Properties

Label 2880.2.w.n.2753.4
Level $2880$
Weight $2$
Character 2880.2753
Analytic conductor $22.997$
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] = [2880,2,Mod(2177,2880)]
 
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, 0, 2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2880.2177");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2880.w (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: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
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: \( 2^{10} \)
Twist minimal: no (minimal twist has level 1440)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 2753.4
Root \(0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 2880.2753
Dual form 2880.2.w.n.2177.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.12132 - 0.707107i) q^{5} +(3.46410 - 3.46410i) q^{7} +O(q^{10})\) \(q+(2.12132 - 0.707107i) q^{5} +(3.46410 - 3.46410i) q^{7} +4.89898i q^{11} +(-1.00000 - 1.00000i) q^{13} +(-2.82843 - 2.82843i) q^{17} -6.92820i q^{19} +(4.00000 - 3.00000i) q^{25} +7.07107 q^{29} -6.92820 q^{31} +(4.89898 - 9.79796i) q^{35} +(-1.00000 + 1.00000i) q^{37} -9.89949i q^{41} +(6.92820 + 6.92820i) q^{43} +(4.89898 + 4.89898i) q^{47} -17.0000i q^{49} +(2.82843 - 2.82843i) q^{53} +(3.46410 + 10.3923i) q^{55} -4.89898 q^{59} +(-2.82843 - 1.41421i) q^{65} +(-6.92820 + 6.92820i) q^{67} +(-7.00000 - 7.00000i) q^{73} +(16.9706 + 16.9706i) q^{77} -6.92820i q^{79} +(-4.89898 + 4.89898i) q^{83} +(-8.00000 - 4.00000i) q^{85} +9.89949 q^{89} -6.92820 q^{91} +(-4.89898 - 14.6969i) q^{95} +(5.00000 - 5.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{13} + 32 q^{25} - 8 q^{37} - 56 q^{73} - 64 q^{85} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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
\(5\) 2.12132 0.707107i 0.948683 0.316228i
\(6\) 0 0
\(7\) 3.46410 3.46410i 1.30931 1.30931i 0.387392 0.921915i \(-0.373376\pi\)
0.921915 0.387392i \(-0.126624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 4.89898i 1.47710i 0.674200 + 0.738549i \(0.264489\pi\)
−0.674200 + 0.738549i \(0.735511\pi\)
\(12\) 0 0
\(13\) −1.00000 1.00000i −0.277350 0.277350i 0.554700 0.832050i \(-0.312833\pi\)
−0.832050 + 0.554700i \(0.812833\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.82843 2.82843i −0.685994 0.685994i 0.275350 0.961344i \(-0.411206\pi\)
−0.961344 + 0.275350i \(0.911206\pi\)
\(18\) 0 0
\(19\) 6.92820i 1.58944i −0.606977 0.794719i \(-0.707618\pi\)
0.606977 0.794719i \(-0.292382\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) 4.00000 3.00000i 0.800000 0.600000i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.07107 1.31306 0.656532 0.754298i \(-0.272023\pi\)
0.656532 + 0.754298i \(0.272023\pi\)
\(30\) 0 0
\(31\) −6.92820 −1.24434 −0.622171 0.782881i \(-0.713749\pi\)
−0.622171 + 0.782881i \(0.713749\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.89898 9.79796i 0.828079 1.65616i
\(36\) 0 0
\(37\) −1.00000 + 1.00000i −0.164399 + 0.164399i −0.784512 0.620113i \(-0.787087\pi\)
0.620113 + 0.784512i \(0.287087\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.89949i 1.54604i −0.634381 0.773021i \(-0.718745\pi\)
0.634381 0.773021i \(-0.281255\pi\)
\(42\) 0 0
\(43\) 6.92820 + 6.92820i 1.05654 + 1.05654i 0.998303 + 0.0582384i \(0.0185483\pi\)
0.0582384 + 0.998303i \(0.481452\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.89898 + 4.89898i 0.714590 + 0.714590i 0.967492 0.252902i \(-0.0813851\pi\)
−0.252902 + 0.967492i \(0.581385\pi\)
\(48\) 0 0
\(49\) 17.0000i 2.42857i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.82843 2.82843i 0.388514 0.388514i −0.485643 0.874157i \(-0.661414\pi\)
0.874157 + 0.485643i \(0.161414\pi\)
\(54\) 0 0
\(55\) 3.46410 + 10.3923i 0.467099 + 1.40130i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.89898 −0.637793 −0.318896 0.947790i \(-0.603312\pi\)
−0.318896 + 0.947790i \(0.603312\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −2.82843 1.41421i −0.350823 0.175412i
\(66\) 0 0
\(67\) −6.92820 + 6.92820i −0.846415 + 0.846415i −0.989684 0.143269i \(-0.954239\pi\)
0.143269 + 0.989684i \(0.454239\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −7.00000 7.00000i −0.819288 0.819288i 0.166717 0.986005i \(-0.446683\pi\)
