Properties

Label 2880.2.t.a.721.1
Level $2880$
Weight $2$
Character 2880.721
Analytic conductor $22.997$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2880,2,Mod(721,2880)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2880, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.721");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.t (of order \(4\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(22.9969157821\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 240)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 721.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2880.721
Dual form 2880.2.t.a.2161.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.707107 + 0.707107i) q^{5} -4.82843i q^{7} +O(q^{10})\) \(q+(-0.707107 + 0.707107i) q^{5} -4.82843i q^{7} +(1.41421 - 1.41421i) q^{11} +(-0.585786 - 0.585786i) q^{13} -5.41421 q^{17} +(-3.82843 - 3.82843i) q^{19} +5.41421i q^{23} -1.00000i q^{25} +(0.585786 + 0.585786i) q^{29} -3.65685 q^{31} +(3.41421 + 3.41421i) q^{35} +(4.58579 - 4.58579i) q^{37} +4.82843i q^{41} +(-3.65685 + 3.65685i) q^{43} -7.07107 q^{47} -16.3137 q^{49} +(4.00000 - 4.00000i) q^{53} +2.00000i q^{55} +(-7.41421 + 7.41421i) q^{59} +(9.48528 + 9.48528i) q^{61} +0.828427 q^{65} +(7.65685 + 7.65685i) q^{67} +8.00000i q^{71} -3.17157i q^{73} +(-6.82843 - 6.82843i) q^{77} +13.6569 q^{79} +(-3.07107 - 3.07107i) q^{83} +(3.82843 - 3.82843i) q^{85} +3.65685i q^{89} +(-2.82843 + 2.82843i) q^{91} +5.41421 q^{95} -13.3137 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{13} - 16 q^{17} - 4 q^{19} + 8 q^{29} + 8 q^{31} + 8 q^{35} + 24 q^{37} + 8 q^{43} - 20 q^{49} + 16 q^{53} - 24 q^{59} + 4 q^{61} - 8 q^{65} + 8 q^{67} - 16 q^{77} + 32 q^{79} + 16 q^{83} + 4 q^{85} + 16 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(641\) \(901\) \(2431\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{3}{4}\right)\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.707107 + 0.707107i −0.316228 + 0.316228i
\(6\) 0 0
\(7\) 4.82843i 1.82497i −0.409106 0.912487i \(-0.634159\pi\)
0.409106 0.912487i \(-0.365841\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421 1.41421i 0.426401 0.426401i −0.460999 0.887401i \(-0.652509\pi\)
0.887401 + 0.460999i \(0.152509\pi\)
\(12\) 0 0
\(13\) −0.585786 0.585786i −0.162468 0.162468i 0.621191 0.783659i \(-0.286649\pi\)
−0.783659 + 0.621191i \(0.786649\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.41421 −1.31314 −0.656570 0.754265i \(-0.727993\pi\)
−0.656570 + 0.754265i \(0.727993\pi\)
\(18\) 0 0
\(19\) −3.82843 3.82843i −0.878301 0.878301i 0.115057 0.993359i \(-0.463295\pi\)
−0.993359 + 0.115057i \(0.963295\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.41421i 1.12894i 0.825453 + 0.564471i \(0.190920\pi\)
−0.825453 + 0.564471i \(0.809080\pi\)
\(24\) 0 0
\(25\) 1.00000i 0.200000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.585786 + 0.585786i 0.108778 + 0.108778i 0.759401 0.650623i \(-0.225492\pi\)
−0.650623 + 0.759401i \(0.725492\pi\)
\(30\) 0 0
\(31\) −3.65685 −0.656790 −0.328395 0.944540i \(-0.606508\pi\)
−0.328395 + 0.944540i \(0.606508\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 3.41421 + 3.41421i 0.577107 + 0.577107i
\(36\) 0 0
\(37\) 4.58579 4.58579i 0.753899 0.753899i −0.221306 0.975204i \(-0.571032\pi\)
0.975204 + 0.221306i \(0.0710319\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.82843i 0.754074i 0.926198 + 0.377037i \(0.123057\pi\)
−0.926198 + 0.377037i \(0.876943\pi\)
\(42\) 0 0
\(43\) −3.65685 + 3.65685i −0.557665 + 0.557665i −0.928642 0.370977i \(-0.879023\pi\)
0.370977 + 0.928642i \(0.379023\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.07107 −1.03142 −0.515711 0.856763i \(-0.672472\pi\)
−0.515711 + 0.856763i \(0.672472\pi\)
\(48\) 0 0
\(49\) −16.3137 −2.33053
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.00000 4.00000i 0.549442 0.549442i −0.376837 0.926279i \(-0.622988\pi\)
0.926279 + 0.376837i \(0.122988\pi\)
\(54\) 0 0
\(55\) 2.00000i 0.269680i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −7.41421 + 7.41421i −0.965248 + 0.965248i −0.999416 0.0341677i \(-0.989122\pi\)
0.0341677 + 0.999416i \(0.489122\pi\)
\(60\) 0 0
\(61\) 9.48528 + 9.48528i 1.21447 + 1.21447i 0.969542 + 0.244923i \(0.0787628\pi\)
0.244923 + 0.969542i \(0.421237\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.828427 0.102754
\(66\) 0 0
\(67\) 7.65685 + 7.65685i 0.935434 + 0.935434i 0.998038 0.0626048i \(-0.0199408\pi\)
−0.0626048 + 0.998038i \(0.519941\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000i 0.949425i 0.880141 + 0.474713i \(0.157448\pi\)
