Properties

Label 2592.2.s.e.863.2
Level $2592$
Weight $2$
Character 2592.863
Analytic conductor $20.697$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2592,2,Mod(863,2592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2592, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2592.863");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.s (of order \(6\), 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: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 863.2
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2592.863
Dual form 2592.2.s.e.1727.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.44949 - 1.41421i) q^{5} +(1.73205 - 1.00000i) q^{7} +O(q^{10})\) \(q+(-2.44949 - 1.41421i) q^{5} +(1.73205 - 1.00000i) q^{7} +(-1.41421 - 2.44949i) q^{11} +(1.00000 - 1.73205i) q^{13} -6.00000i q^{19} +(-2.82843 + 4.89898i) q^{23} +(1.50000 + 2.59808i) q^{25} +(-2.44949 + 1.41421i) q^{29} +(-1.73205 - 1.00000i) q^{31} -5.65685 q^{35} +6.00000 q^{37} +(-4.89898 - 2.82843i) q^{41} +(1.73205 - 1.00000i) q^{43} +(5.65685 + 9.79796i) q^{47} +(-1.50000 + 2.59808i) q^{49} -8.48528i q^{53} +8.00000i q^{55} +(-1.41421 + 2.44949i) q^{59} +(1.00000 + 1.73205i) q^{61} +(-4.89898 + 2.82843i) q^{65} +(-1.73205 - 1.00000i) q^{67} -5.65685 q^{71} -6.00000 q^{73} +(-4.89898 - 2.82843i) q^{77} +(-12.1244 + 7.00000i) q^{79} +(1.41421 + 2.44949i) q^{83} -16.9706i q^{89} -4.00000i q^{91} +(-8.48528 + 14.6969i) q^{95} +(-5.00000 - 8.66025i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{13} + 12 q^{25} + 48 q^{37} - 12 q^{49} + 8 q^{61} - 48 q^{73} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(-1\)

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
\(5\) −2.44949 1.41421i −1.09545 0.632456i −0.160424 0.987048i \(-0.551286\pi\)
−0.935021 + 0.354593i \(0.884620\pi\)
\(6\) 0 0
\(7\) 1.73205 1.00000i 0.654654 0.377964i −0.135583 0.990766i \(-0.543291\pi\)
0.790237 + 0.612801i \(0.209957\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.41421 2.44949i −0.426401 0.738549i 0.570149 0.821541i \(-0.306886\pi\)
−0.996550 + 0.0829925i \(0.973552\pi\)
\(12\) 0 0
\(13\) 1.00000 1.73205i 0.277350 0.480384i −0.693375 0.720577i \(-0.743877\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 6.00000i 1.37649i −0.725476 0.688247i \(-0.758380\pi\)
0.725476 0.688247i \(-0.241620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.82843 + 4.89898i −0.589768 + 1.02151i 0.404495 + 0.914540i \(0.367447\pi\)
−0.994263 + 0.106967i \(0.965886\pi\)
\(24\) 0 0
\(25\) 1.50000 + 2.59808i 0.300000 + 0.519615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.44949 + 1.41421i −0.454859 + 0.262613i −0.709880 0.704323i \(-0.751251\pi\)
0.255021 + 0.966935i \(0.417918\pi\)
\(30\) 0 0
\(31\) −1.73205 1.00000i −0.311086 0.179605i 0.336327 0.941745i \(-0.390815\pi\)
−0.647412 + 0.762140i \(0.724149\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −5.65685 −0.956183
\(36\) 0 0
\(37\) 6.00000 0.986394 0.493197 0.869918i \(-0.335828\pi\)
0.493197 + 0.869918i \(0.335828\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.89898 2.82843i −0.765092 0.441726i 0.0660290 0.997818i \(-0.478967\pi\)
−0.831121 + 0.556092i \(0.812300\pi\)
\(42\) 0 0
\(43\) 1.73205 1.00000i 0.264135 0.152499i −0.362084 0.932145i \(-0.617935\pi\)
0.626219 + 0.779647i \(0.284601\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.65685 + 9.79796i 0.825137 + 1.42918i 0.901815 + 0.432123i \(0.142235\pi\)
−0.0766776 + 0.997056i \(0.524431\pi\)
\(48\) 0 0
\(49\) −1.50000 + 2.59808i −0.214286 + 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.48528i 1.16554i −0.812636 0.582772i \(-0.801968\pi\)
0.812636 0.582772i \(-0.198032\pi\)
\(54\) 0 0
\(55\) 8.00000i 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.41421 + 2.44949i −0.184115 + 0.318896i −0.943278 0.332004i \(-0.892275\pi\)
0.759163 + 0.650901i \(0.225609\pi\)
\(60\) 0 0
\(61\) 1.00000 + 1.73205i 0.128037 + 0.221766i 0.922916 0.385002i \(-0.125799\pi\)
−0.794879 + 0.606768i \(0.792466\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −4.89898 + 2.82843i −0.607644 + 0.350823i
\(66\) 0 0
\(67\) −1.73205 1.00000i −0.211604 0.122169i 0.390453 0.920623i \(-0.372318\pi\)
−0.602056 + 0.798454i \(0.705652\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 0 0
\(73\) −6.00000 −0.702247 −0.351123 0.936329i \(-0.614200\pi\)
−0.351123 + 0.936329i \(0.614200\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −4.89898 2.82843i −0.558291 0.322329i
\(78\) 0 0
\(79\) −12.1244 + 7.00000i −1.36410 + 0.787562i −0.990166 0.139895i \(-0.955323\pi\)
