Properties

Label 720.2.o.b.719.7
Level $720$
Weight $2$
Character 720.719
Analytic conductor $5.749$
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] = [720,2,Mod(719,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.719");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 720.o (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.74922894553\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 719.7
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 720.719
Dual form 720.2.o.b.719.5

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.73205 + 1.41421i) q^{5} -4.24264 q^{7} +O(q^{10})\) \(q+(1.73205 + 1.41421i) q^{5} -4.24264 q^{7} -2.44949 q^{11} +2.44949i q^{13} -6.92820 q^{17} +6.92820i q^{19} -6.00000i q^{23} +(1.00000 + 4.89898i) q^{25} -2.82843i q^{29} +3.46410i q^{31} +(-7.34847 - 6.00000i) q^{35} -2.44949i q^{37} +7.07107i q^{41} -8.48528 q^{43} +11.0000 q^{49} +(-4.24264 - 3.46410i) q^{55} +12.2474 q^{59} -2.00000 q^{61} +(-3.46410 + 4.24264i) q^{65} -9.79796 q^{71} +4.89898i q^{73} +10.3923 q^{77} +10.3923i q^{79} +12.0000i q^{83} +(-12.0000 - 9.79796i) q^{85} -7.07107i q^{89} -10.3923i q^{91} +(-9.79796 + 12.0000i) q^{95} +14.6969i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{25} + 88 q^{49} - 16 q^{61} - 96 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.73205 + 1.41421i 0.774597 + 0.632456i
\(6\) 0 0
\(7\) −4.24264 −1.60357 −0.801784 0.597614i \(-0.796115\pi\)
−0.801784 + 0.597614i \(0.796115\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.44949 −0.738549 −0.369274 0.929320i \(-0.620394\pi\)
−0.369274 + 0.929320i \(0.620394\pi\)
\(12\) 0 0
\(13\) 2.44949i 0.679366i 0.940540 + 0.339683i \(0.110320\pi\)
−0.940540 + 0.339683i \(0.889680\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.92820 −1.68034 −0.840168 0.542326i \(-0.817544\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 0 0
\(19\) 6.92820i 1.58944i 0.606977 + 0.794719i \(0.292382\pi\)
−0.606977 + 0.794719i \(0.707618\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000i 1.25109i −0.780189 0.625543i \(-0.784877\pi\)
0.780189 0.625543i \(-0.215123\pi\)
\(24\) 0 0
\(25\) 1.00000 + 4.89898i 0.200000 + 0.979796i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.82843i 0.525226i −0.964901 0.262613i \(-0.915416\pi\)
0.964901 0.262613i \(-0.0845842\pi\)
\(30\) 0 0
\(31\) 3.46410i 0.622171i 0.950382 + 0.311086i \(0.100693\pi\)
−0.950382 + 0.311086i \(0.899307\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.34847 6.00000i −1.24212 1.01419i
\(36\) 0 0
\(37\) 2.44949i 0.402694i −0.979520 0.201347i \(-0.935468\pi\)
0.979520 0.201347i \(-0.0645318\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.07107i 1.10432i 0.833740 + 0.552158i \(0.186195\pi\)
−0.833740 + 0.552158i \(0.813805\pi\)
\(42\) 0 0
\(43\) −8.48528 −1.29399 −0.646997 0.762493i \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 11.0000 1.57143
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −4.24264 3.46410i −0.572078 0.467099i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.2474 1.59448 0.797241 0.603661i \(-0.206292\pi\)
0.797241 + 0.603661i \(0.206292\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −3.46410 + 4.24264i −0.429669 + 0.526235i
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.79796 −1.16280 −0.581402 0.813617i \(-0.697496\pi\)
−0.581402 + 0.813617i \(0.697496\pi\)
\(72\) 0 0
\(73\) 4.89898i 0.573382i 0.958023 + 0.286691i \(0.0925553\pi\)
−0.958023 + 0.286691i \(0.907445\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3923 1.18431
\(78\) 0 0
\(79\) 10.3923i 1.16923i 0.811312 + 0.584613i \(0.198754\pi\)
−0.811312 + 0.584613i \(0.801246\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 12.0000i 1.31717i 0.752506 + 0.658586i \(0.228845\pi\)
−0.752506 + 0.658586i \(0.771155\pi\)
\(84\) 0 0
\(85\) −12.0000 9.79796i −1.30158 1.06274i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 7.07107i 0.749532i −0.927119 0.374766i \(-0.877723\pi\)
