Properties

Label 6480.2.a.br.1.1
Level $6480$
Weight $2$
Character 6480.1
Self dual yes
Analytic conductor $51.743$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6480 = 2^{4} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6480.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(51.7430605098\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 405)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 6480.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{5} +1.26795 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +1.26795 q^{7} +2.26795 q^{11} -5.46410 q^{13} +0.732051 q^{17} +2.46410 q^{19} -3.46410 q^{23} +1.00000 q^{25} -7.19615 q^{29} +3.00000 q^{31} +1.26795 q^{35} +0.732051 q^{37} +3.19615 q^{41} +10.1962 q^{43} +5.26795 q^{47} -5.39230 q^{49} +3.26795 q^{53} +2.26795 q^{55} +11.7321 q^{59} +4.00000 q^{61} -5.46410 q^{65} +3.46410 q^{67} +0.267949 q^{71} +9.66025 q^{73} +2.87564 q^{77} -8.53590 q^{79} +8.19615 q^{83} +0.732051 q^{85} -5.19615 q^{89} -6.92820 q^{91} +2.46410 q^{95} +7.66025 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 6 q^{7} + 8 q^{11} - 4 q^{13} - 2 q^{17} - 2 q^{19} + 2 q^{25} - 4 q^{29} + 6 q^{31} + 6 q^{35} - 2 q^{37} - 4 q^{41} + 10 q^{43} + 14 q^{47} + 10 q^{49} + 10 q^{53} + 8 q^{55} + 20 q^{59} + 8 q^{61} - 4 q^{65} + 4 q^{71} + 2 q^{73} + 30 q^{77} - 24 q^{79} + 6 q^{83} - 2 q^{85} - 2 q^{95} - 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 1.26795 0.479240 0.239620 0.970867i \(-0.422977\pi\)
0.239620 + 0.970867i \(0.422977\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.26795 0.683812 0.341906 0.939734i \(-0.388927\pi\)
0.341906 + 0.939734i \(0.388927\pi\)
\(12\) 0 0
\(13\) −5.46410 −1.51547 −0.757735 0.652563i \(-0.773694\pi\)
−0.757735 + 0.652563i \(0.773694\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.732051 0.177548 0.0887742 0.996052i \(-0.471705\pi\)
0.0887742 + 0.996052i \(0.471705\pi\)
\(18\) 0 0
\(19\) 2.46410 0.565304 0.282652 0.959223i \(-0.408786\pi\)
0.282652 + 0.959223i \(0.408786\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.19615 −1.33629 −0.668146 0.744030i \(-0.732912\pi\)
−0.668146 + 0.744030i \(0.732912\pi\)
\(30\) 0 0
\(31\) 3.00000 0.538816 0.269408 0.963026i \(-0.413172\pi\)
0.269408 + 0.963026i \(0.413172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.26795 0.214323
\(36\) 0 0
\(37\) 0.732051 0.120348 0.0601742 0.998188i \(-0.480834\pi\)
0.0601742 + 0.998188i \(0.480834\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.19615 0.499155 0.249578 0.968355i \(-0.419708\pi\)
0.249578 + 0.968355i \(0.419708\pi\)
\(42\) 0 0
\(43\) 10.1962 1.55490 0.777449 0.628946i \(-0.216513\pi\)
0.777449 + 0.628946i \(0.216513\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.26795 0.768409 0.384205 0.923248i \(-0.374476\pi\)
0.384205 + 0.923248i \(0.374476\pi\)
\(48\) 0 0
\(49\) −5.39230 −0.770329
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 3.26795 0.448887 0.224444 0.974487i \(-0.427944\pi\)
0.224444 + 0.974487i \(0.427944\pi\)
\(54\) 0 0
\(55\) 2.26795 0.305810
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.7321 1.52738 0.763691 0.645581i \(-0.223385\pi\)
0.763691 + 0.645581i \(0.223385\pi\)
\(60\) 0 0
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.46410 −0.677738
\(66\) 0 0
\(67\) 3.46410 0.423207 0.211604 0.977356i \(-0.432131\pi\)
0.211604 + 0.977356i \(0.432131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0.267949 0.0317997 0.0158999 0.999874i \(-0.494939\pi\)
0.0158999 + 0.999874i \(0.494939\pi\)
\(72\) 0 0
\(73\) 9.66025 1.13065 0.565324 0.824869i \(-0.308751\pi\)
0.565324 + 0.824869i \(0.308751\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.87564 0.327710
\(78\) 0 0
\(79\) −8.53590 −0.960364 −0.480182 0.877169i \(-0.659429\pi\)
−0.480182 + 0.877169i \(0.659429\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.19615 0.899645 0.449822 0.893118i \(-0.351487\pi\)
0.449822 + 0.893118i \(0.351487\pi\)