−0.373930 + 0.927457i \(0.621990\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.41421 + 2.44949i 0.155230 + 0.268866i 0.933143 0.359506i \(-0.117055\pi\)
−0.777913 + 0.628372i \(0.783721\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 16.9706i 1.79888i −0.437048 0.899438i \(-0.643976\pi\)
0.437048 0.899438i \(-0.356024\pi\)
\(90\) 0 0
\(91\) 4.00000i 0.419314i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.48528 + 14.6969i −0.870572 + 1.50787i
\(96\) 0 0
\(97\) −5.00000 8.66025i −0.507673 0.879316i −0.999961 0.00888289i \(-0.997172\pi\)
0.492287 0.870433i \(-0.336161\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.44949 1.41421i 0.243733 0.140720i −0.373158 0.927768i \(-0.621725\pi\)
0.616891 + 0.787048i \(0.288392\pi\)
\(102\) 0 0
\(103\) 12.1244 + 7.00000i 1.19465 + 0.689730i 0.959357 0.282194i \(-0.0910623\pi\)
0.235291 + 0.971925i \(0.424396\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.48528 −0.820303 −0.410152 0.912017i \(-0.634524\pi\)
−0.410152 + 0.912017i \(0.634524\pi\)
\(108\) 0 0
\(109\) −18.0000 −1.72409 −0.862044 0.506834i \(-0.830816\pi\)
−0.862044 + 0.506834i \(0.830816\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.79796 5.65685i −0.921714 0.532152i −0.0375328 0.999295i \(-0.511950\pi\)
−0.884182 + 0.467143i \(0.845283\pi\)
\(114\) 0 0
\(115\) 13.8564 8.00000i 1.29212 0.746004i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.50000 2.59808i 0.136364 0.236189i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 5.65685i 0.505964i
\(126\) 0 0
\(127\) 18.0000i 1.59724i 0.601834 + 0.798621i \(0.294437\pi\)
−0.601834 + 0.798621i \(0.705563\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −9.89949 + 17.1464i −0.864923 + 1.49809i 0.00220084 + 0.999998i \(0.499299\pi\)
−0.867124 + 0.498093i \(0.834034\pi\)
\(132\) 0 0
\(133\) −6.00000 10.3923i −0.520266 0.901127i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.89898 2.82843i 0.418548 0.241649i −0.275908 0.961184i \(-0.588978\pi\)
0.694456 + 0.719535i \(0.255645\pi\)
\(138\) 0 0
\(139\) −8.66025 5.00000i −0.734553 0.424094i 0.0855324 0.996335i \(-0.472741\pi\)
−0.820086 + 0.572241i \(0.806074\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.65685 −0.473050
\(144\) 0 0
\(145\) 8.00000 0.664364
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −12.2474 7.07107i −1.00335 0.579284i −0.0941123 0.995562i \(-0.530001\pi\)
−0.909238 + 0.416277i \(0.863335\pi\)
\(150\) 0 0
\(151\) −12.1244 + 7.00000i −0.986666 + 0.569652i −0.904276 0.426948i \(-0.859589\pi\)
−0.0823900 + 0.996600i \(0.526255\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.82843 + 4.89898i 0.227185 + 0.393496i
\(156\) 0 0
\(157\) −7.00000 + 12.1244i −0.558661 + 0.967629i 0.438948 + 0.898513i \(0.355351\pi\)
−0.997609 + 0.0691164i \(0.977982\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 11.3137i 0.891645i
\(162\) 0 0
\(163\) 6.00000i 0.469956i −0.972001 0.234978i \(-0.924498\pi\)
0.972001 0.234978i \(-0.0755019\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.82843 4.89898i 0.218870 0.379094i −0.735593 0.677424i \(-0.763096\pi\)
0.954463 + 0.298330i \(0.0964295\pi\)
\(168\) 0 0
\(169\) 4.50000 + 7.79423i 0.346154 + 0.599556i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −12.2474 + 7.07107i −0.931156 + 0.537603i −0.887177 0.461429i \(-0.847337\pi\)
−0.0439792 + 0.999032i \(0.514004\pi\)
\(174\) 0 0
\(175\) 5.19615 + 3.00000i 0.392792 + 0.226779i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.48528 0.634220 0.317110 0.948389i \(-0.397288\pi\)
0.317110 + 0.948389i \(0.397288\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −14.6969 8.48528i −1.08054 0.623850i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −11.3137 19.5959i −0.818631 1.41791i −0.906691 0.421796i \(-0.861400\pi\)
0.0880597 0.996115i \(-0.471933\pi\)
\(192\) 0 0
\(193\) 7.00000 12.1244i 0.503871 0.872730i −0.496119 0.868255i \(-0.665242\pi\)
0.999990 0.00447566i \(-0.00142465\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 19.7990i 1.41062i −0.708899 0.705310i \(-0.750808\pi\)
0.708899 0.705310i \(-0.249192\pi\)
\(198\) 0 0
\(199\) 2.00000i 0.141776i 0.997484 + 0.0708881i \(0.0225833\pi\)
−0.997484 + 0.0708881i \(0.977417\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.82843 + 4.89898i −0.198517 + 0.343841i
\(204\) 0 0
\(205\) 8.00000 + 13.8564i 0.558744 + 0.967773i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −14.6969 + 8.48528i −1.01661 + 0.586939i
\(210\) 0 0
\(211\) −1.73205 1.00000i −0.119239 0.0688428i 0.439194 0.898392i \(-0.355264\pi\)
−0.558433 + 0.829549i \(0.688597\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −5.65685 −0.385794
