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Properties

Label 1395.2.a.a.1.1
Level $1395$
Weight $2$
Character 1395.1
Self dual yes
Analytic conductor $11.139$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1395,2,Mod(1,1395)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1395, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1395.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1395 = 3^{2} \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1395.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: \(11.1391310820\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 465)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1395.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{4} -1.00000 q^{5} -2.00000 q^{7} +3.00000 q^{8} +1.00000 q^{10} +4.00000 q^{11} +2.00000 q^{14} -1.00000 q^{16} -2.00000 q^{17} -8.00000 q^{19} +1.00000 q^{20} -4.00000 q^{22} +8.00000 q^{23} +1.00000 q^{25} +2.00000 q^{28} +1.00000 q^{31} -5.00000 q^{32} +2.00000 q^{34} +2.00000 q^{35} +8.00000 q^{37} +8.00000 q^{38} -3.00000 q^{40} +6.00000 q^{41} -4.00000 q^{44} -8.00000 q^{46} -4.00000 q^{47} -3.00000 q^{49} -1.00000 q^{50} -6.00000 q^{53} -4.00000 q^{55} -6.00000 q^{56} -10.0000 q^{59} -14.0000 q^{61} -1.00000 q^{62} +7.00000 q^{64} +2.00000 q^{67} +2.00000 q^{68} -2.00000 q^{70} -6.00000 q^{71} -16.0000 q^{73} -8.00000 q^{74} +8.00000 q^{76} -8.00000 q^{77} +1.00000 q^{80} -6.00000 q^{82} -4.00000 q^{83} +2.00000 q^{85} +12.0000 q^{88} -4.00000 q^{89} -8.00000 q^{92} +4.00000 q^{94} +8.00000 q^{95} +6.00000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107 −0.353553 − 0.935414i \(-0.615027\pi\)
−0.353553 + 0.935414i \(0.615027\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 − 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 3.00000 1.06066
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 4.00000 1.20605 0.603023 − 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 − 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −4.00000 −0.852803
\(23\) 8.00000 1.66812 0.834058 − 0.551677i \(-0.186012\pi\)
0.834058 + 0.551677i \(0.186012\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −5.00000 −0.883883
\(33\) 0 0
\(34\) 2.00000 0.342997
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) 8.00000 1.31519 0.657596 − 0.753371i \(-0.271573\pi\)
0.657596 + 0.753371i \(0.271573\pi\)
\(38\) 8.00000 1.29777
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −4.00000 −0.603023
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) −4.00000 −0.583460 −0.291730 − 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.00000 −0.141421
\(51\) 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
\(55\) −4.00000 −0.539360
\(56\) −6.00000 −0.801784
\(57\) 0 0
\(58\) 0 0
\(59\) −10.0000 −1.30189 −0.650945 − 0.759125i \(-0.725627\pi\)
−0.650945 + 0.759125i \(0.725627\pi\)
\(60\) 0 0
\(61\) −14.0000 −1.79252 −0.896258 − 0.443533i \(-0.853725\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) −1.00000 −0.127000
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 − 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 2.00000 0.242536
\(69\) 0 0
\(70\) −2.00000 −0.239046
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −16.0000 −1.87266 −0.936329 − 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 8.00000 0.917663
\(77\) −8.00000 −0.911685
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 1.00000 0.111803
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) −4.00000 −0.439057 −0.219529 − 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) 0 0
\(88\) 12.0000 1.27920
\(89\) −4.00000 −0.423999 −0.212000 − 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −8.00000 −0.834058
\(93\) 0 0
\(94\) 4.00000 0.412568
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 − 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 3.00000 0.303046
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) −14.0000 −1.39305 −0.696526 − 0.717532i \(-0.745272\pi\)
