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Properties

Label 6240.2.a.m.1.1
Level $6240$
Weight $2$
Character 6240.1
Self dual yes
Analytic conductor $49.827$
Analytic rank $1$
Dimension $1$
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] = [6240,2,Mod(1,6240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6240.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6240 = 2^{5} \cdot 3 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6240.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: \(49.8266508613\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 6240.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +1.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} -1.00000 q^{13} -1.00000 q^{15} -2.00000 q^{17} +2.00000 q^{19} -2.00000 q^{21} -8.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -6.00000 q^{29} -2.00000 q^{31} -2.00000 q^{33} +2.00000 q^{35} +2.00000 q^{37} +1.00000 q^{39} -2.00000 q^{41} +1.00000 q^{45} +6.00000 q^{47} -3.00000 q^{49} +2.00000 q^{51} -10.0000 q^{53} +2.00000 q^{55} -2.00000 q^{57} -14.0000 q^{59} -10.0000 q^{61} +2.00000 q^{63} -1.00000 q^{65} +2.00000 q^{67} +8.00000 q^{69} -6.00000 q^{71} +2.00000 q^{73} -1.00000 q^{75} +4.00000 q^{77} -12.0000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -2.00000 q^{85} +6.00000 q^{87} -18.0000 q^{89} -2.00000 q^{91} +2.00000 q^{93} +2.00000 q^{95} -14.0000 q^{97} +2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 − 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 2.00000 0.458831 0.229416 − 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) 0 0
\(21\) −2.00000 −0.436436
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 − 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 2.00000 0.280056
\(52\) 0 0
\(53\) −10.0000 −1.37361 −0.686803 − 0.726844i \(-0.740986\pi\)
−0.686803 + 0.726844i \(0.740986\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) −2.00000 −0.264906
\(58\) 0 0
\(59\) −14.0000 −1.82264 −0.911322 − 0.411693i \(-0.864937\pi\)
−0.911322 + 0.411693i \(0.864937\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 − 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 − 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 0 0
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −12.0000 −1.35011 −0.675053 − 0.737769i \(-0.735879\pi\)
−0.675053 + 0.737769i \(0.735879\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −2.00000 −0.216930
\(86\) 0 0
\(87\) 6.00000 0.643268
\(88\) 0 0
\(89\) −18.0000 −1.90800 −0.953998 − 0.299813i \(-0.903076\pi\)
−0.953998 + 0.299813i \(0.903076\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 0 0
\(93\) 2.00000 0.207390
\(94\) 0 0
\(95\) 2.00000 0.205196
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 6.00000 0.597022 0.298511 − 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 − 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) −2.00000 −0.195180
\(106\) 0 0
\(107\) 20.0000 1.93347 0.966736 − 0.255774i \(-0.0823304\pi\)
0.966736 + 0.255774i \(0.0823304\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 − 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −2.00000 −0.189832
\(112\) 0 0
\(113\) −2.00000 −0.188144 −0.0940721 − 0.995565i \(-0.529988\pi\)
−0.0940721 + 0.995565i \(0.529988\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −4.00000 −0.366679
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 2.00000 0.180334
