Properties

Label 3240.2.a.p.1.1
Level $3240$
Weight $2$
Character 3240.1
Self dual yes
Analytic conductor $25.872$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3240,2,Mod(1,3240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("3240.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.a (trivial)

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: yes
Analytic conductor: \(25.8715302549\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
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 360)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 3240.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{5} +0.267949 q^{7} +O(q^{10})\) \(q+1.00000 q^{5} +0.267949 q^{7} +1.46410 q^{11} -5.46410 q^{13} -0.535898 q^{17} -2.00000 q^{19} -3.73205 q^{23} +1.00000 q^{25} -1.53590 q^{29} -2.00000 q^{31} +0.267949 q^{35} +10.3923 q^{37} -9.92820 q^{41} -4.53590 q^{43} -0.267949 q^{47} -6.92820 q^{49} -6.00000 q^{53} +1.46410 q^{55} -14.3923 q^{59} -8.46410 q^{61} -5.46410 q^{65} +6.26795 q^{67} -9.46410 q^{71} +6.92820 q^{73} +0.392305 q^{77} +15.4641 q^{79} +13.1962 q^{83} -0.535898 q^{85} +9.92820 q^{89} -1.46410 q^{91} -2.00000 q^{95} -8.92820 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 4 q^{7} - 4 q^{11} - 4 q^{13} - 8 q^{17} - 4 q^{19} - 4 q^{23} + 2 q^{25} - 10 q^{29} - 4 q^{31} + 4 q^{35} - 6 q^{41} - 16 q^{43} - 4 q^{47} - 12 q^{53} - 4 q^{55} - 8 q^{59} - 10 q^{61} - 4 q^{65} + 16 q^{67} - 12 q^{71} - 20 q^{77} + 24 q^{79} + 16 q^{83} - 8 q^{85} + 6 q^{89} + 4 q^{91} - 4 q^{95} - 4 q^{97}+O(q^{100}) \) 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
\(7\) 0.267949 0.101275 0.0506376 0.998717i \(-0.483875\pi\)
0.0506376 + 0.998717i \(0.483875\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.46410 0.441443 0.220722 0.975337i \(-0.429159\pi\)
0.220722 + 0.975337i \(0.429159\pi\)
\(12\) 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
\(17\) −0.535898 −0.129974 −0.0649872 0.997886i \(-0.520701\pi\)
−0.0649872 + 0.997886i \(0.520701\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.73205 −0.778186 −0.389093 0.921198i \(-0.627212\pi\)
−0.389093 + 0.921198i \(0.627212\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.53590 −0.285209 −0.142605 0.989780i \(-0.545548\pi\)
−0.142605 + 0.989780i \(0.545548\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.267949 0.0452917
\(36\) 0 0
\(37\) 10.3923 1.70848 0.854242 0.519875i \(-0.174022\pi\)
0.854242 + 0.519875i \(0.174022\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.92820 −1.55052 −0.775262 0.631639i \(-0.782382\pi\)
−0.775262 + 0.631639i \(0.782382\pi\)
\(42\) 0 0
\(43\) −4.53590 −0.691718 −0.345859 0.938286i \(-0.612412\pi\)
−0.345859 + 0.938286i \(0.612412\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −0.267949 −0.0390844 −0.0195422 0.999809i \(-0.506221\pi\)
−0.0195422 + 0.999809i \(0.506221\pi\)
\(48\) 0 0
\(49\) −6.92820 −0.989743
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 0 0
\(55\) 1.46410 0.197419
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −14.3923 −1.87372 −0.936859 0.349707i \(-0.886281\pi\)
−0.936859 + 0.349707i \(0.886281\pi\)
\(60\) 0 0
\(61\) −8.46410 −1.08372 −0.541859 0.840470i \(-0.682279\pi\)
−0.541859 + 0.840470i \(0.682279\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −5.46410 −0.677738
\(66\) 0 0
\(67\) 6.26795 0.765752 0.382876 0.923800i \(-0.374934\pi\)
0.382876 + 0.923800i \(0.374934\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.46410 −1.12318 −0.561591 0.827415i \(-0.689811\pi\)
−0.561591 + 0.827415i \(0.689811\pi\)
\(72\) 0 0
\(73\) 6.92820 0.810885 0.405442 0.914121i \(-0.367117\pi\)
0.405442 + 0.914121i \(0.367117\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.392305 0.0447073
\(78\) 0 0
\(79\) 15.4641 1.73985 0.869924 0.493186i \(-0.164168\pi\)
0.869924 + 0.493186i \(0.164168\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 13.1962 1.44847 0.724233 0.689555i \(-0.242194\pi\)