\(84\) 0 0
\(85\) 0.732051 0.0794021
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −5.19615 −0.550791 −0.275396 0.961331i \(-0.588809\pi\)
−0.275396 + 0.961331i \(0.588809\pi\)
\(90\) 0 0
\(91\) −6.92820 −0.726273
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.46410 0.252811
\(96\) 0 0
\(97\) 7.66025 0.777781 0.388890 0.921284i \(-0.372858\pi\)
0.388890 + 0.921284i \(0.372858\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 14.6603 1.45875 0.729375 0.684114i \(-0.239811\pi\)
0.729375 + 0.684114i \(0.239811\pi\)
\(102\) 0 0
\(103\) 7.46410 0.735460 0.367730 0.929933i \(-0.380135\pi\)
0.367730 + 0.929933i \(0.380135\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.4641 1.49497 0.747486 0.664278i \(-0.231261\pi\)
0.747486 + 0.664278i \(0.231261\pi\)
\(108\) 0 0
\(109\) −19.9282 −1.90878 −0.954388 0.298570i \(-0.903490\pi\)
−0.954388 + 0.298570i \(0.903490\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 5.12436 0.482059 0.241029 0.970518i \(-0.422515\pi\)
0.241029 + 0.970518i \(0.422515\pi\)
\(114\) 0 0
\(115\) −3.46410 −0.323029
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.928203 0.0850883
\(120\) 0 0
\(121\) −5.85641 −0.532401
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −16.5885 −1.47199 −0.735994 0.676988i \(-0.763285\pi\)
−0.735994 + 0.676988i \(0.763285\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 15.5885 1.36197 0.680985 0.732297i \(-0.261552\pi\)
0.680985 + 0.732297i \(0.261552\pi\)
\(132\) 0 0
\(133\) 3.12436 0.270916
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.46410 0.808573 0.404286 0.914632i \(-0.367520\pi\)
0.404286 + 0.914632i \(0.367520\pi\)
\(138\) 0 0
\(139\) −21.3923 −1.81447 −0.907236 0.420622i \(-0.861812\pi\)
−0.907236 + 0.420622i \(0.861812\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −12.3923 −1.03630
\(144\) 0 0
\(145\) −7.19615 −0.597608
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −8.00000 −0.655386 −0.327693 0.944784i \(-0.606271\pi\)
−0.327693 + 0.944784i \(0.606271\pi\)
\(150\) 0 0
\(151\) 15.3923 1.25261 0.626304 0.779579i \(-0.284567\pi\)
0.626304 + 0.779579i \(0.284567\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.00000 0.240966
\(156\) 0 0
\(157\) 5.12436 0.408968 0.204484 0.978870i \(-0.434448\pi\)
0.204484 + 0.978870i \(0.434448\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −4.39230 −0.346162
\(162\) 0 0
\(163\) −9.26795 −0.725922 −0.362961 0.931804i \(-0.618234\pi\)
−0.362961 + 0.931804i \(0.618234\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0.339746 0.0262903 0.0131452 0.999914i \(-0.495816\pi\)
0.0131452 + 0.999914i \(0.495816\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 15.4641 1.17571 0.587857 0.808965i \(-0.299972\pi\)
0.587857 + 0.808965i \(0.299972\pi\)
\(174\) 0 0
\(175\) 1.26795 0.0958479
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.1244 1.20519 0.602595 0.798047i \(-0.294133\pi\)
0.602595 + 0.798047i \(0.294133\pi\)
\(180\) 0 0
\(181\) 19.5359 1.45209 0.726046 0.687646i \(-0.241356\pi\)
0.726046 + 0.687646i \(0.241356\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.732051 0.0538214
\(186\) 0 0
\(187\) 1.66025 0.121410
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.1244 −1.16672 −0.583359 0.812215i \(-0.698262\pi\)
−0.583359 + 0.812215i \(0.698262\pi\)
\(192\) 0 0
\(193\) −8.73205 −0.628547 −0.314273 0.949333i \(-0.601761\pi\)
−0.314273 + 0.949333i \(0.601761\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 13.8564 0.987228 0.493614 0.869681i \(-0.335676\pi\)
0.493614 + 0.869681i \(0.335676\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −9.12436 −0.640404
\(204\) 0 0
\(205\) 3.19615 0.223229
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.58846 0.386562
\(210\) 0 0
\(211\) −18.8564 −1.29813 −0.649064 0.760734i \(-0.724839\pi\)
−0.649064 + 0.760734i \(0.724839\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 10.1962 0.695372
\(216\) 0 0
\(217\) 3.80385 0.258222