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 20.0000 1.74741 0.873704 − 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 − 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 12.0000 1.01783 0.508913 − 0.860818i \(-0.330047\pi\)
0.508913 + 0.860818i \(0.330047\pi\)
\(140\) 0 0
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0 0
\(147\) 3.00000 0.247436
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 − 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 10.0000 0.813788 0.406894 − 0.913475i \(-0.366612\pi\)
0.406894 + 0.913475i \(0.366612\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −2.00000 −0.160644
\(156\) 0 0
\(157\) 18.0000 1.43656 0.718278 − 0.695756i \(-0.244931\pi\)
0.718278 + 0.695756i \(0.244931\pi\)
\(158\) 0 0
\(159\) 10.0000 0.793052
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −18.0000 −1.40987 −0.704934 − 0.709273i \(-0.749024\pi\)
−0.704934 + 0.709273i \(0.749024\pi\)
\(164\) 0 0
\(165\) −2.00000 −0.155700
\(166\) 0 0
\(167\) −10.0000 −0.773823 −0.386912 − 0.922117i \(-0.626458\pi\)
−0.386912 + 0.922117i \(0.626458\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) 2.00000 0.152944
\(172\) 0 0
\(173\) 14.0000 1.06440 0.532200 − 0.846619i \(-0.321365\pi\)
0.532200 + 0.846619i \(0.321365\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) 14.0000 1.05230
\(178\) 0 0
\(179\) 16.0000 1.19590 0.597948 − 0.801535i \(-0.295983\pi\)
0.597948 + 0.801535i \(0.295983\pi\)
\(180\) 0 0
\(181\) −22.0000 −1.63525 −0.817624 − 0.575753i \(-0.804709\pi\)
−0.817624 + 0.575753i \(0.804709\pi\)
\(182\) 0 0
\(183\) 10.0000 0.739221
\(184\) 0 0
\(185\) 2.00000 0.147043
\(186\) 0 0
\(187\) −4.00000 −0.292509
\(188\) 0 0
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −12.0000 −0.868290 −0.434145 − 0.900843i \(-0.642949\pi\)
−0.434145 + 0.900843i \(0.642949\pi\)
\(192\) 0 0
\(193\) 10.0000 0.719816 0.359908 − 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(194\) 0 0
\(195\) 1.00000 0.0716115
\(196\) 0 0
\(197\) −10.0000 −0.712470 −0.356235 − 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −2.00000 −0.141069
\(202\) 0 0
\(203\) −12.0000 −0.842235
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) −8.00000 −0.556038
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 − 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) −2.00000 −0.135147
\(220\) 0 0
\(221\) 2.00000 0.134535
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 − 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −10.0000 −0.663723 −0.331862 − 0.943328i \(-0.607677\pi\)
−0.331862 + 0.943328i \(0.607677\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 − 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −4.00000 −0.263181
\(232\) 0 0
\(233\) −30.0000 −1.96537 −0.982683 − 0.185296i \(-0.940675\pi\)
−0.982683 + 0.185296i \(0.940675\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 0 0
\(237\) 12.0000 0.779484
\(238\) 0 0
\(239\) 10.0000 0.646846 0.323423 − 0.946254i \(-0.395166\pi\)
0.323423 + 0.946254i \(0.395166\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 − 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) −2.00000 −0.127257
\(248\) 0 0
\(249\) −6.00000 −0.380235
\(250\) 0 0
\(251\) −24.0000 −1.51487 −0.757433 − 0.652913i \(-0.773547\pi\)
−0.757433 + 0.652913i \(0.773547\pi\)
\(252\) 0 0
\(253\) −16.0000 −1.00591
\(254\) 0 0
\(255\) 2.00000 0.125245
\(256\) 0 0
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) −12.0000 −0.739952 −0.369976 − 0.929041i \(-0.620634\pi\)
