0.192719i \(0.938269\pi\)
\(270\) 0 0
\(271\) 173.450i 0.640037i −0.947411 0.320019i \(-0.896311\pi\)
0.947411 0.320019i \(-0.103689\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10.9736 + 10.9736i −0.0399041 + 0.0399041i
\(276\) 0 0
\(277\) −38.4049 + 38.4049i −0.138646 + 0.138646i −0.773023 0.634377i \(-0.781257\pi\)
0.634377 + 0.773023i \(0.281257\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 223.573i 0.795632i 0.917465 + 0.397816i \(0.130232\pi\)
−0.917465 + 0.397816i \(0.869768\pi\)
\(282\) 0 0
\(283\) −247.755 247.755i −0.875459 0.875459i 0.117602 0.993061i \(-0.462479\pi\)
−0.993061 + 0.117602i \(0.962479\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 238.778i 0.831980i
\(288\) 0 0
\(289\) −266.755 −0.923026
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 102.262 102.262i 0.349016 0.349016i −0.510727 0.859743i \(-0.670624\pi\)
0.859743 + 0.510727i \(0.170624\pi\)
\(294\) 0 0
\(295\) 764.729 2.59230
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −217.543 217.543i −0.727570 0.727570i
\(300\) 0 0
\(301\) −49.5275 49.5275i −0.164543 0.164543i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −352.191 −1.15472
\(306\) 0 0
\(307\) −138.292 + 138.292i −0.450463 + 0.450463i −0.895508 0.445045i \(-0.853188\pi\)
0.445045 + 0.895508i \(0.353188\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 205.789 0.661702 0.330851 0.943683i \(-0.392664\pi\)
0.330851 + 0.943683i \(0.392664\pi\)
\(312\) 0 0
\(313\) 223.861i 0.715209i 0.933873 + 0.357605i \(0.116406\pi\)
−0.933873 + 0.357605i \(0.883594\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 176.488 + 176.488i 0.556744 + 0.556744i 0.928379 0.371635i \(-0.121203\pi\)
−0.371635 + 0.928379i \(0.621203\pi\)
\(318\) 0 0
\(319\) 2.31180i 0.00724703i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −11.9612 + 11.9612i −0.0370316 + 0.0370316i
\(324\) 0 0
\(325\) 360.469 360.469i 1.10914 1.10914i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 402.978i 1.22486i
\(330\) 0 0
\(331\) 183.939 + 183.939i 0.555706 + 0.555706i 0.928082 0.372376i \(-0.121457\pi\)
−0.372376 + 0.928082i \(0.621457\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 105.217i 0.314081i
\(336\) 0 0
\(337\) 12.7162 0.0377336 0.0188668 0.999822i \(-0.493994\pi\)
0.0188668 + 0.999822i \(0.493994\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −14.7602 + 14.7602i −0.0432849 + 0.0432849i
\(342\) 0 0
\(343\) 90.0184 0.262444
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −113.546 113.546i −0.327221 0.327221i 0.524308 0.851529i \(-0.324324\pi\)
−0.851529 + 0.524308i \(0.824324\pi\)
\(348\) 0 0
\(349\) 90.9653 + 90.9653i 0.260645 + 0.260645i 0.825316 0.564671i \(-0.190997\pi\)
−0.564671 + 0.825316i \(0.690997\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −36.2208 −0.102609 −0.0513043 0.998683i \(-0.516338\pi\)
−0.0513043 + 0.998683i \(0.516338\pi\)
\(354\) 0 0
\(355\) 252.155 252.155i 0.710294 0.710294i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 142.121 0.395880 0.197940 0.980214i \(-0.436575\pi\)
0.197940 + 0.980214i \(0.436575\pi\)
\(360\) 0 0
\(361\) 348.137i 0.964369i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −324.676 324.676i −0.889523 0.889523i
\(366\) 0 0
\(367\) 654.218i 1.78261i 0.453404 + 0.891305i \(0.350209\pi\)
−0.453404 + 0.891305i \(0.649791\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 289.007 289.007i 0.778995 0.778995i
\(372\) 0 0
\(373\) 335.277 335.277i 0.898867 0.898867i −0.0964690 0.995336i \(-0.530755\pi\)
0.995336 + 0.0964690i \(0.0307549\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 75.9397i 0.201432i