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) 24.7846 1.65970 0.829850 0.557986i \(-0.188426\pi\)
0.829850 + 0.557986i \(0.188426\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −20.0526 −1.33094 −0.665468 0.746427i \(-0.731768\pi\)
−0.665468 + 0.746427i \(0.731768\pi\)
\(228\) 0 0
\(229\) −12.0000 −0.792982 −0.396491 0.918039i \(-0.629772\pi\)
−0.396491 + 0.918039i \(0.629772\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.0526 0.658565 0.329283 0.944231i \(-0.393193\pi\)
0.329283 + 0.944231i \(0.393193\pi\)
\(234\) 0 0
\(235\) 5.26795 0.343643
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.46410 0.482813 0.241406 0.970424i \(-0.422391\pi\)
0.241406 + 0.970424i \(0.422391\pi\)
\(240\) 0 0
\(241\) 18.3205 1.18013 0.590064 0.807357i \(-0.299103\pi\)
0.590064 + 0.807357i \(0.299103\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.39230 −0.344502
\(246\) 0 0
\(247\) −13.4641 −0.856700
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −10.3923 −0.655956 −0.327978 0.944685i \(-0.606367\pi\)
−0.327978 + 0.944685i \(0.606367\pi\)
\(252\) 0 0
\(253\) −7.85641 −0.493928
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −10.3923 −0.648254 −0.324127 0.946014i \(-0.605071\pi\)
−0.324127 + 0.946014i \(0.605071\pi\)
\(258\) 0 0
\(259\) 0.928203 0.0576757
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.3205 0.821378 0.410689 0.911776i \(-0.365288\pi\)
0.410689 + 0.911776i \(0.365288\pi\)
\(264\) 0 0
\(265\) 3.26795 0.200749
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −6.66025 −0.406083 −0.203041 0.979170i \(-0.565083\pi\)
−0.203041 + 0.979170i \(0.565083\pi\)
\(270\) 0 0
\(271\) −10.9282 −0.663841 −0.331921 0.943307i \(-0.607697\pi\)
−0.331921 + 0.943307i \(0.607697\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.26795 0.136762
\(276\) 0 0
\(277\) 14.1962 0.852964 0.426482 0.904496i \(-0.359753\pi\)
0.426482 + 0.904496i \(0.359753\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 8.53590 0.509209 0.254605 0.967045i \(-0.418055\pi\)
0.254605 + 0.967045i \(0.418055\pi\)
\(282\) 0 0
\(283\) −5.32051 −0.316271 −0.158136 0.987417i \(-0.550548\pi\)
−0.158136 + 0.987417i \(0.550548\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.05256 0.239215
\(288\) 0 0
\(289\) −16.4641 −0.968477
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 25.2679 1.47617 0.738085 0.674708i \(-0.235730\pi\)
0.738085 + 0.674708i \(0.235730\pi\)
\(294\) 0 0
\(295\) 11.7321 0.683066
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 18.9282 1.09465
\(300\) 0 0
\(301\) 12.9282 0.745169
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4.00000 0.229039
\(306\) 0 0
\(307\) 24.0526 1.37275 0.686376 0.727247i \(-0.259200\pi\)
0.686376 + 0.727247i \(0.259200\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −16.2679 −0.922471 −0.461235 0.887278i \(-0.652594\pi\)
−0.461235 + 0.887278i \(0.652594\pi\)
\(312\) 0 0
\(313\) −22.9282 −1.29598 −0.647989 0.761649i \(-0.724390\pi\)
−0.647989 + 0.761649i \(0.724390\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −4.19615 −0.235679 −0.117840 0.993033i \(-0.537597\pi\)
−0.117840 + 0.993033i \(0.537597\pi\)
\(318\) 0 0
\(319\) −16.3205 −0.913773
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.80385 0.100369
\(324\) 0 0
\(325\) −5.46410 −0.303094
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 6.67949 0.368252
\(330\) 0 0
\(331\) −6.46410 −0.355299 −0.177650 0.984094i \(-0.556849\pi\)
−0.177650 + 0.984094i \(0.556849\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.46410 0.189264
\(336\) 0 0
\(337\) −7.32051 −0.398773 −0.199387 0.979921i \(-0.563895\pi\)
−0.199387 + 0.979921i \(0.563895\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.80385 0.368449
\(342\) 0 0
\(343\) −15.7128 −0.848412
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.58846 −0.138956 −0.0694778 0.997583i \(-0.522133\pi\)
−0.0694778 + 0.997583i \(0.522133\pi\)
\(348\) 0 0
