Properties

Label 864.2.d.c.433.7
Level $864$
Weight $2$
Character 864.433
Analytic conductor $6.899$
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] = [864,2,Mod(433,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("864.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.d (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: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.629407744.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 2x^{4} - 8x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 216)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 433.7
Root \(-1.38255 + 0.297594i\) of defining polynomial
Character \(\chi\) \(=\) 864.433
Dual form 864.2.d.c.433.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.36028i q^{5} +2.64575 q^{7} +O(q^{10})\) \(q+3.36028i q^{5} +2.64575 q^{7} +2.16991i q^{11} -4.64575i q^{13} +4.55066 q^{17} +6.29150i q^{19} -0.979531 q^{23} -6.29150 q^{25} +4.33981i q^{29} -2.00000 q^{31} +8.89047i q^{35} +1.35425i q^{37} -11.0604 q^{41} -3.29150i q^{43} +10.0808 q^{47} +4.33981i q^{53} -7.29150 q^{55} +11.2712i q^{59} +1.93725i q^{61} +15.6110 q^{65} -3.00000i q^{67} +11.5830 q^{73} +5.74103i q^{77} +8.64575 q^{79} -2.38075i q^{83} +15.2915i q^{85} -4.55066 q^{89} -12.2915i q^{91} -21.1412 q^{95} +2.29150 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{25} - 16 q^{31} - 16 q^{55} + 8 q^{73} + 48 q^{79} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.36028i 1.50276i 0.659867 + 0.751382i \(0.270612\pi\)
−0.659867 + 0.751382i \(0.729388\pi\)
\(6\) 0 0
\(7\) 2.64575 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.16991i 0.654252i 0.944981 + 0.327126i \(0.106080\pi\)
−0.944981 + 0.327126i \(0.893920\pi\)
\(12\) 0 0
\(13\) − 4.64575i − 1.28850i −0.764815 0.644250i \(-0.777170\pi\)
0.764815 0.644250i \(-0.222830\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.55066 1.10370 0.551848 0.833944i \(-0.313923\pi\)
0.551848 + 0.833944i \(0.313923\pi\)
\(18\) 0 0
\(19\) 6.29150i 1.44337i 0.692222 + 0.721685i \(0.256632\pi\)
−0.692222 + 0.721685i \(0.743368\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −0.979531 −0.204246 −0.102123 0.994772i \(-0.532564\pi\)
−0.102123 + 0.994772i \(0.532564\pi\)
\(24\) 0 0
\(25\) −6.29150 −1.25830
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.33981i 0.805883i 0.915226 + 0.402942i \(0.132012\pi\)
−0.915226 + 0.402942i \(0.867988\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 8.89047i 1.50276i
\(36\) 0 0
\(37\) 1.35425i 0.222637i 0.993785 + 0.111319i \(0.0355074\pi\)
−0.993785 + 0.111319i \(0.964493\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.0604 −1.72734 −0.863671 0.504057i \(-0.831840\pi\)
−0.863671 + 0.504057i \(0.831840\pi\)
\(42\) 0 0
\(43\) − 3.29150i − 0.501949i −0.967994 0.250975i \(-0.919249\pi\)
0.967994 0.250975i \(-0.0807511\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.0808 1.47044 0.735222 0.677827i \(-0.237078\pi\)
0.735222 + 0.677827i \(0.237078\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.33981i 0.596119i 0.954547 + 0.298060i \(0.0963394\pi\)
−0.954547 + 0.298060i \(0.903661\pi\)
\(54\) 0 0
\(55\) −7.29150 −0.983186
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.2712i 1.46739i 0.679480 + 0.733694i \(0.262206\pi\)
−0.679480 + 0.733694i \(0.737794\pi\)
\(60\) 0 0
\(61\) 1.93725i 0.248040i 0.992280 + 0.124020i \(0.0395787\pi\)
−0.992280 + 0.124020i \(0.960421\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 15.6110 1.93631
\(66\) 0 0
\(67\) − 3.00000i − 0.366508i −0.983066 0.183254i \(-0.941337\pi\)
0.983066 0.183254i \(-0.0586631\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 11.5830 1.35569 0.677844 0.735206i \(-0.262914\pi\)
0.677844 + 0.735206i \(0.262914\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.74103i 0.654252i
\(78\) 0 0
\(79\) 8.64575 0.972723 0.486362 0.873758i \(-0.338324\pi\)
0.486362 + 0.873758i \(0.338324\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 2.38075i − 0.261321i −0.991427 0.130661i \(-0.958290\pi\)
0.991427 0.130661i \(-0.0417099\pi\)
\(84\) 0 0
\(85\) 15.2915i 1.65860i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.55066 −0.482369 −0.241184 0.970479i \(-0.577536\pi\)