−0.986005 + 0.166717i \(0.946683\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 16.9706 + 16.9706i 1.93398 + 1.93398i
\(78\) 0 0
\(79\) 6.92820i 0.779484i −0.920924 0.389742i \(-0.872564\pi\)
0.920924 0.389742i \(-0.127436\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.89898 + 4.89898i −0.537733 + 0.537733i −0.922862 0.385130i \(-0.874157\pi\)
0.385130 + 0.922862i \(0.374157\pi\)
\(84\) 0 0
\(85\) −8.00000 4.00000i −0.867722 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.89949 1.04934 0.524672 0.851304i \(-0.324188\pi\)
0.524672 + 0.851304i \(0.324188\pi\)
\(90\) 0 0
\(91\) −6.92820 −0.726273
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.89898 14.6969i −0.502625 1.50787i
\(96\) 0 0
\(97\) 5.00000 5.00000i 0.507673 0.507673i −0.406138 0.913812i \(-0.633125\pi\)
0.913812 + 0.406138i \(0.133125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.24264i 0.422159i 0.977469 + 0.211079i \(0.0676978\pi\)
−0.977469 + 0.211079i \(0.932302\pi\)
\(102\) 0 0
\(103\) 3.46410 + 3.46410i 0.341328 + 0.341328i 0.856866 0.515538i \(-0.172408\pi\)
−0.515538 + 0.856866i \(0.672408\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.89898 + 4.89898i 0.473602 + 0.473602i 0.903078 0.429476i \(-0.141302\pi\)
−0.429476 + 0.903078i \(0.641302\pi\)
\(108\) 0 0
\(109\) 8.00000i 0.766261i 0.923694 + 0.383131i \(0.125154\pi\)
−0.923694 + 0.383131i \(0.874846\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.07107 + 7.07107i −0.665190 + 0.665190i −0.956599 0.291409i \(-0.905876\pi\)
0.291409 + 0.956599i \(0.405876\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −19.5959 −1.79635
\(120\) 0 0
\(121\) −13.0000 −1.18182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.36396 9.19239i 0.569210 0.822192i
\(126\) 0 0
\(127\) −10.3923 + 10.3923i −0.922168 + 0.922168i −0.997182 0.0750145i \(-0.976100\pi\)
0.0750145 + 0.997182i \(0.476100\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.89898i 0.428026i −0.976831 0.214013i \(-0.931347\pi\)
0.976831 0.214013i \(-0.0686535\pi\)
\(132\) 0 0
\(133\) −24.0000 24.0000i −2.08106 2.08106i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.41421 + 1.41421i 0.120824 + 0.120824i 0.764934 0.644109i \(-0.222772\pi\)
−0.644109 + 0.764934i \(0.722772\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.89898 4.89898i 0.409673 0.409673i
\(144\) 0 0
\(145\) 15.0000 5.00000i 1.24568 0.415227i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.24264 0.347571 0.173785 0.984784i \(-0.444400\pi\)
0.173785 + 0.984784i \(0.444400\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −14.6969 + 4.89898i −1.18049 + 0.393496i
\(156\) 0 0
\(157\) 13.0000 13.0000i 1.03751 1.03751i 0.0382445 0.999268i \(-0.487823\pi\)
0.999268 0.0382445i \(-0.0121766\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −6.92820 6.92820i −0.542659 0.542659i 0.381649 0.924307i \(-0.375356\pi\)
−0.924307 + 0.381649i \(0.875356\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.79796 + 9.79796i 0.758189 + 0.758189i 0.975993 0.217804i \(-0.0698893\pi\)
−0.217804 + 0.975993i \(0.569889\pi\)
\(168\) 0 0
\(169\) 11.0000i 0.846154i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.41421 1.41421i 0.107521 0.107521i −0.651300 0.758820i \(-0.725776\pi\)
0.758820 + 0.651300i \(0.225776\pi\)
\(174\) 0 0
\(175\) 3.46410 24.2487i 0.261861 1.83303i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.89898 −0.366167 −0.183083 0.983097i \(-0.558608\pi\)
−0.183083 + 0.983097i \(0.558608\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.41421 + 2.82843i −0.103975 + 0.207950i
\(186\) 0 0
\(187\) 13.8564 13.8564i 1.01328 1.01328i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 19.5959i 1.41791i 0.705253 + 0.708955i \(0.250833\pi\)
−0.705253 + 0.708955i \(0.749167\pi\)
\(192\) 0 0
\(193\) −7.00000 7.00000i −0.503871 0.503871i 0.408768 0.912639i \(-0.365959\pi\)
−0.912639 + 0.408768i \(0.865959\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.41421 1.41421i −0.100759 0.100759i 0.654931 0.755689i \(-0.272698\pi\)
−0.755689 + 0.654931i \(0.772698\pi\)
\(198\) 0 0
\(199\) 13.8564i 0.982255i −0.871088 0.491127i \(-0.836585\pi\)
0.871088 0.491127i \(-0.163415\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 24.4949 24.4949i 1.71920 1.71920i
\(204\) 0 0