−0.369976 + 0.929041i \(0.620634\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.614295
\(266\) 0 0
\(267\) 18.0000 1.10158
\(268\) 0 0
\(269\) −6.00000 −0.365826 −0.182913 − 0.983129i \(-0.558553\pi\)
−0.182913 + 0.983129i \(0.558553\pi\)
\(270\) 0 0
\(271\) 2.00000 0.121491 0.0607457 − 0.998153i \(-0.480652\pi\)
0.0607457 + 0.998153i \(0.480652\pi\)
\(272\) 0 0
\(273\) 2.00000 0.121046
\(274\) 0 0
\(275\) 2.00000 0.120605
\(276\) 0 0
\(277\) −26.0000 −1.56219 −0.781094 − 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) −2.00000 −0.119737
\(280\) 0 0
\(281\) 30.0000 1.78965 0.894825 − 0.446417i \(-0.147300\pi\)
0.894825 + 0.446417i \(0.147300\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 0 0
\(285\) −2.00000 −0.118470
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 14.0000 0.820695
\(292\) 0 0
\(293\) 6.00000 0.350524 0.175262 − 0.984522i \(-0.443923\pi\)
0.175262 + 0.984522i \(0.443923\pi\)
\(294\) 0 0
\(295\) −14.0000 −0.815112
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) 8.00000 0.462652
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −6.00000 −0.344691
\(304\) 0 0
\(305\) −10.0000 −0.572598
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 − 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) 0 0
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) −24.0000 −1.36092 −0.680458 − 0.732787i \(-0.738219\pi\)
−0.680458 + 0.732787i \(0.738219\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 − 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 0 0
\(315\) 2.00000 0.112687
\(316\) 0 0
\(317\) 22.0000 1.23564 0.617822 − 0.786318i \(-0.288015\pi\)
0.617822 + 0.786318i \(0.288015\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) −20.0000 −1.11629
\(322\) 0 0
\(323\) −4.00000 −0.222566
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) 0 0
\(327\) −2.00000 −0.110600
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) 18.0000 0.989369 0.494685 − 0.869072i \(-0.335284\pi\)
0.494685 + 0.869072i \(0.335284\pi\)
\(332\) 0 0
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 2.00000 0.109272
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 − 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 2.00000 0.108625
\(340\) 0 0
\(341\) −4.00000 −0.216612
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 8.00000 0.430706
\(346\) 0 0
\(347\) −4.00000 −0.214731 −0.107366 − 0.994220i \(-0.534242\pi\)
−0.107366 + 0.994220i \(0.534242\pi\)
\(348\) 0 0
\(349\) 10.0000 0.535288 0.267644 − 0.963518i \(-0.413755\pi\)
0.267644 + 0.963518i \(0.413755\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 − 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) 0 0
\(357\) 4.00000 0.211702
\(358\) 0 0
\(359\) −10.0000 −0.527780 −0.263890 − 0.964553i \(-0.585006\pi\)
−0.263890 + 0.964553i \(0.585006\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 7.00000 0.367405
\(364\) 0 0
\(365\) 2.00000 0.104685
\(366\) 0 0
\(367\) 28.0000 1.46159 0.730794 − 0.682598i \(-0.239150\pi\)
0.730794 + 0.682598i \(0.239150\pi\)
\(368\) 0 0
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −20.0000 −1.03835
\(372\) 0 0
\(373\) 10.0000 0.517780 0.258890 − 0.965907i \(-0.416643\pi\)
0.258890 + 0.965907i \(0.416643\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) 10.0000 0.513665 0.256833 − 0.966456i \(-0.417321\pi\)
0.256833 + 0.966456i \(0.417321\pi\)
\(380\) 0 0
\(381\) −16.0000 −0.819705
\(382\) 0 0
