Properties

Label 2240.2.n.c.1119.8
Level $2240$
Weight $2$
Character 2240.1119
Analytic conductor $17.886$
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] = [2240,2,Mod(1119,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 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("2240.1119");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.n (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: \(17.8864900528\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1119.8
Root \(-0.258819 - 0.965926i\) of defining polynomial
Character \(\chi\) \(=\) 2240.1119
Dual form 2240.2.n.c.1119.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.41421 + 1.73205i) q^{5} +(2.44949 + 1.00000i) q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+(1.41421 + 1.73205i) q^{5} +(2.44949 + 1.00000i) q^{7} -3.00000 q^{9} -3.46410 q^{11} -3.46410i q^{13} -4.00000 q^{17} -2.82843i q^{19} -4.89898 q^{23} +(-1.00000 + 4.89898i) q^{25} +6.92820i q^{29} -9.79796 q^{31} +(1.73205 + 5.65685i) q^{35} -5.65685 q^{37} +9.79796i q^{41} +2.82843i q^{43} +(-4.24264 - 5.19615i) q^{45} -10.0000i q^{47} +(5.00000 + 4.89898i) q^{49} +11.3137 q^{53} +(-4.89898 - 6.00000i) q^{55} -8.48528i q^{59} -14.1421 q^{61} +(-7.34847 - 3.00000i) q^{63} +(6.00000 - 4.89898i) q^{65} -14.1421i q^{67} +4.00000i q^{71} -12.0000 q^{73} +(-8.48528 - 3.46410i) q^{77} +12.0000i q^{79} +9.00000 q^{81} +13.8564 q^{83} +(-5.65685 - 6.92820i) q^{85} +9.79796i q^{89} +(3.46410 - 8.48528i) q^{91} +(4.89898 - 4.00000i) q^{95} -4.00000 q^{97} +10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 24 q^{9} - 32 q^{17} - 8 q^{25} + 40 q^{49} + 48 q^{65} - 96 q^{73} + 72 q^{81} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) 1.41421 + 1.73205i 0.632456 + 0.774597i
\(6\) 0 0
\(7\) 2.44949 + 1.00000i 0.925820 + 0.377964i
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) 0 0
\(13\) 3.46410i 0.960769i −0.877058 0.480384i \(-0.840497\pi\)
0.877058 0.480384i \(-0.159503\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.00000 −0.970143 −0.485071 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 2.82843i 0.648886i −0.945905 0.324443i \(-0.894823\pi\)
0.945905 0.324443i \(-0.105177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.89898 −1.02151 −0.510754 0.859727i \(-0.670634\pi\)
−0.510754 + 0.859727i \(0.670634\pi\)
\(24\) 0 0
\(25\) −1.00000 + 4.89898i −0.200000 + 0.979796i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 6.92820i 1.28654i 0.765641 + 0.643268i \(0.222422\pi\)
−0.765641 + 0.643268i \(0.777578\pi\)
\(30\) 0 0
\(31\) −9.79796 −1.75977 −0.879883 0.475191i \(-0.842379\pi\)
−0.879883 + 0.475191i \(0.842379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.73205 + 5.65685i 0.292770 + 0.956183i
\(36\) 0 0
\(37\) −5.65685 −0.929981 −0.464991 0.885316i \(-0.653942\pi\)
−0.464991 + 0.885316i \(0.653942\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.79796i 1.53018i 0.643921 + 0.765092i \(0.277307\pi\)
−0.643921 + 0.765092i \(0.722693\pi\)
\(42\) 0 0
\(43\) 2.82843i 0.431331i 0.976467 + 0.215666i \(0.0691921\pi\)
−0.976467 + 0.215666i \(0.930808\pi\)
\(44\) 0 0
\(45\) −4.24264 5.19615i −0.632456 0.774597i
\(46\) 0 0
\(47\) 10.0000i 1.45865i −0.684167 0.729325i \(-0.739834\pi\)
0.684167 0.729325i \(-0.260166\pi\)
\(48\) 0 0
\(49\) 5.00000 + 4.89898i 0.714286 + 0.699854i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.3137 1.55406 0.777029 0.629465i \(-0.216726\pi\)
0.777029 + 0.629465i \(0.216726\pi\)
\(54\) 0 0
\(55\) −4.89898 6.00000i −0.660578 0.809040i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.48528i 1.10469i −0.833616 0.552345i \(-0.813733\pi\)
0.833616 0.552345i \(-0.186267\pi\)
\(60\) 0 0
\(61\) −14.1421 −1.81071 −0.905357 0.424650i \(-0.860397\pi\)
−0.905357 + 0.424650i \(0.860397\pi\)
\(62\) 0 0
\(63\) −7.34847 3.00000i −0.925820 0.377964i
\(64\) 0 0
\(65\) 6.00000 4.89898i 0.744208 0.607644i
\(66\) 0 0
\(67\) 14.1421i 1.72774i −0.503718 0.863868i \(-0.668035\pi\)
0.503718 0.863868i \(-0.331965\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000i 0.474713i 0.971423 + 0.237356i \(0.0762809\pi\)
−0.971423 + 0.237356i \(0.923719\pi\)
\(72\) 0 0
\(73\) −12.0000 −1.40449 −0.702247 0.711934i \(-0.747820\pi\)