\(378\) 0 0
\(379\) 98.7497 + 98.7497i 0.260553 + 0.260553i 0.825279 0.564725i \(-0.191018\pi\)
−0.564725 + 0.825279i \(0.691018\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 156.144i 0.407687i 0.979003 + 0.203844i \(0.0653434\pi\)
−0.979003 + 0.203844i \(0.934657\pi\)
\(384\) 0 0
\(385\) −52.0532 −0.135203
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 391.047 391.047i 1.00526 1.00526i 0.00527486 0.999986i \(-0.498321\pi\)
0.999986 0.00527486i \(-0.00167905\pi\)
\(390\) 0 0
\(391\) 59.3639 0.151826
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −270.349 270.349i −0.684427 0.684427i
\(396\) 0 0
\(397\) 243.862 + 243.862i 0.614262 + 0.614262i 0.944054 0.329791i \(-0.106978\pi\)
−0.329791 + 0.944054i \(0.606978\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 175.261 0.437059 0.218529 0.975830i \(-0.429874\pi\)
0.218529 + 0.975830i \(0.429874\pi\)
\(402\) 0 0
\(403\) 484.852 484.852i 1.20311 1.20311i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 33.7935 0.0830307
\(408\) 0 0
\(409\) 44.4504i 0.108681i −0.998522 0.0543404i \(-0.982694\pi\)
0.998522 0.0543404i \(-0.0173056\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 824.910 + 824.910i 1.99736 + 1.99736i
\(414\) 0 0
\(415\) 558.984i 1.34695i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 14.9985 14.9985i 0.0357959 0.0357959i −0.688982 0.724778i \(-0.741942\pi\)
0.724778 + 0.688982i \(0.241942\pi\)
\(420\) 0 0
\(421\) 312.907 312.907i 0.743247 0.743247i −0.229954 0.973201i \(-0.573858\pi\)
0.973201 + 0.229954i \(0.0738576\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 98.3660i 0.231449i
\(426\) 0 0
\(427\) −379.907 379.907i −0.889712 0.889712i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 532.400i 1.23527i 0.786466 + 0.617633i \(0.211908\pi\)
−0.786466 + 0.617633i \(0.788092\pi\)
\(432\) 0 0
\(433\) 553.451 1.27818 0.639089 0.769133i \(-0.279312\pi\)
0.639089 + 0.769133i \(0.279312\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −31.9197 + 31.9197i −0.0730427 + 0.0730427i
\(438\) 0 0
\(439\) −645.291 −1.46991 −0.734956 0.678115i \(-0.762797\pi\)
−0.734956 + 0.678115i \(0.762797\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 315.833 + 315.833i 0.712941 + 0.712941i 0.967149 0.254208i \(-0.0818149\pi\)
−0.254208 + 0.967149i \(0.581815\pi\)
\(444\) 0 0
\(445\) −629.978 629.978i −1.41568 1.41568i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −218.589 −0.486835 −0.243417 0.969922i \(-0.578268\pi\)
−0.243417 + 0.969922i \(0.578268\pi\)
\(450\) 0 0
\(451\) 12.1621 12.1621i 0.0269670 0.0269670i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1709.88 3.75798
\(456\) 0 0
\(457\) 296.561i 0.648930i 0.945898 + 0.324465i \(0.105184\pi\)
−0.945898 + 0.324465i \(0.894816\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −118.061 118.061i −0.256097 0.256097i 0.567368 0.823465i \(-0.307962\pi\)
−0.823465 + 0.567368i \(0.807962\pi\)
\(462\) 0 0
\(463\) 409.453i 0.884348i −0.896929 0.442174i \(-0.854207\pi\)
0.896929 0.442174i \(-0.145793\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 494.764 494.764i 1.05945 1.05945i 0.0613343 0.998117i \(-0.480464\pi\)
0.998117 0.0613343i \(-0.0195356\pi\)
\(468\) 0 0
\(469\) 113.497 113.497i 0.241999 0.241999i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 5.04534i 0.0106667i
\(474\) 0 0
\(475\) −52.8909 52.8909i −0.111349 0.111349i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 558.806i 1.16661i −0.812254 0.583305i \(-0.801759\pi\)
0.812254 0.583305i \(-0.198241\pi\)
\(480\) 0 0
\(481\) −1110.07 −2.30784