0.927119 0.374766i \(-0.122277\pi\)
\(90\) 0 0
\(91\) 10.3923i 1.08941i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −9.79796 + 12.0000i −1.00525 + 1.23117i
\(96\) 0 0
\(97\) 14.6969i 1.49225i 0.665807 + 0.746124i \(0.268087\pi\)
−0.665807 + 0.746124i \(0.731913\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11.3137i 1.12576i −0.826540 0.562878i \(-0.809694\pi\)
0.826540 0.562878i \(-0.190306\pi\)
\(102\) 0 0
\(103\) 12.7279 1.25412 0.627060 0.778971i \(-0.284258\pi\)
0.627060 + 0.778971i \(0.284258\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.0000i 1.16008i −0.814587 0.580042i \(-0.803036\pi\)
0.814587 0.580042i \(-0.196964\pi\)
\(108\) 0 0
\(109\) −2.00000 −0.191565 −0.0957826 0.995402i \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −6.92820 −0.651751 −0.325875 0.945413i \(-0.605659\pi\)
−0.325875 + 0.945413i \(0.605659\pi\)
\(114\) 0 0
\(115\) 8.48528 10.3923i 0.791257 0.969087i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 29.3939 2.69453
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.19615 + 9.89949i −0.464758 + 0.885438i
\(126\) 0 0
\(127\) 12.7279 1.12942 0.564710 0.825289i \(-0.308988\pi\)
0.564710 + 0.825289i \(0.308988\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.44949 0.214013 0.107006 0.994258i \(-0.465873\pi\)
0.107006 + 0.994258i \(0.465873\pi\)
\(132\) 0 0
\(133\) 29.3939i 2.54877i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 13.8564 1.18383 0.591916 0.805999i \(-0.298372\pi\)
0.591916 + 0.805999i \(0.298372\pi\)
\(138\) 0 0
\(139\) 3.46410i 0.293821i 0.989150 + 0.146911i \(0.0469330\pi\)
−0.989150 + 0.146911i \(0.953067\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.00000i 0.501745i
\(144\) 0 0
\(145\) 4.00000 4.89898i 0.332182 0.406838i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 22.6274i 1.85371i −0.375419 0.926855i \(-0.622501\pi\)
0.375419 0.926855i \(-0.377499\pi\)
\(150\) 0 0
\(151\) 17.3205i 1.40952i −0.709444 0.704761i \(-0.751054\pi\)
0.709444 0.704761i \(-0.248946\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −4.89898 + 6.00000i −0.393496 + 0.481932i
\(156\) 0 0
\(157\) 12.2474i 0.977453i −0.872437 0.488726i \(-0.837462\pi\)
0.872437 0.488726i \(-0.162538\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 25.4558i 2.00620i
\(162\) 0 0
\(163\) 8.48528 0.664619 0.332309 0.943170i \(-0.392172\pi\)
0.332309 + 0.943170i \(0.392172\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.0000i 1.39288i 0.717614 + 0.696441i \(0.245234\pi\)
−0.717614 + 0.696441i \(0.754766\pi\)
\(168\) 0 0
\(169\) 7.00000 0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.8564 1.05348 0.526742 0.850026i \(-0.323414\pi\)
0.526742 + 0.850026i \(0.323414\pi\)
\(174\) 0 0
\(175\) −4.24264 20.7846i −0.320713 1.57117i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.2474 0.915417 0.457709 0.889102i \(-0.348670\pi\)
0.457709 + 0.889102i \(0.348670\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.46410 4.24264i 0.254686 0.311925i
\(186\) 0 0
\(187\) 16.9706 1.24101
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.89898 0.354478 0.177239 0.984168i \(-0.443283\pi\)
0.177239 + 0.984168i \(0.443283\pi\)
\(192\) 0 0
\(193\) 19.5959i 1.41055i 0.708936 + 0.705273i \(0.249175\pi\)
−0.708936 + 0.705273i \(0.750825\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.46410 −0.246807 −0.123404 0.992357i \(-0.539381\pi\)
−0.123404 + 0.992357i \(0.539381\pi\)
\(198\) 0 0
\(199\) 10.3923i 0.736691i 0.929689 + 0.368345i \(0.120076\pi\)
−0.929689 + 0.368345i \(0.879924\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 12.0000i 0.842235i
\(204\) 0 0
\(205\) −10.0000 + 12.2474i −0.698430 + 0.855399i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 16.9706i 1.17388i
\(210\) 0 0
\(211\) 3.46410i 0.238479i 0.992866 + 0.119239i \(0.0380456\pi\)
−0.992866 + 0.119239i \(0.961954\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −14.6969 12.0000i −1.00232 0.818393i
\(216\) 0 0