−0.241184 + 0.970479i \(0.577536\pi\)
\(90\) 0 0
\(91\) − 12.2915i − 1.28850i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −21.1412 −2.16904
\(96\) 0 0
\(97\) 2.29150 0.232667 0.116333 0.993210i \(-0.462886\pi\)
0.116333 + 0.993210i \(0.462886\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 6.72057i − 0.668721i −0.942445 0.334361i \(-0.891480\pi\)
0.942445 0.334361i \(-0.108520\pi\)
\(102\) 0 0
\(103\) 5.35425 0.527570 0.263785 0.964582i \(-0.415029\pi\)
0.263785 + 0.964582i \(0.415029\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 17.9918i − 1.73933i −0.493640 0.869666i \(-0.664334\pi\)
0.493640 0.869666i \(-0.335666\pi\)
\(108\) 0 0
\(109\) − 15.2915i − 1.46466i −0.680950 0.732330i \(-0.738433\pi\)
0.680950 0.732330i \(-0.261567\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.50972 −0.612383 −0.306192 0.951970i \(-0.599055\pi\)
−0.306192 + 0.951970i \(0.599055\pi\)
\(114\) 0 0
\(115\) − 3.29150i − 0.306934i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 12.0399 1.10370
\(120\) 0 0
\(121\) 6.29150 0.571955
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 4.33981i − 0.388165i
\(126\) 0 0
\(127\) −5.29150 −0.469545 −0.234772 0.972050i \(-0.575435\pi\)
−0.234772 + 0.972050i \(0.575435\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 15.4002i − 1.34552i −0.739860 0.672761i \(-0.765108\pi\)
0.739860 0.672761i \(-0.234892\pi\)
\(132\) 0 0
\(133\) 16.6458i 1.44337i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.50972 0.556163 0.278082 0.960557i \(-0.410301\pi\)
0.278082 + 0.960557i \(0.410301\pi\)
\(138\) 0 0
\(139\) 12.2915i 1.04255i 0.853388 + 0.521276i \(0.174544\pi\)
−0.853388 + 0.521276i \(0.825456\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.0808 0.843003
\(144\) 0 0
\(145\) −14.5830 −1.21105
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 8.67963i − 0.711063i −0.934664 0.355531i \(-0.884300\pi\)
0.934664 0.355531i \(-0.115700\pi\)
\(150\) 0 0
\(151\) −12.6458 −1.02910 −0.514548 0.857461i \(-0.672040\pi\)
−0.514548 + 0.857461i \(0.672040\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 6.72057i − 0.539809i
\(156\) 0 0
\(157\) − 8.70850i − 0.695014i −0.937677 0.347507i \(-0.887028\pi\)
0.937677 0.347507i \(-0.112972\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −2.59160 −0.204246
\(162\) 0 0
\(163\) − 6.29150i − 0.492789i −0.969170 0.246394i \(-0.920754\pi\)
0.969170 0.246394i \(-0.0792458\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −19.1822 −1.48436 −0.742180 0.670200i \(-0.766208\pi\)
−0.742180 + 0.670200i \(0.766208\pi\)
\(168\) 0 0
\(169\) −8.58301 −0.660231
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 15.8219i − 1.20292i −0.798905 0.601458i \(-0.794587\pi\)
0.798905 0.601458i \(-0.205413\pi\)
\(174\) 0 0
\(175\) −16.6458 −1.25830
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.76150i 0.355891i 0.984040 + 0.177946i \(0.0569452\pi\)
−0.984040 + 0.177946i \(0.943055\pi\)
\(180\) 0 0
\(181\) 16.6458i 1.23727i 0.785679 + 0.618634i \(0.212314\pi\)
−0.785679 + 0.618634i \(0.787686\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.55066 −0.334571
\(186\) 0 0
\(187\) 9.87451i 0.722096i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.12179 −0.587672 −0.293836 0.955856i \(-0.594932\pi\)
−0.293836 + 0.955856i \(0.594932\pi\)
\(192\) 0 0
\(193\) −4.29150 −0.308909 −0.154455 0.988000i \(-0.549362\pi\)
−0.154455 + 0.988000i \(0.549362\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 7.70010i − 0.548609i −0.961643 0.274305i \(-0.911552\pi\)
0.961643 0.274305i \(-0.0884477\pi\)
\(198\) 0 0
\(199\) 18.5203 1.31287 0.656433 0.754384i \(-0.272064\pi\)
0.656433 + 0.754384i \(0.272064\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 11.4821i 0.805883i
\(204\) 0 0
\(205\) − 37.1660i − 2.59579i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −13.6520 −0.944327
\(210\) 0 0
\(211\) − 0.291503i − 0.0200679i −0.999950 0.0100339i \(-0.996806\pi\)
0.999950 0.0100339i \(-0.00319395\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 11.0604 0.754312
\(216\) 0 0
\(217\) −5.29150 −0.359211