\(205\) −7.00000 21.0000i −0.488901 1.46670i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 33.9411 2.34776
\(210\) 0 0
\(211\) 13.8564 0.953914 0.476957 0.878927i \(-0.341740\pi\)
0.476957 + 0.878927i \(0.341740\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 19.5959 + 9.79796i 1.33643 + 0.668215i
\(216\) 0 0
\(217\) −24.0000 + 24.0000i −1.62923 + 1.62923i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 5.65685i 0.380521i
\(222\) 0 0
\(223\) 10.3923 + 10.3923i 0.695920 + 0.695920i 0.963528 0.267608i \(-0.0862331\pi\)
−0.267608 + 0.963528i \(0.586233\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.79796 + 9.79796i 0.650313 + 0.650313i 0.953068 0.302755i \(-0.0979064\pi\)
−0.302755 + 0.953068i \(0.597906\pi\)
\(228\) 0 0
\(229\) 18.0000i 1.18947i 0.803921 + 0.594737i \(0.202744\pi\)
−0.803921 + 0.594737i \(0.797256\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.82843 2.82843i 0.185296 0.185296i −0.608363 0.793659i \(-0.708173\pi\)
0.793659 + 0.608363i \(0.208173\pi\)
\(234\) 0 0
\(235\) 13.8564 + 6.92820i 0.903892 + 0.451946i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 19.5959 1.26755 0.633777 0.773516i \(-0.281504\pi\)
0.633777 + 0.773516i \(0.281504\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −12.0208 36.0624i −0.767982 2.30395i
\(246\) 0 0
\(247\) −6.92820 + 6.92820i −0.440831 + 0.440831i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.89898i 0.309221i 0.987976 + 0.154610i \(0.0494122\pi\)
−0.987976 + 0.154610i \(0.950588\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 15.5563 + 15.5563i 0.970378 + 0.970378i 0.999574 0.0291953i \(-0.00929448\pi\)
−0.0291953 + 0.999574i \(0.509294\pi\)
\(258\) 0 0
\(259\) 6.92820i 0.430498i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) 0 0
\(265\) 4.00000 8.00000i 0.245718 0.491436i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −21.2132 −1.29339 −0.646696 0.762748i \(-0.723850\pi\)
−0.646696 + 0.762748i \(0.723850\pi\)
\(270\) 0 0
\(271\) 27.7128 1.68343 0.841717 0.539919i \(-0.181545\pi\)
0.841717 + 0.539919i \(0.181545\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.6969 + 19.5959i 0.886259 + 1.18168i
\(276\) 0 0
\(277\) 11.0000 11.0000i 0.660926 0.660926i −0.294672 0.955598i \(-0.595211\pi\)
0.955598 + 0.294672i \(0.0952105\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 24.0416i 1.43420i −0.696969 0.717102i \(-0.745468\pi\)
0.696969 0.717102i \(-0.254532\pi\)
\(282\) 0 0
\(283\) −13.8564 13.8564i −0.823678 0.823678i 0.162956 0.986633i \(-0.447897\pi\)
−0.986633 + 0.162956i \(0.947897\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −34.2929 34.2929i −2.02424 2.02424i
\(288\) 0 0
\(289\) 1.00000i 0.0588235i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9.89949 + 9.89949i −0.578335 + 0.578335i −0.934444 0.356110i \(-0.884103\pi\)
0.356110 + 0.934444i \(0.384103\pi\)
\(294\) 0 0
\(295\) −10.3923 + 3.46410i −0.605063 + 0.201688i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 48.0000 2.76667
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6.92820 + 6.92820i −0.395413 + 0.395413i −0.876612 0.481198i \(-0.840202\pi\)
0.481198 + 0.876612i \(0.340202\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 29.3939i 1.66677i −0.552690 0.833387i \(-0.686399\pi\)
0.552690 0.833387i \(-0.313601\pi\)
\(312\) 0 0
\(313\) 11.0000 + 11.0000i 0.621757 + 0.621757i 0.945980 0.324224i \(-0.105103\pi\)
−0.324224 + 0.945980i \(0.605103\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.82843 + 2.82843i 0.158860 + 0.158860i 0.782062 0.623201i \(-0.214168\pi\)
−0.623201 + 0.782062i \(0.714168\pi\)
\(318\) 0 0
\(319\) 34.6410i 1.93952i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −19.5959 + 19.5959i −1.09035 + 1.09035i
\(324\) 0 0
\(325\) −7.00000 1.00000i −0.388290 0.0554700i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 33.9411 1.87123
\(330\) 0 0
\(331\) −34.6410 −1.90404 −0.952021 0.306032i \(-0.900999\pi\)
−0.952021 + 0.306032i \(0.900999\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −9.79796 + 19.5959i −0.535320 + 1.07064i
\(336\) 0 0
\(337\) −13.0000 + 13.0000i −0.708155 + 0.708155i −0.966147 0.257992i \(-0.916939\pi\)
0.257992 + 0.966147i \(0.416939\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 33.9411i 1.83801i
\(342\) 0 0