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.73205 1.00000i 0.115987 0.0669650i −0.440884 0.897564i \(-0.645335\pi\)
0.556871 + 0.830599i \(0.312002\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.07107 + 12.2474i 0.469323 + 0.812892i 0.999385 0.0350674i \(-0.0111646\pi\)
−0.530062 + 0.847959i \(0.677831\pi\)
\(228\) 0 0
\(229\) 5.00000 8.66025i 0.330409 0.572286i −0.652183 0.758062i \(-0.726147\pi\)
0.982592 + 0.185776i \(0.0594799\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.65685i 0.370593i −0.982683 0.185296i \(-0.940675\pi\)
0.982683 0.185296i \(-0.0593245\pi\)
\(234\) 0 0
\(235\) 32.0000i 2.08745i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.65685 9.79796i 0.365911 0.633777i −0.623011 0.782213i \(-0.714091\pi\)
0.988922 + 0.148436i \(0.0474240\pi\)
\(240\) 0 0
\(241\) −1.00000 1.73205i −0.0644157 0.111571i 0.832019 0.554747i \(-0.187185\pi\)
−0.896435 + 0.443176i \(0.853852\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 7.34847 4.24264i 0.469476 0.271052i
\(246\) 0 0
\(247\) −10.3923 6.00000i −0.661247 0.381771i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −8.48528 −0.535586 −0.267793 0.963476i \(-0.586294\pi\)
−0.267793 + 0.963476i \(0.586294\pi\)
\(252\) 0 0
\(253\) 16.0000 1.00591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 19.5959 + 11.3137i 1.22236 + 0.705730i 0.965420 0.260698i \(-0.0839526\pi\)
0.256939 + 0.966428i \(0.417286\pi\)
\(258\) 0 0
\(259\) 10.3923 6.00000i 0.645746 0.372822i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.82843 + 4.89898i 0.174408 + 0.302084i 0.939956 0.341295i \(-0.110865\pi\)
−0.765548 + 0.643379i \(0.777532\pi\)
\(264\) 0 0
\(265\) −12.0000 + 20.7846i −0.737154 + 1.27679i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.48528i 0.517357i 0.965964 + 0.258678i \(0.0832870\pi\)
−0.965964 + 0.258678i \(0.916713\pi\)
\(270\) 0 0
\(271\) 18.0000i 1.09342i 0.837321 + 0.546711i \(0.184120\pi\)
−0.837321 + 0.546711i \(0.815880\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.24264 7.34847i 0.255841 0.443129i
\(276\) 0 0
\(277\) 5.00000 + 8.66025i 0.300421 + 0.520344i 0.976231 0.216731i \(-0.0695395\pi\)
−0.675810 + 0.737075i \(0.736206\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.89898 2.82843i 0.292249 0.168730i −0.346707 0.937974i \(-0.612700\pi\)
0.638955 + 0.769244i \(0.279367\pi\)
\(282\) 0 0
\(283\) −8.66025 5.00000i −0.514799 0.297219i 0.220005 0.975499i \(-0.429393\pi\)
−0.734804 + 0.678280i \(0.762726\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −11.3137 −0.667827
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −2.44949 1.41421i −0.143101 0.0826192i 0.426740 0.904374i \(-0.359662\pi\)
−0.569841 + 0.821755i \(0.692995\pi\)
\(294\) 0 0
\(295\) 6.92820 4.00000i 0.403376 0.232889i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 5.65685 + 9.79796i 0.327144 + 0.566631i
\(300\) 0 0
\(301\) 2.00000 3.46410i 0.115278 0.199667i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.65685i 0.323911i
\(306\) 0 0
\(307\) 30.0000i 1.71219i −0.516818 0.856095i \(-0.672884\pi\)
0.516818 0.856095i \(-0.327116\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −2.82843 + 4.89898i −0.160385 + 0.277796i −0.935007 0.354629i \(-0.884607\pi\)
0.774622 + 0.632425i \(0.217940\pi\)
\(312\) 0 0
\(313\) −1.00000 1.73205i −0.0565233 0.0979013i 0.836379 0.548151i \(-0.184668\pi\)
−0.892903 + 0.450250i \(0.851335\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.44949 + 1.41421i −0.137577 + 0.0794301i −0.567209 0.823574i \(-0.691977\pi\)
0.429632 + 0.903004i \(0.358643\pi\)
\(318\) 0 0
\(319\) 6.92820 + 4.00000i 0.387905 + 0.223957i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 6.00000 0.332820
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 19.5959 + 11.3137i 1.08036 + 0.623745i
\(330\) 0 0
\(331\) 8.66025 5.00000i 0.476011 0.274825i −0.242742 0.970091i \(-0.578047\pi\)
0.718752 + 0.695266i \(0.244713\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 2.82843 + 4.89898i 0.154533 + 0.267660i
\(336\) 0 0
\(337\) 11.0000 19.0526i 0.599208 1.03786i −0.393730 0.919226i \(-0.628816\pi\)
0.992938 0.118633i \(-0.0378512\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.65685i 0.306336i
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.07107 + 12.2474i −0.379595 + 0.657477i −0.991003 0.133838i \(-0.957270\pi\)
0.611408 + 0.791315i \(0.290603\pi\)
\(348\) 0 0
\(349\) −7.00000 12.1244i −0.374701 0.649002i 0.615581 0.788074i \(-0.288921\pi\)
−0.990282 + 0.139072i \(0.955588\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 19.5959 11.3137i 1.04299 0.602168i 0.122308 0.992492i \(-0.460970\pi\)
0.920677 + 0.390324i \(0.127637\pi\)