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 21.1412i − 1.42211i
\(222\) 0 0
\(223\) 10.0000 0.669650 0.334825 0.942280i \(-0.391323\pi\)
0.334825 + 0.942280i \(0.391323\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 13.4411i − 0.892119i −0.895003 0.446060i \(-0.852827\pi\)
0.895003 0.446060i \(-0.147173\pi\)
\(228\) 0 0
\(229\) − 13.1660i − 0.870034i −0.900422 0.435017i \(-0.856742\pi\)
0.900422 0.435017i \(-0.143258\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.10132 0.596247 0.298124 0.954527i \(-0.403639\pi\)
0.298124 + 0.954527i \(0.403639\pi\)
\(234\) 0 0
\(235\) 33.8745i 2.20973i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.1617 1.30415 0.652076 0.758154i \(-0.273898\pi\)
0.652076 + 0.758154i \(0.273898\pi\)
\(240\) 0 0
\(241\) 26.8745 1.73114 0.865570 0.500789i \(-0.166957\pi\)
0.865570 + 0.500789i \(0.166957\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 29.2288 1.85978
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 24.5015i − 1.54652i −0.634088 0.773261i \(-0.718624\pi\)
0.634088 0.773261i \(-0.281376\pi\)
\(252\) 0 0
\(253\) − 2.12549i − 0.133629i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −20.1617 −1.25765 −0.628826 0.777546i \(-0.716464\pi\)
−0.628826 + 0.777546i \(0.716464\pi\)
\(258\) 0 0
\(259\) 3.58301i 0.222637i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −1.95906 −0.120801 −0.0604005 0.998174i \(-0.519238\pi\)
−0.0604005 + 0.998174i \(0.519238\pi\)
\(264\) 0 0
\(265\) −14.5830 −0.895827
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 5.74103i − 0.350037i −0.984565 0.175019i \(-0.944001\pi\)
0.984565 0.175019i \(-0.0559985\pi\)
\(270\) 0 0
\(271\) 21.2288 1.28956 0.644778 0.764370i \(-0.276950\pi\)
0.644778 + 0.764370i \(0.276950\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 13.6520i − 0.823245i
\(276\) 0 0
\(277\) 8.70850i 0.523243i 0.965171 + 0.261621i \(0.0842572\pi\)
−0.965171 + 0.261621i \(0.915743\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 20.1617 1.20275 0.601373 0.798968i \(-0.294621\pi\)
0.601373 + 0.798968i \(0.294621\pi\)
\(282\) 0 0
\(283\) − 27.2915i − 1.62231i −0.584830 0.811156i \(-0.698839\pi\)
0.584830 0.811156i \(-0.301161\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −29.2630 −1.72734
\(288\) 0 0
\(289\) 3.70850 0.218147
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 14.4207i 0.842464i 0.906953 + 0.421232i \(0.138402\pi\)
−0.906953 + 0.421232i \(0.861598\pi\)
\(294\) 0 0
\(295\) −37.8745 −2.20514
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.55066i 0.263171i
\(300\) 0 0
\(301\) − 8.70850i − 0.501949i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.50972 −0.372746
\(306\) 0 0
\(307\) − 9.87451i − 0.563568i −0.959478 0.281784i \(-0.909074\pi\)
0.959478 0.281784i \(-0.0909261\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −0.979531 −0.0555441 −0.0277721 0.999614i \(-0.508841\pi\)
−0.0277721 + 0.999614i \(0.508841\pi\)
\(312\) 0 0
\(313\) 14.8745 0.840757 0.420378 0.907349i \(-0.361897\pi\)
0.420378 + 0.907349i \(0.361897\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 20.5834i 1.15608i 0.816009 + 0.578039i \(0.196182\pi\)
−0.816009 + 0.578039i \(0.803818\pi\)
\(318\) 0 0
\(319\) −9.41699 −0.527250
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 28.6305i 1.59304i
\(324\) 0 0
\(325\) 29.2288i 1.62132i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 26.6714 1.47044
\(330\) 0 0
\(331\) 5.70850i 0.313767i 0.987617 + 0.156884i \(0.0501448\pi\)
−0.987617 + 0.156884i \(0.949855\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 10.0808 0.550776
\(336\) 0 0
\(337\) 2.29150 0.124826 0.0624131 0.998050i \(-0.480120\pi\)
0.0624131 + 0.998050i \(0.480120\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 4.33981i − 0.235014i
\(342\) 0 0
\(343\) −18.5203 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 0.421689i − 0.0226375i −0.999936 0.0113187i \(-0.996397\pi\)
0.999936 0.0113187i \(-0.00360294\pi\)
\(348\) 0 0
\(349\) − 10.6458i − 0.569854i −0.958549 0.284927i \(-0.908031\pi\)
0.958549 0.284927i \(-0.0919694\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.0194 −0.692955 −0.346478 0.938058i \(-0.612622\pi\)