0.724233 + 0.689555i \(0.242194\pi\)
\(84\) 0 0
\(85\) −0.535898 −0.0581263
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 9.92820 1.05239 0.526194 0.850365i \(-0.323619\pi\)
0.526194 + 0.850365i \(0.323619\pi\)
\(90\) 0 0
\(91\) −1.46410 −0.153480
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.00000 −0.205196
\(96\) 0 0
\(97\) −8.92820 −0.906522 −0.453261 0.891378i \(-0.649739\pi\)
−0.453261 + 0.891378i \(0.649739\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.928203 −0.0923597 −0.0461798 0.998933i \(-0.514705\pi\)
−0.0461798 + 0.998933i \(0.514705\pi\)
\(102\) 0 0
\(103\) 14.3923 1.41812 0.709058 0.705150i \(-0.249120\pi\)
0.709058 + 0.705150i \(0.249120\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.19615 −0.502331 −0.251166 0.967944i \(-0.580814\pi\)
−0.251166 + 0.967944i \(0.580814\pi\)
\(108\) 0 0
\(109\) −17.3923 −1.66588 −0.832940 0.553363i \(-0.813344\pi\)
−0.832940 + 0.553363i \(0.813344\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −13.8564 −1.30350 −0.651751 0.758433i \(-0.725965\pi\)
−0.651751 + 0.758433i \(0.725965\pi\)
\(114\) 0 0
\(115\) −3.73205 −0.348016
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.143594 −0.0131632
\(120\) 0 0
\(121\) −8.85641 −0.805128
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 16.2679 1.44355 0.721774 0.692129i \(-0.243327\pi\)
0.721774 + 0.692129i \(0.243327\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.46410 0.302660 0.151330 0.988483i \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(132\) 0 0
\(133\) −0.535898 −0.0464683
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −21.8564 −1.86732 −0.933659 0.358162i \(-0.883404\pi\)
−0.933659 + 0.358162i \(0.883404\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −8.00000 −0.668994
\(144\) 0 0
\(145\) −1.53590 −0.127549
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2.46410 0.201867 0.100934 0.994893i \(-0.467817\pi\)
0.100934 + 0.994893i \(0.467817\pi\)
\(150\) 0 0
\(151\) 5.46410 0.444662 0.222331 0.974971i \(-0.428633\pi\)
0.222331 + 0.974971i \(0.428633\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 4.92820 0.393313 0.196657 0.980472i \(-0.436992\pi\)
0.196657 + 0.980472i \(0.436992\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −13.3205 −1.04334 −0.521671 0.853147i \(-0.674691\pi\)
−0.521671 + 0.853147i \(0.674691\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 21.5885 1.67056 0.835282 0.549821i \(-0.185304\pi\)
0.835282 + 0.549821i \(0.185304\pi\)
\(168\) 0 0
\(169\) 16.8564 1.29665
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −9.07180 −0.689716 −0.344858 0.938655i \(-0.612073\pi\)
−0.344858 + 0.938655i \(0.612073\pi\)
\(174\) 0 0
\(175\) 0.267949 0.0202551
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −13.4641 −1.00635 −0.503177 0.864183i \(-0.667836\pi\)
−0.503177 + 0.864183i \(0.667836\pi\)
\(180\) 0 0
\(181\) −9.53590 −0.708798 −0.354399 0.935094i \(-0.615314\pi\)
−0.354399 + 0.935094i \(0.615314\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.3923 0.764057
\(186\) 0 0
\(187\) −0.784610 −0.0573763
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −16.9282 −1.22488 −0.612441 0.790516i \(-0.709812\pi\)
−0.612441 + 0.790516i \(0.709812\pi\)
\(192\) 0 0
\(193\) −17.4641 −1.25709 −0.628547 0.777772i \(-0.716350\pi\)
−0.628547 + 0.777772i \(0.716350\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −14.9282 −1.06359 −0.531795 0.846873i \(-0.678482\pi\)
−0.531795 + 0.846873i \(0.678482\pi\)
\(198\) 0 0
\(199\) 7.07180 0.501306 0.250653 0.968077i \(-0.419355\pi\)
0.250653 + 0.968077i \(0.419355\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.411543 −0.0288846
\(204\) 0 0
\(205\) −9.92820 −0.693416
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.92820 −0.202548
\(210\) 0 0
\(211\) 0.392305 0.0270074 0.0135037 0.999909i \(-0.495702\pi\)