−0.696526 + 0.717532i \(0.745272\pi\)
\(102\) 0 0
\(103\) 2.00000 0.197066 0.0985329 − 0.995134i \(-0.468585\pi\)
0.0985329 + 0.995134i \(0.468585\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) −16.0000 −1.54678 −0.773389 − 0.633932i \(-0.781440\pi\)
−0.773389 + 0.633932i \(0.781440\pi\)
\(108\) 0 0
\(109\) −14.0000 −1.34096 −0.670478 − 0.741929i \(-0.733911\pi\)
−0.670478 + 0.741929i \(0.733911\pi\)
\(110\) 4.00000 0.381385
\(111\) 0 0
\(112\) 2.00000 0.188982
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) 0 0
\(118\) 10.0000 0.920575
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 14.0000 1.26750
\(123\) 0 0
\(124\) −1.00000 −0.0898027
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −4.00000 −0.354943 −0.177471 − 0.984126i \(-0.556792\pi\)
−0.177471 + 0.984126i \(0.556792\pi\)
\(128\) 3.00000 0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) 14.0000 1.22319 0.611593 − 0.791173i \(-0.290529\pi\)
0.611593 + 0.791173i \(0.290529\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) −2.00000 −0.172774
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 18.0000 1.53784 0.768922 − 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) −2.00000 −0.169031
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 16.0000 1.32417
\(147\) 0 0
\(148\) −8.00000 −0.657596
\(149\) 18.0000 1.47462 0.737309 − 0.675556i \(-0.236096\pi\)
0.737309 + 0.675556i \(0.236096\pi\)
\(150\) 0 0
\(151\) −24.0000 −1.95309 −0.976546 − 0.215308i \(-0.930924\pi\)
−0.976546 + 0.215308i \(0.930924\pi\)
\(152\) −24.0000 −1.94666
\(153\) 0 0
\(154\) 8.00000 0.644658
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 − 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 − 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) −6.00000 −0.468521
\(165\) 0 0
\(166\) 4.00000 0.310460
\(167\) −16.0000 −1.23812 −0.619059 − 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −2.00000 −0.153393
\(171\) 0 0
\(172\) 0 0
\(173\) −6.00000 −0.456172 −0.228086 − 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) −4.00000 −0.301511
\(177\) 0 0
\(178\) 4.00000 0.299813
\(179\) 4.00000 0.298974 0.149487 − 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 24.0000 1.76930
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) −8.00000 −0.585018
\(188\) 4.00000 0.291730
\(189\) 0 0
\(190\) −8.00000 −0.580381
\(191\) 14.0000 1.01300 0.506502 − 0.862239i \(-0.330938\pi\)
0.506502 + 0.862239i \(0.330938\pi\)
\(192\) 0 0
\(193\) 6.00000 0.431889 0.215945 − 0.976406i \(-0.430717\pi\)
0.215945 + 0.976406i \(0.430717\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) 3.00000 0.214286
\(197\) −2.00000 −0.142494 −0.0712470 − 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 − 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 3.00000 0.212132
\(201\) 0 0
\(202\) 14.0000 0.985037
\(203\) 0 0
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) −2.00000 −0.139347
\(207\) 0 0
\(208\) 0 0
\(209\) −32.0000 −2.21349
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 − 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 6.00000 0.412082
\(213\) 0 0
\(214\) 16.0000 1.09374
\(215\) 0 0
\(216\) 0 0
\(217\) −2.00000 −0.135769
\(218\) 14.0000 0.948200
\(219\) 0 0
\(220\) 4.00000 0.269680
\(221\) 0 0
\(222\) 0 0
\(223\) 16.0000 1.07144 0.535720 − 0.844396i \(-0.320040\pi\)
0.535720 + 0.844396i \(0.320040\pi\)
\(224\) 10.0000 0.668153
\(225\) 0 0
\(226\) −6.00000 −0.399114
\(227\) 20.0000 1.32745 0.663723 − 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) 6.00000 0.396491 0.198246 − 0.980152i \(-0.436476\pi\)
0.198246 + 0.980152i \(0.436476\pi\)
\(230\) 8.00000 0.527504
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 4.00000 0.260931
\(236\) 10.0000 0.650945
\(237\) 0 0
\(238\) −4.00000 −0.259281
\(239\) 20.0000 1.29369 0.646846 − 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 − 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −5.00000 −0.321412