−0.880141 + 0.474713i \(0.842552\pi\)
\(72\) 0 0
\(73\) 3.17157i 0.371205i −0.982625 0.185602i \(-0.940576\pi\)
0.982625 0.185602i \(-0.0594236\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −6.82843 6.82843i −0.778171 0.778171i
\(78\) 0 0
\(79\) 13.6569 1.53652 0.768258 0.640140i \(-0.221124\pi\)
0.768258 + 0.640140i \(0.221124\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.07107 3.07107i −0.337093 0.337093i 0.518179 0.855272i \(-0.326610\pi\)
−0.855272 + 0.518179i \(0.826610\pi\)
\(84\) 0 0
\(85\) 3.82843 3.82843i 0.415251 0.415251i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.65685i 0.387626i 0.981039 + 0.193813i \(0.0620855\pi\)
−0.981039 + 0.193813i \(0.937915\pi\)
\(90\) 0 0
\(91\) −2.82843 + 2.82843i −0.296500 + 0.296500i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.41421 0.555487
\(96\) 0 0
\(97\) −13.3137 −1.35180 −0.675901 0.736992i \(-0.736245\pi\)
−0.675901 + 0.736992i \(0.736245\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.585786 0.585786i 0.0582879 0.0582879i −0.677362 0.735650i \(-0.736877\pi\)
0.735650 + 0.677362i \(0.236877\pi\)
\(102\) 0 0
\(103\) 10.0000i 0.985329i 0.870219 + 0.492665i \(0.163977\pi\)
−0.870219 + 0.492665i \(0.836023\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.2426 12.2426i 1.18354 1.18354i 0.204720 0.978821i \(-0.434372\pi\)
0.978821 0.204720i \(-0.0656285\pi\)
\(108\) 0 0
\(109\) −5.48528 5.48528i −0.525395 0.525395i 0.393801 0.919196i \(-0.371160\pi\)
−0.919196 + 0.393801i \(0.871160\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −17.4142 −1.63819 −0.819096 0.573657i \(-0.805524\pi\)
−0.819096 + 0.573657i \(0.805524\pi\)
\(114\) 0 0
\(115\) −3.82843 3.82843i −0.357003 0.357003i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 26.1421i 2.39645i
\(120\) 0 0
\(121\) 7.00000i 0.636364i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.707107 + 0.707107i 0.0632456 + 0.0632456i
\(126\) 0 0
\(127\) −19.6569 −1.74426 −0.872132 0.489271i \(-0.837263\pi\)
−0.872132 + 0.489271i \(0.837263\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.4142 13.4142i −1.17201 1.17201i −0.981731 0.190274i \(-0.939062\pi\)
−0.190274 0.981731i \(-0.560938\pi\)
\(132\) 0 0
\(133\) −18.4853 + 18.4853i −1.60288 + 1.60288i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.24264i 0.362473i −0.983440 0.181237i \(-0.941990\pi\)
0.983440 0.181237i \(-0.0580100\pi\)
\(138\) 0 0
\(139\) −6.65685 + 6.65685i −0.564627 + 0.564627i −0.930618 0.365991i \(-0.880730\pi\)
0.365991 + 0.930618i \(0.380730\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.65685 −0.138553
\(144\) 0 0
\(145\) −0.828427 −0.0687971
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −7.07107 + 7.07107i −0.579284 + 0.579284i −0.934706 0.355422i \(-0.884337\pi\)
0.355422 + 0.934706i \(0.384337\pi\)
\(150\) 0 0
\(151\) 3.65685i 0.297591i 0.988868 + 0.148795i \(0.0475395\pi\)
−0.988868 + 0.148795i \(0.952460\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.58579 2.58579i 0.207695 0.207695i
\(156\) 0 0
\(157\) −7.07107 7.07107i −0.564333 0.564333i 0.366203 0.930535i \(-0.380658\pi\)
−0.930535 + 0.366203i \(0.880658\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 26.1421 2.06029
\(162\) 0 0
\(163\) −2.34315 2.34315i −0.183529 0.183529i 0.609362 0.792892i \(-0.291425\pi\)
−0.792892 + 0.609362i \(0.791425\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.07107i 0.237646i 0.992915 + 0.118823i \(0.0379122\pi\)
−0.992915 + 0.118823i \(0.962088\pi\)
\(168\) 0 0
\(169\) 12.3137i 0.947208i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −11.3137 11.3137i −0.860165 0.860165i 0.131192 0.991357i \(-0.458120\pi\)
−0.991357 + 0.131192i \(0.958120\pi\)
\(174\) 0 0
\(175\) −4.82843 −0.364995
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.5858 10.5858i −0.791219 0.791219i 0.190474 0.981692i \(-0.438998\pi\)
−0.981692 + 0.190474i \(0.938998\pi\)
\(180\) 0 0
\(181\) 8.17157 8.17157i 0.607388 0.607388i −0.334875 0.942263i \(-0.608694\pi\)
0.942263 + 0.334875i \(0.108694\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.48528i 0.476807i
\(186\) 0 0
\(187\) −7.65685 + 7.65685i −0.559925 + 0.559925i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.48528 −0.324544 −0.162272 0.986746i \(-0.551882\pi\)
−0.162272 + 0.986746i \(0.551882\pi\)
\(192\) 0 0
\(193\) 0.828427 0.0596315 0.0298157 0.999555i \(-0.490508\pi\)
0.0298157 + 0.999555i \(0.490508\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.75736 7.75736i 0.552689 0.552689i −0.374527 0.927216i \(-0.622195\pi\)