\(217\) 14.6969i 0.997693i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 16.9706i 1.14156i
\(222\) 0 0
\(223\) −4.24264 −0.284108 −0.142054 0.989859i \(-0.545371\pi\)
−0.142054 + 0.989859i \(0.545371\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.0000i 1.59294i 0.604681 + 0.796468i \(0.293301\pi\)
−0.604681 + 0.796468i \(0.706699\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.92820 −0.453882 −0.226941 0.973909i \(-0.572872\pi\)
−0.226941 + 0.973909i \(0.572872\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 14.6969 0.950666 0.475333 0.879806i \(-0.342328\pi\)
0.475333 + 0.879806i \(0.342328\pi\)
\(240\) 0 0
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 19.0526 + 15.5563i 1.21722 + 0.993859i
\(246\) 0 0
\(247\) −16.9706 −1.07981
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −22.0454 −1.39149 −0.695747 0.718287i \(-0.744926\pi\)
−0.695747 + 0.718287i \(0.744926\pi\)
\(252\) 0 0
\(253\) 14.6969i 0.923989i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −27.7128 −1.72868 −0.864339 0.502910i \(-0.832263\pi\)
−0.864339 + 0.502910i \(0.832263\pi\)
\(258\) 0 0
\(259\) 10.3923i 0.645746i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.00000i 0.369976i 0.982741 + 0.184988i \(0.0592246\pi\)
−0.982741 + 0.184988i \(0.940775\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 19.7990i 1.20717i 0.797300 + 0.603583i \(0.206261\pi\)
−0.797300 + 0.603583i \(0.793739\pi\)
\(270\) 0 0
\(271\) 10.3923i 0.631288i −0.948878 0.315644i \(-0.897780\pi\)
0.948878 0.315644i \(-0.102220\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −2.44949 12.0000i −0.147710 0.723627i
\(276\) 0 0
\(277\) 7.34847i 0.441527i −0.975327 0.220763i \(-0.929145\pi\)
0.975327 0.220763i \(-0.0708548\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.07107i 0.421825i 0.977505 + 0.210912i \(0.0676434\pi\)
−0.977505 + 0.210912i \(0.932357\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 30.0000i 1.77084i
\(288\) 0 0
\(289\) 31.0000 1.82353
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 17.3205 1.01187 0.505937 0.862570i \(-0.331147\pi\)
0.505937 + 0.862570i \(0.331147\pi\)
\(294\) 0 0
\(295\) 21.2132 + 17.3205i 1.23508 + 1.00844i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 14.6969 0.849946
\(300\) 0 0
\(301\) 36.0000 2.07501
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.46410 2.82843i −0.198354 0.161955i
\(306\) 0 0
\(307\) −33.9411 −1.93712 −0.968561 0.248776i \(-0.919972\pi\)
−0.968561 + 0.248776i \(0.919972\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4.89898 −0.277796 −0.138898 0.990307i \(-0.544356\pi\)
−0.138898 + 0.990307i \(0.544356\pi\)
\(312\) 0 0
\(313\) 19.5959i 1.10763i −0.832641 0.553813i \(-0.813172\pi\)
0.832641 0.553813i \(-0.186828\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −24.2487 −1.36194 −0.680972 0.732310i \(-0.738442\pi\)
−0.680972 + 0.732310i \(0.738442\pi\)
\(318\) 0 0
\(319\) 6.92820i 0.387905i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 48.0000i 2.67079i
\(324\) 0 0
\(325\) −12.0000 + 2.44949i −0.665640 + 0.135873i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 13.8564i 0.761617i 0.924654 + 0.380808i \(0.124354\pi\)
−0.924654 + 0.380808i \(0.875646\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 14.6969i 0.800593i 0.916386 + 0.400297i \(0.131093\pi\)
−0.916386 + 0.400297i \(0.868907\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 8.48528i 0.459504i
\(342\) 0 0
\(343\) −16.9706 −0.916324
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 12.0000i 0.644194i −0.946707 0.322097i \(-0.895612\pi\)
0.946707 0.322097i \(-0.104388\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) −16.9706 13.8564i −0.900704 0.735422i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −34.2929 −1.80991 −0.904954 0.425510i \(-0.860095\pi\)
−0.904954 + 0.425510i \(0.860095\pi\)
\(360\) 0 0
\(361\) −29.0000 −1.52632
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.92820 + 8.48528i −0.362639 + 0.444140i