0.0135037 + 0.999909i \(0.495702\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −4.53590 −0.309346
\(216\) 0 0
\(217\) −0.535898 −0.0363792
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.92820 0.196972
\(222\) 0 0
\(223\) 22.6603 1.51744 0.758721 0.651415i \(-0.225824\pi\)
0.758721 + 0.651415i \(0.225824\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −24.2487 −1.60944 −0.804722 0.593652i \(-0.797686\pi\)
−0.804722 + 0.593652i \(0.797686\pi\)
\(228\) 0 0
\(229\) 25.2487 1.66848 0.834241 0.551400i \(-0.185906\pi\)
0.834241 + 0.551400i \(0.185906\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 10.3923 0.680823 0.340411 0.940277i \(-0.389434\pi\)
0.340411 + 0.940277i \(0.389434\pi\)
\(234\) 0 0
\(235\) −0.267949 −0.0174791
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −20.7846 −1.34444 −0.672222 0.740349i \(-0.734660\pi\)
−0.672222 + 0.740349i \(0.734660\pi\)
\(240\) 0 0
\(241\) −6.07180 −0.391119 −0.195559 0.980692i \(-0.562652\pi\)
−0.195559 + 0.980692i \(0.562652\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −6.92820 −0.442627
\(246\) 0 0
\(247\) 10.9282 0.695345
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 7.85641 0.495892 0.247946 0.968774i \(-0.420244\pi\)
0.247946 + 0.968774i \(0.420244\pi\)
\(252\) 0 0
\(253\) −5.46410 −0.343525
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.928203 0.0578997 0.0289499 0.999581i \(-0.490784\pi\)
0.0289499 + 0.999581i \(0.490784\pi\)
\(258\) 0 0
\(259\) 2.78461 0.173027
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −22.3923 −1.38077 −0.690384 0.723443i \(-0.742559\pi\)
−0.690384 + 0.723443i \(0.742559\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −11.3923 −0.694601 −0.347301 0.937754i \(-0.612902\pi\)
−0.347301 + 0.937754i \(0.612902\pi\)
\(270\) 0 0
\(271\) −19.8564 −1.20619 −0.603095 0.797669i \(-0.706066\pi\)
−0.603095 + 0.797669i \(0.706066\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.46410 0.0882886
\(276\) 0 0
\(277\) −14.3923 −0.864750 −0.432375 0.901694i \(-0.642324\pi\)
−0.432375 + 0.901694i \(0.642324\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 14.8564 0.886259 0.443129 0.896458i \(-0.353868\pi\)
0.443129 + 0.896458i \(0.353868\pi\)
\(282\) 0 0
\(283\) 21.7321 1.29184 0.645918 0.763407i \(-0.276475\pi\)
0.645918 + 0.763407i \(0.276475\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.66025 −0.157030
\(288\) 0 0
\(289\) −16.7128 −0.983107
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.46410 0.436057 0.218029 0.975942i \(-0.430037\pi\)
0.218029 + 0.975942i \(0.430037\pi\)
\(294\) 0 0
\(295\) −14.3923 −0.837952
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 20.3923 1.17932
\(300\) 0 0
\(301\) −1.21539 −0.0700539
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −8.46410 −0.484653
\(306\) 0 0
\(307\) 4.12436 0.235389 0.117695 0.993050i \(-0.462450\pi\)
0.117695 + 0.993050i \(0.462450\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.07180 −0.174186 −0.0870928 0.996200i \(-0.527758\pi\)
−0.0870928 + 0.996200i \(0.527758\pi\)
\(312\) 0 0
\(313\) 12.3923 0.700454 0.350227 0.936665i \(-0.386104\pi\)
0.350227 + 0.936665i \(0.386104\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.53590 −0.142430 −0.0712151 0.997461i \(-0.522688\pi\)
−0.0712151 + 0.997461i \(0.522688\pi\)
\(318\) 0 0
\(319\) −2.24871 −0.125904
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.07180 0.0596364
\(324\) 0 0
\(325\) −5.46410 −0.303094
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −0.0717968 −0.00395828
\(330\) 0 0
\(331\) −3.46410 −0.190404 −0.0952021 0.995458i \(-0.530350\pi\)
−0.0952021 + 0.995458i \(0.530350\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 6.26795 0.342455
\(336\) 0 0
\(337\) 9.07180 0.494172 0.247086 0.968994i \(-0.420527\pi\)
0.247086 + 0.968994i \(0.420527\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.92820 −0.158571
\(342\) 0 0
\(343\) −3.73205 −0.201512