\(383\) −22.0000 −1.12415 −0.562074 − 0.827087i \(-0.689996\pi\)
−0.562074 + 0.827087i \(0.689996\pi\)
\(384\) 0 0
\(385\) 4.00000 0.203859
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −22.0000 −1.11544 −0.557722 − 0.830028i \(-0.688325\pi\)
−0.557722 + 0.830028i \(0.688325\pi\)
\(390\) 0 0
\(391\) 16.0000 0.809155
\(392\) 0 0
\(393\) −20.0000 −1.00887
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) 18.0000 0.903394 0.451697 − 0.892171i \(-0.350819\pi\)
0.451697 + 0.892171i \(0.350819\pi\)
\(398\) 0 0
\(399\) −4.00000 −0.200250
\(400\) 0 0
\(401\) −34.0000 −1.69788 −0.848939 − 0.528490i \(-0.822758\pi\)
−0.848939 + 0.528490i \(0.822758\pi\)
\(402\) 0 0
\(403\) 2.00000 0.0996271
\(404\) 0 0
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) −6.00000 −0.296681 −0.148340 − 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 0 0
\(411\) 10.0000 0.493264
\(412\) 0 0
\(413\) −28.0000 −1.37779
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 0 0
\(417\) −12.0000 −0.587643
\(418\) 0 0
\(419\) −20.0000 −0.977064 −0.488532 − 0.872546i \(-0.662467\pi\)
−0.488532 + 0.872546i \(0.662467\pi\)
\(420\) 0 0
\(421\) −14.0000 −0.682318 −0.341159 − 0.940006i \(-0.610819\pi\)
−0.341159 + 0.940006i \(0.610819\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −20.0000 −0.967868
\(428\) 0 0
\(429\) 2.00000 0.0965609
\(430\) 0 0
\(431\) 18.0000 0.867029 0.433515 − 0.901146i \(-0.357273\pi\)
0.433515 + 0.901146i \(0.357273\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 − 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 6.00000 0.287678
\(436\) 0 0
\(437\) −16.0000 −0.765384
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 − 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) −18.0000 −0.853282
\(446\) 0 0
\(447\) −6.00000 −0.283790
\(448\) 0 0
\(449\) 30.0000 1.41579 0.707894 − 0.706319i \(-0.249646\pi\)
0.707894 + 0.706319i \(0.249646\pi\)
\(450\) 0 0
\(451\) −4.00000 −0.188353
\(452\) 0 0
\(453\) −10.0000 −0.469841
\(454\) 0 0
\(455\) −2.00000 −0.0937614
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 − 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 0 0
\(459\) 2.00000 0.0933520
\(460\) 0 0
\(461\) −26.0000 −1.21094 −0.605470 − 0.795868i \(-0.707015\pi\)
−0.605470 + 0.795868i \(0.707015\pi\)
\(462\) 0 0
\(463\) −30.0000 −1.39422 −0.697109 − 0.716965i \(-0.745531\pi\)
−0.697109 + 0.716965i \(0.745531\pi\)
\(464\) 0 0
\(465\) 2.00000 0.0927478
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 4.00000 0.184703
\(470\) 0 0
\(471\) −18.0000 −0.829396
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 2.00000 0.0917663
\(476\) 0 0
\(477\) −10.0000 −0.457869
\(478\) 0 0
\(479\) −42.0000 −1.91903 −0.959514 − 0.281659i \(-0.909115\pi\)
−0.959514 + 0.281659i \(0.909115\pi\)
\(480\) 0 0
\(481\) −2.00000 −0.0911922
\(482\) 0 0
\(483\) 16.0000 0.728025
\(484\) 0 0
\(485\) −14.0000 −0.635707
\(486\) 0 0
\(487\) −26.0000 −1.17817 −0.589086 − 0.808070i \(-0.700512\pi\)
−0.589086 + 0.808070i \(0.700512\pi\)
\(488\) 0 0
\(489\) 18.0000 0.813988
\(490\) 0 0
\(491\) 8.00000 0.361035 0.180517 − 0.983572i \(-0.442223\pi\)
0.180517 + 0.983572i \(0.442223\pi\)
\(492\) 0 0
\(493\) 12.0000 0.540453
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) 2.00000 0.0895323 0.0447661 − 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) 0 0
\(501\) 10.0000 0.446767
\(502\) 0 0
\(503\) −4.00000 −0.178351 −0.0891756 − 0.996016i \(-0.528423\pi\)
−0.0891756 + 0.996016i \(0.528423\pi\)
\(504\) 0 0