0.927216 + 0.374527i \(0.122195\pi\)
\(198\) 0 0
\(199\) 6.34315i 0.449654i 0.974399 + 0.224827i \(0.0721816\pi\)
−0.974399 + 0.224827i \(0.927818\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.82843 2.82843i 0.198517 0.198517i
\(204\) 0 0
\(205\) −3.41421 3.41421i −0.238459 0.238459i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −10.8284 −0.749018
\(210\) 0 0
\(211\) −2.65685 2.65685i −0.182905 0.182905i 0.609715 0.792621i \(-0.291284\pi\)
−0.792621 + 0.609715i \(0.791284\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.17157i 0.352698i
\(216\) 0 0
\(217\) 17.6569i 1.19863i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.17157 + 3.17157i 0.213343 + 0.213343i
\(222\) 0 0
\(223\) 15.1716 1.01596 0.507982 0.861368i \(-0.330392\pi\)
0.507982 + 0.861368i \(0.330392\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.5858 14.5858i −0.968093 0.968093i 0.0314138 0.999506i \(-0.489999\pi\)
−0.999506 + 0.0314138i \(0.989999\pi\)
\(228\) 0 0
\(229\) −13.0000 + 13.0000i −0.859064 + 0.859064i −0.991228 0.132164i \(-0.957808\pi\)
0.132164 + 0.991228i \(0.457808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.55635i 0.232984i −0.993192 0.116492i \(-0.962835\pi\)
0.993192 0.116492i \(-0.0371650\pi\)
\(234\) 0 0
\(235\) 5.00000 5.00000i 0.326164 0.326164i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.9706 1.35647 0.678236 0.734844i \(-0.262745\pi\)
0.678236 + 0.734844i \(0.262745\pi\)
\(240\) 0 0
\(241\) −10.3431 −0.666261 −0.333130 0.942881i \(-0.608105\pi\)
−0.333130 + 0.942881i \(0.608105\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 11.5355 11.5355i 0.736978 0.736978i
\(246\) 0 0
\(247\) 4.48528i 0.285392i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.89949 + 3.89949i −0.246134 + 0.246134i −0.819382 0.573248i \(-0.805683\pi\)
0.573248 + 0.819382i \(0.305683\pi\)
\(252\) 0 0
\(253\) 7.65685 + 7.65685i 0.481382 + 0.481382i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.3848 −0.647785 −0.323892 0.946094i \(-0.604992\pi\)
−0.323892 + 0.946094i \(0.604992\pi\)
\(258\) 0 0
\(259\) −22.1421 22.1421i −1.37585 1.37585i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 22.3848i 1.38030i −0.723664 0.690152i \(-0.757544\pi\)
0.723664 0.690152i \(-0.242456\pi\)
\(264\) 0 0
\(265\) 5.65685i 0.347498i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 18.7279 + 18.7279i 1.14186 + 1.14186i 0.988109 + 0.153752i \(0.0491357\pi\)
0.153752 + 0.988109i \(0.450864\pi\)
\(270\) 0 0
\(271\) −23.3137 −1.41621 −0.708103 0.706109i \(-0.750449\pi\)
−0.708103 + 0.706109i \(0.750449\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.41421 1.41421i −0.0852803 0.0852803i
\(276\) 0 0
\(277\) 16.5858 16.5858i 0.996543 0.996543i −0.00345072 0.999994i \(-0.501098\pi\)
0.999994 + 0.00345072i \(0.00109840\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.1421i 0.962959i −0.876457 0.481480i \(-0.840100\pi\)
0.876457 0.481480i \(-0.159900\pi\)
\(282\) 0 0
\(283\) 11.6569 11.6569i 0.692928 0.692928i −0.269947 0.962875i \(-0.587006\pi\)
0.962875 + 0.269947i \(0.0870062\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 23.3137 1.37616
\(288\) 0 0
\(289\) 12.3137 0.724336
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.34315 2.34315i 0.136888 0.136888i −0.635342 0.772231i \(-0.719141\pi\)
0.772231 + 0.635342i \(0.219141\pi\)
\(294\) 0 0
\(295\) 10.4853i 0.610477i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.17157 3.17157i 0.183417 0.183417i
\(300\) 0 0
\(301\) 17.6569 + 17.6569i 1.01772 + 1.01772i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −13.4142 −0.768096
\(306\) 0 0
\(307\) 3.65685 + 3.65685i 0.208708 + 0.208708i 0.803718 0.595010i \(-0.202852\pi\)
−0.595010 + 0.803718i \(0.702852\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0.970563i 0.0550356i −0.999621 0.0275178i \(-0.991240\pi\)
0.999621 0.0275178i \(-0.00876029\pi\)
\(312\) 0 0
\(313\) 11.1716i 0.631455i 0.948850 + 0.315727i \(0.102248\pi\)
−0.948850 + 0.315727i \(0.897752\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 6.58579 + 6.58579i 0.369895 + 0.369895i 0.867439 0.497544i \(-0.165765\pi\)
−0.497544 + 0.867439i \(0.665765\pi\)
\(318\) 0 0
\(319\) 1.65685 0.0927660
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 20.7279 + 20.7279i 1.15333 + 1.15333i
\(324\) 0 0
\(325\) −0.585786 + 0.585786i −0.0324936 + 0.0324936i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 34.1421i 1.88232i
\(330\) 0 0
\(331\) −10.1716 + 10.1716i −0.559080 + 0.559080i −0.929046 0.369965i \(-0.879370\pi\)