\(349\) 8.85641 0.474073 0.237036 0.971501i \(-0.423824\pi\)
0.237036 + 0.971501i \(0.423824\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −19.5167 −1.03877 −0.519384 0.854541i \(-0.673838\pi\)
−0.519384 + 0.854541i \(0.673838\pi\)
\(354\) 0 0
\(355\) 0.267949 0.0142213
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −18.1244 −0.956567 −0.478283 0.878206i \(-0.658741\pi\)
−0.478283 + 0.878206i \(0.658741\pi\)
\(360\) 0 0
\(361\) −12.9282 −0.680432
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 9.66025 0.505641
\(366\) 0 0
\(367\) 31.1769 1.62742 0.813711 0.581270i \(-0.197444\pi\)
0.813711 + 0.581270i \(0.197444\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.14359 0.215125
\(372\) 0 0
\(373\) −18.0526 −0.934726 −0.467363 0.884065i \(-0.654796\pi\)
−0.467363 + 0.884065i \(0.654796\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 39.3205 2.02511
\(378\) 0 0
\(379\) 18.3923 0.944749 0.472375 0.881398i \(-0.343397\pi\)
0.472375 + 0.881398i \(0.343397\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −9.46410 −0.483593 −0.241797 0.970327i \(-0.577737\pi\)
−0.241797 + 0.970327i \(0.577737\pi\)
\(384\) 0 0
\(385\) 2.87564 0.146556
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 20.5359 1.04121 0.520606 0.853797i \(-0.325706\pi\)
0.520606 + 0.853797i \(0.325706\pi\)
\(390\) 0 0
\(391\) −2.53590 −0.128246
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.53590 −0.429488
\(396\) 0 0
\(397\) −6.39230 −0.320821 −0.160410 0.987050i \(-0.551282\pi\)
−0.160410 + 0.987050i \(0.551282\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.0718 0.552899 0.276450 0.961028i \(-0.410842\pi\)
0.276450 + 0.961028i \(0.410842\pi\)
\(402\) 0 0
\(403\) −16.3923 −0.816559
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.66025 0.0822957
\(408\) 0 0
\(409\) 9.85641 0.487368 0.243684 0.969855i \(-0.421644\pi\)
0.243684 + 0.969855i \(0.421644\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14.8756 0.731983
\(414\) 0 0
\(415\) 8.19615 0.402333
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.392305 0.0191653 0.00958267 0.999954i \(-0.496950\pi\)
0.00958267 + 0.999954i \(0.496950\pi\)
\(420\) 0 0
\(421\) −7.78461 −0.379399 −0.189699 0.981842i \(-0.560751\pi\)
−0.189699 + 0.981842i \(0.560751\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.732051 0.0355097
\(426\) 0 0
\(427\) 5.07180 0.245441
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 38.6603 1.86220 0.931099 0.364765i \(-0.118851\pi\)
0.931099 + 0.364765i \(0.118851\pi\)
\(432\) 0 0
\(433\) −28.5359 −1.37135 −0.685674 0.727909i \(-0.740492\pi\)
−0.685674 + 0.727909i \(0.740492\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.53590 −0.408327
\(438\) 0 0
\(439\) 15.3923 0.734635 0.367317 0.930096i \(-0.380276\pi\)
0.367317 + 0.930096i \(0.380276\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 17.6603 0.839064 0.419532 0.907741i \(-0.362194\pi\)
0.419532 + 0.907741i \(0.362194\pi\)
\(444\) 0 0
\(445\) −5.19615 −0.246321
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −16.1244 −0.760955 −0.380478 0.924790i \(-0.624240\pi\)
−0.380478 + 0.924790i \(0.624240\pi\)
\(450\) 0 0
\(451\) 7.24871 0.341328
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −6.92820 −0.324799
\(456\) 0 0
\(457\) −0.732051 −0.0342439 −0.0171219 0.999853i \(-0.505450\pi\)
−0.0171219 + 0.999853i \(0.505450\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.05256 0.0490226 0.0245113 0.999700i \(-0.492197\pi\)
0.0245113 + 0.999700i \(0.492197\pi\)
\(462\) 0 0
\(463\) −10.3923 −0.482971 −0.241486 0.970404i \(-0.577635\pi\)
−0.241486 + 0.970404i \(0.577635\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 35.3731 1.63687 0.818435 0.574599i \(-0.194842\pi\)
0.818435 + 0.574599i \(0.194842\pi\)
\(468\) 0 0
\(469\) 4.39230 0.202818
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 23.1244 1.06326
\(474\) 0 0
\(475\) 2.46410 0.113061
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 36.1244 1.65056 0.825282 0.564721i \(-0.191016\pi\)