\(354\) 0 0
\(355\) 13.8564 + 8.00000i 0.735422 + 0.424596i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.9706 0.895672 0.447836 0.894116i \(-0.352195\pi\)
0.447836 + 0.894116i \(0.352195\pi\)
\(360\) 0 0
\(361\) −17.0000 −0.894737
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 14.6969 + 8.48528i 0.769273 + 0.444140i
\(366\) 0 0
\(367\) −12.1244 + 7.00000i −0.632886 + 0.365397i −0.781869 0.623443i \(-0.785733\pi\)
0.148983 + 0.988840i \(0.452400\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −8.48528 14.6969i −0.440534 0.763027i
\(372\) 0 0
\(373\) 13.0000 22.5167i 0.673114 1.16587i −0.303902 0.952703i \(-0.598289\pi\)
0.977016 0.213165i \(-0.0683772\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.65685i 0.291343i
\(378\) 0 0
\(379\) 2.00000i 0.102733i 0.998680 + 0.0513665i \(0.0163577\pi\)
−0.998680 + 0.0513665i \(0.983642\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 11.3137 19.5959i 0.578103 1.00130i −0.417593 0.908634i \(-0.637126\pi\)
0.995697 0.0926706i \(-0.0295404\pi\)
\(384\) 0 0
\(385\) 8.00000 + 13.8564i 0.407718 + 0.706188i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.44949 1.41421i 0.124194 0.0717035i −0.436616 0.899648i \(-0.643823\pi\)
0.560810 + 0.827945i \(0.310490\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 39.5980 1.99239
\(396\) 0 0
\(397\) −18.0000 −0.903394 −0.451697 0.892171i \(-0.649181\pi\)
−0.451697 + 0.892171i \(0.649181\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −19.5959 11.3137i −0.978573 0.564980i −0.0767343 0.997052i \(-0.524449\pi\)
−0.901839 + 0.432072i \(0.857783\pi\)
\(402\) 0 0
\(403\) −3.46410 + 2.00000i −0.172559 + 0.0996271i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −8.48528 14.6969i −0.420600 0.728500i
\(408\) 0 0
\(409\) −13.0000 + 22.5167i −0.642809 + 1.11338i 0.341994 + 0.939702i \(0.388898\pi\)
−0.984803 + 0.173675i \(0.944436\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 5.65685i 0.278356i
\(414\) 0 0
\(415\) 8.00000i 0.392705i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.3848 31.8434i 0.898155 1.55565i 0.0683046 0.997665i \(-0.478241\pi\)
0.829851 0.557986i \(-0.188426\pi\)
\(420\) 0 0
\(421\) −11.0000 19.0526i −0.536107 0.928565i −0.999109 0.0422075i \(-0.986561\pi\)
0.463002 0.886357i \(-0.346772\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3.46410 + 2.00000i 0.167640 + 0.0967868i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 33.9411 1.63489 0.817443 0.576009i \(-0.195391\pi\)
0.817443 + 0.576009i \(0.195391\pi\)
\(432\) 0 0
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 29.3939 + 16.9706i 1.40610 + 0.811812i
\(438\) 0 0
\(439\) −12.1244 + 7.00000i −0.578664 + 0.334092i −0.760602 0.649218i \(-0.775096\pi\)
0.181938 + 0.983310i \(0.441763\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.07107 12.2474i −0.335957 0.581894i 0.647712 0.761886i \(-0.275726\pi\)
−0.983668 + 0.179992i \(0.942393\pi\)
\(444\) 0 0
\(445\) −24.0000 + 41.5692i −1.13771 + 1.97057i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 33.9411i 1.60178i 0.598811 + 0.800890i \(0.295640\pi\)
−0.598811 + 0.800890i \(0.704360\pi\)
\(450\) 0 0
\(451\) 16.0000i 0.753411i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.65685 + 9.79796i −0.265197 + 0.459335i
\(456\) 0 0
\(457\) −5.00000 8.66025i −0.233890 0.405110i 0.725059 0.688686i \(-0.241812\pi\)
−0.958950 + 0.283577i \(0.908479\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 26.9444 15.5563i 1.25493 0.724531i 0.282842 0.959167i \(-0.408723\pi\)
0.972083 + 0.234635i \(0.0753896\pi\)
\(462\) 0 0
\(463\) −1.73205 1.00000i −0.0804952 0.0464739i 0.459212 0.888327i \(-0.348132\pi\)
−0.539707 + 0.841853i \(0.681465\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −25.4558 −1.17796 −0.588978 0.808149i \(-0.700470\pi\)
−0.588978 + 0.808149i \(0.700470\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.89898 2.82843i −0.225255 0.130051i
\(474\) 0 0
\(475\) 15.5885 9.00000i 0.715247 0.412948i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11.3137 + 19.5959i 0.516937 + 0.895360i 0.999807 + 0.0196683i \(0.00626102\pi\)
−0.482870 + 0.875692i \(0.660406\pi\)
\(480\) 0 0
\(481\) 6.00000 10.3923i 0.273576 0.473848i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 28.2843i 1.28432i
\(486\) 0 0
\(487\) 18.0000i 0.815658i 0.913058 + 0.407829i \(0.133714\pi\)
−0.913058 + 0.407829i \(0.866286\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.5563 26.9444i 0.702048 1.21598i −0.265698 0.964056i \(-0.585602\pi\)
0.967746 0.251927i \(-0.0810642\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.79796 + 5.65685i −0.439499 + 0.253745i