0.369965 + 0.929046i \(0.379370\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −10.8284 −0.591620
\(336\) 0 0
\(337\) 2.48528 0.135382 0.0676910 0.997706i \(-0.478437\pi\)
0.0676910 + 0.997706i \(0.478437\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5.17157 + 5.17157i −0.280056 + 0.280056i
\(342\) 0 0
\(343\) 44.9706i 2.42818i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.17157 9.17157i 0.492356 0.492356i −0.416692 0.909048i \(-0.636811\pi\)
0.909048 + 0.416692i \(0.136811\pi\)
\(348\) 0 0
\(349\) −7.48528 7.48528i −0.400678 0.400678i 0.477794 0.878472i \(-0.341437\pi\)
−0.878472 + 0.477794i \(0.841437\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −21.4142 −1.13976 −0.569882 0.821727i \(-0.693011\pi\)
−0.569882 + 0.821727i \(0.693011\pi\)
\(354\) 0 0
\(355\) −5.65685 5.65685i −0.300235 0.300235i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.8284i 0.993726i 0.867829 + 0.496863i \(0.165515\pi\)
−0.867829 + 0.496863i \(0.834485\pi\)
\(360\) 0 0
\(361\) 10.3137i 0.542827i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 2.24264 + 2.24264i 0.117385 + 0.117385i
\(366\) 0 0
\(367\) 34.9706 1.82545 0.912724 0.408576i \(-0.133975\pi\)
0.912724 + 0.408576i \(0.133975\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −19.3137 19.3137i −1.00272 1.00272i
\(372\) 0 0
\(373\) −11.0711 + 11.0711i −0.573238 + 0.573238i −0.933032 0.359794i \(-0.882847\pi\)
0.359794 + 0.933032i \(0.382847\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.686292i 0.0353458i
\(378\) 0 0
\(379\) −5.34315 + 5.34315i −0.274459 + 0.274459i −0.830892 0.556433i \(-0.812169\pi\)
0.556433 + 0.830892i \(0.312169\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −23.0711 −1.17888 −0.589438 0.807813i \(-0.700651\pi\)
−0.589438 + 0.807813i \(0.700651\pi\)
\(384\) 0 0
\(385\) 9.65685 0.492159
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 10.3848 10.3848i 0.526529 0.526529i −0.393007 0.919536i \(-0.628565\pi\)
0.919536 + 0.393007i \(0.128565\pi\)
\(390\) 0 0
\(391\) 29.3137i 1.48246i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.65685 + 9.65685i −0.485889 + 0.485889i
\(396\) 0 0
\(397\) −9.89949 9.89949i −0.496841 0.496841i 0.413612 0.910453i \(-0.364267\pi\)
−0.910453 + 0.413612i \(0.864267\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −26.9706 −1.34685 −0.673423 0.739258i \(-0.735177\pi\)
−0.673423 + 0.739258i \(0.735177\pi\)
\(402\) 0 0
\(403\) 2.14214 + 2.14214i 0.106707 + 0.106707i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.9706i 0.642927i
\(408\) 0 0
\(409\) 30.6274i 1.51443i 0.653167 + 0.757214i \(0.273440\pi\)
−0.653167 + 0.757214i \(0.726560\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 35.7990 + 35.7990i 1.76155 + 1.76155i
\(414\) 0 0
\(415\) 4.34315 0.213197
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.2426 + 10.2426i 0.500386 + 0.500386i 0.911558 0.411172i \(-0.134880\pi\)
−0.411172 + 0.911558i \(0.634880\pi\)
\(420\) 0 0
\(421\) 3.00000 3.00000i 0.146211 0.146211i −0.630212 0.776423i \(-0.717032\pi\)
0.776423 + 0.630212i \(0.217032\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.41421i 0.262628i
\(426\) 0 0
\(427\) 45.7990 45.7990i 2.21637 2.21637i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.4853 −0.601395 −0.300697 0.953720i \(-0.597219\pi\)
−0.300697 + 0.953720i \(0.597219\pi\)
\(432\) 0 0
\(433\) 36.8284 1.76986 0.884931 0.465723i \(-0.154206\pi\)
0.884931 + 0.465723i \(0.154206\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 20.7279 20.7279i 0.991551 0.991551i
\(438\) 0 0
\(439\) 6.34315i 0.302742i −0.988477 0.151371i \(-0.951631\pi\)
0.988477 0.151371i \(-0.0483688\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10.5858 + 10.5858i −0.502946 + 0.502946i −0.912352 0.409406i \(-0.865736\pi\)
0.409406 + 0.912352i \(0.365736\pi\)
\(444\) 0 0
\(445\) −2.58579 2.58579i −0.122578 0.122578i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −8.82843 −0.416639 −0.208320 0.978061i \(-0.566799\pi\)
−0.208320 + 0.978061i \(0.566799\pi\)
\(450\) 0 0
\(451\) 6.82843 + 6.82843i 0.321538 + 0.321538i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.00000i 0.187523i
\(456\) 0 0
\(457\) 23.6569i 1.10662i −0.832975 0.553310i \(-0.813364\pi\)
0.832975 0.553310i \(-0.186636\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.72792 + 8.72792i 0.406500 + 0.406500i 0.880516 0.474016i \(-0.157196\pi\)
−0.474016 + 0.880516i \(0.657196\pi\)
\(462\) 0 0
\(463\) −11.6569 −0.541740 −0.270870 0.962616i \(-0.587311\pi\)