\(243\) 0 0
\(244\) 14.0000 0.896258
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 3.00000 0.190500
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −20.0000 −1.26239 −0.631194 − 0.775625i \(-0.717435\pi\)
−0.631194 + 0.775625i \(0.717435\pi\)
\(252\) 0 0
\(253\) 32.0000 2.01182
\(254\) 4.00000 0.250982
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 18.0000 1.12281 0.561405 − 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) −16.0000 −0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) −14.0000 −0.864923
\(263\) 24.0000 1.47990 0.739952 − 0.672660i \(-0.234848\pi\)
0.739952 + 0.672660i \(0.234848\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) −16.0000 −0.981023
\(267\) 0 0
\(268\) −2.00000 −0.122169
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 8.00000 0.485965 0.242983 − 0.970031i \(-0.421874\pi\)
0.242983 + 0.970031i \(0.421874\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −18.0000 −1.08742
\(275\) 4.00000 0.241209
\(276\) 0 0
\(277\) −28.0000 −1.68236 −0.841178 − 0.540758i \(-0.818138\pi\)
−0.841178 + 0.540758i \(0.818138\pi\)
\(278\) 12.0000 0.719712
\(279\) 0 0
\(280\) 6.00000 0.358569
\(281\) −30.0000 −1.78965 −0.894825 − 0.446417i \(-0.852700\pi\)
−0.894825 + 0.446417i \(0.852700\pi\)
\(282\) 0 0
\(283\) 10.0000 0.594438 0.297219 − 0.954809i \(-0.403941\pi\)
0.297219 + 0.954809i \(0.403941\pi\)
\(284\) 6.00000 0.356034
\(285\) 0 0
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) 16.0000 0.936329
\(293\) −26.0000 −1.51894 −0.759468 − 0.650545i \(-0.774541\pi\)
−0.759468 + 0.650545i \(0.774541\pi\)
\(294\) 0 0
\(295\) 10.0000 0.582223
\(296\) 24.0000 1.39497
\(297\) 0 0
\(298\) −18.0000 −1.04271
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 24.0000 1.38104
\(303\) 0 0
\(304\) 8.00000 0.458831
\(305\) 14.0000 0.801638
\(306\) 0 0
\(307\) 22.0000 1.25561 0.627803 − 0.778372i \(-0.283954\pi\)
0.627803 + 0.778372i \(0.283954\pi\)
\(308\) 8.00000 0.455842
\(309\) 0 0
\(310\) 1.00000 0.0567962
\(311\) −18.0000 −1.02069 −0.510343 − 0.859971i \(-0.670482\pi\)
−0.510343 + 0.859971i \(0.670482\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 − 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) −2.00000 −0.112867
\(315\) 0 0
\(316\) 0 0
\(317\) 6.00000 0.336994 0.168497 − 0.985702i \(-0.446109\pi\)
0.168497 + 0.985702i \(0.446109\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −7.00000 −0.391312
\(321\) 0 0
\(322\) 16.0000 0.891645
\(323\) 16.0000 0.890264
\(324\) 0 0
\(325\) 0 0
\(326\) −2.00000 −0.110770
\(327\) 0 0
\(328\) 18.0000 0.993884
\(329\) 8.00000 0.441054
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 4.00000 0.219529
\(333\) 0 0
\(334\) 16.0000 0.875481
\(335\) −2.00000 −0.109272
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 − 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 13.0000 0.707107
\(339\) 0 0
\(340\) −2.00000 −0.108465
\(341\) 4.00000 0.216612
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 0 0
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 − 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 2.00000 0.106904
\(351\) 0 0
\(352\) −20.0000 −1.06600
\(353\) 14.0000 0.745145 0.372572 − 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 4.00000 0.212000
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) −26.0000 −1.37223 −0.686114 − 0.727494i \(-0.740685\pi\)
−0.686114 + 0.727494i \(0.740685\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 6.00000 0.315353
\(363\) 0 0
\(364\) 0 0
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) −32.0000 −1.67039 −0.835193 − 0.549957i \(-0.814644\pi\)
−0.835193 + 0.549957i \(0.814644\pi\)
\(368\) −8.00000 −0.417029
\(369\) 0 0
\(370\) 8.00000 0.415900
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 − 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 0 0
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 − 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) −8.00000 −0.410391