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 291.806 291.806i 0.601661 0.601661i
\(486\) 0 0
\(487\) 361.328 0.741946 0.370973 0.928644i \(-0.379024\pi\)
0.370973 + 0.928644i \(0.379024\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −488.975 488.975i −0.995876 0.995876i 0.00411514 0.999992i \(-0.498690\pi\)
−0.999992 + 0.00411514i \(0.998690\pi\)
\(492\) 0 0
\(493\) −10.3613 10.3613i −0.0210169 0.0210169i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 543.996 1.09456
\(498\) 0 0
\(499\) 102.895 102.895i 0.206203 0.206203i −0.596448 0.802652i \(-0.703422\pi\)
0.802652 + 0.596448i \(0.203422\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −881.975 −1.75343 −0.876715 0.481011i \(-0.840270\pi\)
−0.876715 + 0.481011i \(0.840270\pi\)
\(504\) 0 0
\(505\) 1053.29i 2.08572i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −161.639 161.639i −0.317563 0.317563i 0.530268 0.847830i \(-0.322091\pi\)
−0.847830 + 0.530268i \(0.822091\pi\)
\(510\) 0 0
\(511\) 700.454i 1.37075i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −829.791 + 829.791i −1.61124 + 1.61124i
\(516\) 0 0
\(517\) −20.5256 + 20.5256i −0.0397013 + 0.0397013i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 763.931i 1.46628i −0.680078 0.733140i \(-0.738054\pi\)
0.680078 0.733140i \(-0.261946\pi\)
\(522\) 0 0
\(523\) −295.573 295.573i −0.565150 0.565150i 0.365616 0.930766i \(-0.380858\pi\)
−0.930766 + 0.365616i \(0.880858\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 132.308i 0.251059i
\(528\) 0 0
\(529\) −370.582 −0.700532
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −399.509 + 399.509i −0.749549 + 0.749549i
\(534\) 0 0
\(535\) −243.961 −0.456002
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −30.3673 30.3673i −0.0563401 0.0563401i
\(540\) 0 0
\(541\) 243.037 + 243.037i 0.449236 + 0.449236i 0.895100 0.445865i \(-0.147104\pi\)
−0.445865 + 0.895100i \(0.647104\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 316.607 0.580931
\(546\) 0 0
\(547\) −424.574 + 424.574i −0.776187 + 0.776187i −0.979180 0.202993i \(-0.934933\pi\)
0.202993 + 0.979180i \(0.434933\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.1425 0.0202223
\(552\) 0 0
\(553\) 583.248i 1.05470i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 445.773 + 445.773i 0.800311 + 0.800311i 0.983144 0.182833i \(-0.0585268\pi\)
−0.182833 + 0.983144i \(0.558527\pi\)
\(558\) 0 0
\(559\) 165.733i 0.296481i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −529.295 + 529.295i −0.940133 + 0.940133i −0.998307 0.0581732i \(-0.981472\pi\)
0.0581732 + 0.998307i \(0.481472\pi\)
\(564\) 0 0
\(565\) −671.123 + 671.123i −1.18783 + 1.18783i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 346.814i 0.609516i 0.952430 + 0.304758i \(0.0985755\pi\)
−0.952430 + 0.304758i \(0.901424\pi\)
\(570\) 0 0
\(571\) 155.711 + 155.711i 0.272699 + 0.272699i 0.830186 0.557487i \(-0.188234\pi\)
−0.557487 + 0.830186i \(0.688234\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 262.499i 0.456520i
\(576\) 0 0
\(577\) 620.510 1.07541 0.537704 0.843134i \(-0.319292\pi\)
0.537704 + 0.843134i \(0.319292\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −602.974 + 602.974i −1.03782 + 1.03782i
\(582\) 0 0
\(583\) −29.4410 −0.0504992
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 561.656 + 561.656i 0.956825 + 0.956825i 0.999106 0.0422810i \(-0.0134625\pi\)
−0.0422810 + 0.999106i \(0.513462\pi\)
\(588\) 0 0
\(589\) −71.1413 71.1413i −0.120783 0.120783i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −851.739 −1.43632 −0.718161 0.695877i \(-0.755016\pi\)