−0.270870 + 0.962616i \(0.587311\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 7.31371 + 7.31371i 0.338438 + 0.338438i 0.855779 0.517341i \(-0.173078\pi\)
−0.517341 + 0.855779i \(0.673078\pi\)
\(468\) 0 0
\(469\) 36.9706 36.9706i 1.70714 1.70714i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 10.3431i 0.475578i
\(474\) 0 0
\(475\) −3.82843 + 3.82843i −0.175660 + 0.175660i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.00000 −0.182765 −0.0913823 0.995816i \(-0.529129\pi\)
−0.0913823 + 0.995816i \(0.529129\pi\)
\(480\) 0 0
\(481\) −5.37258 −0.244969
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.41421 9.41421i 0.427477 0.427477i
\(486\) 0 0
\(487\) 26.0000i 1.17817i −0.808070 0.589086i \(-0.799488\pi\)
0.808070 0.589086i \(-0.200512\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.24264 2.24264i 0.101209 0.101209i −0.654689 0.755898i \(-0.727200\pi\)
0.755898 + 0.654689i \(0.227200\pi\)
\(492\) 0 0
\(493\) −3.17157 3.17157i −0.142840 0.142840i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 38.6274 1.73268
\(498\) 0 0
\(499\) 4.51472 + 4.51472i 0.202107 + 0.202107i 0.800902 0.598795i \(-0.204354\pi\)
−0.598795 + 0.800902i \(0.704354\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 20.2426i 0.902575i −0.892379 0.451287i \(-0.850965\pi\)
0.892379 0.451287i \(-0.149035\pi\)
\(504\) 0 0
\(505\) 0.828427i 0.0368645i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.58579 + 8.58579i 0.380558 + 0.380558i 0.871303 0.490745i \(-0.163275\pi\)
−0.490745 + 0.871303i \(0.663275\pi\)
\(510\) 0 0
\(511\) −15.3137 −0.677439
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −7.07107 7.07107i −0.311588 0.311588i
\(516\) 0 0
\(517\) −10.0000 + 10.0000i −0.439799 + 0.439799i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18.4853i 0.809855i 0.914349 + 0.404927i \(0.132703\pi\)
−0.914349 + 0.404927i \(0.867297\pi\)
\(522\) 0 0
\(523\) 16.6274 16.6274i 0.727066 0.727066i −0.242968 0.970034i \(-0.578121\pi\)
0.970034 + 0.242968i \(0.0781210\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 19.7990 0.862458
\(528\) 0 0
\(529\) −6.31371 −0.274509
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 2.82843 2.82843i 0.122513 0.122513i
\(534\) 0 0
\(535\) 17.3137i 0.748537i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −23.0711 + 23.0711i −0.993741 + 0.993741i
\(540\) 0 0
\(541\) −17.4853 17.4853i −0.751751 0.751751i 0.223055 0.974806i \(-0.428397\pi\)
−0.974806 + 0.223055i \(0.928397\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.75736 0.332289
\(546\) 0 0
\(547\) 14.4853 + 14.4853i 0.619346 + 0.619346i 0.945364 0.326018i \(-0.105707\pi\)
−0.326018 + 0.945364i \(0.605707\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.48528i 0.191079i
\(552\) 0 0
\(553\) 65.9411i 2.80410i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −13.1716 13.1716i −0.558097 0.558097i 0.370668 0.928765i \(-0.379129\pi\)
−0.928765 + 0.370668i \(0.879129\pi\)
\(558\) 0 0
\(559\) 4.28427 0.181205
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.48528 + 8.48528i 0.357612 + 0.357612i 0.862932 0.505320i \(-0.168626\pi\)
−0.505320 + 0.862932i \(0.668626\pi\)
\(564\) 0 0
\(565\) 12.3137 12.3137i 0.518042 0.518042i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.6863i 0.447993i 0.974590 + 0.223996i \(0.0719104\pi\)
−0.974590 + 0.223996i \(0.928090\pi\)
\(570\) 0 0
\(571\) 18.1716 18.1716i 0.760457 0.760457i −0.215948 0.976405i \(-0.569284\pi\)
0.976405 + 0.215948i \(0.0692842\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 5.41421 0.225788
\(576\) 0 0
\(577\) −27.6569 −1.15137 −0.575685 0.817672i \(-0.695265\pi\)
−0.575685 + 0.817672i \(0.695265\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −14.8284 + 14.8284i −0.615187 + 0.615187i
\(582\) 0 0
\(583\) 11.3137i 0.468566i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −3.07107 + 3.07107i −0.126757 + 0.126757i −0.767639 0.640882i \(-0.778569\pi\)
0.640882 + 0.767639i \(0.278569\pi\)
\(588\) 0 0
\(589\) 14.0000 + 14.0000i 0.576860 + 0.576860i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −33.8995 −1.39209 −0.696043 0.718000i \(-0.745058\pi\)
−0.696043 + 0.718000i \(0.745058\pi\)
\(594\) 0 0
\(595\) −18.4853 18.4853i −0.757823 0.757823i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.85786i 0.0759103i 0.999279 + 0.0379551i \(0.0120844\pi\)
−0.999279 + 0.0379551i \(0.987916\pi\)
\(600\) 0 0
\(601\) 4.97056i 0.202753i −0.994848 0.101377i \(-0.967675\pi\)
0.994848 0.101377i \(-0.0323247\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −4.94975 4.94975i −0.201236 0.201236i