\(498\) 0 0
\(499\) −29.4449 17.0000i −1.31813 0.761025i −0.334705 0.942323i \(-0.608637\pi\)
−0.983428 + 0.181298i \(0.941970\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.65685 0.252227 0.126113 0.992016i \(-0.459750\pi\)
0.126113 + 0.992016i \(0.459750\pi\)
\(504\) 0 0
\(505\) −8.00000 −0.355995
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.44949 + 1.41421i 0.108572 + 0.0626839i 0.553303 0.832980i \(-0.313367\pi\)
−0.444731 + 0.895664i \(0.646701\pi\)
\(510\) 0 0
\(511\) −10.3923 + 6.00000i −0.459728 + 0.265424i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −19.7990 34.2929i −0.872448 1.51112i
\(516\) 0 0
\(517\) 16.0000 27.7128i 0.703679 1.21881i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5.65685i 0.247831i 0.992293 + 0.123916i \(0.0395452\pi\)
−0.992293 + 0.123916i \(0.960455\pi\)
\(522\) 0 0
\(523\) 34.0000i 1.48672i 0.668894 + 0.743358i \(0.266768\pi\)
−0.668894 + 0.743358i \(0.733232\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −4.50000 7.79423i −0.195652 0.338880i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −9.79796 + 5.65685i −0.424397 + 0.245026i
\(534\) 0 0
\(535\) 20.7846 + 12.0000i 0.898597 + 0.518805i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 8.48528 0.365487
\(540\) 0 0
\(541\) −2.00000 −0.0859867 −0.0429934 0.999075i \(-0.513689\pi\)
−0.0429934 + 0.999075i \(0.513689\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 44.0908 + 25.4558i 1.88864 + 1.09041i
\(546\) 0 0
\(547\) −32.9090 + 19.0000i −1.40709 + 0.812381i −0.995106 0.0988117i \(-0.968496\pi\)
−0.411980 + 0.911193i \(0.635163\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.48528 + 14.6969i 0.361485 + 0.626111i
\(552\) 0 0
\(553\) −14.0000 + 24.2487i −0.595341 + 1.03116i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.48528i 0.359533i 0.983709 + 0.179766i \(0.0575342\pi\)
−0.983709 + 0.179766i \(0.942466\pi\)
\(558\) 0 0
\(559\) 4.00000i 0.169182i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.41421 2.44949i 0.0596020 0.103234i −0.834685 0.550728i \(-0.814350\pi\)
0.894287 + 0.447494i \(0.147684\pi\)
\(564\) 0 0
\(565\) 16.0000 + 27.7128i 0.673125 + 1.16589i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.89898 + 2.82843i −0.205376 + 0.118574i −0.599161 0.800629i \(-0.704499\pi\)
0.393785 + 0.919203i \(0.371166\pi\)
\(570\) 0 0
\(571\) 19.0526 + 11.0000i 0.797325 + 0.460336i 0.842535 0.538642i \(-0.181062\pi\)
−0.0452101 + 0.998978i \(0.514396\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −16.9706 −0.707721
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.89898 + 2.82843i 0.203244 + 0.117343i
\(582\) 0 0
\(583\) −20.7846 + 12.0000i −0.860811 + 0.496989i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −7.07107 12.2474i −0.291854 0.505506i 0.682394 0.730985i \(-0.260939\pi\)
−0.974248 + 0.225478i \(0.927606\pi\)
\(588\) 0 0
\(589\) −6.00000 + 10.3923i −0.247226 + 0.428207i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 33.9411i 1.39379i −0.717171 0.696897i \(-0.754563\pi\)
0.717171 0.696897i \(-0.245437\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 19.7990 34.2929i 0.808965 1.40117i −0.104617 0.994513i \(-0.533362\pi\)
0.913582 0.406656i \(-0.133305\pi\)
\(600\) 0 0
\(601\) −5.00000 8.66025i −0.203954 0.353259i 0.745845 0.666120i \(-0.232046\pi\)
−0.949799 + 0.312861i \(0.898713\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −7.34847 + 4.24264i −0.298758 + 0.172488i
\(606\) 0 0
\(607\) 12.1244 + 7.00000i 0.492112 + 0.284121i 0.725450 0.688274i \(-0.241632\pi\)
−0.233338 + 0.972396i \(0.574965\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 22.6274 0.915407
\(612\) 0 0
\(613\) 6.00000 0.242338 0.121169 0.992632i \(-0.461336\pi\)
0.121169 + 0.992632i \(0.461336\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 24.4949 + 14.1421i 0.986127 + 0.569341i 0.904114 0.427290i \(-0.140532\pi\)
0.0820130 + 0.996631i \(0.473865\pi\)
\(618\) 0 0
\(619\) 8.66025 5.00000i 0.348085 0.200967i −0.315757 0.948840i \(-0.602258\pi\)
0.663842 + 0.747873i \(0.268925\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −16.9706 29.3939i −0.679911 1.17764i
\(624\) 0 0
\(625\) 15.5000 26.8468i 0.620000 1.07387i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 30.0000i 1.19428i −0.802137 0.597141i \(-0.796303\pi\)
0.802137 0.597141i \(-0.203697\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 25.4558 44.0908i 1.01018 1.74969i
\(636\) 0 0
\(637\) 3.00000 + 5.19615i 0.118864 + 0.205879i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.79796 5.65685i 0.386996 0.223432i −0.293862 0.955848i \(-0.594941\pi\)
0.680858 + 0.732416i \(0.261607\pi\)