\(381\) 0 0
\(382\) −14.0000 −0.716302
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 8.00000 0.407718
\(386\) −6.00000 −0.305392
\(387\) 0 0
\(388\) −6.00000 −0.304604
\(389\) −16.0000 −0.811232 −0.405616 − 0.914044i \(-0.632943\pi\)
−0.405616 + 0.914044i \(0.632943\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) −9.00000 −0.454569
\(393\) 0 0
\(394\) 2.00000 0.100759
\(395\) 0 0
\(396\) 0 0
\(397\) 14.0000 0.702640 0.351320 − 0.936255i \(-0.385733\pi\)
0.351320 + 0.936255i \(0.385733\pi\)
\(398\) 8.00000 0.401004
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) −16.0000 −0.799002 −0.399501 − 0.916733i \(-0.630817\pi\)
−0.399501 + 0.916733i \(0.630817\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 14.0000 0.696526
\(405\) 0 0
\(406\) 0 0
\(407\) 32.0000 1.58618
\(408\) 0 0
\(409\) −18.0000 −0.890043 −0.445021 − 0.895520i \(-0.646804\pi\)
−0.445021 + 0.895520i \(0.646804\pi\)
\(410\) 6.00000 0.296319
\(411\) 0 0
\(412\) −2.00000 −0.0985329
\(413\) 20.0000 0.984136
\(414\) 0 0
\(415\) 4.00000 0.196352
\(416\) 0 0
\(417\) 0 0
\(418\) 32.0000 1.56517
\(419\) −14.0000 −0.683945 −0.341972 − 0.939710i \(-0.611095\pi\)
−0.341972 + 0.939710i \(0.611095\pi\)
\(420\) 0 0
\(421\) 18.0000 0.877266 0.438633 − 0.898666i \(-0.355463\pi\)
0.438633 + 0.898666i \(0.355463\pi\)
\(422\) −8.00000 −0.389434
\(423\) 0 0
\(424\) −18.0000 −0.874157
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) 28.0000 1.35501
\(428\) 16.0000 0.773389
\(429\) 0 0
\(430\) 0 0
\(431\) −2.00000 −0.0963366 −0.0481683 − 0.998839i \(-0.515338\pi\)
−0.0481683 + 0.998839i \(0.515338\pi\)
\(432\) 0 0
\(433\) 24.0000 1.15337 0.576683 − 0.816968i \(-0.304347\pi\)
0.576683 + 0.816968i \(0.304347\pi\)
\(434\) 2.00000 0.0960031
\(435\) 0 0
\(436\) 14.0000 0.670478
\(437\) −64.0000 −3.06154
\(438\) 0 0
\(439\) 24.0000 1.14546 0.572729 − 0.819745i \(-0.305885\pi\)
0.572729 + 0.819745i \(0.305885\pi\)
\(440\) −12.0000 −0.572078
\(441\) 0 0
\(442\) 0 0
\(443\) 20.0000 0.950229 0.475114 − 0.879924i \(-0.342407\pi\)
0.475114 + 0.879924i \(0.342407\pi\)
\(444\) 0 0
\(445\) 4.00000 0.189618
\(446\) −16.0000 −0.757622
\(447\) 0 0
\(448\) −14.0000 −0.661438
\(449\) 8.00000 0.377543 0.188772 − 0.982021i \(-0.439549\pi\)
0.188772 + 0.982021i \(0.439549\pi\)
\(450\) 0 0
\(451\) 24.0000 1.13012
\(452\) −6.00000 −0.282216
\(453\) 0 0
\(454\) −20.0000 −0.938647
\(455\) 0 0
\(456\) 0 0
\(457\) −8.00000 −0.374224 −0.187112 − 0.982339i \(-0.559913\pi\)
−0.187112 + 0.982339i \(0.559913\pi\)
\(458\) −6.00000 −0.280362
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −12.0000 −0.558896 −0.279448 − 0.960161i \(-0.590151\pi\)
−0.279448 + 0.960161i \(0.590151\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 − 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 6.00000 0.277945
\(467\) −40.0000 −1.85098 −0.925490 − 0.378773i \(-0.876346\pi\)
−0.925490 + 0.378773i \(0.876346\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) −4.00000 −0.184506
\(471\) 0 0
\(472\) −30.0000 −1.38086
\(473\) 0 0
\(474\) 0 0
\(475\) −8.00000 −0.367065
\(476\) −4.00000 −0.183340
\(477\) 0 0
\(478\) −20.0000 −0.914779
\(479\) 10.0000 0.456912 0.228456 − 0.973554i \(-0.426632\pi\)
0.228456 + 0.973554i \(0.426632\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) −5.00000 −0.227273
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 − 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) −42.0000 −1.90125
\(489\) 0 0
\(490\) −3.00000 −0.135526
\(491\) 24.0000 1.08310 0.541552 − 0.840667i \(-0.317837\pi\)
0.541552 + 0.840667i \(0.317837\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) 20.0000 0.892644
\(503\) 12.0000 0.535054 0.267527 − 0.963550i \(-0.413794\pi\)
0.267527 + 0.963550i \(0.413794\pi\)
\(504\) 0 0
\(505\) 14.0000 0.622992
\(506\) −32.0000 −1.42257
\(507\) 0 0
\(508\) 4.00000 0.177471
\(509\) 28.0000 1.24108 0.620539 − 0.784176i \(-0.286914\pi\)