−0.346478 + 0.938058i \(0.612622\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −21.1412 −1.11579 −0.557896 0.829911i \(-0.688391\pi\)
−0.557896 + 0.829911i \(0.688391\pi\)
\(360\) 0 0
\(361\) −20.5830 −1.08332
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 38.9222i 2.03728i
\(366\) 0 0
\(367\) −4.52026 −0.235956 −0.117978 0.993016i \(-0.537641\pi\)
−0.117978 + 0.993016i \(0.537641\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.4821i 0.596119i
\(372\) 0 0
\(373\) 38.5203i 1.99450i 0.0740880 + 0.997252i \(0.476395\pi\)
−0.0740880 + 0.997252i \(0.523605\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 20.1617 1.03838
\(378\) 0 0
\(379\) − 21.5830i − 1.10864i −0.832302 0.554322i \(-0.812978\pi\)
0.832302 0.554322i \(-0.187022\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 20.1617 1.03021 0.515107 0.857126i \(-0.327752\pi\)
0.515107 + 0.857126i \(0.327752\pi\)
\(384\) 0 0
\(385\) −19.2915 −0.983186
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.31935i 0.269702i 0.990866 + 0.134851i \(0.0430555\pi\)
−0.990866 + 0.134851i \(0.956944\pi\)
\(390\) 0 0
\(391\) −4.45751 −0.225426
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 29.0522i 1.46177i
\(396\) 0 0
\(397\) − 15.2915i − 0.767459i −0.923446 0.383729i \(-0.874640\pi\)
0.923446 0.383729i \(-0.125360\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.0194 0.650160 0.325080 0.945687i \(-0.394609\pi\)
0.325080 + 0.945687i \(0.394609\pi\)
\(402\) 0 0
\(403\) 9.29150i 0.462843i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.93859 −0.145661
\(408\) 0 0
\(409\) 8.29150 0.409988 0.204994 0.978763i \(-0.434282\pi\)
0.204994 + 0.978763i \(0.434282\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 29.8209i 1.46739i
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.16991i 0.106007i 0.998594 + 0.0530035i \(0.0168794\pi\)
−0.998594 + 0.0530035i \(0.983121\pi\)
\(420\) 0 0
\(421\) 22.6458i 1.10369i 0.833948 + 0.551843i \(0.186075\pi\)
−0.833948 + 0.551843i \(0.813925\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −28.6305 −1.38878
\(426\) 0 0
\(427\) 5.12549i 0.248040i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 30.2425 1.45673 0.728366 0.685188i \(-0.240280\pi\)
0.728366 + 0.685188i \(0.240280\pi\)
\(432\) 0 0
\(433\) −0.125492 −0.00603077 −0.00301538 0.999995i \(-0.500960\pi\)
−0.00301538 + 0.999995i \(0.500960\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 6.16272i − 0.294803i
\(438\) 0 0
\(439\) 11.1660 0.532925 0.266462 0.963845i \(-0.414145\pi\)
0.266462 + 0.963845i \(0.414145\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 28.8413i 1.37029i 0.728405 + 0.685146i \(0.240262\pi\)
−0.728405 + 0.685146i \(0.759738\pi\)
\(444\) 0 0
\(445\) − 15.2915i − 0.724887i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −19.5292 −0.921638 −0.460819 0.887494i \(-0.652444\pi\)
−0.460819 + 0.887494i \(0.652444\pi\)
\(450\) 0 0
\(451\) − 24.0000i − 1.13012i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 41.3029 1.93631
\(456\) 0 0
\(457\) 23.8745 1.11680 0.558401 0.829571i \(-0.311415\pi\)
0.558401 + 0.829571i \(0.311415\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 18.7605i − 0.873763i −0.899519 0.436881i \(-0.856083\pi\)
0.899519 0.436881i \(-0.143917\pi\)
\(462\) 0 0
\(463\) −31.2288 −1.45132 −0.725662 0.688052i \(-0.758466\pi\)
−0.725662 + 0.688052i \(0.758466\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0.210845i 0.00975672i 0.999988 + 0.00487836i \(0.00155284\pi\)
−0.999988 + 0.00487836i \(0.998447\pi\)
\(468\) 0 0
\(469\) − 7.93725i − 0.366508i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.14226 0.328401
\(474\) 0 0
\(475\) − 39.5830i − 1.81619i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −35.1402 −1.60560 −0.802798 0.596250i \(-0.796657\pi\)
−0.802798 + 0.596250i \(0.796657\pi\)
\(480\) 0 0
\(481\) 6.29150 0.286868
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.70010i 0.349643i
\(486\) 0 0
\(487\) −6.06275 −0.274729 −0.137365 0.990521i \(-0.543863\pi\)
−0.137365 + 0.990521i \(0.543863\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.12897i 0.186338i 0.995650 + 0.0931689i \(0.0296997\pi\)