\(642\) 0 0
\(643\) −22.5167 13.0000i −0.887970 0.512670i −0.0146923 0.999892i \(-0.504677\pi\)
−0.873278 + 0.487222i \(0.838010\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.9706 0.667182 0.333591 0.942718i \(-0.391740\pi\)
0.333591 + 0.942718i \(0.391740\pi\)
\(648\) 0 0
\(649\) 8.00000 0.314027
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 12.2474 + 7.07107i 0.479280 + 0.276712i 0.720116 0.693853i \(-0.244088\pi\)
−0.240837 + 0.970566i \(0.577422\pi\)
\(654\) 0 0
\(655\) 48.4974 28.0000i 1.89495 1.09405i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7.07107 + 12.2474i 0.275450 + 0.477093i 0.970248 0.242111i \(-0.0778399\pi\)
−0.694799 + 0.719204i \(0.744507\pi\)
\(660\) 0 0
\(661\) 5.00000 8.66025i 0.194477 0.336845i −0.752252 0.658876i \(-0.771032\pi\)
0.946729 + 0.322031i \(0.104366\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 33.9411i 1.31618i
\(666\) 0 0
\(667\) 16.0000i 0.619522i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.82843 4.89898i 0.109190 0.189123i
\(672\) 0 0
\(673\) −5.00000 8.66025i −0.192736 0.333828i 0.753420 0.657539i \(-0.228403\pi\)
−0.946156 + 0.323711i \(0.895069\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.44949 1.41421i 0.0941415 0.0543526i −0.452190 0.891922i \(-0.649357\pi\)
0.546332 + 0.837569i \(0.316024\pi\)
\(678\) 0 0
\(679\) −17.3205 10.0000i −0.664700 0.383765i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −42.4264 −1.62340 −0.811701 0.584074i \(-0.801458\pi\)
−0.811701 + 0.584074i \(0.801458\pi\)
\(684\) 0 0
\(685\) −16.0000 −0.611329
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −14.6969 8.48528i −0.559909 0.323263i
\(690\) 0 0
\(691\) 22.5167 13.0000i 0.856574 0.494543i −0.00628943 0.999980i \(-0.502002\pi\)
0.862864 + 0.505437i \(0.168669\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 14.1421 + 24.4949i 0.536442 + 0.929144i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 25.4558i 0.961454i −0.876870 0.480727i \(-0.840373\pi\)
0.876870 0.480727i \(-0.159627\pi\)
\(702\) 0 0
\(703\) 36.0000i 1.35777i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.82843 4.89898i 0.106374 0.184245i
\(708\) 0 0
\(709\) −19.0000 32.9090i −0.713560 1.23592i −0.963512 0.267664i \(-0.913748\pi\)
0.249952 0.968258i \(-0.419585\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 9.79796 5.65685i 0.366936 0.211851i
\(714\) 0 0
\(715\) 13.8564 + 8.00000i 0.518200 + 0.299183i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 33.9411 1.26579 0.632895 0.774237i \(-0.281866\pi\)
0.632895 + 0.774237i \(0.281866\pi\)
\(720\) 0 0
\(721\) 28.0000 1.04277
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7.34847 4.24264i −0.272915 0.157568i
\(726\) 0 0
\(727\) 29.4449 17.0000i 1.09205 0.630495i 0.157928 0.987451i \(-0.449519\pi\)
0.934121 + 0.356956i \(0.116185\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −23.0000 + 39.8372i −0.849524 + 1.47142i 0.0321090 + 0.999484i \(0.489778\pi\)
−0.881633 + 0.471935i \(0.843556\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.65685i 0.208373i
\(738\) 0 0
\(739\) 34.0000i 1.25071i 0.780340 + 0.625355i \(0.215046\pi\)
−0.780340 + 0.625355i \(0.784954\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.82843 4.89898i 0.103765 0.179726i −0.809468 0.587164i \(-0.800244\pi\)
0.913233 + 0.407438i \(0.133578\pi\)
\(744\) 0 0
\(745\) 20.0000 + 34.6410i 0.732743 + 1.26915i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −14.6969 + 8.48528i −0.537014 + 0.310045i
\(750\) 0 0
\(751\) −29.4449 17.0000i −1.07446 0.620339i −0.145062 0.989423i \(-0.546338\pi\)
−0.929396 + 0.369084i \(0.879672\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 39.5980 1.44112
\(756\) 0 0
\(757\) 38.0000 1.38113 0.690567 0.723269i \(-0.257361\pi\)
0.690567 + 0.723269i \(0.257361\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 24.4949 + 14.1421i 0.887939 + 0.512652i 0.873268 0.487240i \(-0.161996\pi\)
0.0146714 + 0.999892i \(0.495330\pi\)
\(762\) 0 0
\(763\) −31.1769 + 18.0000i −1.12868 + 0.651644i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.82843 + 4.89898i 0.102129 + 0.176892i
\(768\) 0 0
\(769\) 23.0000 39.8372i 0.829401 1.43657i −0.0691074 0.997609i \(-0.522015\pi\)
0.898509 0.438956i \(-0.144652\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 25.4558i 0.915583i 0.889060 + 0.457792i \(0.151359\pi\)
−0.889060 + 0.457792i \(0.848641\pi\)
\(774\) 0 0
\(775\) 6.00000i 0.215526i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −16.9706 + 29.3939i −0.608034 + 1.05314i
\(780\) 0 0
\(781\) 8.00000 + 13.8564i 0.286263 + 0.495821i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 34.2929 19.7990i 1.22396 0.706656i