−0.718161 + 0.695877i \(0.755016\pi\)
\(594\) 0 0
\(595\) −233.299 + 233.299i −0.392099 + 0.392099i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1001.69 −1.67228 −0.836138 0.548519i \(-0.815192\pi\)
−0.836138 + 0.548519i \(0.815192\pi\)
\(600\) 0 0
\(601\) 955.182i 1.58932i −0.607054 0.794661i \(-0.707649\pi\)
0.607054 0.794661i \(-0.292351\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −576.734 576.734i −0.953279 0.953279i
\(606\) 0 0
\(607\) 291.885i 0.480865i −0.970666 0.240432i \(-0.922711\pi\)
0.970666 0.240432i \(-0.0772892\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 674.238 674.238i 1.10350 1.10350i
\(612\) 0 0
\(613\) −332.933 + 332.933i −0.543121 + 0.543121i −0.924442 0.381322i \(-0.875469\pi\)
0.381322 + 0.924442i \(0.375469\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 970.864i 1.57352i 0.617257 + 0.786762i \(0.288244\pi\)
−0.617257 + 0.786762i \(0.711756\pi\)
\(618\) 0 0
\(619\) 696.761 + 696.761i 1.12562 + 1.12562i 0.990881 + 0.134744i \(0.0430210\pi\)
0.134744 + 0.990881i \(0.456979\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1359.11i 2.18156i
\(624\) 0 0
\(625\) 711.432 1.13829
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 151.460 151.460i 0.240795 0.240795i
\(630\) 0 0
\(631\) 377.591 0.598401 0.299200 0.954190i \(-0.403280\pi\)
0.299200 + 0.954190i \(0.403280\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 195.582 + 195.582i 0.308003 + 0.308003i
\(636\) 0 0
\(637\) 997.527 + 997.527i 1.56598 + 1.56598i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −729.200 −1.13760 −0.568799 0.822477i \(-0.692592\pi\)
−0.568799 + 0.822477i \(0.692592\pi\)
\(642\) 0 0
\(643\) −243.958 + 243.958i −0.379406 + 0.379406i −0.870888 0.491482i \(-0.836455\pi\)
0.491482 + 0.870888i \(0.336455\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −281.594 −0.435230 −0.217615 0.976035i \(-0.569828\pi\)
−0.217615 + 0.976035i \(0.569828\pi\)
\(648\) 0 0
\(649\) 84.0331i 0.129481i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −323.704 323.704i −0.495718 0.495718i 0.414384 0.910102i \(-0.363997\pi\)
−0.910102 + 0.414384i \(0.863997\pi\)
\(654\) 0 0
\(655\) 718.407i 1.09680i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 507.811 507.811i 0.770578 0.770578i −0.207629 0.978208i \(-0.566575\pi\)
0.978208 + 0.207629i \(0.0665748\pi\)
\(660\) 0 0
\(661\) 57.1593 57.1593i 0.0864741 0.0864741i −0.662547 0.749021i \(-0.730524\pi\)
0.749021 + 0.662547i \(0.230524\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 250.887i 0.377274i
\(666\) 0 0
\(667\) −27.6502 27.6502i −0.0414546 0.0414546i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 38.7009i 0.0576765i
\(672\) 0 0
\(673\) 1110.84 1.65059 0.825293 0.564705i \(-0.191010\pi\)
0.825293 + 0.564705i \(0.191010\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −397.465 + 397.465i −0.587097 + 0.587097i −0.936844 0.349747i \(-0.886268\pi\)
0.349747 + 0.936844i \(0.386268\pi\)
\(678\) 0 0
\(679\) 629.539 0.927156
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −238.015 238.015i −0.348485 0.348485i 0.511060 0.859545i \(-0.329253\pi\)
−0.859545 + 0.511060i \(0.829253\pi\)
\(684\) 0 0
\(685\) −641.768 641.768i −0.936887 0.936887i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 967.099 1.40363
\(690\) 0 0
\(691\) 685.172 685.172i 0.991565 0.991565i −0.00839951 0.999965i \(-0.502674\pi\)
0.999965 + 0.00839951i \(0.00267368\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −219.111 −0.315267
\(696\) 0 0