\(606\) 0 0
\(607\) 18.4853 0.750294 0.375147 0.926965i \(-0.377592\pi\)
0.375147 + 0.926965i \(0.377592\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 4.14214 + 4.14214i 0.167573 + 0.167573i
\(612\) 0 0
\(613\) −1.41421 + 1.41421i −0.0571195 + 0.0571195i −0.735090 0.677970i \(-0.762860\pi\)
0.677970 + 0.735090i \(0.262860\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 40.7279i 1.63964i −0.572618 0.819822i \(-0.694072\pi\)
0.572618 0.819822i \(-0.305928\pi\)
\(618\) 0 0
\(619\) 16.3137 16.3137i 0.655703 0.655703i −0.298657 0.954360i \(-0.596539\pi\)
0.954360 + 0.298657i \(0.0965387\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 17.6569 0.707407
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −24.8284 + 24.8284i −0.989974 + 0.989974i
\(630\) 0 0
\(631\) 11.3137i 0.450392i −0.974314 0.225196i \(-0.927698\pi\)
0.974314 0.225196i \(-0.0723022\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 13.8995 13.8995i 0.551585 0.551585i
\(636\) 0 0
\(637\) 9.55635 + 9.55635i 0.378636 + 0.378636i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −9.31371 −0.367869 −0.183935 0.982938i \(-0.558883\pi\)
−0.183935 + 0.982938i \(0.558883\pi\)
\(642\) 0 0
\(643\) 0.970563 + 0.970563i 0.0382753 + 0.0382753i 0.725985 0.687710i \(-0.241384\pi\)
−0.687710 + 0.725985i \(0.741384\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 7.07107i 0.277992i 0.990293 + 0.138996i \(0.0443876\pi\)
−0.990293 + 0.138996i \(0.955612\pi\)
\(648\) 0 0
\(649\) 20.9706i 0.823167i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9.65685 9.65685i −0.377902 0.377902i 0.492443 0.870345i \(-0.336104\pi\)
−0.870345 + 0.492443i \(0.836104\pi\)
\(654\) 0 0
\(655\) 18.9706 0.741241
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 9.27208 + 9.27208i 0.361189 + 0.361189i 0.864251 0.503062i \(-0.167793\pi\)
−0.503062 + 0.864251i \(0.667793\pi\)
\(660\) 0 0
\(661\) 11.8284 11.8284i 0.460072 0.460072i −0.438607 0.898679i \(-0.644528\pi\)
0.898679 + 0.438607i \(0.144528\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 26.1421i 1.01375i
\(666\) 0 0
\(667\) −3.17157 + 3.17157i −0.122804 + 0.122804i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 26.8284 1.03570
\(672\) 0 0
\(673\) 25.7990 0.994478 0.497239 0.867614i \(-0.334347\pi\)
0.497239 + 0.867614i \(0.334347\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 20.0000 20.0000i 0.768662 0.768662i −0.209209 0.977871i \(-0.567089\pi\)
0.977871 + 0.209209i \(0.0670888\pi\)
\(678\) 0 0
\(679\) 64.2843i 2.46700i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −14.3431 + 14.3431i −0.548825 + 0.548825i −0.926101 0.377276i \(-0.876861\pi\)
0.377276 + 0.926101i \(0.376861\pi\)
\(684\) 0 0
\(685\) 3.00000 + 3.00000i 0.114624 + 0.114624i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.68629 −0.178533
\(690\) 0 0
\(691\) 4.17157 + 4.17157i 0.158694 + 0.158694i 0.781988 0.623294i \(-0.214206\pi\)
−0.623294 + 0.781988i \(0.714206\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 9.41421i 0.357101i
\(696\) 0 0
\(697\) 26.1421i 0.990204i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −29.0711 29.0711i −1.09800 1.09800i −0.994645 0.103354i \(-0.967042\pi\)
−0.103354 0.994645i \(-0.532958\pi\)
\(702\) 0 0
\(703\) −35.1127 −1.32430
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.82843 2.82843i −0.106374 0.106374i
\(708\) 0 0
\(709\) 29.2843 29.2843i 1.09979 1.09979i 0.105360 0.994434i \(-0.466401\pi\)
0.994434 0.105360i \(-0.0335994\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 19.7990i 0.741478i
\(714\) 0 0
\(715\) 1.17157 1.17157i 0.0438143 0.0438143i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.97056 0.334546 0.167273 0.985911i \(-0.446504\pi\)
0.167273 + 0.985911i \(0.446504\pi\)
\(720\) 0 0
\(721\) 48.2843 1.79820
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.585786 0.585786i 0.0217556 0.0217556i
\(726\) 0 0
\(727\) 17.3137i 0.642130i 0.947057 + 0.321065i \(0.104041\pi\)
−0.947057 + 0.321065i \(0.895959\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 19.7990 19.7990i 0.732292 0.732292i
\(732\) 0 0
\(733\) −7.41421 7.41421i −0.273850 0.273850i 0.556798 0.830648i \(-0.312030\pi\)
−0.830648 + 0.556798i \(0.812030\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 21.6569 0.797740
\(738\) 0 0
\(739\) 7.82843 + 7.82843i 0.287973 + 0.287973i 0.836278 0.548305i \(-0.184727\pi\)
−0.548305 + 0.836278i \(0.684727\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 46.1838i 1.69432i 0.531339 + 0.847159i \(0.321689\pi\)