\(343\) −34.6410 34.6410i −1.87044 1.87044i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −19.5959 19.5959i −1.05196 1.05196i −0.998574 0.0533903i \(-0.982997\pi\)
−0.0533903 0.998574i \(-0.517003\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −19.7990 + 19.7990i −1.05379 + 1.05379i −0.0553255 + 0.998468i \(0.517620\pi\)
−0.998468 + 0.0553255i \(0.982380\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.79796 0.517116 0.258558 0.965996i \(-0.416753\pi\)
0.258558 + 0.965996i \(0.416753\pi\)
\(360\) 0 0
\(361\) −29.0000 −1.52632
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −19.7990 9.89949i −1.03633 0.518163i
\(366\) 0 0
\(367\) 10.3923 10.3923i 0.542474 0.542474i −0.381780 0.924253i \(-0.624689\pi\)
0.924253 + 0.381780i \(0.124689\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 19.5959i 1.01737i
\(372\) 0 0
\(373\) −7.00000 7.00000i −0.362446 0.362446i 0.502267 0.864713i \(-0.332500\pi\)
−0.864713 + 0.502267i \(0.832500\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −7.07107 7.07107i −0.364179 0.364179i
\(378\) 0 0
\(379\) 27.7128i 1.42351i 0.702427 + 0.711756i \(0.252100\pi\)
−0.702427 + 0.711756i \(0.747900\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14.6969 + 14.6969i −0.750978 + 0.750978i −0.974662 0.223683i \(-0.928192\pi\)
0.223683 + 0.974662i \(0.428192\pi\)
\(384\) 0 0
\(385\) 48.0000 + 24.0000i 2.44631 + 1.22315i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −4.24264 −0.215110 −0.107555 0.994199i \(-0.534302\pi\)
−0.107555 + 0.994199i \(0.534302\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.89898 14.6969i −0.246494 0.739483i
\(396\) 0 0
\(397\) 5.00000 5.00000i 0.250943 0.250943i −0.570414 0.821357i \(-0.693217\pi\)
0.821357 + 0.570414i \(0.193217\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 21.2132i 1.05934i 0.848205 + 0.529668i \(0.177684\pi\)
−0.848205 + 0.529668i \(0.822316\pi\)
\(402\) 0 0
\(403\) 6.92820 + 6.92820i 0.345118 + 0.345118i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.89898 4.89898i −0.242833 0.242833i
\(408\) 0 0
\(409\) 8.00000i 0.395575i 0.980245 + 0.197787i \(0.0633755\pi\)
−0.980245 + 0.197787i \(0.936624\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −16.9706 + 16.9706i −0.835067 + 0.835067i
\(414\) 0 0
\(415\) −6.92820 + 13.8564i −0.340092 + 0.680184i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.89898 −0.239331 −0.119665 0.992814i \(-0.538182\pi\)
−0.119665 + 0.992814i \(0.538182\pi\)
\(420\) 0 0
\(421\) −30.0000 −1.46211 −0.731055 0.682318i \(-0.760972\pi\)
−0.731055 + 0.682318i \(0.760972\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −19.7990 2.82843i −0.960392 0.137199i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 19.5959i 0.943902i 0.881625 + 0.471951i \(0.156450\pi\)
−0.881625 + 0.471951i \(0.843550\pi\)
\(432\) 0 0
\(433\) 7.00000 + 7.00000i 0.336399 + 0.336399i 0.855010 0.518611i \(-0.173551\pi\)
−0.518611 + 0.855010i \(0.673551\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 27.7128i 1.32266i 0.750095 + 0.661330i \(0.230008\pi\)
−0.750095 + 0.661330i \(0.769992\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −9.79796 + 9.79796i −0.465515 + 0.465515i −0.900458 0.434943i \(-0.856769\pi\)
0.434943 + 0.900458i \(0.356769\pi\)
\(444\) 0 0
\(445\) 21.0000 7.00000i 0.995495 0.331832i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 21.2132 1.00111 0.500556 0.865704i \(-0.333129\pi\)
0.500556 + 0.865704i \(0.333129\pi\)
\(450\) 0 0
\(451\) 48.4974 2.28365
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −14.6969 + 4.89898i −0.689003 + 0.229668i
\(456\) 0 0
\(457\) −23.0000 + 23.0000i −1.07589 + 1.07589i −0.0790217 + 0.996873i \(0.525180\pi\)
−0.996873 + 0.0790217i \(0.974820\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.7279i 0.592798i 0.955064 + 0.296399i \(0.0957859\pi\)
−0.955064 + 0.296399i \(0.904214\pi\)
\(462\) 0 0
\(463\) 3.46410 + 3.46410i 0.160990 + 0.160990i 0.783005 0.622015i \(-0.213686\pi\)
−0.622015 + 0.783005i \(0.713686\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −24.4949 24.4949i −1.13349 1.13349i −0.989593 0.143896i \(-0.954037\pi\)
−0.143896 0.989593i \(-0.545963\pi\)
\(468\) 0 0
\(469\) 48.0000i 2.21643i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −33.9411 + 33.9411i −1.56061 + 1.56061i
\(474\) 0 0