−0.995650 + 0.0931689i \(0.970300\pi\)
\(492\) 0 0
\(493\) 19.7490i 0.889451i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 43.7490i 1.95847i 0.202717 + 0.979237i \(0.435023\pi\)
−0.202717 + 0.979237i \(0.564977\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −2.93859 −0.131025 −0.0655127 0.997852i \(-0.520868\pi\)
−0.0655127 + 0.997852i \(0.520868\pi\)
\(504\) 0 0
\(505\) 22.5830 1.00493
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 33.0450i − 1.46469i −0.680932 0.732347i \(-0.738425\pi\)
0.680932 0.732347i \(-0.261575\pi\)
\(510\) 0 0
\(511\) 30.6458 1.35569
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 17.9918i 0.792813i
\(516\) 0 0
\(517\) 21.8745i 0.962040i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −13.6520 −0.598104 −0.299052 0.954237i \(-0.596670\pi\)
−0.299052 + 0.954237i \(0.596670\pi\)
\(522\) 0 0
\(523\) − 0.874508i − 0.0382396i −0.999817 0.0191198i \(-0.993914\pi\)
0.999817 0.0191198i \(-0.00608639\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −9.10132 −0.396460
\(528\) 0 0
\(529\) −22.0405 −0.958283
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 51.3838i 2.22568i
\(534\) 0 0
\(535\) 60.4575 2.61381
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) − 26.5203i − 1.14019i −0.821577 0.570097i \(-0.806905\pi\)
0.821577 0.570097i \(-0.193095\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 51.3838 2.20104
\(546\) 0 0
\(547\) − 9.58301i − 0.409740i −0.978789 0.204870i \(-0.934323\pi\)
0.978789 0.204870i \(-0.0656771\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −27.3040 −1.16319
\(552\) 0 0
\(553\) 22.8745 0.972723
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.40122i 0.0593716i 0.999559 + 0.0296858i \(0.00945067\pi\)
−0.999559 + 0.0296858i \(0.990549\pi\)
\(558\) 0 0
\(559\) −15.2915 −0.646762
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 41.8608i 1.76422i 0.471042 + 0.882111i \(0.343878\pi\)
−0.471042 + 0.882111i \(0.656122\pi\)
\(564\) 0 0
\(565\) − 21.8745i − 0.920267i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.6714 −1.11812 −0.559062 0.829126i \(-0.688839\pi\)
−0.559062 + 0.829126i \(0.688839\pi\)
\(570\) 0 0
\(571\) − 5.70850i − 0.238893i −0.992841 0.119447i \(-0.961888\pi\)
0.992841 0.119447i \(-0.0381120\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.16272 0.257003
\(576\) 0 0
\(577\) 23.5830 0.981773 0.490887 0.871223i \(-0.336673\pi\)
0.490887 + 0.871223i \(0.336673\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 6.29888i − 0.261321i
\(582\) 0 0
\(583\) −9.41699 −0.390012
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 21.9099i − 0.904319i −0.891937 0.452160i \(-0.850654\pi\)
0.891937 0.452160i \(-0.149346\pi\)
\(588\) 0 0
\(589\) − 12.5830i − 0.518474i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 33.1811 1.36259 0.681293 0.732011i \(-0.261418\pi\)
0.681293 + 0.732011i \(0.261418\pi\)
\(594\) 0 0
\(595\) 40.4575i 1.65860i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 35.1402 1.43579 0.717895 0.696151i \(-0.245106\pi\)
0.717895 + 0.696151i \(0.245106\pi\)
\(600\) 0 0
\(601\) −31.8745 −1.30019 −0.650094 0.759854i \(-0.725271\pi\)
−0.650094 + 0.759854i \(0.725271\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 21.1412i 0.859513i
\(606\) 0 0
\(607\) −33.3542 −1.35381 −0.676904 0.736072i \(-0.736679\pi\)
−0.676904 + 0.736072i \(0.736679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 46.8331i − 1.89467i
\(612\) 0 0
\(613\) − 21.1033i − 0.852353i −0.904640 0.426176i \(-0.859860\pi\)
0.904640 0.426176i \(-0.140140\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 17.5701 0.707346 0.353673 0.935369i \(-0.384933\pi\)
0.353673 + 0.935369i \(0.384933\pi\)
\(618\) 0 0
\(619\) 39.0000i 1.56754i 0.621050 + 0.783771i \(0.286706\pi\)
−0.621050 + 0.783771i \(0.713294\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −12.0399 −0.482369
\(624\) 0 0
\(625\) −16.8745 −0.674980
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.16272i 0.245724i
\(630\) 0 0
\(631\) −19.2288 −0.765485 −0.382742 0.923855i \(-0.625020\pi\)