0.825282 + 0.564721i \(0.191016\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.66025 0.347834
\(486\) 0 0
\(487\) −3.60770 −0.163480 −0.0817401 0.996654i \(-0.526048\pi\)
−0.0817401 + 0.996654i \(0.526048\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 38.1244 1.72053 0.860264 0.509849i \(-0.170299\pi\)
0.860264 + 0.509849i \(0.170299\pi\)
\(492\) 0 0
\(493\) −5.26795 −0.237256
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0.339746 0.0152397
\(498\) 0 0
\(499\) 10.3205 0.462009 0.231005 0.972953i \(-0.425799\pi\)
0.231005 + 0.972953i \(0.425799\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −27.3205 −1.21816 −0.609081 0.793108i \(-0.708461\pi\)
−0.609081 + 0.793108i \(0.708461\pi\)
\(504\) 0 0
\(505\) 14.6603 0.652373
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 34.7846 1.54180 0.770900 0.636956i \(-0.219807\pi\)
0.770900 + 0.636956i \(0.219807\pi\)
\(510\) 0 0
\(511\) 12.2487 0.541851
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 7.46410 0.328908
\(516\) 0 0
\(517\) 11.9474 0.525448
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.5359 0.549208 0.274604 0.961557i \(-0.411453\pi\)
0.274604 + 0.961557i \(0.411453\pi\)
\(522\) 0 0
\(523\) 26.2487 1.14778 0.573888 0.818934i \(-0.305434\pi\)
0.573888 + 0.818934i \(0.305434\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.19615 0.0956659
\(528\) 0 0
\(529\) −11.0000 −0.478261
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −17.4641 −0.756454
\(534\) 0 0
\(535\) 15.4641 0.668571
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −12.2295 −0.526761
\(540\) 0 0
\(541\) −17.5359 −0.753927 −0.376964 0.926228i \(-0.623032\pi\)
−0.376964 + 0.926228i \(0.623032\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −19.9282 −0.853630
\(546\) 0 0
\(547\) 6.14359 0.262681 0.131341 0.991337i \(-0.458072\pi\)
0.131341 + 0.991337i \(0.458072\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −17.7321 −0.755411
\(552\) 0 0
\(553\) −10.8231 −0.460244
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −9.46410 −0.401007 −0.200503 0.979693i \(-0.564258\pi\)
−0.200503 + 0.979693i \(0.564258\pi\)
\(558\) 0 0
\(559\) −55.7128 −2.35640
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 13.2679 0.559177 0.279589 0.960120i \(-0.409802\pi\)
0.279589 + 0.960120i \(0.409802\pi\)
\(564\) 0 0
\(565\) 5.12436 0.215583
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 32.9090 1.37962 0.689808 0.723993i \(-0.257695\pi\)
0.689808 + 0.723993i \(0.257695\pi\)
\(570\) 0 0
\(571\) −1.78461 −0.0746836 −0.0373418 0.999303i \(-0.511889\pi\)
−0.0373418 + 0.999303i \(0.511889\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.46410 −0.144463
\(576\) 0 0
\(577\) 18.7321 0.779825 0.389913 0.920852i \(-0.372505\pi\)
0.389913 + 0.920852i \(0.372505\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 10.3923 0.431145
\(582\) 0 0
\(583\) 7.41154 0.306955
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.66025 −0.233624 −0.116812 0.993154i \(-0.537267\pi\)
−0.116812 + 0.993154i \(0.537267\pi\)
\(588\) 0 0
\(589\) 7.39230 0.304595
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −27.8564 −1.14393 −0.571963 0.820280i \(-0.693818\pi\)
−0.571963 + 0.820280i \(0.693818\pi\)
\(594\) 0 0
\(595\) 0.928203 0.0380526
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −16.8038 −0.686587 −0.343293 0.939228i \(-0.611542\pi\)
−0.343293 + 0.939228i \(0.611542\pi\)
\(600\) 0 0
\(601\) 17.2487 0.703590 0.351795 0.936077i \(-0.385571\pi\)
0.351795 + 0.936077i \(0.385571\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.85641 −0.238097
\(606\) 0 0
\(607\) 5.80385 0.235571 0.117785 0.993039i \(-0.462420\pi\)
0.117785 + 0.993039i \(0.462420\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −28.7846 −1.16450
\(612\) 0 0
\(613\) −5.46410 −0.220693 −0.110346 0.993893i \(-0.535196\pi\)