\(505\) 6.00000 0.266996
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 22.0000 0.975133 0.487566 − 0.873086i \(-0.337885\pi\)
0.487566 + 0.873086i \(0.337885\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 0 0
\(513\) −2.00000 −0.0883022
\(514\) 0 0
\(515\) 8.00000 0.352522
\(516\) 0 0
\(517\) 12.0000 0.527759
\(518\) 0 0
\(519\) −14.0000 −0.614532
\(520\) 0 0
\(521\) −10.0000 −0.438108 −0.219054 − 0.975713i \(-0.570297\pi\)
−0.219054 + 0.975713i \(0.570297\pi\)
\(522\) 0 0
\(523\) 12.0000 0.524723 0.262362 − 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 0 0
\(525\) −2.00000 −0.0872872
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) 0 0
\(531\) −14.0000 −0.607548
\(532\) 0 0
\(533\) 2.00000 0.0866296
\(534\) 0 0
\(535\) 20.0000 0.864675
\(536\) 0 0
\(537\) −16.0000 −0.690451
\(538\) 0 0
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −22.0000 −0.945854 −0.472927 − 0.881102i \(-0.656803\pi\)
−0.472927 + 0.881102i \(0.656803\pi\)
\(542\) 0 0
\(543\) 22.0000 0.944110
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) 12.0000 0.513083 0.256541 − 0.966533i \(-0.417417\pi\)
0.256541 + 0.966533i \(0.417417\pi\)
\(548\) 0 0
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) −12.0000 −0.511217
\(552\) 0 0
\(553\) −24.0000 −1.02058
\(554\) 0 0
\(555\) −2.00000 −0.0848953
\(556\) 0 0
\(557\) 30.0000 1.27114 0.635570 − 0.772043i \(-0.280765\pi\)
0.635570 + 0.772043i \(0.280765\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 4.00000 0.168880
\(562\) 0 0
\(563\) 24.0000 1.01148 0.505740 − 0.862686i \(-0.331220\pi\)
0.505740 + 0.862686i \(0.331220\pi\)
\(564\) 0 0
\(565\) −2.00000 −0.0841406
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 − 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 − 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) 0 0
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) −6.00000 −0.249783 −0.124892 − 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 0 0
\(579\) −10.0000 −0.415586
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −20.0000 −0.828315
\(584\) 0 0
\(585\) −1.00000 −0.0413449
\(586\) 0 0
\(587\) −18.0000 −0.742940 −0.371470 − 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 0 0
\(589\) −4.00000 −0.164817
\(590\) 0 0
\(591\) 10.0000 0.411345
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 − 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) 0 0
\(597\) 8.00000 0.327418
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 − 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) 0 0
\(605\) −7.00000 −0.284590
\(606\) 0 0
\(607\) 20.0000 0.811775 0.405887 − 0.913923i \(-0.366962\pi\)
0.405887 + 0.913923i \(0.366962\pi\)
\(608\) 0 0
\(609\) 12.0000 0.486265
\(610\) 0 0
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) −30.0000 −1.21169 −0.605844 − 0.795583i \(-0.707165\pi\)
−0.605844 + 0.795583i \(0.707165\pi\)
\(614\) 0 0
\(615\) 2.00000 0.0806478
\(616\) 0 0
\(617\) 14.0000 0.563619 0.281809 − 0.959470i \(-0.409065\pi\)
0.281809 + 0.959470i \(0.409065\pi\)
\(618\) 0 0
\(619\) −2.00000 −0.0803868 −0.0401934 − 0.999192i \(-0.512797\pi\)
−0.0401934 + 0.999192i \(0.512797\pi\)
\(620\) 0 0
\(621\) 8.00000 0.321029
\(622\) 0 0
\(623\) −36.0000 −1.44231
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −4.00000 −0.159745
\(628\) 0 0
\(629\) −4.00000 −0.159490
\(630\) 0 0
\(631\) −22.0000 −0.875806 −0.437903 − 0.899022i \(-0.644279\pi\)