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.3923 0.987351 0.493675 0.869646i \(-0.335653\pi\)
0.493675 + 0.869646i \(0.335653\pi\)
\(348\) 0 0
\(349\) 0.464102 0.0248428 0.0124214 0.999923i \(-0.496046\pi\)
0.0124214 + 0.999923i \(0.496046\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 21.8564 1.16330 0.581650 0.813439i \(-0.302408\pi\)
0.581650 + 0.813439i \(0.302408\pi\)
\(354\) 0 0
\(355\) −9.46410 −0.502302
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 12.9282 0.682324 0.341162 0.940004i \(-0.389179\pi\)
0.341162 + 0.940004i \(0.389179\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.92820 0.362639
\(366\) 0 0
\(367\) −14.3923 −0.751272 −0.375636 0.926767i \(-0.622576\pi\)
−0.375636 + 0.926767i \(0.622576\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.60770 −0.0834674
\(372\) 0 0
\(373\) −8.92820 −0.462285 −0.231142 0.972920i \(-0.574246\pi\)
−0.231142 + 0.972920i \(0.574246\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 8.39230 0.432226
\(378\) 0 0
\(379\) 35.1769 1.80692 0.903458 0.428676i \(-0.141020\pi\)
0.903458 + 0.428676i \(0.141020\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −3.46410 −0.177007 −0.0885037 0.996076i \(-0.528208\pi\)
−0.0885037 + 0.996076i \(0.528208\pi\)
\(384\) 0 0
\(385\) 0.392305 0.0199937
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 23.5359 1.19332 0.596659 0.802495i \(-0.296495\pi\)
0.596659 + 0.802495i \(0.296495\pi\)
\(390\) 0 0
\(391\) 2.00000 0.101144
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 15.4641 0.778083
\(396\) 0 0
\(397\) 10.0000 0.501886 0.250943 0.968002i \(-0.419259\pi\)
0.250943 + 0.968002i \(0.419259\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −14.7846 −0.738308 −0.369154 0.929368i \(-0.620353\pi\)
−0.369154 + 0.929368i \(0.620353\pi\)
\(402\) 0 0
\(403\) 10.9282 0.544373
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.2154 0.754199
\(408\) 0 0
\(409\) 4.92820 0.243684 0.121842 0.992550i \(-0.461120\pi\)
0.121842 + 0.992550i \(0.461120\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.85641 −0.189761
\(414\) 0 0
\(415\) 13.1962 0.647774
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −30.2487 −1.47775 −0.738873 0.673845i \(-0.764642\pi\)
−0.738873 + 0.673845i \(0.764642\pi\)
\(420\) 0 0
\(421\) −36.9282 −1.79977 −0.899885 0.436127i \(-0.856350\pi\)
−0.899885 + 0.436127i \(0.856350\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −0.535898 −0.0259949
\(426\) 0 0
\(427\) −2.26795 −0.109754
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4.53590 0.218487 0.109243 0.994015i \(-0.465157\pi\)
0.109243 + 0.994015i \(0.465157\pi\)
\(432\) 0 0
\(433\) −12.5359 −0.602437 −0.301218 0.953555i \(-0.597393\pi\)
−0.301218 + 0.953555i \(0.597393\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.46410 0.357056
\(438\) 0 0
\(439\) 21.0718 1.00570 0.502851 0.864373i \(-0.332284\pi\)
0.502851 + 0.864373i \(0.332284\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −34.5167 −1.63994 −0.819968 0.572409i \(-0.806009\pi\)
−0.819968 + 0.572409i \(0.806009\pi\)
\(444\) 0 0
\(445\) 9.92820 0.470642
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −4.14359 −0.195548 −0.0977741 0.995209i \(-0.531172\pi\)
−0.0977741 + 0.995209i \(0.531172\pi\)
\(450\) 0 0
\(451\) −14.5359 −0.684469
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.46410 −0.0686381
\(456\) 0 0
\(457\) 5.60770 0.262317 0.131158 0.991361i \(-0.458130\pi\)
0.131158 + 0.991361i \(0.458130\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −4.60770 −0.214602 −0.107301 0.994227i \(-0.534221\pi\)
−0.107301 + 0.994227i \(0.534221\pi\)
\(462\) 0 0
\(463\) −3.46410 −0.160990 −0.0804952 0.996755i \(-0.525650\pi\)
−0.0804952 + 0.996755i \(0.525650\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 30.3923 1.40639 0.703194 0.710998i \(-0.251757\pi\)
0.703194 + 0.710998i \(0.251757\pi\)
\(468\) 0 0
\(469\) 1.67949 0.0775517