−0.110346 + 0.993893i \(0.535196\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.92820 0.278919 0.139459 0.990228i \(-0.455464\pi\)
0.139459 + 0.990228i \(0.455464\pi\)
\(618\) 0 0
\(619\) −15.8564 −0.637323 −0.318661 0.947869i \(-0.603233\pi\)
−0.318661 + 0.947869i \(0.603233\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6.58846 −0.263961
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0.535898 0.0213677
\(630\) 0 0
\(631\) −22.7128 −0.904183 −0.452091 0.891972i \(-0.649322\pi\)
−0.452091 + 0.891972i \(0.649322\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −16.5885 −0.658293
\(636\) 0 0
\(637\) 29.4641 1.16741
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 26.6603 1.05302 0.526508 0.850170i \(-0.323501\pi\)
0.526508 + 0.850170i \(0.323501\pi\)
\(642\) 0 0
\(643\) −24.3397 −0.959866 −0.479933 0.877305i \(-0.659339\pi\)
−0.479933 + 0.877305i \(0.659339\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.53590 −0.178325 −0.0891623 0.996017i \(-0.528419\pi\)
−0.0891623 + 0.996017i \(0.528419\pi\)
\(648\) 0 0
\(649\) 26.6077 1.04444
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −10.5359 −0.412302 −0.206151 0.978520i \(-0.566094\pi\)
−0.206151 + 0.978520i \(0.566094\pi\)
\(654\) 0 0
\(655\) 15.5885 0.609091
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −5.46410 −0.212851 −0.106426 0.994321i \(-0.533941\pi\)
−0.106426 + 0.994321i \(0.533941\pi\)
\(660\) 0 0
\(661\) −16.3205 −0.634794 −0.317397 0.948293i \(-0.602809\pi\)
−0.317397 + 0.948293i \(0.602809\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.12436 0.121157
\(666\) 0 0
\(667\) 24.9282 0.965224
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 9.07180 0.350213
\(672\) 0 0
\(673\) 10.3923 0.400594 0.200297 0.979735i \(-0.435809\pi\)
0.200297 + 0.979735i \(0.435809\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −18.0000 −0.691796 −0.345898 0.938272i \(-0.612426\pi\)
−0.345898 + 0.938272i \(0.612426\pi\)
\(678\) 0 0
\(679\) 9.71281 0.372744
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 40.3923 1.54557 0.772784 0.634669i \(-0.218863\pi\)
0.772784 + 0.634669i \(0.218863\pi\)
\(684\) 0 0
\(685\) 9.46410 0.361605
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −17.8564 −0.680275
\(690\) 0 0
\(691\) −37.7128 −1.43466 −0.717332 0.696732i \(-0.754637\pi\)
−0.717332 + 0.696732i \(0.754637\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −21.3923 −0.811456
\(696\) 0 0
\(697\) 2.33975 0.0886242
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −31.1962 −1.17826 −0.589131 0.808037i \(-0.700530\pi\)
−0.589131 + 0.808037i \(0.700530\pi\)
\(702\) 0 0
\(703\) 1.80385 0.0680334
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.5885 0.699091
\(708\) 0 0
\(709\) −29.4641 −1.10655 −0.553274 0.832999i \(-0.686622\pi\)
−0.553274 + 0.832999i \(0.686622\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −10.3923 −0.389195
\(714\) 0 0
\(715\) −12.3923 −0.463446
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −39.5885 −1.47640 −0.738200 0.674582i \(-0.764324\pi\)
−0.738200 + 0.674582i \(0.764324\pi\)
\(720\) 0 0
\(721\) 9.46410 0.352462
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −7.19615 −0.267258
\(726\) 0 0
\(727\) −12.3923 −0.459605 −0.229803 0.973237i \(-0.573808\pi\)
−0.229803 + 0.973237i \(0.573808\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.46410 0.276070
\(732\) 0 0
\(733\) −6.78461 −0.250595 −0.125298 0.992119i \(-0.539989\pi\)
−0.125298 + 0.992119i \(0.539989\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.85641 0.289394
\(738\) 0 0
\(739\) −15.5359 −0.571497 −0.285749 0.958305i \(-0.592242\pi\)
−0.285749 + 0.958305i \(0.592242\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 45.9090 1.68424 0.842118 0.539293i \(-0.181308\pi\)
0.842118 + 0.539293i \(0.181308\pi\)
\(744\) 0 0
\(745\) −8.00000 −0.293097
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19.6077 0.716450
\(750\) 0 0
\(751\) −7.21539 −0.263293 −0.131647 0.991297i \(-0.542026\pi\)