−0.531339 + 0.847159i \(0.678311\pi\)
\(744\) 0 0
\(745\) 10.0000i 0.366372i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −59.1127 59.1127i −2.15993 2.15993i
\(750\) 0 0
\(751\) 11.6569 0.425365 0.212682 0.977121i \(-0.431780\pi\)
0.212682 + 0.977121i \(0.431780\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −2.58579 2.58579i −0.0941064 0.0941064i
\(756\) 0 0
\(757\) −4.58579 + 4.58579i −0.166673 + 0.166673i −0.785515 0.618842i \(-0.787602\pi\)
0.618842 + 0.785515i \(0.287602\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 44.8284i 1.62503i −0.582941 0.812515i \(-0.698098\pi\)
0.582941 0.812515i \(-0.301902\pi\)
\(762\) 0 0
\(763\) −26.4853 + 26.4853i −0.958832 + 0.958832i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.68629 0.313644
\(768\) 0 0
\(769\) −40.9706 −1.47744 −0.738718 0.674014i \(-0.764569\pi\)
−0.738718 + 0.674014i \(0.764569\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 3.27208 3.27208i 0.117688 0.117688i −0.645810 0.763498i \(-0.723480\pi\)
0.763498 + 0.645810i \(0.223480\pi\)
\(774\) 0 0
\(775\) 3.65685i 0.131358i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 18.4853 18.4853i 0.662304 0.662304i
\(780\) 0 0
\(781\) 11.3137 + 11.3137i 0.404836 + 0.404836i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 10.0000 0.356915
\(786\) 0 0
\(787\) −16.1421 16.1421i −0.575405 0.575405i 0.358229 0.933634i \(-0.383381\pi\)
−0.933634 + 0.358229i \(0.883381\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 84.0833i 2.98966i
\(792\) 0 0
\(793\) 11.1127i 0.394623i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.89949 + 9.89949i 0.350658 + 0.350658i 0.860354 0.509696i \(-0.170242\pi\)
−0.509696 + 0.860354i \(0.670242\pi\)
\(798\) 0 0
\(799\) 38.2843 1.35440
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −4.48528 4.48528i −0.158282 0.158282i
\(804\) 0 0
\(805\) −18.4853 + 18.4853i −0.651521 + 0.651521i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 22.9706i 0.807602i 0.914847 + 0.403801i \(0.132311\pi\)
−0.914847 + 0.403801i \(0.867689\pi\)
\(810\) 0 0
\(811\) −21.0000 + 21.0000i −0.737410 + 0.737410i −0.972076 0.234666i \(-0.924600\pi\)
0.234666 + 0.972076i \(0.424600\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 3.31371 0.116074
\(816\) 0 0
\(817\) 28.0000 0.979596
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.92893 + 2.92893i −0.102220 + 0.102220i −0.756367 0.654147i \(-0.773028\pi\)
0.654147 + 0.756367i \(0.273028\pi\)
\(822\) 0 0
\(823\) 54.0833i 1.88522i −0.333890 0.942612i \(-0.608361\pi\)
0.333890 0.942612i \(-0.391639\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 26.1421 26.1421i 0.909051 0.909051i −0.0871446 0.996196i \(-0.527774\pi\)
0.996196 + 0.0871446i \(0.0277742\pi\)
\(828\) 0 0
\(829\) −12.5147 12.5147i −0.434654 0.434654i 0.455554 0.890208i \(-0.349441\pi\)
−0.890208 + 0.455554i \(0.849441\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 88.3259 3.06031
\(834\) 0 0
\(835\) −2.17157 2.17157i −0.0751504 0.0751504i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 15.7990i 0.545442i −0.962093 0.272721i \(-0.912076\pi\)
0.962093 0.272721i \(-0.0879235\pi\)
\(840\) 0 0
\(841\) 28.3137i 0.976335i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 8.70711 + 8.70711i 0.299534 + 0.299534i
\(846\) 0 0
\(847\) 33.7990 1.16135
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 24.8284 + 24.8284i 0.851108 + 0.851108i
\(852\) 0 0
\(853\) −40.0416 + 40.0416i −1.37100 + 1.37100i −0.512034 + 0.858965i \(0.671108\pi\)
−0.858965 + 0.512034i \(0.828892\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 51.5563i 1.76113i −0.473924 0.880566i \(-0.657163\pi\)
0.473924 0.880566i \(-0.342837\pi\)
\(858\) 0 0
\(859\) −29.2843 + 29.2843i −0.999166 + 0.999166i −1.00000 0.000833212i \(-0.999735\pi\)
0.000833212 1.00000i \(0.499735\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −29.4142 −1.00127 −0.500636 0.865658i \(-0.666900\pi\)
−0.500636 + 0.865658i \(0.666900\pi\)
\(864\) 0 0
\(865\) 16.0000 0.544016
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 19.3137 19.3137i 0.655173 0.655173i
\(870\) 0 0
\(871\) 8.97056i 0.303956i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.41421 3.41421i 0.115421 0.115421i
\(876\) 0 0
\(877\) 19.0711 + 19.0711i 0.643984 + 0.643984i 0.951532 0.307548i \(-0.0995086\pi\)
−0.307548 + 0.951532i \(0.599509\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 39.9411 1.34565 0.672825 0.739801i \(-0.265081\pi\)
0.672825 + 0.739801i \(0.265081\pi\)
\(882\) 0 0