−0.437903 + 0.899022i \(0.644279\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) 16.0000 0.634941
\(636\) 0 0
\(637\) 3.00000 0.118864
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 − 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) 10.0000 0.394362 0.197181 − 0.980367i \(-0.436821\pi\)
0.197181 + 0.980367i \(0.436821\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −32.0000 −1.25805 −0.629025 − 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) −28.0000 −1.09910
\(650\) 0 0
\(651\) 4.00000 0.156772
\(652\) 0 0
\(653\) 10.0000 0.391330 0.195665 − 0.980671i \(-0.437313\pi\)
0.195665 + 0.980671i \(0.437313\pi\)
\(654\) 0 0
\(655\) 20.0000 0.781465
\(656\) 0 0
\(657\) 2.00000 0.0780274
\(658\) 0 0
\(659\) −12.0000 −0.467454 −0.233727 − 0.972302i \(-0.575092\pi\)
−0.233727 + 0.972302i \(0.575092\pi\)
\(660\) 0 0
\(661\) 2.00000 0.0777910 0.0388955 − 0.999243i \(-0.487616\pi\)
0.0388955 + 0.999243i \(0.487616\pi\)
\(662\) 0 0
\(663\) −2.00000 −0.0776736
\(664\) 0 0
\(665\) 4.00000 0.155113
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) 0 0
\(669\) 10.0000 0.386622
\(670\) 0 0
\(671\) −20.0000 −0.772091
\(672\) 0 0
\(673\) 6.00000 0.231283 0.115642 − 0.993291i \(-0.463108\pi\)
0.115642 + 0.993291i \(0.463108\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 0 0
\(677\) 22.0000 0.845529 0.422764 − 0.906240i \(-0.361060\pi\)
0.422764 + 0.906240i \(0.361060\pi\)
\(678\) 0 0
\(679\) −28.0000 −1.07454
\(680\) 0 0
\(681\) 10.0000 0.383201
\(682\) 0 0
\(683\) 26.0000 0.994862 0.497431 − 0.867503i \(-0.334277\pi\)
0.497431 + 0.867503i \(0.334277\pi\)
\(684\) 0 0
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) 22.0000 0.839352
\(688\) 0 0
\(689\) 10.0000 0.380970
\(690\) 0 0
\(691\) 2.00000 0.0760836 0.0380418 − 0.999276i \(-0.487888\pi\)
0.0380418 + 0.999276i \(0.487888\pi\)
\(692\) 0 0
\(693\) 4.00000 0.151947
\(694\) 0 0
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) 30.0000 1.13470
\(700\) 0 0
\(701\) 2.00000 0.0755390 0.0377695 − 0.999286i \(-0.487975\pi\)
0.0377695 + 0.999286i \(0.487975\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 0 0
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) 12.0000 0.451306
\(708\) 0 0
\(709\) −46.0000 −1.72757 −0.863783 − 0.503864i \(-0.831911\pi\)
−0.863783 + 0.503864i \(0.831911\pi\)
\(710\) 0 0
\(711\) −12.0000 −0.450035
\(712\) 0 0
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) −2.00000 −0.0747958
\(716\) 0 0
\(717\) −10.0000 −0.373457
\(718\) 0 0
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 0 0
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 2.00000 0.0738717 0.0369358 − 0.999318i \(-0.488240\pi\)
0.0369358 + 0.999318i \(0.488240\pi\)
\(734\) 0 0
\(735\) 3.00000 0.110657
\(736\) 0 0
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) 30.0000 1.10357 0.551784 − 0.833987i \(-0.313947\pi\)
0.551784 + 0.833987i \(0.313947\pi\)
\(740\) 0 0
\(741\) 2.00000 0.0734718
\(742\) 0 0
\(743\) −6.00000 −0.220119 −0.110059 − 0.993925i \(-0.535104\pi\)
−0.110059 + 0.993925i \(0.535104\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) 0 0
\(749\) 40.0000 1.46157
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 − 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 0 0
\(753\) 24.0000 0.874609
\(754\) 0 0
\(755\) 10.0000 0.363937
\(756\) 0 0
\(757\) −18.0000 −0.654221 −0.327111 − 0.944986i \(-0.606075\pi\)
−0.327111 + 0.944986i \(0.606075\pi\)
\(758\) 0 0
\(759\) 16.0000 0.580763
\(760\) 0 0