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −6.64102 −0.305354
\(474\) 0 0
\(475\) −2.00000 −0.0917663
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 14.3923 0.657601 0.328801 0.944399i \(-0.393356\pi\)
0.328801 + 0.944399i \(0.393356\pi\)
\(480\) 0 0
\(481\) −56.7846 −2.58916
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −8.92820 −0.405409
\(486\) 0 0
\(487\) 40.2487 1.82384 0.911922 0.410364i \(-0.134599\pi\)
0.911922 + 0.410364i \(0.134599\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 25.8564 1.16688 0.583442 0.812155i \(-0.301706\pi\)
0.583442 + 0.812155i \(0.301706\pi\)
\(492\) 0 0
\(493\) 0.823085 0.0370699
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.53590 −0.113751
\(498\) 0 0
\(499\) −17.8564 −0.799363 −0.399681 0.916654i \(-0.630879\pi\)
−0.399681 + 0.916654i \(0.630879\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.1244 0.808125 0.404063 0.914731i \(-0.367598\pi\)
0.404063 + 0.914731i \(0.367598\pi\)
\(504\) 0 0
\(505\) −0.928203 −0.0413045
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 20.6077 0.913420 0.456710 0.889616i \(-0.349028\pi\)
0.456710 + 0.889616i \(0.349028\pi\)
\(510\) 0 0
\(511\) 1.85641 0.0821226
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 14.3923 0.634201
\(516\) 0 0
\(517\) −0.392305 −0.0172535
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 13.0000 0.569540 0.284770 0.958596i \(-0.408083\pi\)
0.284770 + 0.958596i \(0.408083\pi\)
\(522\) 0 0
\(523\) 10.8038 0.472419 0.236210 0.971702i \(-0.424095\pi\)
0.236210 + 0.971702i \(0.424095\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.07180 0.0466882
\(528\) 0 0
\(529\) −9.07180 −0.394426
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 54.2487 2.34977
\(534\) 0 0
\(535\) −5.19615 −0.224649
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −10.1436 −0.436916
\(540\) 0 0
\(541\) −18.6077 −0.800007 −0.400004 0.916514i \(-0.630991\pi\)
−0.400004 + 0.916514i \(0.630991\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −17.3923 −0.745004
\(546\) 0 0
\(547\) 45.9808 1.96600 0.982998 0.183618i \(-0.0587809\pi\)
0.982998 + 0.183618i \(0.0587809\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 3.07180 0.130863
\(552\) 0 0
\(553\) 4.14359 0.176204
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −32.6410 −1.38304 −0.691522 0.722355i \(-0.743060\pi\)
−0.691522 + 0.722355i \(0.743060\pi\)
\(558\) 0 0
\(559\) 24.7846 1.04828
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 12.4115 0.523084 0.261542 0.965192i \(-0.415769\pi\)
0.261542 + 0.965192i \(0.415769\pi\)
\(564\) 0 0
\(565\) −13.8564 −0.582943
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −40.7846 −1.70678 −0.853391 0.521271i \(-0.825458\pi\)
−0.853391 + 0.521271i \(0.825458\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −3.73205 −0.155637
\(576\) 0 0
\(577\) 32.9282 1.37082 0.685410 0.728158i \(-0.259623\pi\)
0.685410 + 0.728158i \(0.259623\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.53590 0.146694
\(582\) 0 0
\(583\) −8.78461 −0.363821
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.41154 −0.347182 −0.173591 0.984818i \(-0.555537\pi\)
−0.173591 + 0.984818i \(0.555537\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 39.4641 1.62060 0.810298 0.586018i \(-0.199305\pi\)
0.810298 + 0.586018i \(0.199305\pi\)
\(594\) 0 0
\(595\) −0.143594 −0.00588676
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.60770 −0.0656886 −0.0328443 0.999460i \(-0.510457\pi\)
−0.0328443 + 0.999460i \(0.510457\pi\)
\(600\) 0 0
\(601\) 30.7846 1.25573 0.627865 0.778322i \(-0.283929\pi\)
0.627865 + 0.778322i \(0.283929\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −8.85641 −0.360064
\(606\) 0 0
\(607\) 40.2679 1.63443 0.817213 0.576336i \(-0.195518\pi\)
0.817213 + 0.576336i \(0.195518\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.46410 0.0592312
\(612\) 0 0