−0.382742 + 0.923855i \(0.625020\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 17.7809i − 0.705615i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −22.1208 −0.873718 −0.436859 0.899530i \(-0.643909\pi\)
−0.436859 + 0.899530i \(0.643909\pi\)
\(642\) 0 0
\(643\) − 27.2915i − 1.07627i −0.842858 0.538136i \(-0.819129\pi\)
0.842858 0.538136i \(-0.180871\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 33.1811 1.30449 0.652243 0.758010i \(-0.273828\pi\)
0.652243 + 0.758010i \(0.273828\pi\)
\(648\) 0 0
\(649\) −24.4575 −0.960041
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 24.5015i 0.958818i 0.877592 + 0.479409i \(0.159149\pi\)
−0.877592 + 0.479409i \(0.840851\pi\)
\(654\) 0 0
\(655\) 51.7490 2.02200
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 11.4821i − 0.447278i −0.974672 0.223639i \(-0.928206\pi\)
0.974672 0.223639i \(-0.0717937\pi\)
\(660\) 0 0
\(661\) − 14.5203i − 0.564773i −0.959301 0.282386i \(-0.908874\pi\)
0.959301 0.282386i \(-0.0911260\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −55.9344 −2.16904
\(666\) 0 0
\(667\) − 4.25098i − 0.164599i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −4.20366 −0.162281
\(672\) 0 0
\(673\) −40.8745 −1.57560 −0.787798 0.615933i \(-0.788779\pi\)
−0.787798 + 0.615933i \(0.788779\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 31.7799i − 1.22140i −0.791861 0.610701i \(-0.790888\pi\)
0.791861 0.610701i \(-0.209112\pi\)
\(678\) 0 0
\(679\) 6.06275 0.232667
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 17.7809i 0.680369i 0.940359 + 0.340185i \(0.110490\pi\)
−0.940359 + 0.340185i \(0.889510\pi\)
\(684\) 0 0
\(685\) 21.8745i 0.835782i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 20.1617 0.768100
\(690\) 0 0
\(691\) 27.2915i 1.03822i 0.854708 + 0.519109i \(0.173736\pi\)
−0.854708 + 0.519109i \(0.826264\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −41.3029 −1.56671
\(696\) 0 0
\(697\) −50.3320 −1.90646
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 16.8014i − 0.634581i −0.948328 0.317290i \(-0.897227\pi\)
0.948328 0.317290i \(-0.102773\pi\)
\(702\) 0 0
\(703\) −8.52026 −0.321348
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 17.7809i − 0.668721i
\(708\) 0 0
\(709\) − 3.47974i − 0.130684i −0.997863 0.0653422i \(-0.979186\pi\)
0.997863 0.0653422i \(-0.0208139\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.95906 0.0733675
\(714\) 0 0
\(715\) 33.8745i 1.26683i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −51.3838 −1.91629 −0.958146 0.286281i \(-0.907581\pi\)
−0.958146 + 0.286281i \(0.907581\pi\)
\(720\) 0 0
\(721\) 14.1660 0.527570
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 27.3040i − 1.01404i
\(726\) 0 0
\(727\) −44.5830 −1.65349 −0.826746 0.562575i \(-0.809811\pi\)
−0.826746 + 0.562575i \(0.809811\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 14.9785i − 0.554000i
\(732\) 0 0
\(733\) − 28.4575i − 1.05110i −0.850762 0.525551i \(-0.823859\pi\)
0.850762 0.525551i \(-0.176141\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.50972 0.239789
\(738\) 0 0
\(739\) 40.4575i 1.48825i 0.668038 + 0.744127i \(0.267134\pi\)
−0.668038 + 0.744127i \(0.732866\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −25.0594 −0.919339 −0.459669 0.888090i \(-0.652032\pi\)
−0.459669 + 0.888090i \(0.652032\pi\)
\(744\) 0 0
\(745\) 29.1660 1.06856
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 47.6018i − 1.73933i
\(750\) 0 0
\(751\) −24.0627 −0.878062 −0.439031 0.898472i \(-0.644678\pi\)
−0.439031 + 0.898472i \(0.644678\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 42.4933i − 1.54649i
\(756\) 0 0
\(757\) 13.9373i 0.506558i 0.967393 + 0.253279i \(0.0815091\pi\)
−0.967393 + 0.253279i \(0.918491\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −9.73385 −0.352852 −0.176426 0.984314i \(-0.556454\pi\)
−0.176426 + 0.984314i \(0.556454\pi\)
\(762\) 0 0
\(763\) − 40.4575i − 1.46466i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 52.3633 1.89073
\(768\) 0 0
\(769\) 36.1660 1.30418 0.652090 0.758142i \(-0.273892\pi\)
0.652090 + 0.758142i \(0.273892\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 6.29888i 0.226555i 0.993563 + 0.113277i \(0.0361349\pi\)