\(475\) −20.7846 27.7128i −0.953663 1.27155i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 19.5959 0.895360 0.447680 0.894194i \(-0.352250\pi\)
0.447680 + 0.894194i \(0.352250\pi\)
\(480\) 0 0
\(481\) 2.00000 0.0911922
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.07107 14.1421i 0.321081 0.642161i
\(486\) 0 0
\(487\) 17.3205 17.3205i 0.784867 0.784867i −0.195781 0.980648i \(-0.562724\pi\)
0.980648 + 0.195781i \(0.0627240\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 24.4949i 1.10544i 0.833367 + 0.552720i \(0.186410\pi\)
−0.833367 + 0.552720i \(0.813590\pi\)
\(492\) 0 0
\(493\) −20.0000 20.0000i −0.900755 0.900755i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 6.92820i 0.310149i −0.987903 0.155074i \(-0.950438\pi\)
0.987903 0.155074i \(-0.0495618\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −4.89898 + 4.89898i −0.218435 + 0.218435i −0.807839 0.589404i \(-0.799363\pi\)
0.589404 + 0.807839i \(0.299363\pi\)
\(504\) 0 0
\(505\) 3.00000 + 9.00000i 0.133498 + 0.400495i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7.07107 −0.313420 −0.156710 0.987645i \(-0.550089\pi\)
−0.156710 + 0.987645i \(0.550089\pi\)
\(510\) 0 0
\(511\) −48.4974 −2.14540
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 9.79796 + 4.89898i 0.431750 + 0.215875i
\(516\) 0 0
\(517\) −24.0000 + 24.0000i −1.05552 + 1.05552i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 38.1838i 1.67286i −0.548073 0.836431i \(-0.684638\pi\)
0.548073 0.836431i \(-0.315362\pi\)
\(522\) 0 0
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 19.5959 + 19.5959i 0.853612 + 0.853612i
\(528\) 0 0
\(529\) 23.0000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −9.89949 + 9.89949i −0.428795 + 0.428795i
\(534\) 0 0
\(535\) 13.8564 + 6.92820i 0.599065 + 0.299532i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 83.2827 3.58724
\(540\) 0 0
\(541\) 40.0000 1.71973 0.859867 0.510518i \(-0.170546\pi\)
0.859867 + 0.510518i \(0.170546\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.65685 + 16.9706i 0.242313 + 0.726939i
\(546\) 0 0
\(547\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 48.9898i 2.08704i
\(552\) 0 0
\(553\) −24.0000 24.0000i −1.02058 1.02058i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.1127 + 31.1127i 1.31829 + 1.31829i 0.915131 + 0.403156i \(0.132087\pi\)
0.403156 + 0.915131i \(0.367913\pi\)
\(558\) 0 0
\(559\) 13.8564i 0.586064i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) −10.0000 + 20.0000i −0.420703 + 0.841406i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −38.1838 −1.60075 −0.800373 0.599502i \(-0.795365\pi\)
−0.800373 + 0.599502i \(0.795365\pi\)
\(570\) 0 0
\(571\) 20.7846 0.869809 0.434904 0.900477i \(-0.356782\pi\)
0.434904 + 0.900477i \(0.356782\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.00000 + 1.00000i −0.0416305 + 0.0416305i −0.727616 0.685985i \(-0.759372\pi\)
0.685985 + 0.727616i \(0.259372\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 33.9411i 1.40812i
\(582\) 0 0
\(583\) 13.8564 + 13.8564i 0.573874 + 0.573874i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4.89898 + 4.89898i 0.202203 + 0.202203i 0.800943 0.598741i \(-0.204332\pi\)
−0.598741 + 0.800943i \(0.704332\pi\)
\(588\) 0 0
\(589\) 48.0000i 1.97781i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −18.3848 + 18.3848i −0.754972 + 0.754972i −0.975403 0.220430i \(-0.929254\pi\)
0.220430 + 0.975403i \(0.429254\pi\)
\(594\) 0 0
\(595\) −41.5692 + 13.8564i −1.70417 + 0.568057i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 24.0000 0.978980 0.489490 0.872009i \(-0.337183\pi\)
0.489490 + 0.872009i \(0.337183\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −27.5772 + 9.19239i −1.12117 + 0.373724i
\(606\) 0 0
\(607\) 31.1769 31.1769i 1.26543 1.26543i 0.317010 0.948422i \(-0.397321\pi\)
0.948422 0.317010i \(-0.102679\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.79796i 0.396383i
\(612\) 0 0
\(613\) 7.00000 + 7.00000i 0.282727 + 0.282727i 0.834196 0.551468i \(-0.185932\pi\)
−0.551468 + 0.834196i \(0.685932\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.1421 + 14.1421i 0.569341 + 0.569341i 0.931944 0.362603i \(-0.118112\pi\)
−0.362603 + 0.931944i \(0.618112\pi\)
\(618\) 0 0
\(619\) 13.8564i 0.556936i −0.960446 0.278468i \(-0.910173\pi\)