\(366\) 0 0
\(367\) −4.24264 −0.221464 −0.110732 0.993850i \(-0.535320\pi\)
−0.110732 + 0.993850i \(0.535320\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 22.0454i 1.14147i 0.821135 + 0.570734i \(0.193341\pi\)
−0.821135 + 0.570734i \(0.806659\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.92820 0.356821
\(378\) 0 0
\(379\) 10.3923i 0.533817i 0.963722 + 0.266908i \(0.0860021\pi\)
−0.963722 + 0.266908i \(0.913998\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 24.0000i 1.22634i 0.789950 + 0.613171i \(0.210106\pi\)
−0.789950 + 0.613171i \(0.789894\pi\)
\(384\) 0 0
\(385\) 18.0000 + 14.6969i 0.917365 + 0.749025i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.82843i 0.143407i 0.997426 + 0.0717035i \(0.0228435\pi\)
−0.997426 + 0.0717035i \(0.977156\pi\)
\(390\) 0 0
\(391\) 41.5692i 2.10225i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −14.6969 + 18.0000i −0.739483 + 0.905678i
\(396\) 0 0
\(397\) 31.8434i 1.59817i 0.601216 + 0.799086i \(0.294683\pi\)
−0.601216 + 0.799086i \(0.705317\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 35.3553i 1.76556i 0.469785 + 0.882781i \(0.344331\pi\)
−0.469785 + 0.882781i \(0.655669\pi\)
\(402\) 0 0
\(403\) −8.48528 −0.422682
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.00000i 0.297409i
\(408\) 0 0
\(409\) −26.0000 −1.28562 −0.642809 0.766027i \(-0.722231\pi\)
−0.642809 + 0.766027i \(0.722231\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −51.9615 −2.55686
\(414\) 0 0
\(415\) −16.9706 + 20.7846i −0.833052 + 1.02028i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −7.34847 −0.358996 −0.179498 0.983758i \(-0.557447\pi\)
−0.179498 + 0.983758i \(0.557447\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.92820 33.9411i −0.336067 1.64639i
\(426\) 0 0
\(427\) 8.48528 0.410632
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.89898 −0.235976 −0.117988 0.993015i \(-0.537644\pi\)
−0.117988 + 0.993015i \(0.537644\pi\)
\(432\) 0 0
\(433\) 4.89898i 0.235430i −0.993047 0.117715i \(-0.962443\pi\)
0.993047 0.117715i \(-0.0375569\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 41.5692 1.98853
\(438\) 0 0
\(439\) 3.46410i 0.165333i −0.996577 0.0826663i \(-0.973656\pi\)
0.996577 0.0826663i \(-0.0263436\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.0000i 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 0 0
\(445\) 10.0000 12.2474i 0.474045 0.580585i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 9.89949i 0.467186i −0.972334 0.233593i \(-0.924952\pi\)
0.972334 0.233593i \(-0.0750483\pi\)
\(450\) 0 0
\(451\) 17.3205i 0.815591i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 14.6969 18.0000i 0.689003 0.843853i
\(456\) 0 0
\(457\) 4.89898i 0.229165i −0.993414 0.114582i \(-0.963447\pi\)
0.993414 0.114582i \(-0.0365530\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 11.3137i 0.526932i −0.964669 0.263466i \(-0.915134\pi\)
0.964669 0.263466i \(-0.0848657\pi\)
\(462\) 0 0
\(463\) −21.2132 −0.985861 −0.492931 0.870069i \(-0.664074\pi\)
−0.492931 + 0.870069i \(0.664074\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 20.7846 0.955677
\(474\) 0 0
\(475\) −33.9411 + 6.92820i −1.55733 + 0.317888i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −19.5959 −0.895360 −0.447680 0.894194i \(-0.647750\pi\)
−0.447680 + 0.894194i \(0.647750\pi\)
\(480\) 0 0
\(481\) 6.00000 0.273576
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −20.7846 + 25.4558i −0.943781 + 1.15589i
\(486\) 0 0
\(487\) 21.2132 0.961262 0.480631 0.876923i \(-0.340408\pi\)
0.480631 + 0.876923i \(0.340408\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 17.1464 0.773807 0.386904 0.922120i \(-0.373545\pi\)
0.386904 + 0.922120i \(0.373545\pi\)
\(492\) 0 0
\(493\) 19.5959i 0.882556i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 41.5692 1.86463
\(498\) 0 0
\(499\) 13.8564i 0.620298i −0.950688 0.310149i \(-0.899621\pi\)