\(786\) 0 0
\(787\) 32.9090 + 19.0000i 1.17308 + 0.677277i 0.954403 0.298521i \(-0.0964933\pi\)
0.218675 + 0.975798i \(0.429827\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −22.6274 −0.804538
\(792\) 0 0
\(793\) 4.00000 0.142044
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.44949 + 1.41421i 0.0867654 + 0.0500940i 0.542755 0.839891i \(-0.317381\pi\)
−0.455990 + 0.889985i \(0.650715\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 8.48528 + 14.6969i 0.299439 + 0.518644i
\(804\) 0 0
\(805\) 16.0000 27.7128i 0.563926 0.976748i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 50.9117i 1.78996i 0.446107 + 0.894980i \(0.352810\pi\)
−0.446107 + 0.894980i \(0.647190\pi\)
\(810\) 0 0
\(811\) 30.0000i 1.05344i −0.850038 0.526721i \(-0.823421\pi\)
0.850038 0.526721i \(-0.176579\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −8.48528 + 14.6969i −0.297226 + 0.514811i
\(816\) 0 0
\(817\) −6.00000 10.3923i −0.209913 0.363581i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 12.2474 7.07107i 0.427439 0.246782i −0.270816 0.962631i \(-0.587294\pi\)
0.698255 + 0.715849i \(0.253960\pi\)
\(822\) 0 0
\(823\) −1.73205 1.00000i −0.0603755 0.0348578i 0.469508 0.882928i \(-0.344431\pi\)
−0.529884 + 0.848070i \(0.677765\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.4558 0.885186 0.442593 0.896723i \(-0.354059\pi\)
0.442593 + 0.896723i \(0.354059\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −13.8564 + 8.00000i −0.479521 + 0.276851i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −19.7990 34.2929i −0.683537 1.18392i −0.973894 0.227003i \(-0.927107\pi\)
0.290357 0.956918i \(-0.406226\pi\)
\(840\) 0 0
\(841\) −10.5000 + 18.1865i −0.362069 + 0.627122i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 25.4558i 0.875708i
\(846\) 0 0
\(847\) 6.00000i 0.206162i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −16.9706 + 29.3939i −0.581743 + 1.00761i
\(852\) 0 0
\(853\) 13.0000 + 22.5167i 0.445112 + 0.770956i 0.998060 0.0622597i \(-0.0198307\pi\)
−0.552948 + 0.833215i \(0.686497\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −24.4949 + 14.1421i −0.836730 + 0.483086i −0.856151 0.516725i \(-0.827151\pi\)
0.0194215 + 0.999811i \(0.493818\pi\)
\(858\) 0 0
\(859\) −1.73205 1.00000i −0.0590968 0.0341196i 0.470160 0.882581i \(-0.344196\pi\)
−0.529257 + 0.848461i \(0.677529\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 40.0000 1.36004
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 34.2929 + 19.7990i 1.16331 + 0.671635i
\(870\) 0 0
\(871\) −3.46410 + 2.00000i −0.117377 + 0.0677674i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 5.65685 + 9.79796i 0.191237 + 0.331231i
\(876\) 0 0
\(877\) 1.00000 1.73205i 0.0337676 0.0584872i −0.848648 0.528958i \(-0.822583\pi\)
0.882415 + 0.470471i \(0.155916\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 33.9411i 1.14351i −0.820426 0.571753i \(-0.806264\pi\)
0.820426 0.571753i \(-0.193736\pi\)
\(882\) 0 0
\(883\) 6.00000i 0.201916i −0.994891 0.100958i \(-0.967809\pi\)
0.994891 0.100958i \(-0.0321908\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.82843 + 4.89898i −0.0949693 + 0.164492i −0.909596 0.415494i \(-0.863609\pi\)
0.814627 + 0.579986i \(0.196942\pi\)
\(888\) 0 0
\(889\) 18.0000 + 31.1769i 0.603701 + 1.04564i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 58.7878 33.9411i 1.96726 1.13580i
\(894\) 0 0
\(895\) −20.7846 12.0000i −0.694753 0.401116i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5.65685 0.188667
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −14.6969 8.48528i −0.488543 0.282060i
\(906\) 0 0
\(907\) 1.73205 1.00000i 0.0575118 0.0332045i −0.470968 0.882150i \(-0.656095\pi\)
0.528480 + 0.848946i \(0.322762\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.65685 9.79796i −0.187420 0.324621i 0.756969 0.653450i \(-0.226679\pi\)
−0.944389 + 0.328830i \(0.893346\pi\)
\(912\) 0 0
\(913\) 4.00000 6.92820i 0.132381 0.229290i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 39.5980i 1.30764i
\(918\) 0 0
\(919\) 30.0000i 0.989609i −0.869004 0.494804i \(-0.835240\pi\)
0.869004 0.494804i \(-0.164760\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −5.65685 + 9.79796i −0.186198 + 0.322504i
\(924\) 0 0
\(925\) 9.00000 + 15.5885i 0.295918 + 0.512545i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −9.79796 + 5.65685i −0.321461 + 0.185595i −0.652043 0.758182i \(-0.726088\pi\)
0.330583 + 0.943777i \(0.392755\pi\)
\(930\) 0 0
\(931\) 15.5885 + 9.00000i 0.510891 + 0.294963i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −30.0000 −0.980057 −0.490029 0.871706i \(-0.663014\pi\)