−0.993563 + 0.113277i \(0.963865\pi\)
\(774\) 0 0
\(775\) 12.5830 0.451995
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 69.5864i − 2.49319i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 29.2630 1.04444
\(786\) 0 0
\(787\) − 50.6235i − 1.80453i −0.431178 0.902267i \(-0.641902\pi\)
0.431178 0.902267i \(-0.358098\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −17.2231 −0.612383
\(792\) 0 0
\(793\) 9.00000 0.319599
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 50.9621i − 1.80517i −0.430512 0.902585i \(-0.641667\pi\)
0.430512 0.902585i \(-0.358333\pi\)
\(798\) 0 0
\(799\) 45.8745 1.62292
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 25.1340i 0.886961i
\(804\) 0 0
\(805\) − 8.70850i − 0.306934i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 20.7085i 0.727174i 0.931560 + 0.363587i \(0.118448\pi\)
−0.931560 + 0.363587i \(0.881552\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 21.1412 0.740545
\(816\) 0 0
\(817\) 20.7085 0.724499
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 49.9826i − 1.74440i −0.489146 0.872202i \(-0.662692\pi\)
0.489146 0.872202i \(-0.337308\pi\)
\(822\) 0 0
\(823\) 20.0627 0.699343 0.349672 0.936872i \(-0.386293\pi\)
0.349672 + 0.936872i \(0.386293\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 49.6356i 1.72600i 0.505206 + 0.862999i \(0.331417\pi\)
−0.505206 + 0.862999i \(0.668583\pi\)
\(828\) 0 0
\(829\) 6.77124i 0.235175i 0.993063 + 0.117588i \(0.0375161\pi\)
−0.993063 + 0.117588i \(0.962484\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) − 64.4575i − 2.23064i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 31.2221 1.07791 0.538953 0.842336i \(-0.318820\pi\)
0.538953 + 0.842336i \(0.318820\pi\)
\(840\) 0 0
\(841\) 10.1660 0.350552
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 28.8413i − 0.992172i
\(846\) 0 0
\(847\) 16.6458 0.571955
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 1.32653i − 0.0454728i
\(852\) 0 0
\(853\) 29.8118i 1.02074i 0.859956 + 0.510368i \(0.170491\pi\)
−0.859956 + 0.510368i \(0.829509\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −40.3234 −1.37742 −0.688711 0.725036i \(-0.741823\pi\)
−0.688711 + 0.725036i \(0.741823\pi\)
\(858\) 0 0
\(859\) − 26.4170i − 0.901336i −0.892692 0.450668i \(-0.851186\pi\)
0.892692 0.450668i \(-0.148814\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 36.1197 1.22953 0.614765 0.788710i \(-0.289251\pi\)
0.614765 + 0.788710i \(0.289251\pi\)
\(864\) 0 0
\(865\) 53.1660 1.80770
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 18.7605i 0.636406i
\(870\) 0 0
\(871\) −13.9373 −0.472246
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 11.4821i − 0.388165i
\(876\) 0 0
\(877\) 48.3948i 1.63418i 0.576514 + 0.817088i \(0.304413\pi\)
−0.576514 + 0.817088i \(0.695587\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −31.8546 −1.07321 −0.536605 0.843834i \(-0.680293\pi\)
−0.536605 + 0.843834i \(0.680293\pi\)
\(882\) 0 0
\(883\) − 16.1660i − 0.544030i −0.962293 0.272015i \(-0.912310\pi\)
0.962293 0.272015i \(-0.0876900\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −7.14226 −0.239813 −0.119907 0.992785i \(-0.538260\pi\)
−0.119907 + 0.992785i \(0.538260\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 63.4237i 2.12239i
\(894\) 0 0
\(895\) −16.0000 −0.534821
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 8.67963i − 0.289482i
\(900\) 0 0
\(901\) 19.7490i 0.657935i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −55.9344 −1.85932
\(906\) 0 0
\(907\) 56.0405i 1.86079i 0.366552 + 0.930397i \(0.380538\pi\)
−0.366552 + 0.930397i \(0.619462\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −47.4657 −1.57261 −0.786304 0.617840i \(-0.788008\pi\)
−0.786304 + 0.617840i \(0.788008\pi\)
\(912\) 0 0
\(913\) 5.16601 0.170970
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 40.7451i − 1.34552i
\(918\) 0 0
\(919\) −20.5830 −0.678971 −0.339485 0.940611i \(-0.610253\pi\)
−0.339485 + 0.940611i \(0.610253\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) − 8.52026i − 0.280144i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 16.9376 0.555704 0.277852 0.960624i \(-0.410378\pi\)