−0.702247 + 0.711934i \(0.747820\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −8.48528 3.46410i −0.966988 0.394771i
\(78\) 0 0
\(79\) 12.0000i 1.35011i 0.737769 + 0.675053i \(0.235879\pi\)
−0.737769 + 0.675053i \(0.764121\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) 13.8564 1.52094 0.760469 0.649374i \(-0.224969\pi\)
0.760469 + 0.649374i \(0.224969\pi\)
\(84\) 0 0
\(85\) −5.65685 6.92820i −0.613572 0.751469i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.79796i 1.03858i 0.854598 + 0.519291i \(0.173804\pi\)
−0.854598 + 0.519291i \(0.826196\pi\)
\(90\) 0 0
\(91\) 3.46410 8.48528i 0.363137 0.889499i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.89898 4.00000i 0.502625 0.410391i
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) 10.3923 1.04447
\(100\) 0 0
\(101\) 8.48528 0.844317 0.422159 0.906522i \(-0.361273\pi\)
0.422159 + 0.906522i \(0.361273\pi\)
\(102\) 0 0
\(103\) 6.00000i 0.591198i −0.955312 0.295599i \(-0.904481\pi\)
0.955312 0.295599i \(-0.0955191\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.82843i 0.273434i 0.990610 + 0.136717i \(0.0436552\pi\)
−0.990610 + 0.136717i \(0.956345\pi\)
\(108\) 0 0
\(109\) 6.92820i 0.663602i −0.943349 0.331801i \(-0.892344\pi\)
0.943349 0.331801i \(-0.107656\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.79796i 0.921714i 0.887474 + 0.460857i \(0.152458\pi\)
−0.887474 + 0.460857i \(0.847542\pi\)
\(114\) 0 0
\(115\) −6.92820 8.48528i −0.646058 0.791257i
\(116\) 0 0
\(117\) 10.3923i 0.960769i
\(118\) 0 0
\(119\) −9.79796 4.00000i −0.898177 0.366679i
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.89949 + 5.19615i −0.885438 + 0.464758i
\(126\) 0 0
\(127\) −4.89898 −0.434714 −0.217357 0.976092i \(-0.569744\pi\)
−0.217357 + 0.976092i \(0.569744\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.82843i 0.247121i −0.992337 0.123560i \(-0.960569\pi\)
0.992337 0.123560i \(-0.0394313\pi\)
\(132\) 0 0
\(133\) 2.82843 6.92820i 0.245256 0.600751i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 2.82843i 0.239904i 0.992780 + 0.119952i \(0.0382741\pi\)
−0.992780 + 0.119952i \(0.961726\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.0000i 1.00349i
\(144\) 0 0
\(145\) −12.0000 + 9.79796i −0.996546 + 0.813676i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.92820i 0.567581i 0.958886 + 0.283790i \(0.0915919\pi\)
−0.958886 + 0.283790i \(0.908408\pi\)
\(150\) 0 0
\(151\) 12.0000i 0.976546i 0.872691 + 0.488273i \(0.162373\pi\)
−0.872691 + 0.488273i \(0.837627\pi\)
\(152\) 0 0
\(153\) 12.0000 0.970143
\(154\) 0 0
\(155\) −13.8564 16.9706i −1.11297 1.36311i
\(156\) 0 0
\(157\) 10.3923i 0.829396i −0.909959 0.414698i \(-0.863887\pi\)
0.909959 0.414698i \(-0.136113\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −12.0000 4.89898i −0.945732 0.386094i
\(162\) 0 0
\(163\) 19.7990i 1.55078i 0.631485 + 0.775388i \(0.282446\pi\)
−0.631485 + 0.775388i \(0.717554\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 10.0000i 0.773823i −0.922117 0.386912i \(-0.873542\pi\)
0.922117 0.386912i \(-0.126458\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 8.48528i 0.648886i
\(172\) 0 0
\(173\) 17.3205i 1.31685i −0.752645 0.658427i \(-0.771222\pi\)
0.752645 0.658427i \(-0.228778\pi\)
\(174\) 0 0
\(175\) −7.34847 + 11.0000i −0.555492 + 0.831522i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.3923 −0.776757 −0.388379 0.921500i \(-0.626965\pi\)
−0.388379 + 0.921500i \(0.626965\pi\)
\(180\) 0 0
\(181\) −2.82843 −0.210235 −0.105118 0.994460i \(-0.533522\pi\)
−0.105118 + 0.994460i \(0.533522\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.00000 9.79796i −0.588172 0.720360i
\(186\) 0 0
\(187\) 13.8564 1.01328
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 20.0000i 1.44715i −0.690246 0.723575i \(-0.742498\pi\)
0.690246 0.723575i \(-0.257502\pi\)
\(192\) 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
\(197\) 11.3137 0.806068 0.403034 0.915185i \(-0.367956\pi\)
0.403034 + 0.915185i \(0.367956\pi\)
\(198\) 0 0
\(199\) −19.5959 −1.38912 −0.694559 0.719436i \(-0.744400\pi\)
−0.694559 + 0.719436i \(0.744400\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.92820 + 16.9706i −0.486265 + 1.19110i
\(204\) 0 0
\(205\) −16.9706 + 13.8564i −1.18528 + 0.967773i
\(206\) 0 0
\(207\) 14.6969 1.02151
\(208\) 0 0
\(209\) 9.79796i 0.677739i