−0.490029 + 0.871706i \(0.663014\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −46.5403 26.8701i −1.51717 0.875939i −0.999796 0.0201796i \(-0.993576\pi\)
−0.517374 0.855759i \(-0.673090\pi\)
\(942\) 0 0
\(943\) 27.7128 16.0000i 0.902453 0.521032i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.07107 + 12.2474i 0.229779 + 0.397989i 0.957742 0.287627i \(-0.0928664\pi\)
−0.727964 + 0.685616i \(0.759533\pi\)
\(948\) 0 0
\(949\) −6.00000 + 10.3923i −0.194768 + 0.337348i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 50.9117i 1.64919i −0.565723 0.824596i \(-0.691403\pi\)
0.565723 0.824596i \(-0.308597\pi\)
\(954\) 0 0
\(955\) 64.0000i 2.07099i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 5.65685 9.79796i 0.182669 0.316393i
\(960\) 0 0
\(961\) −13.5000 23.3827i −0.435484 0.754280i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −34.2929 + 19.7990i −1.10393 + 0.637352i
\(966\) 0 0
\(967\) −43.3013 25.0000i −1.39247 0.803946i −0.398886 0.917000i \(-0.630603\pi\)
−0.993589 + 0.113055i \(0.963936\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −19.7990 −0.635380 −0.317690 0.948195i \(-0.602907\pi\)
−0.317690 + 0.948195i \(0.602907\pi\)
\(972\) 0 0
\(973\) −20.0000 −0.641171
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −39.1918 22.6274i −1.25386 0.723915i −0.281984 0.959419i \(-0.590993\pi\)
−0.971873 + 0.235504i \(0.924326\pi\)
\(978\) 0 0
\(979\) −41.5692 + 24.0000i −1.32856 + 0.767043i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −14.1421 24.4949i −0.451064 0.781266i 0.547388 0.836879i \(-0.315622\pi\)
−0.998452 + 0.0556128i \(0.982289\pi\)
\(984\) 0 0
\(985\) −28.0000 + 48.4974i −0.892154 + 1.54526i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 11.3137i 0.359755i
\(990\) 0 0
\(991\) 18.0000i 0.571789i 0.958261 + 0.285894i \(0.0922907\pi\)
−0.958261 + 0.285894i \(0.907709\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.82843 4.89898i 0.0896672 0.155308i
\(996\) 0 0
\(997\) −19.0000 32.9090i −0.601736 1.04224i −0.992558 0.121771i \(-0.961143\pi\)
0.390822 0.920466i \(-0.372191\pi\)
\(998\) 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 2592.2.s.e.863.2 8
3.2 odd 2 inner 2592.2.s.e.863.4 8
4.3 odd 2 inner 2592.2.s.e.863.1 8
9.2 odd 6 inner 2592.2.s.e.1727.1 8
9.4 even 3 96.2.c.a.95.2 yes 4
9.5 odd 6 96.2.c.a.95.4 yes 4
9.7 even 3 inner 2592.2.s.e.1727.3 8
12.11 even 2 inner 2592.2.s.e.863.3 8
36.7 odd 6 inner 2592.2.s.e.1727.4 8
36.11 even 6 inner 2592.2.s.e.1727.2 8
36.23 even 6 96.2.c.a.95.1 4
36.31 odd 6 96.2.c.a.95.3 yes 4
45.4 even 6 2400.2.h.c.1151.3 4
45.13 odd 12 2400.2.o.a.2399.2 4
45.14 odd 6 2400.2.h.c.1151.1 4
45.22 odd 12 2400.2.o.h.2399.4 4
45.23 even 12 2400.2.o.a.2399.3 4
45.32 even 12 2400.2.o.h.2399.1 4
72.5 odd 6 192.2.c.b.191.1 4
72.13 even 6 192.2.c.b.191.3 4
72.59 even 6 192.2.c.b.191.4 4
72.67 odd 6 192.2.c.b.191.2 4
144.5 odd 12 768.2.f.g.383.2 4
144.13 even 12 768.2.f.a.383.2 4
144.59 even 12 768.2.f.a.383.4 4
144.67 odd 12 768.2.f.g.383.4 4
144.77 odd 12 768.2.f.a.383.3 4
144.85 even 12 768.2.f.g.383.3 4
144.131 even 12 768.2.f.g.383.1 4
144.139 odd 12 768.2.f.a.383.1 4
180.23 odd 12 2400.2.o.h.2399.2 4
180.59 even 6 2400.2.h.c.1151.4 4
180.67 even 12 2400.2.o.a.2399.1 4
180.103 even 12 2400.2.o.h.2399.3 4
180.139 odd 6 2400.2.h.c.1151.2 4
180.167 odd 12 2400.2.o.a.2399.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
96.2.c.a.95.1 4 36.23 even 6
96.2.c.a.95.2 yes 4 9.4 even 3
96.2.c.a.95.3 yes 4 36.31 odd 6
96.2.c.a.95.4 yes 4 9.5 odd 6
192.2.c.b.191.1 4 72.5 odd 6
192.2.c.b.191.2 4 72.67 odd 6
192.2.c.b.191.3 4 72.13 even 6
192.2.c.b.191.4 4 72.59 even 6
768.2.f.a.383.1 4 144.139 odd 12
768.2.f.a.383.2 4 144.13 even 12
768.2.f.a.383.3 4 144.77 odd 12
768.2.f.a.383.4 4 144.59 even 12
768.2.f.g.383.1 4 144.131 even 12
768.2.f.g.383.2 4 144.5 odd 12
768.2.f.g.383.3 4 144.85 even 12
768.2.f.g.383.4 4 144.67 odd 12
2400.2.h.c.1151.1 4 45.14 odd 6
2400.2.h.c.1151.2 4 180.139 odd 6
2400.2.h.c.1151.3 4 45.4 even 6
2400.2.h.c.1151.4 4 180.59 even 6
2400.2.o.a.2399.1 4 180.67 even 12
2400.2.o.a.2399.2 4 45.13 odd 12
2400.2.o.a.2399.3 4 45.23 even 12
2400.2.o.a.2399.4 4 180.167 odd 12
2400.2.o.h.2399.1 4 45.32 even 12
2400.2.o.h.2399.2 4 180.23 odd 12
2400.2.o.h.2399.3 4 180.103 even 12
2400.2.o.h.2399.4 4 45.22 odd 12
2592.2.s.e.863.1 8 4.3 odd 2 inner
2592.2.s.e.863.2 8 1.1 even 1 trivial
2592.2.s.e.863.3 8 12.11 even 2 inner
2592.2.s.e.863.4 8 3.2 odd 2 inner
2592.2.s.e.1727.1 8 9.2 odd 6 inner
2592.2.s.e.1727.2 8 36.11 even 6 inner
2592.2.s.e.1727.3 8 9.7 even 3 inner
2592.2.s.e.1727.4 8 36.7 odd 6 inner