\(697\) 109.019i 0.156412i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −543.074 543.074i −0.774713 0.774713i 0.204214 0.978926i \(-0.434536\pi\)
−0.978926 + 0.204214i \(0.934536\pi\)
\(702\) 0 0
\(703\) 162.879i 0.231691i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1136.18 + 1136.18i −1.60704 + 1.60704i
\(708\) 0 0
\(709\) −488.019 + 488.019i −0.688320 + 0.688320i −0.961860 0.273541i \(-0.911805\pi\)
0.273541 + 0.961860i \(0.411805\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 353.076i 0.495198i
\(714\) 0 0
\(715\) −87.0923 87.0923i −0.121807 0.121807i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 297.369i 0.413587i 0.978385 + 0.206793i \(0.0663028\pi\)
−0.978385 + 0.206793i \(0.933697\pi\)
\(720\) 0 0
\(721\) −1790.18 −2.48292
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 45.8164 45.8164i 0.0631950 0.0631950i
\(726\) 0 0
\(727\) −1158.85 −1.59402 −0.797009 0.603967i \(-0.793586\pi\)
−0.797009 + 0.603967i \(0.793586\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 22.6128 + 22.6128i 0.0309341 + 0.0309341i
\(732\) 0 0
\(733\) −348.835 348.835i −0.475901 0.475901i 0.427917 0.903818i \(-0.359248\pi\)
−0.903818 + 0.427917i \(0.859248\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.5619 −0.0156878
\(738\) 0 0
\(739\) −825.489 + 825.489i −1.11703 + 1.11703i −0.124860 + 0.992174i \(0.539848\pi\)
−0.992174 + 0.124860i \(0.960152\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −899.725 −1.21094 −0.605468 0.795870i \(-0.707014\pi\)
−0.605468 + 0.795870i \(0.707014\pi\)
\(744\) 0 0
\(745\) 89.2659i 0.119820i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −263.160 263.160i −0.351348 0.351348i
\(750\) 0 0
\(751\) 80.4386i 0.107109i 0.998565 + 0.0535543i \(0.0170550\pi\)
−0.998565 + 0.0535543i \(0.982945\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 242.003 242.003i 0.320533 0.320533i
\(756\) 0 0
\(757\) 233.298 233.298i 0.308187 0.308187i −0.536019 0.844206i \(-0.680072\pi\)
0.844206 + 0.536019i \(0.180072\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 56.1906i 0.0738378i 0.999318 + 0.0369189i \(0.0117543\pi\)
−0.999318 + 0.0369189i \(0.988246\pi\)
\(762\) 0 0
\(763\) 341.523 + 341.523i 0.447606 + 0.447606i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2760.38i 3.59893i
\(768\) 0 0
\(769\) 517.343 0.672748 0.336374 0.941728i \(-0.390799\pi\)
0.336374 + 0.941728i \(0.390799\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 523.925 523.925i 0.677781 0.677781i −0.281716 0.959498i \(-0.590904\pi\)
0.959498 + 0.281716i \(0.0909037\pi\)
\(774\) 0 0
\(775\) −585.048 −0.754900
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 58.6192 + 58.6192i 0.0752492 + 0.0752492i
\(780\) 0 0
\(781\) −27.7083 27.7083i −0.0354780 0.0354780i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −918.248 −1.16974
\(786\) 0 0
\(787\) 46.6965 46.6965i 0.0593348 0.0593348i −0.676817 0.736152i \(-0.736641\pi\)
0.736152 + 0.676817i \(0.236641\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1447.87 −1.83044
\(792\) 0 0
\(793\) 1271.27i 1.60312i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −127.126 127.126i −0.159505 0.159505i 0.622842 0.782348i \(-0.285978\pi\)
−0.782348 + 0.622842i \(0.785978\pi\)
\(798\) 0 0
\(799\) 183.988i 0.230273i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −35.6774 + 35.6774i −0.0444302 + 0.0444302i
\(804\) 0 0
\(805\) −622.580 + 622.580i −0.773392 + 0.773392i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1047.16i 1.29439i −0.762325 0.647194i \(-0.775942\pi\)
0.762325 0.647194i \(-0.224058\pi\)
\(810\) 0 0
\(811\) 112.206 + 112.206i 0.138356 + 0.138356i 0.772893 0.634537i \(-0.218809\pi\)