\(210\) 0 0
\(211\) −10.3923 −0.715436 −0.357718 0.933830i \(-0.616445\pi\)
−0.357718 + 0.933830i \(0.616445\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.89898 + 4.00000i −0.334108 + 0.272798i
\(216\) 0 0
\(217\) −24.0000 9.79796i −1.62923 0.665129i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 13.8564i 0.932083i
\(222\) 0 0
\(223\) 18.0000i 1.20537i −0.797980 0.602685i \(-0.794098\pi\)
0.797980 0.602685i \(-0.205902\pi\)
\(224\) 0 0
\(225\) 3.00000 14.6969i 0.200000 0.979796i
\(226\) 0 0
\(227\) −13.8564 −0.919682 −0.459841 0.888001i \(-0.652094\pi\)
−0.459841 + 0.888001i \(0.652094\pi\)
\(228\) 0 0
\(229\) 2.82843 0.186908 0.0934539 0.995624i \(-0.470209\pi\)
0.0934539 + 0.995624i \(0.470209\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.79796i 0.641886i 0.947099 + 0.320943i \(0.104000\pi\)
−0.947099 + 0.320943i \(0.896000\pi\)
\(234\) 0 0
\(235\) 17.3205 14.1421i 1.12987 0.922531i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.00000i 0.258738i 0.991596 + 0.129369i \(0.0412952\pi\)
−0.991596 + 0.129369i \(0.958705\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.41421 + 15.5885i −0.0903508 + 0.995910i
\(246\) 0 0
\(247\) −9.79796 −0.623429
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 2.82843i 0.178529i 0.996008 + 0.0892644i \(0.0284516\pi\)
−0.996008 + 0.0892644i \(0.971548\pi\)
\(252\) 0 0
\(253\) 16.9706 1.06693
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 20.0000 1.24757 0.623783 0.781598i \(-0.285595\pi\)
0.623783 + 0.781598i \(0.285595\pi\)
\(258\) 0 0
\(259\) −13.8564 5.65685i −0.860995 0.351500i
\(260\) 0 0
\(261\) 20.7846i 1.28654i
\(262\) 0 0
\(263\) −14.6969 −0.906252 −0.453126 0.891446i \(-0.649691\pi\)
−0.453126 + 0.891446i \(0.649691\pi\)
\(264\) 0 0
\(265\) 16.0000 + 19.5959i 0.982872 + 1.20377i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −8.48528 −0.517357 −0.258678 0.965964i \(-0.583287\pi\)
−0.258678 + 0.965964i \(0.583287\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.46410 16.9706i 0.208893 1.02336i
\(276\) 0 0
\(277\) −5.65685 −0.339887 −0.169944 0.985454i \(-0.554359\pi\)
−0.169944 + 0.985454i \(0.554359\pi\)
\(278\) 0 0
\(279\) 29.3939 1.75977
\(280\) 0 0
\(281\) 2.00000 0.119310 0.0596550 0.998219i \(-0.481000\pi\)
0.0596550 + 0.998219i \(0.481000\pi\)
\(282\) 0 0
\(283\) 27.7128 1.64736 0.823678 0.567058i \(-0.191918\pi\)
0.823678 + 0.567058i \(0.191918\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.79796 + 24.0000i −0.578355 + 1.41668i
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 24.2487i 1.41662i 0.705899 + 0.708312i \(0.250543\pi\)
−0.705899 + 0.708312i \(0.749457\pi\)
\(294\) 0 0
\(295\) 14.6969 12.0000i 0.855689 0.698667i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 16.9706i 0.981433i
\(300\) 0 0
\(301\) −2.82843 + 6.92820i −0.163028 + 0.399335i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −20.0000 24.4949i −1.14520 1.40257i
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.79796 0.555591 0.277796 0.960640i \(-0.410396\pi\)
0.277796 + 0.960640i \(0.410396\pi\)
\(312\) 0 0
\(313\) 12.0000 0.678280 0.339140 0.940736i \(-0.389864\pi\)
0.339140 + 0.940736i \(0.389864\pi\)
\(314\) 0 0
\(315\) −5.19615 16.9706i −0.292770 0.956183i
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 24.0000i 1.34374i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 11.3137i 0.629512i
\(324\) 0 0
\(325\) 16.9706 + 3.46410i 0.941357 + 0.192154i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 10.0000 24.4949i 0.551318 1.35045i
\(330\) 0 0
\(331\) −3.46410 −0.190404 −0.0952021 0.995458i \(-0.530350\pi\)
−0.0952021 + 0.995458i \(0.530350\pi\)
\(332\) 0 0
\(333\) 16.9706 0.929981
\(334\) 0 0
\(335\) 24.4949 20.0000i 1.33830 1.09272i
\(336\) 0 0
\(337\) 9.79796i 0.533729i 0.963734 + 0.266864i \(0.0859876\pi\)
−0.963734 + 0.266864i \(0.914012\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 33.9411 1.83801
\(342\) 0 0
\(343\) 7.34847 + 17.0000i 0.396780 + 0.917914i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.48528i 0.455514i 0.973718 + 0.227757i \(0.0731391\pi\)
−0.973718 + 0.227757i \(0.926861\pi\)
\(348\) 0 0
\(349\) −19.7990 −1.05982 −0.529908 0.848055i \(-0.677773\pi\)