0.960446 0.278468i \(-0.0898266\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 34.2929 34.2929i 1.37391 1.37391i
\(624\) 0 0
\(625\) 7.00000 24.0000i 0.280000 0.960000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.65685 0.225554
\(630\) 0 0
\(631\) 34.6410 1.37904 0.689519 0.724268i \(-0.257822\pi\)
0.689519 + 0.724268i \(0.257822\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −14.6969 + 29.3939i −0.583230 + 1.16646i
\(636\) 0 0
\(637\) −17.0000 + 17.0000i −0.673565 + 0.673565i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 15.5563i 0.614439i −0.951639 0.307219i \(-0.900601\pi\)
0.951639 0.307219i \(-0.0993986\pi\)
\(642\) 0 0
\(643\) 13.8564 + 13.8564i 0.546443 + 0.546443i 0.925410 0.378967i \(-0.123721\pi\)
−0.378967 + 0.925410i \(0.623721\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.4949 + 24.4949i 0.962994 + 0.962994i 0.999339 0.0363455i \(-0.0115717\pi\)
−0.0363455 + 0.999339i \(0.511572\pi\)
\(648\) 0 0
\(649\) 24.0000i 0.942082i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −31.1127 + 31.1127i −1.21753 + 1.21753i −0.249041 + 0.968493i \(0.580115\pi\)
−0.968493 + 0.249041i \(0.919885\pi\)
\(654\) 0 0
\(655\) −3.46410 10.3923i −0.135354 0.406061i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.89898 0.190837 0.0954186 0.995437i \(-0.469581\pi\)
0.0954186 + 0.995437i \(0.469581\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −67.8823 33.9411i −2.63236 1.31618i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 25.0000 + 25.0000i 0.963679 + 0.963679i 0.999363 0.0356839i \(-0.0113610\pi\)
−0.0356839 + 0.999363i \(0.511361\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2.82843 2.82843i −0.108705 0.108705i 0.650662 0.759367i \(-0.274491\pi\)
−0.759367 + 0.650662i \(0.774491\pi\)
\(678\) 0 0
\(679\) 34.6410i 1.32940i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −24.4949 + 24.4949i −0.937271 + 0.937271i −0.998145 0.0608742i \(-0.980611\pi\)
0.0608742 + 0.998145i \(0.480611\pi\)
\(684\) 0 0
\(685\) 4.00000 + 2.00000i 0.152832 + 0.0764161i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.65685 −0.215509
\(690\) 0 0
\(691\) −20.7846 −0.790684 −0.395342 0.918534i \(-0.629374\pi\)
−0.395342 + 0.918534i \(0.629374\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −28.0000 + 28.0000i −1.06058 + 1.06058i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 26.8701i 1.01487i −0.861691 0.507434i \(-0.830594\pi\)
0.861691 0.507434i \(-0.169406\pi\)
\(702\) 0 0
\(703\) 6.92820 + 6.92820i 0.261302 + 0.261302i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 14.6969 + 14.6969i 0.552735 + 0.552735i
\(708\) 0 0
\(709\) 30.0000i 1.12667i 0.826227 + 0.563337i \(0.190483\pi\)
−0.826227 + 0.563337i \(0.809517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 6.92820 13.8564i 0.259100 0.518200i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −39.1918 −1.46161 −0.730804 0.682587i \(-0.760855\pi\)
−0.730804 + 0.682587i \(0.760855\pi\)
\(720\) 0 0
\(721\) 24.0000 0.893807
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 28.2843 21.2132i 1.05045 0.787839i
\(726\) 0 0
\(727\) −17.3205 + 17.3205i −0.642382 + 0.642382i −0.951140 0.308758i \(-0.900087\pi\)
0.308758 + 0.951140i \(0.400087\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 39.1918i 1.44956i
\(732\) 0 0
\(733\) 31.0000 + 31.0000i 1.14501 + 1.14501i 0.987520 + 0.157491i \(0.0503404\pi\)
0.157491 + 0.987520i \(0.449660\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −33.9411 33.9411i −1.25024 1.25024i
\(738\) 0 0
\(739\) 48.4974i 1.78401i −0.452029 0.892003i \(-0.649300\pi\)
0.452029 0.892003i \(-0.350700\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34.2929 + 34.2929i −1.25808 + 1.25808i −0.306076 + 0.952007i \(0.599016\pi\)
−0.952007 + 0.306076i \(0.900984\pi\)
\(744\) 0 0
\(745\) 9.00000 3.00000i 0.329734 0.109911i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 33.9411 1.24018
\(750\) 0 0
\(751\) −20.7846 −0.758441 −0.379221 0.925306i \(-0.623808\pi\)
−0.379221 + 0.925306i \(0.623808\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −19.0000 + 19.0000i −0.690567 + 0.690567i −0.962357 0.271790i \(-0.912384\pi\)
0.271790 + 0.962357i \(0.412384\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 4.24264i 0.153796i −0.997039 0.0768978i \(-0.975498\pi\)