−0.634537 + 0.772893i \(0.718809\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1349.20i 1.65546i
\(816\) 0 0
\(817\) −24.3176 −0.0297645
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 7.63080 7.63080i 0.00929452 0.00929452i −0.702444 0.711739i \(-0.747908\pi\)
0.711739 + 0.702444i \(0.247908\pi\)
\(822\) 0 0
\(823\) −1316.28 −1.59937 −0.799687 0.600417i \(-0.795001\pi\)
−0.799687 + 0.600417i \(0.795001\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 341.515 + 341.515i 0.412957 + 0.412957i 0.882767 0.469810i \(-0.155678\pi\)
−0.469810 + 0.882767i \(0.655678\pi\)
\(828\) 0 0
\(829\) −621.672 621.672i −0.749905 0.749905i 0.224556 0.974461i \(-0.427907\pi\)
−0.974461 + 0.224556i \(0.927907\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −272.208 −0.326781
\(834\) 0 0
\(835\) −516.412 + 516.412i −0.618457 + 0.618457i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1440.49 1.71692 0.858459 0.512883i \(-0.171422\pi\)
0.858459 + 0.512883i \(0.171422\pi\)
\(840\) 0 0
\(841\) 831.348i 0.988523i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2051.65 + 2051.65i 2.42798 + 2.42798i
\(846\) 0 0
\(847\) 1244.24i 1.46900i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 404.186 404.186i 0.474954 0.474954i
\(852\) 0 0
\(853\) −625.193 + 625.193i −0.732934 + 0.732934i −0.971200 0.238266i \(-0.923421\pi\)
0.238266 + 0.971200i \(0.423421\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1105.18i 1.28959i −0.764356 0.644794i \(-0.776943\pi\)
0.764356 0.644794i \(-0.223057\pi\)
\(858\) 0 0
\(859\) −379.841 379.841i −0.442190 0.442190i 0.450558 0.892747i \(-0.351225\pi\)
−0.892747 + 0.450558i \(0.851225\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 381.969i 0.442606i −0.975205 0.221303i \(-0.928969\pi\)
0.975205 0.221303i \(-0.0710311\pi\)
\(864\) 0 0
\(865\) 516.031 0.596567
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −29.7076 + 29.7076i −0.0341859 + 0.0341859i
\(870\) 0 0
\(871\) 379.794 0.436044
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 204.994 + 204.994i 0.234279 + 0.234279i
\(876\) 0 0
\(877\) 638.602 + 638.602i 0.728166 + 0.728166i 0.970254 0.242088i \(-0.0778323\pi\)
−0.242088 + 0.970254i \(0.577832\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1362.97 1.54707 0.773533 0.633756i \(-0.218488\pi\)
0.773533 + 0.633756i \(0.218488\pi\)
\(882\) 0 0
\(883\) 897.988 897.988i 1.01697 1.01697i 0.0171209 0.999853i \(-0.494550\pi\)
0.999853 0.0171209i \(-0.00545001\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1343.56 1.51472 0.757359 0.652998i \(-0.226489\pi\)
0.757359 + 0.652998i \(0.226489\pi\)
\(888\) 0 0
\(889\) 421.947i 0.474631i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −98.9295 98.9295i −0.110783 0.110783i
\(894\) 0 0
\(895\) 997.676i 1.11472i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 61.6257 61.6257i 0.0685491 0.0685491i
\(900\) 0 0
\(901\) −131.952 + 131.952i −0.146451 + 0.146451i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1967.97i 2.17456i
\(906\) 0 0
\(907\) 671.651 + 671.651i 0.740519 + 0.740519i 0.972678 0.232159i \(-0.0745789\pi\)
−0.232159 + 0.972678i \(0.574579\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 770.729i 0.846025i −0.906124 0.423012i \(-0.860973\pi\)
0.906124 0.423012i \(-0.139027\pi\)
\(912\) 0 0
\(913\) 61.4246 0.0672778
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −774.942 + 774.942i −0.845084 + 0.845084i
\(918\) 0 0
\(919\) 1153.98 1.25569 0.627843 0.778340i \(-0.283938\pi\)
0.627843 + 0.778340i \(0.283938\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 910.182 + 910.182i 0.986112 + 0.986112i