−0.529908 + 0.848055i \(0.677773\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.00000 0.212899 0.106449 0.994318i \(-0.466052\pi\)
0.106449 + 0.994318i \(0.466052\pi\)
\(354\) 0 0
\(355\) −6.92820 + 5.65685i −0.367711 + 0.300235i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.00000i 0.211112i −0.994413 0.105556i \(-0.966338\pi\)
0.994413 0.105556i \(-0.0336622\pi\)
\(360\) 0 0
\(361\) 11.0000 0.578947
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −16.9706 20.7846i −0.888280 1.08792i
\(366\) 0 0
\(367\) 2.00000i 0.104399i 0.998637 + 0.0521996i \(0.0166232\pi\)
−0.998637 + 0.0521996i \(0.983377\pi\)
\(368\) 0 0
\(369\) 29.3939i 1.53018i
\(370\) 0 0
\(371\) 27.7128 + 11.3137i 1.43878 + 0.587378i
\(372\) 0 0
\(373\) 28.2843 1.46450 0.732252 0.681034i \(-0.238469\pi\)
0.732252 + 0.681034i \(0.238469\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 3.46410 0.177939 0.0889695 0.996034i \(-0.471643\pi\)
0.0889695 + 0.996034i \(0.471643\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 22.0000i 1.12415i 0.827087 + 0.562074i \(0.189996\pi\)
−0.827087 + 0.562074i \(0.810004\pi\)
\(384\) 0 0
\(385\) −6.00000 19.5959i −0.305788 0.998700i
\(386\) 0 0
\(387\) 8.48528i 0.431331i
\(388\) 0 0
\(389\) 20.7846i 1.05382i −0.849921 0.526911i \(-0.823350\pi\)
0.849921 0.526911i \(-0.176650\pi\)
\(390\) 0 0
\(391\) 19.5959 0.991008
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −20.7846 + 16.9706i −1.04579 + 0.853882i
\(396\) 0 0
\(397\) 3.46410i 0.173858i 0.996214 + 0.0869291i \(0.0277054\pi\)
−0.996214 + 0.0869291i \(0.972295\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 0 0
\(403\) 33.9411i 1.69073i
\(404\) 0 0
\(405\) 12.7279 + 15.5885i 0.632456 + 0.774597i
\(406\) 0 0
\(407\) 19.5959 0.971334
\(408\) 0 0
\(409\) 9.79796i 0.484478i −0.970217 0.242239i \(-0.922118\pi\)
0.970217 0.242239i \(-0.0778818\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.48528 20.7846i 0.417533 1.02274i
\(414\) 0 0
\(415\) 19.5959 + 24.0000i 0.961926 + 1.17811i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 25.4558i 1.24360i 0.783176 + 0.621800i \(0.213598\pi\)
−0.783176 + 0.621800i \(0.786402\pi\)
\(420\) 0 0
\(421\) 6.92820i 0.337660i 0.985645 + 0.168830i \(0.0539989\pi\)
−0.985645 + 0.168830i \(0.946001\pi\)
\(422\) 0 0
\(423\) 30.0000i 1.45865i
\(424\) 0 0
\(425\) 4.00000 19.5959i 0.194029 0.950542i
\(426\) 0 0
\(427\) −34.6410 14.1421i −1.67640 0.684386i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 20.0000i 0.963366i 0.876346 + 0.481683i \(0.159974\pi\)
−0.876346 + 0.481683i \(0.840026\pi\)
\(432\) 0 0
\(433\) 28.0000 1.34559 0.672797 0.739827i \(-0.265093\pi\)
0.672797 + 0.739827i \(0.265093\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 13.8564i 0.662842i
\(438\) 0 0
\(439\) 9.79796 0.467631 0.233816 0.972281i \(-0.424879\pi\)
0.233816 + 0.972281i \(0.424879\pi\)
\(440\) 0 0
\(441\) −15.0000 14.6969i −0.714286 0.699854i
\(442\) 0 0
\(443\) 8.48528i 0.403148i 0.979473 + 0.201574i \(0.0646056\pi\)
−0.979473 + 0.201574i \(0.935394\pi\)
\(444\) 0 0
\(445\) −16.9706 + 13.8564i −0.804482 + 0.656857i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −22.0000 −1.03824 −0.519122 0.854700i \(-0.673741\pi\)
−0.519122 + 0.854700i \(0.673741\pi\)
\(450\) 0 0
\(451\) 33.9411i 1.59823i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 19.5959 6.00000i 0.918671 0.281284i
\(456\) 0 0
\(457\) 29.3939i 1.37499i −0.726190 0.687494i \(-0.758711\pi\)
0.726190 0.687494i \(-0.241289\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −25.4558 −1.18560 −0.592798 0.805351i \(-0.701977\pi\)
−0.592798 + 0.805351i \(0.701977\pi\)
\(462\) 0 0
\(463\) −34.2929 −1.59372 −0.796862 0.604161i \(-0.793508\pi\)
−0.796862 + 0.604161i \(0.793508\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 14.1421 34.6410i 0.653023 1.59957i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.79796i 0.450511i
\(474\) 0 0
\(475\) 13.8564 + 2.82843i 0.635776 + 0.129777i
\(476\) 0 0
\(477\) −33.9411 −1.55406
\(478\) 0 0
\(479\) 9.79796 0.447680 0.223840 0.974626i \(-0.428141\pi\)
0.223840 + 0.974626i \(0.428141\pi\)
\(480\) 0 0
\(481\) 19.5959i 0.893497i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −5.65685 6.92820i −0.256865 0.314594i
\(486\) 0 0
\(487\) −4.89898 −0.221994 −0.110997 0.993821i \(-0.535404\pi\)