0.950688 0.310149i \(-0.100379\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 16.0000 19.5959i 0.711991 0.872007i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 39.5980i 1.75515i 0.479440 + 0.877575i \(0.340840\pi\)
−0.479440 + 0.877575i \(0.659160\pi\)
\(510\) 0 0
\(511\) 20.7846i 0.919457i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 22.0454 + 18.0000i 0.971437 + 0.793175i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.07107i 0.309789i −0.987931 0.154895i \(-0.950496\pi\)
0.987931 0.154895i \(-0.0495038\pi\)
\(522\) 0 0
\(523\) 25.4558 1.11311 0.556553 0.830812i \(-0.312124\pi\)
0.556553 + 0.830812i \(0.312124\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24.0000i 1.04546i
\(528\) 0 0
\(529\) −13.0000 −0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −17.3205 −0.750234
\(534\) 0 0
\(535\) 16.9706 20.7846i 0.733701 0.898597i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −26.9444 −1.16058
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.46410 2.82843i −0.148386 0.121157i
\(546\) 0 0
\(547\) 8.48528 0.362804 0.181402 0.983409i \(-0.441936\pi\)
0.181402 + 0.983409i \(0.441936\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 19.5959 0.834814
\(552\) 0 0
\(553\) 44.0908i 1.87493i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −27.7128 −1.17423 −0.587115 0.809504i \(-0.699736\pi\)
−0.587115 + 0.809504i \(0.699736\pi\)
\(558\) 0 0
\(559\) 20.7846i 0.879095i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.0000i 0.505740i 0.967500 + 0.252870i \(0.0813744\pi\)
−0.967500 + 0.252870i \(0.918626\pi\)
\(564\) 0 0
\(565\) −12.0000 9.79796i −0.504844 0.412203i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.89949i 0.415008i −0.978234 0.207504i \(-0.933466\pi\)
0.978234 0.207504i \(-0.0665341\pi\)
\(570\) 0 0
\(571\) 27.7128i 1.15975i 0.814707 + 0.579873i \(0.196898\pi\)
−0.814707 + 0.579873i \(0.803102\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 29.3939 6.00000i 1.22581 0.250217i
\(576\) 0 0
\(577\) 4.89898i 0.203947i −0.994787 0.101974i \(-0.967484\pi\)
0.994787 0.101974i \(-0.0325157\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 50.9117i 2.11217i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000i 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) −24.0000 −0.988903
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 27.7128 1.13803 0.569014 0.822328i \(-0.307325\pi\)
0.569014 + 0.822328i \(0.307325\pi\)
\(594\) 0 0
\(595\) 50.9117 + 41.5692i 2.08718 + 1.70417i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.89898 0.200167 0.100083 0.994979i \(-0.468089\pi\)
0.100083 + 0.994979i \(0.468089\pi\)
\(600\) 0 0
\(601\) −44.0000 −1.79480 −0.897399 0.441221i \(-0.854546\pi\)
−0.897399 + 0.441221i \(0.854546\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8.66025 7.07107i −0.352089 0.287480i
\(606\) 0 0
\(607\) 21.2132 0.861017 0.430509 0.902586i \(-0.358334\pi\)
0.430509 + 0.902586i \(0.358334\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 22.0454i 0.890406i −0.895430 0.445203i \(-0.853132\pi\)
0.895430 0.445203i \(-0.146868\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.8564 −0.557838 −0.278919 0.960315i \(-0.589976\pi\)
−0.278919 + 0.960315i \(0.589976\pi\)
\(618\) 0 0
\(619\) 31.1769i 1.25311i −0.779379 0.626553i \(-0.784465\pi\)
0.779379 0.626553i \(-0.215535\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 30.0000i 1.20192i
\(624\) 0 0
\(625\) −23.0000 + 9.79796i −0.920000 + 0.391918i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.9706i 0.676661i
\(630\) 0 0
\(631\) 17.3205i 0.689519i −0.938691 0.344759i \(-0.887961\pi\)
0.938691 0.344759i \(-0.112039\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.0454 + 18.0000i 0.874845 + 0.714308i
\(636\) 0 0
\(637\) 26.9444i 1.06758i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.5563i 0.614439i 0.951639 + 0.307219i \(0.0993986\pi\)
−0.951639 + 0.307219i \(0.900601\pi\)
\(642\) 0 0
\(643\) 25.4558 1.00388 0.501940 0.864902i \(-0.332620\pi\)