\(924\) 0 0
\(925\) 669.736 + 669.736i 0.724039 + 0.724039i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 652.736 0.702622 0.351311 0.936259i \(-0.385736\pi\)
0.351311 + 0.936259i \(0.385736\pi\)
\(930\) 0 0
\(931\) 146.365 146.365i 0.157213 0.157213i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 23.7660 0.0254182
\(936\) 0 0
\(937\) 644.074i 0.687378i 0.939083 + 0.343689i \(0.111677\pi\)
−0.939083 + 0.343689i \(0.888323\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −171.348 171.348i −0.182092 0.182092i 0.610175 0.792267i \(-0.291099\pi\)
−0.792267 + 0.610175i \(0.791099\pi\)
\(942\) 0 0
\(943\) 290.929i 0.308514i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 731.249 731.249i 0.772174 0.772174i −0.206312 0.978486i \(-0.566146\pi\)
0.978486 + 0.206312i \(0.0661461\pi\)
\(948\) 0 0
\(949\) 1171.96 1171.96i 1.23494 1.23494i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1745.08i 1.83115i −0.402152 0.915573i \(-0.631738\pi\)
0.402152 0.915573i \(-0.368262\pi\)
\(954\) 0 0
\(955\) 1188.72 + 1188.72i 1.24473 + 1.24473i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1384.54i 1.44374i
\(960\) 0 0
\(961\) 174.077 0.181142
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −622.102 + 622.102i −0.644665 + 0.644665i
\(966\) 0 0
\(967\) −904.237 −0.935095 −0.467548 0.883968i \(-0.654862\pi\)
−0.467548 + 0.883968i \(0.654862\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −1010.37 1010.37i −1.04055 1.04055i −0.999143 0.0414029i \(-0.986817\pi\)
−0.0414029 0.999143i \(-0.513183\pi\)
\(972\) 0 0
\(973\) −236.354 236.354i −0.242913 0.242913i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 396.922 0.406266 0.203133 0.979151i \(-0.434888\pi\)
0.203133 + 0.979151i \(0.434888\pi\)
\(978\) 0 0
\(979\) −69.2259 + 69.2259i −0.0707108 + 0.0707108i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1672.52 1.70145 0.850724 0.525612i \(-0.176164\pi\)
0.850724 + 0.525612i \(0.176164\pi\)
\(984\) 0 0
\(985\) 2269.66i 2.30423i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 60.3446 + 60.3446i 0.0610157 + 0.0610157i
\(990\) 0 0
\(991\) 775.801i 0.782847i 0.920211 + 0.391423i \(0.128017\pi\)
−0.920211 + 0.391423i \(0.871983\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1180.35 + 1180.35i −1.18629 + 1.18629i
\(996\) 0 0
\(997\) 201.495 201.495i 0.202101 0.202101i −0.598799 0.800900i \(-0.704355\pi\)
0.800900 + 0.598799i \(0.204355\pi\)
\(998\) 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 576.3.m.c.559.7 16
3.2 odd 2 192.3.l.a.175.5 16
4.3 odd 2 144.3.m.c.19.8 16
8.3 odd 2 1152.3.m.f.991.2 16
8.5 even 2 1152.3.m.c.991.2 16
12.11 even 2 48.3.l.a.19.1 16
16.3 odd 4 1152.3.m.c.415.2 16
16.5 even 4 144.3.m.c.91.8 16
16.11 odd 4 inner 576.3.m.c.271.7 16
16.13 even 4 1152.3.m.f.415.2 16
24.5 odd 2 384.3.l.b.223.4 16
24.11 even 2 384.3.l.a.223.8 16
48.5 odd 4 48.3.l.a.43.1 yes 16
48.11 even 4 192.3.l.a.79.5 16
48.29 odd 4 384.3.l.a.31.8 16
48.35 even 4 384.3.l.b.31.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.3.l.a.19.1 16 12.11 even 2
48.3.l.a.43.1 yes 16 48.5 odd 4
144.3.m.c.19.8 16 4.3 odd 2
144.3.m.c.91.8 16 16.5 even 4
192.3.l.a.79.5 16 48.11 even 4
192.3.l.a.175.5 16 3.2 odd 2
384.3.l.a.31.8 16 48.29 odd 4
384.3.l.a.223.8 16 24.11 even 2
384.3.l.b.31.4 16 48.35 even 4
384.3.l.b.223.4 16 24.5 odd 2
576.3.m.c.271.7 16 16.11 odd 4 inner
576.3.m.c.559.7 16 1.1 even 1 trivial
1152.3.m.c.415.2 16 16.3 odd 4
1152.3.m.c.991.2 16 8.5 even 2
1152.3.m.f.415.2 16 16.13 even 4
1152.3.m.f.991.2 16 8.3 odd 2