−0.110997 + 0.993821i \(0.535404\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 31.1769 1.40699 0.703497 0.710698i \(-0.251621\pi\)
0.703497 + 0.710698i \(0.251621\pi\)
\(492\) 0 0
\(493\) 27.7128i 1.24812i
\(494\) 0 0
\(495\) 14.6969 + 18.0000i 0.660578 + 0.809040i
\(496\) 0 0
\(497\) −4.00000 + 9.79796i −0.179425 + 0.439499i
\(498\) 0 0
\(499\) 17.3205 0.775372 0.387686 0.921791i \(-0.373274\pi\)
0.387686 + 0.921791i \(0.373274\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.0000i 0.624229i −0.950044 0.312115i \(-0.898963\pi\)
0.950044 0.312115i \(-0.101037\pi\)
\(504\) 0 0
\(505\) 12.0000 + 14.6969i 0.533993 + 0.654005i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −14.1421 −0.626839 −0.313420 0.949615i \(-0.601475\pi\)
−0.313420 + 0.949615i \(0.601475\pi\)
\(510\) 0 0
\(511\) −29.3939 12.0000i −1.30031 0.530849i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 10.3923 8.48528i 0.457940 0.373906i
\(516\) 0 0
\(517\) 34.6410i 1.52351i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 39.1918i 1.71703i −0.512792 0.858513i \(-0.671389\pi\)
0.512792 0.858513i \(-0.328611\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 39.1918 1.70722
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 25.4558i 1.10469i
\(532\) 0 0
\(533\) 33.9411 1.47015
\(534\) 0 0
\(535\) −4.89898 + 4.00000i −0.211801 + 0.172935i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −17.3205 16.9706i −0.746047 0.730974i
\(540\) 0 0
\(541\) 20.7846i 0.893600i −0.894634 0.446800i \(-0.852564\pi\)
0.894634 0.446800i \(-0.147436\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 12.0000 9.79796i 0.514024 0.419698i
\(546\) 0 0
\(547\) 14.1421i 0.604674i −0.953201 0.302337i \(-0.902233\pi\)
0.953201 0.302337i \(-0.0977668\pi\)
\(548\) 0 0
\(549\) 42.4264 1.81071
\(550\) 0 0
\(551\) 19.5959 0.834814
\(552\) 0 0
\(553\) −12.0000 + 29.3939i −0.510292 + 1.24995i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −16.9706 −0.719066 −0.359533 0.933132i \(-0.617064\pi\)
−0.359533 + 0.933132i \(0.617064\pi\)
\(558\) 0 0
\(559\) 9.79796 0.414410
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −41.5692 −1.75193 −0.875967 0.482371i \(-0.839776\pi\)
−0.875967 + 0.482371i \(0.839776\pi\)
\(564\) 0 0
\(565\) −16.9706 + 13.8564i −0.713957 + 0.582943i
\(566\) 0 0
\(567\) 22.0454 + 9.00000i 0.925820 + 0.377964i
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −24.2487 −1.01478 −0.507388 0.861717i \(-0.669389\pi\)
−0.507388 + 0.861717i \(0.669389\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.89898 24.0000i 0.204302 1.00087i
\(576\) 0 0
\(577\) −12.0000 −0.499567 −0.249783 0.968302i \(-0.580359\pi\)
−0.249783 + 0.968302i \(0.580359\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 33.9411 + 13.8564i 1.40812 + 0.574861i
\(582\) 0 0
\(583\) −39.1918 −1.62316
\(584\) 0 0
\(585\) −18.0000 + 14.6969i −0.744208 + 0.607644i
\(586\) 0 0
\(587\) 13.8564 0.571915 0.285958 0.958242i \(-0.407688\pi\)
0.285958 + 0.958242i \(0.407688\pi\)
\(588\) 0 0
\(589\) 27.7128i 1.14189i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 4.00000 0.164260 0.0821302 0.996622i \(-0.473828\pi\)
0.0821302 + 0.996622i \(0.473828\pi\)
\(594\) 0 0
\(595\) −6.92820 22.6274i −0.284029 0.927634i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 4.00000i 0.163436i 0.996656 + 0.0817178i \(0.0260406\pi\)
−0.996656 + 0.0817178i \(0.973959\pi\)
\(600\) 0 0
\(601\) 19.5959i 0.799334i 0.916660 + 0.399667i \(0.130874\pi\)
−0.916660 + 0.399667i \(0.869126\pi\)
\(602\) 0 0
\(603\) 42.4264i 1.72774i
\(604\) 0 0
\(605\) 1.41421 + 1.73205i 0.0574960 + 0.0704179i
\(606\) 0 0
\(607\) 30.0000i 1.21766i −0.793300 0.608831i \(-0.791639\pi\)
0.793300 0.608831i \(-0.208361\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −34.6410 −1.40143
\(612\) 0 0
\(613\) 11.3137 0.456956 0.228478 0.973549i \(-0.426625\pi\)
0.228478 + 0.973549i \(0.426625\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 39.1918i 1.57780i −0.614519 0.788902i \(-0.710650\pi\)
0.614519 0.788902i \(-0.289350\pi\)
\(618\) 0 0
\(619\) 31.1127i 1.25052i 0.780415 + 0.625262i \(0.215008\pi\)
−0.780415 + 0.625262i \(0.784992\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −9.79796 + 24.0000i −0.392547 + 0.961540i
\(624\) 0 0