−0.131647 + 0.991297i \(0.542026\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 15.3923 0.560183
\(756\) 0 0
\(757\) 53.1769 1.93275 0.966374 0.257141i \(-0.0827805\pi\)
0.966374 + 0.257141i \(0.0827805\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −35.4449 −1.28488 −0.642438 0.766338i \(-0.722077\pi\)
−0.642438 + 0.766338i \(0.722077\pi\)
\(762\) 0 0
\(763\) −25.2679 −0.914761
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −64.1051 −2.31470
\(768\) 0 0
\(769\) −16.4641 −0.593711 −0.296855 0.954922i \(-0.595938\pi\)
−0.296855 + 0.954922i \(0.595938\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −43.5167 −1.56519 −0.782593 0.622534i \(-0.786103\pi\)
−0.782593 + 0.622534i \(0.786103\pi\)
\(774\) 0 0
\(775\) 3.00000 0.107763
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.87564 0.282174
\(780\) 0 0
\(781\) 0.607695 0.0217450
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 5.12436 0.182896
\(786\) 0 0
\(787\) 9.94744 0.354588 0.177294 0.984158i \(-0.443266\pi\)
0.177294 + 0.984158i \(0.443266\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.49742 0.231022
\(792\) 0 0
\(793\) −21.8564 −0.776144
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −36.3923 −1.28908 −0.644541 0.764570i \(-0.722951\pi\)
−0.644541 + 0.764570i \(0.722951\pi\)
\(798\) 0 0
\(799\) 3.85641 0.136430
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 21.9090 0.773151
\(804\) 0 0
\(805\) −4.39230 −0.154808
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −13.4449 −0.472696 −0.236348 0.971668i \(-0.575951\pi\)
−0.236348 + 0.971668i \(0.575951\pi\)
\(810\) 0 0
\(811\) 11.5359 0.405080 0.202540 0.979274i \(-0.435080\pi\)
0.202540 + 0.979274i \(0.435080\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −9.26795 −0.324642
\(816\) 0 0
\(817\) 25.1244 0.878990
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −34.2679 −1.19596 −0.597980 0.801511i \(-0.704030\pi\)
−0.597980 + 0.801511i \(0.704030\pi\)
\(822\) 0 0
\(823\) 23.8564 0.831582 0.415791 0.909460i \(-0.363505\pi\)
0.415791 + 0.909460i \(0.363505\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 32.3923 1.12639 0.563195 0.826324i \(-0.309572\pi\)
0.563195 + 0.826324i \(0.309572\pi\)
\(828\) 0 0
\(829\) 17.7846 0.617685 0.308843 0.951113i \(-0.400058\pi\)
0.308843 + 0.951113i \(0.400058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.94744 −0.136771
\(834\) 0 0
\(835\) 0.339746 0.0117574
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −22.8038 −0.787276 −0.393638 0.919265i \(-0.628784\pi\)
−0.393638 + 0.919265i \(0.628784\pi\)
\(840\) 0 0
\(841\) 22.7846 0.785676
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 16.8564 0.579878
\(846\) 0 0
\(847\) −7.42563 −0.255148
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −2.53590 −0.0869295
\(852\) 0 0
\(853\) 55.5167 1.90085 0.950427 0.310947i \(-0.100646\pi\)
0.950427 + 0.310947i \(0.100646\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −52.4974 −1.79328 −0.896639 0.442763i \(-0.853998\pi\)
−0.896639 + 0.442763i \(0.853998\pi\)
\(858\) 0 0
\(859\) 11.0000 0.375315 0.187658 0.982235i \(-0.439910\pi\)
0.187658 + 0.982235i \(0.439910\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.12436 −0.0382735 −0.0191368 0.999817i \(-0.506092\pi\)
−0.0191368 + 0.999817i \(0.506092\pi\)
\(864\) 0 0
\(865\) 15.4641 0.525795
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −19.3590 −0.656709
\(870\) 0 0
\(871\) −18.9282 −0.641358
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.26795 0.0428645
\(876\) 0 0
\(877\) 26.5359 0.896054 0.448027 0.894020i \(-0.352127\pi\)
0.448027 + 0.894020i \(0.352127\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −8.94744 −0.301447 −0.150723 0.988576i \(-0.548160\pi\)
−0.150723 + 0.988576i \(0.548160\pi\)
\(882\) 0 0
\(883\) 19.8038 0.666453 0.333226 0.942847i \(-0.391863\pi\)
0.333226 + 0.942847i \(0.391863\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −44.1962 −1.48396 −0.741981 0.670421i \(-0.766113\pi\)