0.997039 0.0768978i \(-0.0245015\pi\)
\(762\) 0 0
\(763\) 27.7128 + 27.7128i 1.00327 + 1.00327i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.89898 + 4.89898i 0.176892 + 0.176892i
\(768\) 0 0
\(769\) 24.0000i 0.865462i 0.901523 + 0.432731i \(0.142450\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −19.7990 + 19.7990i −0.712120 + 0.712120i −0.966978 0.254858i \(-0.917971\pi\)
0.254858 + 0.966978i \(0.417971\pi\)
\(774\) 0 0
\(775\) −27.7128 + 20.7846i −0.995474 + 0.746605i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −68.5857 −2.45734
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 18.3848 36.7696i 0.656181 1.31236i
\(786\) 0 0
\(787\) 20.7846 20.7846i 0.740891 0.740891i −0.231858 0.972750i \(-0.574481\pi\)
0.972750 + 0.231858i \(0.0744806\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 48.9898i 1.74188i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35.3553 + 35.3553i 1.25235 + 1.25235i 0.954664 + 0.297687i \(0.0962151\pi\)
0.297687 + 0.954664i \(0.403785\pi\)
\(798\) 0 0
\(799\) 27.7128i 0.980409i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 34.2929 34.2929i 1.21017 1.21017i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −9.89949 −0.348048 −0.174024 0.984741i \(-0.555677\pi\)
−0.174024 + 0.984741i \(0.555677\pi\)
\(810\) 0 0
\(811\) 13.8564 0.486564 0.243282 0.969956i \(-0.421776\pi\)
0.243282 + 0.969956i \(0.421776\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −19.5959 9.79796i −0.686415 0.343208i
\(816\) 0 0
\(817\) 48.0000 48.0000i 1.67931 1.67931i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.41421i 0.0493564i −0.999695 0.0246782i \(-0.992144\pi\)
0.999695 0.0246782i \(-0.00785611\pi\)
\(822\) 0 0
\(823\) 10.3923 + 10.3923i 0.362253 + 0.362253i 0.864642 0.502389i \(-0.167545\pi\)
−0.502389 + 0.864642i \(0.667545\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 9.79796 + 9.79796i 0.340708 + 0.340708i 0.856634 0.515925i \(-0.172552\pi\)
−0.515925 + 0.856634i \(0.672552\pi\)
\(828\) 0 0
\(829\) 8.00000i 0.277851i −0.990303 0.138926i \(-0.955635\pi\)
0.990303 0.138926i \(-0.0443649\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −48.0833 + 48.0833i −1.66599 + 1.66599i
\(834\) 0 0
\(835\) 27.7128 + 13.8564i 0.959041 + 0.479521i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.79796 0.338263 0.169132 0.985593i \(-0.445904\pi\)
0.169132 + 0.985593i \(0.445904\pi\)
\(840\) 0 0
\(841\) 21.0000 0.724138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −7.77817 23.3345i −0.267577 0.802732i
\(846\) 0 0
\(847\) −45.0333 + 45.0333i −1.54736 + 1.54736i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −23.0000 23.0000i −0.787505 0.787505i 0.193580 0.981085i \(-0.437990\pi\)
−0.981085 + 0.193580i \(0.937990\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −14.1421 14.1421i −0.483086 0.483086i 0.423030 0.906116i \(-0.360967\pi\)
−0.906116 + 0.423030i \(0.860967\pi\)
\(858\) 0 0
\(859\) 41.5692i 1.41832i 0.705046 + 0.709162i \(0.250926\pi\)
−0.705046 + 0.709162i \(0.749074\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.79796 9.79796i 0.333526 0.333526i −0.520398 0.853924i \(-0.674216\pi\)
0.853924 + 0.520398i \(0.174216\pi\)
\(864\) 0 0
\(865\) 2.00000 4.00000i 0.0680020 0.136004i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 33.9411 1.15137
\(870\) 0 0
\(871\) 13.8564 0.469506
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −9.79796 53.8888i −0.331231 1.82177i
\(876\) 0 0
\(877\) −5.00000 + 5.00000i −0.168838 + 0.168838i −0.786468 0.617630i \(-0.788093\pi\)
0.617630 + 0.786468i \(0.288093\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 15.5563i 0.524107i 0.965053 + 0.262053i \(0.0843996\pi\)
−0.965053 + 0.262053i \(0.915600\pi\)
\(882\) 0 0
\(883\) −34.6410 34.6410i −1.16576 1.16576i −0.983193 0.182570i \(-0.941558\pi\)
−0.182570 0.983193i \(-0.558442\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −29.3939 29.3939i −0.986950 0.986950i 0.0129661 0.999916i \(-0.495873\pi\)
−0.999916 + 0.0129661i \(0.995873\pi\)
\(888\) 0 0
\(889\) 72.0000i 2.41480i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 33.9411 33.9411i 1.13580 1.13580i
\(894\) 0 0
\(895\) −10.3923 + 3.46410i −0.347376 + 0.115792i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −48.9898 −1.63390