\(625\) −23.0000 9.79796i −0.920000 0.391918i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 22.6274 0.902214
\(630\) 0 0
\(631\) 4.00000i 0.159237i 0.996825 + 0.0796187i \(0.0253703\pi\)
−0.996825 + 0.0796187i \(0.974630\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.92820 8.48528i −0.274937 0.336728i
\(636\) 0 0
\(637\) 16.9706 17.3205i 0.672398 0.686264i
\(638\) 0 0
\(639\) 12.0000i 0.474713i
\(640\) 0 0
\(641\) −2.00000 −0.0789953 −0.0394976 0.999220i \(-0.512576\pi\)
−0.0394976 + 0.999220i \(0.512576\pi\)
\(642\) 0 0
\(643\) 41.5692 1.63933 0.819665 0.572843i \(-0.194160\pi\)
0.819665 + 0.572843i \(0.194160\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 38.0000i 1.49393i 0.664861 + 0.746967i \(0.268491\pi\)
−0.664861 + 0.746967i \(0.731509\pi\)
\(648\) 0 0
\(649\) 29.3939i 1.15381i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 33.9411 1.32822 0.664109 0.747635i \(-0.268811\pi\)
0.664109 + 0.747635i \(0.268811\pi\)
\(654\) 0 0
\(655\) 4.89898 4.00000i 0.191419 0.156293i
\(656\) 0 0
\(657\) 36.0000 1.40449
\(658\) 0 0
\(659\) 10.3923 0.404827 0.202413 0.979300i \(-0.435122\pi\)
0.202413 + 0.979300i \(0.435122\pi\)
\(660\) 0 0
\(661\) −14.1421 −0.550065 −0.275033 0.961435i \(-0.588689\pi\)
−0.275033 + 0.961435i \(0.588689\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 16.0000 4.89898i 0.620453 0.189974i
\(666\) 0 0
\(667\) 33.9411i 1.31421i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 48.9898 1.89123
\(672\) 0 0
\(673\) 9.79796i 0.377684i −0.982008 0.188842i \(-0.939527\pi\)
0.982008 0.188842i \(-0.0604733\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.3205i 0.665681i 0.942983 + 0.332841i \(0.108007\pi\)
−0.942983 + 0.332841i \(0.891993\pi\)
\(678\) 0 0
\(679\) −9.79796 4.00000i −0.376011 0.153506i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 36.7696i 1.40695i 0.710721 + 0.703474i \(0.248369\pi\)
−0.710721 + 0.703474i \(0.751631\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 39.1918i 1.49309i
\(690\) 0 0
\(691\) 31.1127i 1.18358i 0.806091 + 0.591791i \(0.201579\pi\)
−0.806091 + 0.591791i \(0.798421\pi\)
\(692\) 0 0
\(693\) 25.4558 + 10.3923i 0.966988 + 0.394771i
\(694\) 0 0
\(695\) −4.89898 + 4.00000i −0.185829 + 0.151729i
\(696\) 0 0
\(697\) 39.1918i 1.48450i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 20.7846i 0.785024i 0.919747 + 0.392512i \(0.128394\pi\)
−0.919747 + 0.392512i \(0.871606\pi\)
\(702\) 0 0
\(703\) 16.0000i 0.603451i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 20.7846 + 8.48528i 0.781686 + 0.319122i
\(708\) 0 0
\(709\) 48.4974i 1.82136i 0.413114 + 0.910679i \(0.364441\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 36.0000i 1.35011i
\(712\) 0 0
\(713\) 48.0000 1.79761
\(714\) 0 0
\(715\) −20.7846 + 16.9706i −0.777300 + 0.634663i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9.79796 0.365402 0.182701 0.983169i \(-0.441516\pi\)
0.182701 + 0.983169i \(0.441516\pi\)
\(720\) 0 0
\(721\) 6.00000 14.6969i 0.223452 0.547343i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −33.9411 6.92820i −1.26054 0.257307i
\(726\) 0 0
\(727\) 42.0000i 1.55769i −0.627214 0.778847i \(-0.715805\pi\)
0.627214 0.778847i \(-0.284195\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) 11.3137i 0.418453i
\(732\) 0 0
\(733\) 17.3205i 0.639748i 0.947460 + 0.319874i \(0.103641\pi\)
−0.947460 + 0.319874i \(0.896359\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 48.9898i 1.80456i
\(738\) 0 0
\(739\) 10.3923 0.382287 0.191144 0.981562i \(-0.438780\pi\)
0.191144 + 0.981562i \(0.438780\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.89898 0.179726 0.0898631 0.995954i \(-0.471357\pi\)
0.0898631 + 0.995954i \(0.471357\pi\)
\(744\) 0 0
\(745\) −12.0000 + 9.79796i −0.439646 + 0.358969i
\(746\) 0 0
\(747\) −41.5692 −1.52094
\(748\) 0 0
\(749\) −2.82843 + 6.92820i −0.103348 + 0.253151i
\(750\) 0 0
\(751\) 44.0000i 1.60558i −0.596260 0.802791i \(-0.703347\pi\)
0.596260 0.802791i \(-0.296653\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −20.7846 + 16.9706i −0.756429 + 0.617622i
\(756\) 0 0
\(757\) −39.5980 −1.43921 −0.719607 0.694382i \(-0.755678\pi\)
−0.719607 + 0.694382i \(0.755678\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 19.5959i 0.710351i −0.934800 0.355176i \(-0.884421\pi\)