0.501940 + 0.864902i \(0.332620\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.0000i 0.943537i 0.881722 + 0.471769i \(0.156384\pi\)
−0.881722 + 0.471769i \(0.843616\pi\)
\(648\) 0 0
\(649\) −30.0000 −1.17760
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 10.3923 0.406682 0.203341 0.979108i \(-0.434820\pi\)
0.203341 + 0.979108i \(0.434820\pi\)
\(654\) 0 0
\(655\) 4.24264 + 3.46410i 0.165774 + 0.135354i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −36.7423 −1.43128 −0.715639 0.698470i \(-0.753865\pi\)
−0.715639 + 0.698470i \(0.753865\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 41.5692 50.9117i 1.61199 1.97427i
\(666\) 0 0
\(667\) −16.9706 −0.657103
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.89898 0.189123
\(672\) 0 0
\(673\) 34.2929i 1.32189i 0.750433 + 0.660946i \(0.229845\pi\)
−0.750433 + 0.660946i \(0.770155\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −10.3923 −0.399409 −0.199704 0.979856i \(-0.563998\pi\)
−0.199704 + 0.979856i \(0.563998\pi\)
\(678\) 0 0
\(679\) 62.3538i 2.39292i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000i 0.918334i 0.888350 + 0.459167i \(0.151852\pi\)
−0.888350 + 0.459167i \(0.848148\pi\)
\(684\) 0 0
\(685\) 24.0000 + 19.5959i 0.916993 + 0.748722i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 20.7846i 0.790684i −0.918534 0.395342i \(-0.870626\pi\)
0.918534 0.395342i \(-0.129374\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.89898 + 6.00000i −0.185829 + 0.227593i
\(696\) 0 0
\(697\) 48.9898i 1.85562i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.65685i 0.213656i −0.994277 0.106828i \(-0.965931\pi\)
0.994277 0.106828i \(-0.0340695\pi\)
\(702\) 0 0
\(703\) 16.9706 0.640057
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 48.0000i 1.80523i
\(708\) 0 0
\(709\) 2.00000 0.0751116 0.0375558 0.999295i \(-0.488043\pi\)
0.0375558 + 0.999295i \(0.488043\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 20.7846 0.778390
\(714\) 0 0
\(715\) 8.48528 10.3923i 0.317332 0.388650i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.89898 0.182701 0.0913506 0.995819i \(-0.470882\pi\)
0.0913506 + 0.995819i \(0.470882\pi\)
\(720\) 0 0
\(721\) −54.0000 −2.01107
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 13.8564 2.82843i 0.514614 0.105045i
\(726\) 0 0
\(727\) −21.2132 −0.786754 −0.393377 0.919377i \(-0.628693\pi\)
−0.393377 + 0.919377i \(0.628693\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 58.7878 2.17434
\(732\) 0 0
\(733\) 7.34847i 0.271422i −0.990748 0.135711i \(-0.956668\pi\)
0.990748 0.135711i \(-0.0433318\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 27.7128i 1.01943i −0.860343 0.509716i \(-0.829750\pi\)
0.860343 0.509716i \(-0.170250\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.0000i 0.880475i −0.897881 0.440237i \(-0.854894\pi\)
0.897881 0.440237i \(-0.145106\pi\)
\(744\) 0 0
\(745\) 32.0000 39.1918i 1.17239 1.43588i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 50.9117i 1.86027i
\(750\) 0 0
\(751\) 51.9615i 1.89610i 0.318117 + 0.948051i \(0.396950\pi\)
−0.318117 + 0.948051i \(0.603050\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 24.4949 30.0000i 0.891461 1.09181i
\(756\) 0 0
\(757\) 36.7423i 1.33542i −0.744420 0.667712i \(-0.767274\pi\)
0.744420 0.667712i \(-0.232726\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.41421i 0.0512652i 0.999671 + 0.0256326i \(0.00816000\pi\)
−0.999671 + 0.0256326i \(0.991840\pi\)
\(762\) 0 0
\(763\) 8.48528 0.307188
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 30.0000i 1.08324i
\(768\) 0 0
\(769\) 32.0000 1.15395 0.576975 0.816762i \(-0.304233\pi\)
0.576975 + 0.816762i \(0.304233\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 38.1051 1.37055 0.685273 0.728286i \(-0.259683\pi\)
0.685273 + 0.728286i \(0.259683\pi\)
\(774\) 0 0
\(775\) −16.9706 + 3.46410i −0.609601 + 0.124434i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −48.9898 −1.75524
\(780\) 0 0