\(613\) 22.3923 0.904417 0.452208 0.891912i \(-0.350636\pi\)
0.452208 + 0.891912i \(0.350636\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.9282 −1.24512 −0.622561 0.782571i \(-0.713908\pi\)
−0.622561 + 0.782571i \(0.713908\pi\)
\(618\) 0 0
\(619\) −37.7128 −1.51581 −0.757903 0.652367i \(-0.773776\pi\)
−0.757903 + 0.652367i \(0.773776\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.66025 0.106581
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −5.56922 −0.222059
\(630\) 0 0
\(631\) 31.7128 1.26247 0.631234 0.775593i \(-0.282549\pi\)
0.631234 + 0.775593i \(0.282549\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 16.2679 0.645574
\(636\) 0 0
\(637\) 37.8564 1.49993
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 34.8564 1.37675 0.688373 0.725357i \(-0.258325\pi\)
0.688373 + 0.725357i \(0.258325\pi\)
\(642\) 0 0
\(643\) −34.2679 −1.35140 −0.675698 0.737179i \(-0.736158\pi\)
−0.675698 + 0.737179i \(0.736158\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.5167 0.649337 0.324668 0.945828i \(-0.394747\pi\)
0.324668 + 0.945828i \(0.394747\pi\)
\(648\) 0 0
\(649\) −21.0718 −0.827140
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −31.4641 −1.23129 −0.615643 0.788025i \(-0.711104\pi\)
−0.615643 + 0.788025i \(0.711104\pi\)
\(654\) 0 0
\(655\) 3.46410 0.135354
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 20.7846 0.809653 0.404827 0.914393i \(-0.367332\pi\)
0.404827 + 0.914393i \(0.367332\pi\)
\(660\) 0 0
\(661\) 21.7128 0.844531 0.422265 0.906472i \(-0.361235\pi\)
0.422265 + 0.906472i \(0.361235\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.535898 −0.0207812
\(666\) 0 0
\(667\) 5.73205 0.221946
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −12.3923 −0.478400
\(672\) 0 0
\(673\) −12.6795 −0.488758 −0.244379 0.969680i \(-0.578584\pi\)
−0.244379 + 0.969680i \(0.578584\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.1769 0.660162 0.330081 0.943953i \(-0.392924\pi\)
0.330081 + 0.943953i \(0.392924\pi\)
\(678\) 0 0
\(679\) −2.39230 −0.0918082
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −22.3923 −0.856818 −0.428409 0.903585i \(-0.640926\pi\)
−0.428409 + 0.903585i \(0.640926\pi\)
\(684\) 0 0
\(685\) −21.8564 −0.835090
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 32.7846 1.24899
\(690\) 0 0
\(691\) −50.3923 −1.91701 −0.958507 0.285070i \(-0.907983\pi\)
−0.958507 + 0.285070i \(0.907983\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 5.32051 0.201529
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 24.1769 0.913149 0.456575 0.889685i \(-0.349076\pi\)
0.456575 + 0.889685i \(0.349076\pi\)
\(702\) 0 0
\(703\) −20.7846 −0.783906
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.248711 −0.00935375
\(708\) 0 0
\(709\) 15.3923 0.578070 0.289035 0.957319i \(-0.406666\pi\)
0.289035 + 0.957319i \(0.406666\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 7.46410 0.279533
\(714\) 0 0
\(715\) −8.00000 −0.299183
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 33.3205 1.24265 0.621323 0.783555i \(-0.286596\pi\)
0.621323 + 0.783555i \(0.286596\pi\)
\(720\) 0 0
\(721\) 3.85641 0.143620
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.53590 −0.0570418
\(726\) 0 0
\(727\) 15.9808 0.592694 0.296347 0.955080i \(-0.404232\pi\)
0.296347 + 0.955080i \(0.404232\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.43078 0.0899057
\(732\) 0 0
\(733\) 16.1436 0.596277 0.298139 0.954523i \(-0.403634\pi\)
0.298139 + 0.954523i \(0.403634\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 9.17691 0.338036
\(738\) 0 0
\(739\) −0.248711 −0.00914899 −0.00457450 0.999990i \(-0.501456\pi\)
−0.00457450 + 0.999990i \(0.501456\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 33.8372 1.24137 0.620683 0.784062i \(-0.286856\pi\)
0.620683 + 0.784062i \(0.286856\pi\)
\(744\) 0 0
\(745\) 2.46410 0.0902777
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.39230 −0.0508737