\(900\) 0 0
\(901\) −16.0000 −0.533037
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 16.9706 5.65685i 0.564121 0.188040i
\(906\) 0 0
\(907\) −20.7846 + 20.7846i −0.690142 + 0.690142i −0.962263 0.272121i \(-0.912275\pi\)
0.272121 + 0.962263i \(0.412275\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 29.3939i 0.973863i −0.873440 0.486931i \(-0.838116\pi\)
0.873440 0.486931i \(-0.161884\pi\)
\(912\) 0 0
\(913\) −24.0000 24.0000i −0.794284 0.794284i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −16.9706 16.9706i −0.560417 0.560417i
\(918\) 0 0
\(919\) 20.7846i 0.685621i 0.939405 + 0.342811i \(0.111379\pi\)
−0.939405 + 0.342811i \(0.888621\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.00000 + 7.00000i −0.0328798 + 0.230159i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −12.7279 −0.417590 −0.208795 0.977959i \(-0.566954\pi\)
−0.208795 + 0.977959i \(0.566954\pi\)
\(930\) 0 0
\(931\) −117.779 −3.86007
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 19.5959 39.1918i 0.640855 1.28171i
\(936\) 0 0
\(937\) 5.00000 5.00000i 0.163343 0.163343i −0.620703 0.784046i \(-0.713153\pi\)
0.784046 + 0.620703i \(0.213153\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 21.2132i 0.691531i 0.938321 + 0.345765i \(0.112381\pi\)
−0.938321 + 0.345765i \(0.887619\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 29.3939 + 29.3939i 0.955173 + 0.955173i 0.999037 0.0438648i \(-0.0139671\pi\)
−0.0438648 + 0.999037i \(0.513967\pi\)
\(948\) 0 0
\(949\) 14.0000i 0.454459i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 35.3553 35.3553i 1.14527 1.14527i 0.157801 0.987471i \(-0.449560\pi\)
0.987471 0.157801i \(-0.0504404\pi\)
\(954\) 0 0
\(955\) 13.8564 + 41.5692i 0.448383 + 1.34515i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 9.79796 0.316393
\(960\) 0 0
\(961\) 17.0000 0.548387
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −19.7990 9.89949i −0.637352 0.318676i
\(966\) 0 0
\(967\) −24.2487 + 24.2487i −0.779786 + 0.779786i −0.979794 0.200008i \(-0.935903\pi\)
0.200008 + 0.979794i \(0.435903\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 44.0908i 1.41494i 0.706743 + 0.707471i \(0.250164\pi\)
−0.706743 + 0.707471i \(0.749836\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 19.7990 + 19.7990i 0.633426 + 0.633426i 0.948926 0.315500i \(-0.102172\pi\)
−0.315500 + 0.948926i \(0.602172\pi\)
\(978\) 0 0
\(979\) 48.4974i 1.54998i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.89898 + 4.89898i −0.156253 + 0.156253i −0.780904 0.624651i \(-0.785241\pi\)
0.624651 + 0.780904i \(0.285241\pi\)
\(984\) 0 0
\(985\) −4.00000 2.00000i −0.127451 0.0637253i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 27.7128 0.880327 0.440163 0.897918i \(-0.354921\pi\)
0.440163 + 0.897918i \(0.354921\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −9.79796 29.3939i −0.310616 0.931849i
\(996\) 0 0
\(997\) 7.00000 7.00000i 0.221692 0.221692i −0.587519 0.809211i \(-0.699895\pi\)
0.809211 + 0.587519i \(0.199895\pi\)
\(998\) 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.w.n.2753.4 8
3.2 odd 2 inner 2880.2.w.n.2753.2 8
4.3 odd 2 inner 2880.2.w.n.2753.3 8
5.2 odd 4 inner 2880.2.w.n.2177.2 8
8.3 odd 2 1440.2.w.e.1313.1 yes 8
8.5 even 2 1440.2.w.e.1313.2 yes 8
12.11 even 2 inner 2880.2.w.n.2753.1 8
15.2 even 4 inner 2880.2.w.n.2177.4 8
20.7 even 4 inner 2880.2.w.n.2177.1 8
24.5 odd 2 1440.2.w.e.1313.4 yes 8
24.11 even 2 1440.2.w.e.1313.3 yes 8
40.27 even 4 1440.2.w.e.737.3 yes 8
40.37 odd 4 1440.2.w.e.737.4 yes 8
60.47 odd 4 inner 2880.2.w.n.2177.3 8
120.77 even 4 1440.2.w.e.737.2 yes 8
120.107 odd 4 1440.2.w.e.737.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1440.2.w.e.737.1 8 120.107 odd 4
1440.2.w.e.737.2 yes 8 120.77 even 4
1440.2.w.e.737.3 yes 8 40.27 even 4
1440.2.w.e.737.4 yes 8 40.37 odd 4
1440.2.w.e.1313.1 yes 8 8.3 odd 2
1440.2.w.e.1313.2 yes 8 8.5 even 2
1440.2.w.e.1313.3 yes 8 24.11 even 2
1440.2.w.e.1313.4 yes 8 24.5 odd 2
2880.2.w.n.2177.1 8 20.7 even 4 inner
2880.2.w.n.2177.2 8 5.2 odd 4 inner
2880.2.w.n.2177.3 8 60.47 odd 4 inner
2880.2.w.n.2177.4 8 15.2 even 4 inner
2880.2.w.n.2753.1 8 12.11 even 2 inner
2880.2.w.n.2753.2 8 3.2 odd 2 inner
2880.2.w.n.2753.3 8 4.3 odd 2 inner
2880.2.w.n.2753.4 8 1.1 even 1 trivial