\(781\) 24.0000 0.858788
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 17.3205 21.2132i 0.618195 0.757132i
\(786\) 0 0
\(787\) 25.4558 0.907403 0.453701 0.891154i \(-0.350103\pi\)
0.453701 + 0.891154i \(0.350103\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 29.3939 1.04513
\(792\) 0 0
\(793\) 4.89898i 0.173968i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −13.8564 −0.490819 −0.245410 0.969419i \(-0.578922\pi\)
−0.245410 + 0.969419i \(0.578922\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.0000i 0.423471i
\(804\) 0 0
\(805\) −36.0000 + 44.0908i −1.26883 + 1.55400i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18.3848i 0.646374i 0.946335 + 0.323187i \(0.104754\pi\)
−0.946335 + 0.323187i \(0.895246\pi\)
\(810\) 0 0
\(811\) 38.1051i 1.33805i −0.743239 0.669026i \(-0.766712\pi\)
0.743239 0.669026i \(-0.233288\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 14.6969 + 12.0000i 0.514811 + 0.420342i
\(816\) 0 0
\(817\) 58.7878i 2.05672i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 2.82843i 0.0987128i 0.998781 + 0.0493564i \(0.0157170\pi\)
−0.998781 + 0.0493564i \(0.984283\pi\)
\(822\) 0 0
\(823\) 21.2132 0.739446 0.369723 0.929142i \(-0.379453\pi\)
0.369723 + 0.929142i \(0.379453\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 12.0000i 0.417281i 0.977992 + 0.208640i \(0.0669038\pi\)
−0.977992 + 0.208640i \(0.933096\pi\)
\(828\) 0 0
\(829\) −26.0000 −0.903017 −0.451509 0.892267i \(-0.649114\pi\)
−0.451509 + 0.892267i \(0.649114\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −76.2102 −2.64053
\(834\) 0 0
\(835\) −25.4558 + 31.1769i −0.880936 + 1.07892i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −34.2929 −1.18392 −0.591960 0.805967i \(-0.701646\pi\)
−0.591960 + 0.805967i \(0.701646\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 12.1244 + 9.89949i 0.417091 + 0.340553i
\(846\) 0 0
\(847\) 21.2132 0.728894
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −14.6969 −0.503805
\(852\) 0 0
\(853\) 36.7423i 1.25803i 0.777392 + 0.629017i \(0.216542\pi\)
−0.777392 + 0.629017i \(0.783458\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −13.8564 −0.473326 −0.236663 0.971592i \(-0.576054\pi\)
−0.236663 + 0.971592i \(0.576054\pi\)
\(858\) 0 0
\(859\) 45.0333i 1.53652i −0.640140 0.768259i \(-0.721124\pi\)
0.640140 0.768259i \(-0.278876\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 18.0000i 0.612727i −0.951915 0.306364i \(-0.900888\pi\)
0.951915 0.306364i \(-0.0991123\pi\)
\(864\) 0 0
\(865\) 24.0000 + 19.5959i 0.816024 + 0.666281i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 25.4558i 0.863530i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 22.0454 42.0000i 0.745271 1.41986i
\(876\) 0 0
\(877\) 41.6413i 1.40613i 0.711127 + 0.703064i \(0.248185\pi\)
−0.711127 + 0.703064i \(0.751815\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.89949i 0.333522i −0.985997 0.166761i \(-0.946669\pi\)
0.985997 0.166761i \(-0.0533309\pi\)
\(882\) 0 0
\(883\) −42.4264 −1.42776 −0.713881 0.700267i \(-0.753064\pi\)
−0.713881 + 0.700267i \(0.753064\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.00000i 0.201460i −0.994914 0.100730i \(-0.967882\pi\)
0.994914 0.100730i \(-0.0321179\pi\)
\(888\) 0 0
\(889\) −54.0000 −1.81110
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 21.2132 + 17.3205i 0.709079 + 0.578961i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.79796 0.326780
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −17.3205 14.1421i −0.575753 0.470100i
\(906\) 0 0
\(907\) 8.48528 0.281749 0.140875 0.990027i \(-0.455009\pi\)
0.140875 + 0.990027i \(0.455009\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −48.9898 −1.62310 −0.811552 0.584280i \(-0.801377\pi\)
−0.811552 + 0.584280i \(0.801377\pi\)
\(912\) 0 0
\(913\) 29.3939i 0.972795i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −10.3923 −0.343184
\(918\) 0 0
\(919\) 45.0333i 1.48551i 0.669562 + 0.742756i \(0.266482\pi\)