\(750\) 0 0
\(751\) 16.3923 0.598164 0.299082 0.954227i \(-0.403320\pi\)
0.299082 + 0.954227i \(0.403320\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.46410 0.198859
\(756\) 0 0
\(757\) −20.3923 −0.741171 −0.370585 0.928798i \(-0.620843\pi\)
−0.370585 + 0.928798i \(0.620843\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.0000 0.761249 0.380625 0.924730i \(-0.375709\pi\)
0.380625 + 0.924730i \(0.375709\pi\)
\(762\) 0 0
\(763\) −4.66025 −0.168713
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 78.6410 2.83956
\(768\) 0 0
\(769\) 33.7846 1.21830 0.609152 0.793053i \(-0.291510\pi\)
0.609152 + 0.793053i \(0.291510\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 8.78461 0.315960 0.157980 0.987442i \(-0.449502\pi\)
0.157980 + 0.987442i \(0.449502\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 19.8564 0.711430
\(780\) 0 0
\(781\) −13.8564 −0.495821
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.92820 0.175895
\(786\) 0 0
\(787\) 1.32051 0.0470710 0.0235355 0.999723i \(-0.492508\pi\)
0.0235355 + 0.999723i \(0.492508\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.71281 −0.132012
\(792\) 0 0
\(793\) 46.2487 1.64234
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.60770 0.269478 0.134739 0.990881i \(-0.456980\pi\)
0.134739 + 0.990881i \(0.456980\pi\)
\(798\) 0 0
\(799\) 0.143594 0.00507997
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 10.1436 0.357960
\(804\) 0 0
\(805\) −1.00000 −0.0352454
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 48.9282 1.72022 0.860112 0.510105i \(-0.170394\pi\)
0.860112 + 0.510105i \(0.170394\pi\)
\(810\) 0 0
\(811\) −50.3923 −1.76951 −0.884757 0.466053i \(-0.845675\pi\)
−0.884757 + 0.466053i \(0.845675\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −13.3205 −0.466597
\(816\) 0 0
\(817\) 9.07180 0.317382
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −23.2487 −0.811386 −0.405693 0.914009i \(-0.632970\pi\)
−0.405693 + 0.914009i \(0.632970\pi\)
\(822\) 0 0
\(823\) 23.7321 0.827247 0.413624 0.910448i \(-0.364263\pi\)
0.413624 + 0.910448i \(0.364263\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 16.9474 0.589320 0.294660 0.955602i \(-0.404794\pi\)
0.294660 + 0.955602i \(0.404794\pi\)
\(828\) 0 0
\(829\) 11.3923 0.395671 0.197836 0.980235i \(-0.436609\pi\)
0.197836 + 0.980235i \(0.436609\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.71281 0.128641
\(834\) 0 0
\(835\) 21.5885 0.747099
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −35.8564 −1.23790 −0.618950 0.785430i \(-0.712442\pi\)
−0.618950 + 0.785430i \(0.712442\pi\)
\(840\) 0 0
\(841\) −26.6410 −0.918656
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 16.8564 0.579878
\(846\) 0 0
\(847\) −2.37307 −0.0815395
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −38.7846 −1.32952
\(852\) 0 0
\(853\) −49.4641 −1.69362 −0.846809 0.531897i \(-0.821479\pi\)
−0.846809 + 0.531897i \(0.821479\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.2487 −0.623364 −0.311682 0.950186i \(-0.600892\pi\)
−0.311682 + 0.950186i \(0.600892\pi\)
\(858\) 0 0
\(859\) −20.3923 −0.695776 −0.347888 0.937536i \(-0.613101\pi\)
−0.347888 + 0.937536i \(0.613101\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26.6603 −0.907526 −0.453763 0.891123i \(-0.649919\pi\)
−0.453763 + 0.891123i \(0.649919\pi\)
\(864\) 0 0
\(865\) −9.07180 −0.308450
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 22.6410 0.768044
\(870\) 0 0
\(871\) −34.2487 −1.16047
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0.267949 0.00905834
\(876\) 0 0
\(877\) 11.6077 0.391964 0.195982 0.980607i \(-0.437211\pi\)
0.195982 + 0.980607i \(0.437211\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −37.6410 −1.26816 −0.634079 0.773268i \(-0.718621\pi\)
−0.634079 + 0.773268i \(0.718621\pi\)
\(882\) 0 0
\(883\) −9.19615 −0.309475 −0.154738 0.987956i \(-0.549453\pi\)