\(761\) −26.0000 −0.942499 −0.471250 − 0.882000i \(-0.656197\pi\)
−0.471250 + 0.882000i \(0.656197\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 0 0
\(765\) −2.00000 −0.0723102
\(766\) 0 0
\(767\) 14.0000 0.505511
\(768\) 0 0
\(769\) −14.0000 −0.504853 −0.252426 − 0.967616i \(-0.581229\pi\)
−0.252426 + 0.967616i \(0.581229\pi\)
\(770\) 0 0
\(771\) 6.00000 0.216085
\(772\) 0 0
\(773\) 30.0000 1.07903 0.539513 − 0.841978i \(-0.318609\pi\)
0.539513 + 0.841978i \(0.318609\pi\)
\(774\) 0 0
\(775\) −2.00000 −0.0718421
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) −4.00000 −0.143315
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 0 0
\(783\) 6.00000 0.214423
\(784\) 0 0
\(785\) 18.0000 0.642448
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 − 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 0 0
\(789\) 12.0000 0.427211
\(790\) 0 0
\(791\) −4.00000 −0.142224
\(792\) 0 0
\(793\) 10.0000 0.355110
\(794\) 0 0
\(795\) 10.0000 0.354663
\(796\) 0 0
\(797\) 18.0000 0.637593 0.318796 − 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −18.0000 −0.635999
\(802\) 0 0
\(803\) 4.00000 0.141157
\(804\) 0 0
\(805\) −16.0000 −0.563926
\(806\) 0 0
\(807\) 6.00000 0.211210
\(808\) 0 0
\(809\) 22.0000 0.773479 0.386739 − 0.922189i \(-0.373601\pi\)
0.386739 + 0.922189i \(0.373601\pi\)
\(810\) 0 0
\(811\) 42.0000 1.47482 0.737410 − 0.675446i \(-0.236049\pi\)
0.737410 + 0.675446i \(0.236049\pi\)
\(812\) 0 0
\(813\) −2.00000 −0.0701431
\(814\) 0 0
\(815\) −18.0000 −0.630512
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −2.00000 −0.0698857
\(820\) 0 0
\(821\) −50.0000 −1.74501 −0.872506 − 0.488603i \(-0.837507\pi\)
−0.872506 + 0.488603i \(0.837507\pi\)
\(822\) 0 0
\(823\) 40.0000 1.39431 0.697156 − 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) 0 0
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) −30.0000 −1.04320 −0.521601 − 0.853189i \(-0.674665\pi\)
−0.521601 + 0.853189i \(0.674665\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 − 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) 26.0000 0.901930
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) −10.0000 −0.346064
\(836\) 0 0
\(837\) 2.00000 0.0691301
\(838\) 0 0
\(839\) 6.00000 0.207143 0.103572 − 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −30.0000 −1.03325
\(844\) 0 0
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −14.0000 −0.481046
\(848\) 0 0
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) −16.0000 −0.548473
\(852\) 0 0
\(853\) −6.00000 −0.205436 −0.102718 − 0.994711i \(-0.532754\pi\)
−0.102718 + 0.994711i \(0.532754\pi\)
\(854\) 0 0
\(855\) 2.00000 0.0683986
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 − 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 20.0000 0.682391 0.341196 − 0.939992i \(-0.389168\pi\)
0.341196 + 0.939992i \(0.389168\pi\)
\(860\) 0 0
\(861\) 4.00000 0.136320
\(862\) 0 0
\(863\) 10.0000 0.340404 0.170202 − 0.985409i \(-0.445558\pi\)
0.170202 + 0.985409i \(0.445558\pi\)
\(864\) 0 0
\(865\) 14.0000 0.476014
\(866\) 0 0
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) −2.00000 −0.0677674
\(872\) 0 0
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −14.0000 −0.472746 −0.236373 − 0.971662i \(-0.575959\pi\)
−0.236373 + 0.971662i \(0.575959\pi\)
\(878\) 0 0
\(879\) −6.00000 −0.202375
\(880\) 0 0
\(881\) 10.0000 0.336909 0.168454 − 0.985709i \(-0.446122\pi\)
0.168454 + 0.985709i \(0.446122\pi\)
\(882\) 0 0