0.277852 + 0.960624i \(0.410378\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −33.1811 −1.08514
\(936\) 0 0
\(937\) −9.12549 −0.298117 −0.149058 0.988828i \(-0.547624\pi\)
−0.149058 + 0.988828i \(0.547624\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.5629i 0.702931i 0.936201 + 0.351466i \(0.114317\pi\)
−0.936201 + 0.351466i \(0.885683\pi\)
\(942\) 0 0
\(943\) 10.8340 0.352803
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 31.0112i − 1.00773i −0.863782 0.503865i \(-0.831911\pi\)
0.863782 0.503865i \(-0.168089\pi\)
\(948\) 0 0
\(949\) − 53.8118i − 1.74680i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 40.9559 1.32669 0.663346 0.748312i \(-0.269136\pi\)
0.663346 + 0.748312i \(0.269136\pi\)
\(954\) 0 0
\(955\) − 27.2915i − 0.883132i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 17.2231 0.556163
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 14.4207i − 0.464218i
\(966\) 0 0
\(967\) 38.0627 1.22402 0.612008 0.790852i \(-0.290362\pi\)
0.612008 + 0.790852i \(0.290362\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 17.9918i − 0.577384i −0.957422 0.288692i \(-0.906780\pi\)
0.957422 0.288692i \(-0.0932204\pi\)
\(972\) 0 0
\(973\) 32.5203i 1.04255i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −29.2630 −0.936207 −0.468103 0.883674i \(-0.655063\pi\)
−0.468103 + 0.883674i \(0.655063\pi\)
\(978\) 0 0
\(979\) − 9.87451i − 0.315591i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 28.2835 0.902103 0.451052 0.892498i \(-0.351049\pi\)
0.451052 + 0.892498i \(0.351049\pi\)
\(984\) 0 0
\(985\) 25.8745 0.824430
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3.22413i 0.102521i
\(990\) 0 0
\(991\) −21.9373 −0.696860 −0.348430 0.937335i \(-0.613285\pi\)
−0.348430 + 0.937335i \(0.613285\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 62.2333i 1.97293i
\(996\) 0 0
\(997\) − 21.8745i − 0.692773i −0.938092 0.346386i \(-0.887409\pi\)
0.938092 0.346386i \(-0.112591\pi\)
\(998\) 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 864.2.d.c.433.7 8
3.2 odd 2 inner 864.2.d.c.433.1 8
4.3 odd 2 216.2.d.c.109.8 yes 8
8.3 odd 2 216.2.d.c.109.7 yes 8
8.5 even 2 inner 864.2.d.c.433.2 8
9.2 odd 6 2592.2.r.q.433.8 16
9.4 even 3 2592.2.r.q.2161.7 16
9.5 odd 6 2592.2.r.q.2161.1 16
9.7 even 3 2592.2.r.q.433.2 16
12.11 even 2 216.2.d.c.109.1 8
16.3 odd 4 6912.2.a.cj.1.4 4
16.5 even 4 6912.2.a.cd.1.1 4
16.11 odd 4 6912.2.a.cc.1.1 4
16.13 even 4 6912.2.a.ci.1.4 4
24.5 odd 2 inner 864.2.d.c.433.8 8
24.11 even 2 216.2.d.c.109.2 yes 8
36.7 odd 6 648.2.n.q.109.4 16
36.11 even 6 648.2.n.q.109.5 16
36.23 even 6 648.2.n.q.541.7 16
36.31 odd 6 648.2.n.q.541.2 16
48.5 odd 4 6912.2.a.cd.1.4 4
48.11 even 4 6912.2.a.cc.1.4 4
48.29 odd 4 6912.2.a.ci.1.1 4
48.35 even 4 6912.2.a.cj.1.1 4
72.5 odd 6 2592.2.r.q.2161.8 16
72.11 even 6 648.2.n.q.109.7 16
72.13 even 6 2592.2.r.q.2161.2 16
72.29 odd 6 2592.2.r.q.433.1 16
72.43 odd 6 648.2.n.q.109.2 16
72.59 even 6 648.2.n.q.541.5 16
72.61 even 6 2592.2.r.q.433.7 16
72.67 odd 6 648.2.n.q.541.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.2.d.c.109.1 8 12.11 even 2
216.2.d.c.109.2 yes 8 24.11 even 2
216.2.d.c.109.7 yes 8 8.3 odd 2
216.2.d.c.109.8 yes 8 4.3 odd 2
648.2.n.q.109.2 16 72.43 odd 6
648.2.n.q.109.4 16 36.7 odd 6
648.2.n.q.109.5 16 36.11 even 6
648.2.n.q.109.7 16 72.11 even 6
648.2.n.q.541.2 16 36.31 odd 6
648.2.n.q.541.4 16 72.67 odd 6
648.2.n.q.541.5 16 72.59 even 6
648.2.n.q.541.7 16 36.23 even 6
864.2.d.c.433.1 8 3.2 odd 2 inner
864.2.d.c.433.2 8 8.5 even 2 inner
864.2.d.c.433.7 8 1.1 even 1 trivial
864.2.d.c.433.8 8 24.5 odd 2 inner
2592.2.r.q.433.1 16 72.29 odd 6
2592.2.r.q.433.2 16 9.7 even 3
2592.2.r.q.433.7 16 72.61 even 6
2592.2.r.q.433.8 16 9.2 odd 6
2592.2.r.q.2161.1 16 9.5 odd 6
2592.2.r.q.2161.2 16 72.13 even 6
2592.2.r.q.2161.7 16 9.4 even 3
2592.2.r.q.2161.8 16 72.5 odd 6
6912.2.a.cc.1.1 4 16.11 odd 4
6912.2.a.cc.1.4 4 48.11 even 4
6912.2.a.cd.1.1 4 16.5 even 4
6912.2.a.cd.1.4 4 48.5 odd 4
6912.2.a.ci.1.1 4 48.29 odd 4
6912.2.a.ci.1.4 4 16.13 even 4
6912.2.a.cj.1.1 4 48.35 even 4
6912.2.a.cj.1.4 4 16.3 odd 4