−0.741981 + 0.670421i \(0.766113\pi\)
\(888\) 0 0
\(889\) −21.0333 −0.705435
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 12.9808 0.434385
\(894\) 0 0
\(895\) 16.1244 0.538978
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −21.5885 −0.720015
\(900\) 0 0
\(901\) 2.39230 0.0796992
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 19.5359 0.649395
\(906\) 0 0
\(907\) 30.0000 0.996134 0.498067 0.867139i \(-0.334043\pi\)
0.498067 + 0.867139i \(0.334043\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 7.58846 0.251417 0.125708 0.992067i \(-0.459880\pi\)
0.125708 + 0.992067i \(0.459880\pi\)
\(912\) 0 0
\(913\) 18.5885 0.615188
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 19.7654 0.652710
\(918\) 0 0
\(919\) −46.9615 −1.54912 −0.774559 0.632502i \(-0.782028\pi\)
−0.774559 + 0.632502i \(0.782028\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.46410 −0.0481915
\(924\) 0 0
\(925\) 0.732051 0.0240697
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 28.3731 0.930890 0.465445 0.885077i \(-0.345894\pi\)
0.465445 + 0.885077i \(0.345894\pi\)
\(930\) 0 0
\(931\) −13.2872 −0.435470
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.66025 0.0542961
\(936\) 0 0
\(937\) −31.8564 −1.04070 −0.520352 0.853952i \(-0.674199\pi\)
−0.520352 + 0.853952i \(0.674199\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −27.1769 −0.885942 −0.442971 0.896536i \(-0.646076\pi\)
−0.442971 + 0.896536i \(0.646076\pi\)
\(942\) 0 0
\(943\) −11.0718 −0.360547
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 2.28719 0.0743236 0.0371618 0.999309i \(-0.488168\pi\)
0.0371618 + 0.999309i \(0.488168\pi\)
\(948\) 0 0
\(949\) −52.7846 −1.71346
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 15.6077 0.505583 0.252791 0.967521i \(-0.418651\pi\)
0.252791 + 0.967521i \(0.418651\pi\)
\(954\) 0 0
\(955\) −16.1244 −0.521772
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −8.73205 −0.281095
\(966\) 0 0
\(967\) 36.1962 1.16399 0.581995 0.813192i \(-0.302272\pi\)
0.581995 + 0.813192i \(0.302272\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 41.4449 1.33003 0.665014 0.746830i \(-0.268425\pi\)
0.665014 + 0.746830i \(0.268425\pi\)
\(972\) 0 0
\(973\) −27.1244 −0.869567
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.46410 −0.0468408 −0.0234204 0.999726i \(-0.507456\pi\)
−0.0234204 + 0.999726i \(0.507456\pi\)
\(978\) 0 0
\(979\) −11.7846 −0.376638
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −17.4115 −0.555342 −0.277671 0.960676i \(-0.589563\pi\)
−0.277671 + 0.960676i \(0.589563\pi\)
\(984\) 0 0
\(985\) 13.8564 0.441502
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −35.3205 −1.12313
\(990\) 0 0
\(991\) −3.14359 −0.0998595 −0.0499298 0.998753i \(-0.515900\pi\)
−0.0499298 + 0.998753i \(0.515900\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2.00000 0.0634043
\(996\) 0 0
\(997\) −19.5167 −0.618099 −0.309049 0.951046i \(-0.600011\pi\)
−0.309049 + 0.951046i \(0.600011\pi\)
\(998\) 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 6480.2.a.br.1.1 2
3.2 odd 2 6480.2.a.bi.1.1 2
4.3 odd 2 405.2.a.g.1.1 2
12.11 even 2 405.2.a.h.1.2 yes 2
20.3 even 4 2025.2.b.g.649.4 4
20.7 even 4 2025.2.b.g.649.1 4
20.19 odd 2 2025.2.a.m.1.2 2
36.7 odd 6 405.2.e.l.271.2 4
36.11 even 6 405.2.e.i.271.1 4
36.23 even 6 405.2.e.i.136.1 4
36.31 odd 6 405.2.e.l.136.2 4
60.23 odd 4 2025.2.b.h.649.1 4
60.47 odd 4 2025.2.b.h.649.4 4
60.59 even 2 2025.2.a.g.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.2.a.g.1.1 2 4.3 odd 2
405.2.a.h.1.2 yes 2 12.11 even 2
405.2.e.i.136.1 4 36.23 even 6
405.2.e.i.271.1 4 36.11 even 6
405.2.e.l.136.2 4 36.31 odd 6
405.2.e.l.271.2 4 36.7 odd 6
2025.2.a.g.1.1 2 60.59 even 2
2025.2.a.m.1.2 2 20.19 odd 2
2025.2.b.g.649.1 4 20.7 even 4
2025.2.b.g.649.4 4 20.3 even 4
2025.2.b.h.649.1 4 60.23 odd 4
2025.2.b.h.649.4 4 60.47 odd 4
6480.2.a.bi.1.1 2 3.2 odd 2
6480.2.a.br.1.1 2 1.1 even 1 trivial