0.934800 0.355176i \(-0.115579\pi\)
\(762\) 0 0
\(763\) 6.92820 16.9706i 0.250818 0.614376i
\(764\) 0 0
\(765\) 16.9706 + 20.7846i 0.613572 + 0.751469i
\(766\) 0 0
\(767\) −29.3939 −1.06135
\(768\) 0 0
\(769\) 48.9898i 1.76662i 0.468792 + 0.883309i \(0.344689\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 10.3923i 0.373785i −0.982380 0.186893i \(-0.940158\pi\)
0.982380 0.186893i \(-0.0598416\pi\)
\(774\) 0 0
\(775\) 9.79796 48.0000i 0.351953 1.72421i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 27.7128 0.992915
\(780\) 0 0
\(781\) 13.8564i 0.495821i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 18.0000 14.6969i 0.642448 0.524556i
\(786\) 0 0
\(787\) −27.7128 −0.987855 −0.493928 0.869503i \(-0.664439\pi\)
−0.493928 + 0.869503i \(0.664439\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −9.79796 + 24.0000i −0.348375 + 0.853342i
\(792\) 0 0
\(793\) 48.9898i 1.73968i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.46410i 0.122705i 0.998116 + 0.0613524i \(0.0195413\pi\)
−0.998116 + 0.0613524i \(0.980459\pi\)
\(798\) 0 0
\(799\) 40.0000i 1.41510i
\(800\) 0 0
\(801\) 29.3939i 1.03858i
\(802\) 0 0
\(803\) 41.5692 1.46695
\(804\) 0 0
\(805\) −8.48528 27.7128i −0.299067 0.976748i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.00000 0.0703163 0.0351581 0.999382i \(-0.488807\pi\)
0.0351581 + 0.999382i \(0.488807\pi\)
\(810\) 0 0
\(811\) 19.7990i 0.695237i 0.937636 + 0.347618i \(0.113009\pi\)
−0.937636 + 0.347618i \(0.886991\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −34.2929 + 28.0000i −1.20123 + 0.980797i
\(816\) 0 0
\(817\) 8.00000 0.279885
\(818\) 0 0
\(819\) −10.3923 + 25.4558i −0.363137 + 0.889499i
\(820\) 0 0
\(821\) 48.4974i 1.69257i −0.532729 0.846286i \(-0.678834\pi\)
0.532729 0.846286i \(-0.321166\pi\)
\(822\) 0 0
\(823\) −4.89898 −0.170768 −0.0853838 0.996348i \(-0.527212\pi\)
−0.0853838 + 0.996348i \(0.527212\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.4558i 0.885186i −0.896723 0.442593i \(-0.854059\pi\)
0.896723 0.442593i \(-0.145941\pi\)
\(828\) 0 0
\(829\) 2.82843 0.0982353 0.0491177 0.998793i \(-0.484359\pi\)
0.0491177 + 0.998793i \(0.484359\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −20.0000 19.5959i −0.692959 0.678958i
\(834\) 0 0
\(835\) 17.3205 14.1421i 0.599401 0.489409i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −19.0000 −0.655172
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.41421 + 1.73205i 0.0486504 + 0.0595844i
\(846\) 0 0
\(847\) 2.44949 + 1.00000i 0.0841655 + 0.0343604i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 27.7128 0.949983
\(852\) 0 0
\(853\) 38.1051i 1.30469i −0.757920 0.652347i \(-0.773784\pi\)
0.757920 0.652347i \(-0.226216\pi\)
\(854\) 0 0
\(855\) −14.6969 + 12.0000i −0.502625 + 0.410391i
\(856\) 0 0
\(857\) −52.0000 −1.77629 −0.888143 0.459567i \(-0.848005\pi\)
−0.888143 + 0.459567i \(0.848005\pi\)
\(858\) 0 0
\(859\) 53.7401i 1.83359i −0.399359 0.916795i \(-0.630767\pi\)
0.399359 0.916795i \(-0.369233\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 34.2929 1.16734 0.583671 0.811990i \(-0.301616\pi\)
0.583671 + 0.811990i \(0.301616\pi\)
\(864\) 0 0
\(865\) 30.0000 24.4949i 1.02003 0.832851i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 41.5692i 1.41014i
\(870\) 0 0
\(871\) −48.9898 −1.65996
\(872\) 0 0
\(873\) 12.0000 0.406138
\(874\) 0 0
\(875\) −29.4449 + 2.82843i −0.995418 + 0.0956183i
\(876\) 0 0
\(877\) −28.2843 −0.955092 −0.477546 0.878607i \(-0.658474\pi\)
−0.477546 + 0.878607i \(0.658474\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 29.3939i 0.990305i 0.868806 + 0.495152i \(0.164888\pi\)
−0.868806 + 0.495152i \(0.835112\pi\)
\(882\) 0 0
\(883\) 14.1421i 0.475921i 0.971275 + 0.237960i \(0.0764788\pi\)
−0.971275 + 0.237960i \(0.923521\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 26.0000i 0.872995i 0.899706 + 0.436497i \(0.143781\pi\)
−0.899706 + 0.436497i \(0.856219\pi\)
\(888\) 0 0
\(889\) −12.0000 4.89898i −0.402467 0.164306i
\(890\) 0 0
\(891\) −31.1769 −1.04447
\(892\) 0 0
\(893\) −28.2843 −0.946497
\(894\) 0 0
\(895\) −14.6969 18.0000i −0.491264 0.601674i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 67.8823i 2.26400i
\(900\) 0 0
\(901\) −45.2548 −1.50766
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.00000 4.89898i −0.132964 0.162848i