−0.154738 + 0.987956i \(0.549453\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.32051 0.312952 0.156476 0.987682i \(-0.449987\pi\)
0.156476 + 0.987682i \(0.449987\pi\)
\(888\) 0 0
\(889\) 4.35898 0.146196
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0.535898 0.0179332
\(894\) 0 0
\(895\) −13.4641 −0.450055
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.07180 0.102450
\(900\) 0 0
\(901\) 3.21539 0.107120
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −9.53590 −0.316984
\(906\) 0 0
\(907\) −13.7321 −0.455965 −0.227983 0.973665i \(-0.573213\pi\)
−0.227983 + 0.973665i \(0.573213\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 12.5359 0.415333 0.207666 0.978200i \(-0.433413\pi\)
0.207666 + 0.978200i \(0.433413\pi\)
\(912\) 0 0
\(913\) 19.3205 0.639415
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0.928203 0.0306520
\(918\) 0 0
\(919\) −17.6077 −0.580824 −0.290412 0.956902i \(-0.593792\pi\)
−0.290412 + 0.956902i \(0.593792\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 51.7128 1.70215
\(924\) 0 0
\(925\) 10.3923 0.341697
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −24.6410 −0.808446 −0.404223 0.914661i \(-0.632458\pi\)
−0.404223 + 0.914661i \(0.632458\pi\)
\(930\) 0 0
\(931\) 13.8564 0.454125
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −0.784610 −0.0256595
\(936\) 0 0
\(937\) −11.7128 −0.382641 −0.191320 0.981528i \(-0.561277\pi\)
−0.191320 + 0.981528i \(0.561277\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 23.3923 0.762567 0.381284 0.924458i \(-0.375482\pi\)
0.381284 + 0.924458i \(0.375482\pi\)
\(942\) 0 0
\(943\) 37.0526 1.20660
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 14.5167 0.471728 0.235864 0.971786i \(-0.424208\pi\)
0.235864 + 0.971786i \(0.424208\pi\)
\(948\) 0 0
\(949\) −37.8564 −1.22887
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −40.3923 −1.30844 −0.654218 0.756306i \(-0.727002\pi\)
−0.654218 + 0.756306i \(0.727002\pi\)
\(954\) 0 0
\(955\) −16.9282 −0.547784
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −5.85641 −0.189113
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −17.4641 −0.562189
\(966\) 0 0
\(967\) 21.3397 0.686240 0.343120 0.939292i \(-0.388516\pi\)
0.343120 + 0.939292i \(0.388516\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −25.7128 −0.825163 −0.412582 0.910921i \(-0.635373\pi\)
−0.412582 + 0.910921i \(0.635373\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −17.1769 −0.549538 −0.274769 0.961510i \(-0.588601\pi\)
−0.274769 + 0.961510i \(0.588601\pi\)
\(978\) 0 0
\(979\) 14.5359 0.464569
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −40.5167 −1.29228 −0.646140 0.763219i \(-0.723618\pi\)
−0.646140 + 0.763219i \(0.723618\pi\)
\(984\) 0 0
\(985\) −14.9282 −0.475652
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.9282 0.538286
\(990\) 0 0
\(991\) −37.0333 −1.17640 −0.588201 0.808715i \(-0.700164\pi\)
−0.588201 + 0.808715i \(0.700164\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 7.07180 0.224191
\(996\) 0 0
\(997\) 47.5692 1.50653 0.753266 0.657716i \(-0.228477\pi\)
0.753266 + 0.657716i \(0.228477\pi\)
\(998\) 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 3240.2.a.p.1.1 2
3.2 odd 2 3240.2.a.k.1.1 2
4.3 odd 2 6480.2.a.bk.1.2 2
9.2 odd 6 360.2.q.b.121.2 4
9.4 even 3 1080.2.q.b.721.2 4
9.5 odd 6 360.2.q.b.241.2 yes 4
9.7 even 3 1080.2.q.b.361.2 4
12.11 even 2 6480.2.a.ba.1.2 2
36.7 odd 6 2160.2.q.h.1441.1 4
36.11 even 6 720.2.q.h.481.1 4
36.23 even 6 720.2.q.h.241.1 4
36.31 odd 6 2160.2.q.h.721.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.q.b.121.2 4 9.2 odd 6
360.2.q.b.241.2 yes 4 9.5 odd 6
720.2.q.h.241.1 4 36.23 even 6
720.2.q.h.481.1 4 36.11 even 6
1080.2.q.b.361.2 4 9.7 even 3
1080.2.q.b.721.2 4 9.4 even 3
2160.2.q.h.721.1 4 36.31 odd 6
2160.2.q.h.1441.1 4 36.7 odd 6
3240.2.a.k.1.1 2 3.2 odd 2
3240.2.a.p.1.1 2 1.1 even 1 trivial
6480.2.a.ba.1.2 2 12.11 even 2
6480.2.a.bk.1.2 2 4.3 odd 2