\(883\) −32.0000 −1.07689 −0.538443 − 0.842662i \(-0.680987\pi\)
−0.538443 + 0.842662i \(0.680987\pi\)
\(884\) 0 0
\(885\) 14.0000 0.470605
\(886\) 0 0
\(887\) −12.0000 −0.402921 −0.201460 − 0.979497i \(-0.564569\pi\)
−0.201460 + 0.979497i \(0.564569\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) 12.0000 0.401565
\(894\) 0 0
\(895\) 16.0000 0.534821
\(896\) 0 0
\(897\) −8.00000 −0.267112
\(898\) 0 0
\(899\) 12.0000 0.400222
\(900\) 0 0
\(901\) 20.0000 0.666297
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −22.0000 −0.731305
\(906\) 0 0
\(907\) 32.0000 1.06254 0.531271 − 0.847202i \(-0.321714\pi\)
0.531271 + 0.847202i \(0.321714\pi\)
\(908\) 0 0
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 28.0000 0.927681 0.463841 − 0.885919i \(-0.346471\pi\)
0.463841 + 0.885919i \(0.346471\pi\)
\(912\) 0 0
\(913\) 12.0000 0.397142
\(914\) 0 0
\(915\) 10.0000 0.330590
\(916\) 0 0
\(917\) 40.0000 1.32092
\(918\) 0 0
\(919\) −44.0000 −1.45143 −0.725713 − 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(920\) 0 0
\(921\) 2.00000 0.0659022
\(922\) 0 0
\(923\) 6.00000 0.197492
\(924\) 0 0
\(925\) 2.00000 0.0657596
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 − 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −6.00000 −0.196642
\(932\) 0 0
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) −4.00000 −0.130814
\(936\) 0 0
\(937\) 42.0000 1.37208 0.686040 − 0.727564i \(-0.259347\pi\)
0.686040 + 0.727564i \(0.259347\pi\)
\(938\) 0 0
\(939\) −22.0000 −0.717943
\(940\) 0 0
\(941\) −42.0000 −1.36916 −0.684580 − 0.728937i \(-0.740015\pi\)
−0.684580 + 0.728937i \(0.740015\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 0 0
\(945\) −2.00000 −0.0650600
\(946\) 0 0
\(947\) −30.0000 −0.974869 −0.487435 − 0.873160i \(-0.662067\pi\)
−0.487435 + 0.873160i \(0.662067\pi\)
\(948\) 0 0
\(949\) −2.00000 −0.0649227
\(950\) 0 0
\(951\) −22.0000 −0.713399
\(952\) 0 0
\(953\) −30.0000 −0.971795 −0.485898 − 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) 0 0
\(957\) 12.0000 0.387905
\(958\) 0 0
\(959\) −20.0000 −0.645834
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 20.0000 0.644491
\(964\) 0 0
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 38.0000 1.22200 0.610999 − 0.791632i \(-0.290768\pi\)
0.610999 + 0.791632i \(0.290768\pi\)
\(968\) 0 0
\(969\) 4.00000 0.128499
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 − 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 24.0000 0.769405
\(974\) 0 0
\(975\) 1.00000 0.0320256
\(976\) 0 0
\(977\) 54.0000 1.72761 0.863807 − 0.503824i \(-0.168074\pi\)
0.863807 + 0.503824i \(0.168074\pi\)
\(978\) 0 0
\(979\) −36.0000 −1.15056
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) 6.00000 0.191370 0.0956851 − 0.995412i \(-0.469496\pi\)
0.0956851 + 0.995412i \(0.469496\pi\)
\(984\) 0 0
\(985\) −10.0000 −0.318626
\(986\) 0 0
\(987\) −12.0000 −0.381964
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 36.0000 1.14358 0.571789 − 0.820401i \(-0.306250\pi\)
0.571789 + 0.820401i \(0.306250\pi\)
\(992\) 0 0
\(993\) −18.0000 −0.571213
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) 0 0
\(997\) 14.0000 0.443384 0.221692 − 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 0 0
\(999\) −2.00000 −0.0632772
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 6240.2.a.m.1.1 ✓ 1
4.3 odd 2 6240.2.a.bb.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6240.2.a.m.1.1 ✓ 1 1.1 even 1 trivial
6240.2.a.bb.1.1 yes 1 4.3 odd 2