\(906\) 0 0
\(907\) 2.82843i 0.0939164i −0.998897 0.0469582i \(-0.985047\pi\)
0.998897 0.0469582i \(-0.0149528\pi\)
\(908\) 0 0
\(909\) −25.4558 −0.844317
\(910\) 0 0
\(911\) 20.0000i 0.662630i 0.943520 + 0.331315i \(0.107492\pi\)
−0.943520 + 0.331315i \(0.892508\pi\)
\(912\) 0 0
\(913\) −48.0000 −1.58857
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.82843 6.92820i 0.0934029 0.228789i
\(918\) 0 0
\(919\) 20.0000i 0.659739i 0.944027 + 0.329870i \(0.107005\pi\)
−0.944027 + 0.329870i \(0.892995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 13.8564 0.456089
\(924\) 0 0
\(925\) 5.65685 27.7128i 0.185996 0.911192i
\(926\) 0 0
\(927\) 18.0000i 0.591198i
\(928\) 0 0
\(929\) 19.5959i 0.642921i 0.946923 + 0.321461i \(0.104174\pi\)
−0.946923 + 0.321461i \(0.895826\pi\)
\(930\) 0 0
\(931\) 13.8564 14.1421i 0.454125 0.463490i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 19.5959 + 24.0000i 0.640855 + 0.784884i
\(936\) 0 0
\(937\) −28.0000 −0.914720 −0.457360 0.889282i \(-0.651205\pi\)
−0.457360 + 0.889282i \(0.651205\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 25.4558 0.829837 0.414918 0.909859i \(-0.363810\pi\)
0.414918 + 0.909859i \(0.363810\pi\)
\(942\) 0 0
\(943\) 48.0000i 1.56310i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.1421i 0.459558i 0.973243 + 0.229779i \(0.0738003\pi\)
−0.973243 + 0.229779i \(0.926200\pi\)
\(948\) 0 0
\(949\) 41.5692i 1.34939i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 58.7878i 1.90432i 0.305597 + 0.952161i \(0.401144\pi\)
−0.305597 + 0.952161i \(0.598856\pi\)
\(954\) 0 0
\(955\) 34.6410 28.2843i 1.12096 0.915258i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 65.0000 2.09677
\(962\) 0 0
\(963\) 8.48528i 0.273434i
\(964\) 0 0
\(965\) −33.9411 + 27.7128i −1.09260 + 0.892107i
\(966\) 0 0
\(967\) 4.89898 0.157541 0.0787703 0.996893i \(-0.474901\pi\)
0.0787703 + 0.996893i \(0.474901\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.82843i 0.0907685i −0.998970 0.0453843i \(-0.985549\pi\)
0.998970 0.0453843i \(-0.0144512\pi\)
\(972\) 0 0
\(973\) −2.82843 + 6.92820i −0.0906752 + 0.222108i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 29.3939i 0.940393i 0.882562 + 0.470197i \(0.155817\pi\)
−0.882562 + 0.470197i \(0.844183\pi\)
\(978\) 0 0
\(979\) 33.9411i 1.08476i
\(980\) 0 0
\(981\) 20.7846i 0.663602i
\(982\) 0 0
\(983\) 26.0000i 0.829271i −0.909988 0.414636i \(-0.863909\pi\)
0.909988 0.414636i \(-0.136091\pi\)
\(984\) 0 0
\(985\) 16.0000 + 19.5959i 0.509802 + 0.624378i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 13.8564i 0.440608i
\(990\) 0 0
\(991\) 44.0000i 1.39771i 0.715265 + 0.698853i \(0.246306\pi\)
−0.715265 + 0.698853i \(0.753694\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −27.7128 33.9411i −0.878555 1.07601i
\(996\) 0 0
\(997\) 45.0333i 1.42622i −0.701052 0.713110i \(-0.747286\pi\)
0.701052 0.713110i \(-0.252714\pi\)
\(998\) 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 2240.2.n.c.1119.8 yes 8
4.3 odd 2 inner 2240.2.n.c.1119.7 yes 8
5.4 even 2 2240.2.n.e.1119.5 yes 8
7.6 odd 2 2240.2.n.e.1119.2 yes 8
8.3 odd 2 inner 2240.2.n.c.1119.1 8
8.5 even 2 inner 2240.2.n.c.1119.2 yes 8
20.19 odd 2 2240.2.n.e.1119.6 yes 8
28.27 even 2 2240.2.n.e.1119.1 yes 8
35.34 odd 2 inner 2240.2.n.c.1119.3 yes 8
40.19 odd 2 2240.2.n.e.1119.4 yes 8
40.29 even 2 2240.2.n.e.1119.3 yes 8
56.13 odd 2 2240.2.n.e.1119.8 yes 8
56.27 even 2 2240.2.n.e.1119.7 yes 8
140.139 even 2 inner 2240.2.n.c.1119.4 yes 8
280.69 odd 2 inner 2240.2.n.c.1119.5 yes 8
280.139 even 2 inner 2240.2.n.c.1119.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2240.2.n.c.1119.1 8 8.3 odd 2 inner
2240.2.n.c.1119.2 yes 8 8.5 even 2 inner
2240.2.n.c.1119.3 yes 8 35.34 odd 2 inner
2240.2.n.c.1119.4 yes 8 140.139 even 2 inner
2240.2.n.c.1119.5 yes 8 280.69 odd 2 inner
2240.2.n.c.1119.6 yes 8 280.139 even 2 inner
2240.2.n.c.1119.7 yes 8 4.3 odd 2 inner
2240.2.n.c.1119.8 yes 8 1.1 even 1 trivial
2240.2.n.e.1119.1 yes 8 28.27 even 2
2240.2.n.e.1119.2 yes 8 7.6 odd 2
2240.2.n.e.1119.3 yes 8 40.29 even 2
2240.2.n.e.1119.4 yes 8 40.19 odd 2
2240.2.n.e.1119.5 yes 8 5.4 even 2
2240.2.n.e.1119.6 yes 8 20.19 odd 2
2240.2.n.e.1119.7 yes 8 56.27 even 2
2240.2.n.e.1119.8 yes 8 56.13 odd 2