−0.669562 + 0.742756i \(0.733518\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 24.0000i 0.789970i
\(924\) 0 0
\(925\) 12.0000 2.44949i 0.394558 0.0805387i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 9.89949i 0.324792i −0.986726 0.162396i \(-0.948078\pi\)
0.986726 0.162396i \(-0.0519222\pi\)
\(930\) 0 0
\(931\) 76.2102i 2.49769i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 29.3939 + 24.0000i 0.961283 + 0.784884i
\(936\) 0 0
\(937\) 48.9898i 1.60043i 0.599715 + 0.800213i \(0.295280\pi\)
−0.599715 + 0.800213i \(0.704720\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 53.7401i 1.75188i −0.482422 0.875939i \(-0.660243\pi\)
0.482422 0.875939i \(-0.339757\pi\)
\(942\) 0 0
\(943\) 42.4264 1.38159
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 60.0000i 1.94974i −0.222779 0.974869i \(-0.571513\pi\)
0.222779 0.974869i \(-0.428487\pi\)
\(948\) 0 0
\(949\) −12.0000 −0.389536
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 27.7128 0.897706 0.448853 0.893606i \(-0.351833\pi\)
0.448853 + 0.893606i \(0.351833\pi\)
\(954\) 0 0
\(955\) 8.48528 + 6.92820i 0.274577 + 0.224191i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −58.7878 −1.89836
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −27.7128 + 33.9411i −0.892107 + 1.09260i
\(966\) 0 0
\(967\) −46.6690 −1.50078 −0.750388 0.660998i \(-0.770133\pi\)
−0.750388 + 0.660998i \(0.770133\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 12.2474 0.393039 0.196520 0.980500i \(-0.437036\pi\)
0.196520 + 0.980500i \(0.437036\pi\)
\(972\) 0 0
\(973\) 14.6969i 0.471162i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 17.3205i 0.553566i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000i 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 0 0
\(985\) −6.00000 4.89898i −0.191176 0.156094i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 50.9117i 1.61890i
\(990\) 0 0
\(991\) 17.3205i 0.550204i −0.961415 0.275102i \(-0.911288\pi\)
0.961415 0.275102i \(-0.0887116\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −14.6969 + 18.0000i −0.465924 + 0.570638i
\(996\) 0 0
\(997\) 2.44949i 0.0775761i −0.999247 0.0387881i \(-0.987650\pi\)
0.999247 0.0387881i \(-0.0123497\pi\)
\(998\) 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 720.2.o.b.719.7 yes 8
3.2 odd 2 inner 720.2.o.b.719.1 8
4.3 odd 2 inner 720.2.o.b.719.8 yes 8
5.2 odd 4 3600.2.h.f.1151.1 4
5.3 odd 4 3600.2.h.g.1151.3 4
5.4 even 2 inner 720.2.o.b.719.4 yes 8
8.3 odd 2 2880.2.o.d.2879.2 8
8.5 even 2 2880.2.o.d.2879.1 8
12.11 even 2 inner 720.2.o.b.719.2 yes 8
15.2 even 4 3600.2.h.g.1151.2 4
15.8 even 4 3600.2.h.f.1151.4 4
15.14 odd 2 inner 720.2.o.b.719.6 yes 8
20.3 even 4 3600.2.h.f.1151.2 4
20.7 even 4 3600.2.h.g.1151.4 4
20.19 odd 2 inner 720.2.o.b.719.3 yes 8
24.5 odd 2 2880.2.o.d.2879.7 8
24.11 even 2 2880.2.o.d.2879.8 8
40.19 odd 2 2880.2.o.d.2879.5 8
40.29 even 2 2880.2.o.d.2879.6 8
60.23 odd 4 3600.2.h.g.1151.1 4
60.47 odd 4 3600.2.h.f.1151.3 4
60.59 even 2 inner 720.2.o.b.719.5 yes 8
120.29 odd 2 2880.2.o.d.2879.4 8
120.59 even 2 2880.2.o.d.2879.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
720.2.o.b.719.1 8 3.2 odd 2 inner
720.2.o.b.719.2 yes 8 12.11 even 2 inner
720.2.o.b.719.3 yes 8 20.19 odd 2 inner
720.2.o.b.719.4 yes 8 5.4 even 2 inner
720.2.o.b.719.5 yes 8 60.59 even 2 inner
720.2.o.b.719.6 yes 8 15.14 odd 2 inner
720.2.o.b.719.7 yes 8 1.1 even 1 trivial
720.2.o.b.719.8 yes 8 4.3 odd 2 inner
2880.2.o.d.2879.1 8 8.5 even 2
2880.2.o.d.2879.2 8 8.3 odd 2
2880.2.o.d.2879.3 8 120.59 even 2
2880.2.o.d.2879.4 8 120.29 odd 2
2880.2.o.d.2879.5 8 40.19 odd 2
2880.2.o.d.2879.6 8 40.29 even 2
2880.2.o.d.2879.7 8 24.5 odd 2
2880.2.o.d.2879.8 8 24.11 even 2
3600.2.h.f.1151.1 4 5.2 odd 4
3600.2.h.f.1151.2 4 20.3 even 4
3600.2.h.f.1151.3 4 60.47 odd 4
3600.2.h.f.1151.4 4 15.8 even 4
3600.2.h.g.1151.1 4 60.23 odd 4
3600.2.h.g.1151.2 4 15.2 even 4
3600.2.h.g.1151.3 4 5.3 odd 4
3600.2.h.g.1151.4 4 20.7 even 4