\(883\) 19.5147 + 19.5147i 0.656723 + 0.656723i 0.954603 0.297881i \(-0.0962797\pi\)
−0.297881 + 0.954603i \(0.596280\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 14.7868i 0.496492i −0.968697 0.248246i \(-0.920146\pi\)
0.968697 0.248246i \(-0.0798541\pi\)
\(888\) 0 0
\(889\) 94.9117i 3.18324i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 27.0711 + 27.0711i 0.905899 + 0.905899i
\(894\) 0 0
\(895\) 14.9706 0.500411
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.14214 2.14214i −0.0714442 0.0714442i
\(900\) 0 0
\(901\) −21.6569 + 21.6569i −0.721494 + 0.721494i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 11.5563i 0.384146i
\(906\) 0 0
\(907\) −25.1716 + 25.1716i −0.835808 + 0.835808i −0.988304 0.152496i \(-0.951269\pi\)
0.152496 + 0.988304i \(0.451269\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 6.82843 0.226236 0.113118 0.993582i \(-0.463916\pi\)
0.113118 + 0.993582i \(0.463916\pi\)
\(912\) 0 0
\(913\) −8.68629 −0.287474
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −64.7696 + 64.7696i −2.13888 + 2.13888i
\(918\) 0 0
\(919\) 16.6274i 0.548488i 0.961660 + 0.274244i \(0.0884276\pi\)
−0.961660 + 0.274244i \(0.911572\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 4.68629 4.68629i 0.154251 0.154251i
\(924\) 0 0
\(925\) −4.58579 4.58579i −0.150780 0.150780i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −3.45584 −0.113383 −0.0566913 0.998392i \(-0.518055\pi\)
−0.0566913 + 0.998392i \(0.518055\pi\)
\(930\) 0 0
\(931\) 62.4558 + 62.4558i 2.04691 + 2.04691i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 10.8284i 0.354127i
\(936\) 0 0
\(937\) 36.6274i 1.19657i −0.801285 0.598283i \(-0.795850\pi\)
0.801285 0.598283i \(-0.204150\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.2132 + 21.2132i 0.691531 + 0.691531i 0.962569 0.271038i \(-0.0873669\pi\)
−0.271038 + 0.962569i \(0.587367\pi\)
\(942\) 0 0
\(943\) −26.1421 −0.851305
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −11.0711 11.0711i −0.359761 0.359761i 0.503964 0.863725i \(-0.331875\pi\)
−0.863725 + 0.503964i \(0.831875\pi\)
\(948\) 0 0
\(949\) −1.85786 + 1.85786i −0.0603088 + 0.0603088i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 57.9828i 1.87825i 0.343582 + 0.939123i \(0.388360\pi\)
−0.343582 + 0.939123i \(0.611640\pi\)
\(954\) 0 0
\(955\) 3.17157 3.17157i 0.102630 0.102630i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −20.4853 −0.661504
\(960\) 0 0
\(961\) −17.6274 −0.568626
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −0.585786 + 0.585786i −0.0188571 + 0.0188571i
\(966\) 0 0
\(967\) 19.4558i 0.625658i −0.949810 0.312829i \(-0.898723\pi\)
0.949810 0.312829i \(-0.101277\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 22.2426 22.2426i 0.713800 0.713800i −0.253528 0.967328i \(-0.581591\pi\)
0.967328 + 0.253528i \(0.0815909\pi\)
\(972\) 0 0
\(973\) 32.1421 + 32.1421i 1.03043 + 1.03043i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 11.5563 0.369720 0.184860 0.982765i \(-0.440817\pi\)
0.184860 + 0.982765i \(0.440817\pi\)
\(978\) 0 0
\(979\) 5.17157 + 5.17157i 0.165284 + 0.165284i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 20.7279i 0.661118i −0.943785 0.330559i \(-0.892763\pi\)
0.943785 0.330559i \(-0.107237\pi\)
\(984\) 0 0
\(985\) 10.9706i 0.349551i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −19.7990 19.7990i −0.629571 0.629571i
\(990\) 0 0
\(991\) 4.00000 0.127064 0.0635321 0.997980i \(-0.479763\pi\)
0.0635321 + 0.997980i \(0.479763\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −4.48528 4.48528i −0.142193 0.142193i
\(996\) 0 0
\(997\) 9.75736 9.75736i 0.309019 0.309019i −0.535510 0.844529i \(-0.679881\pi\)
0.844529 + 0.535510i \(0.179881\pi\)
\(998\) 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 2880.2.t.a.721.1 4
3.2 odd 2 960.2.s.a.721.2 4
4.3 odd 2 720.2.t.a.541.2 4
12.11 even 2 240.2.s.a.61.1 4
16.5 even 4 inner 2880.2.t.a.2161.1 4
16.11 odd 4 720.2.t.a.181.2 4
24.5 odd 2 1920.2.s.b.1441.1 4
24.11 even 2 1920.2.s.a.1441.2 4
48.5 odd 4 960.2.s.a.241.2 4
48.11 even 4 240.2.s.a.181.1 yes 4
48.29 odd 4 1920.2.s.b.481.1 4
48.35 even 4 1920.2.s.a.481.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
240.2.s.a.61.1 4 12.11 even 2
240.2.s.a.181.1 yes 4 48.11 even 4
720.2.t.a.181.2 4 16.11 odd 4
720.2.t.a.541.2 4 4.3 odd 2
960.2.s.a.241.2 4 48.5 odd 4
960.2.s.a.721.2 4 3.2 odd 2
1920.2.s.a.481.2 4 48.35 even 4
1920.2.s.a.1441.2 4 24.11 even 2
1920.2.s.b.481.1 4 48.29 odd 4
1920.2.s.b.1441.1 4 24.5 odd 2
2880.2.t.a.721.1 4 1.1 even 1 trivial
2880.2.t.a.2161.1 4 16.5 even 4 inner