0.620539 + 0.784176i \(0.286914\pi\)
\(510\) 0 0
\(511\) 32.0000 1.41560
\(512\) 11.0000 0.486136
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) −2.00000 −0.0881305
\(516\) 0 0
\(517\) −16.0000 −0.703679
\(518\) 16.0000 0.703000
\(519\) 0 0
\(520\) 0 0
\(521\) −18.0000 −0.788594 −0.394297 − 0.918983i \(-0.629012\pi\)
−0.394297 + 0.918983i \(0.629012\pi\)
\(522\) 0 0
\(523\) −8.00000 −0.349816 −0.174908 − 0.984585i \(-0.555963\pi\)
−0.174908 + 0.984585i \(0.555963\pi\)
\(524\) −14.0000 −0.611593
\(525\) 0 0
\(526\) −24.0000 −1.04645
\(527\) −2.00000 −0.0871214
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) −16.0000 −0.693688
\(533\) 0 0
\(534\) 0 0
\(535\) 16.0000 0.691740
\(536\) 6.00000 0.259161
\(537\) 0 0
\(538\) 0 0
\(539\) −12.0000 −0.516877
\(540\) 0 0
\(541\) 34.0000 1.46177 0.730887 − 0.682498i \(-0.239107\pi\)
0.730887 + 0.682498i \(0.239107\pi\)
\(542\) −8.00000 −0.343629
\(543\) 0 0
\(544\) 10.0000 0.428746
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) −22.0000 −0.940652 −0.470326 − 0.882493i \(-0.655864\pi\)
−0.470326 + 0.882493i \(0.655864\pi\)
\(548\) −18.0000 −0.768922
\(549\) 0 0
\(550\) −4.00000 −0.170561
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 28.0000 1.18961
\(555\) 0 0
\(556\) 12.0000 0.508913
\(557\) −22.0000 −0.932170 −0.466085 − 0.884740i \(-0.654336\pi\)
−0.466085 + 0.884740i \(0.654336\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −2.00000 −0.0845154
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −6.00000 −0.252422
\(566\) −10.0000 −0.420331
\(567\) 0 0
\(568\) −18.0000 −0.755263
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 4.00000 0.167395 0.0836974 − 0.996491i \(-0.473327\pi\)
0.0836974 + 0.996491i \(0.473327\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 8.00000 0.333623
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 − 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 13.0000 0.540729
\(579\) 0 0
\(580\) 0 0
\(581\) 8.00000 0.331896
\(582\) 0 0
\(583\) −24.0000 −0.993978
\(584\) −48.0000 −1.98625
\(585\) 0 0
\(586\) 26.0000 1.07405
\(587\) 12.0000 0.495293 0.247647 − 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) −10.0000 −0.411693
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) −18.0000 −0.739171 −0.369586 − 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) −4.00000 −0.163984
\(596\) −18.0000 −0.737309
\(597\) 0 0
\(598\) 0 0
\(599\) 14.0000 0.572024 0.286012 − 0.958226i \(-0.407670\pi\)
0.286012 + 0.958226i \(0.407670\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 − 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 24.0000 0.976546
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) −46.0000 −1.86708 −0.933541 − 0.358470i \(-0.883298\pi\)
−0.933541 + 0.358470i \(0.883298\pi\)
\(608\) 40.0000 1.62221
\(609\) 0 0
\(610\) −14.0000 −0.566843
\(611\) 0 0
\(612\) 0 0
\(613\) −44.0000 −1.77714 −0.888572 − 0.458738i \(-0.848302\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(614\) −22.0000 −0.887848
\(615\) 0 0
\(616\) −24.0000 −0.966988
\(617\) 38.0000 1.52982 0.764911 − 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 − 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 1.00000 0.0401610
\(621\) 0 0
\(622\) 18.0000 0.721734
\(623\) 8.00000 0.320513
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 8.00000 0.319744
\(627\) 0 0
\(628\) −2.00000 −0.0798087
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 − 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) 4.00000 0.158735
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −40.0000 −1.57991 −0.789953 − 0.613168i \(-0.789895\pi\)
−0.789953 + 0.613168i \(0.789895\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 − 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 16.0000 0.630488
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) 16.0000 0.629025 0.314512 − 0.949253i \(-0.398159\pi\)
0.314512 + 0.949253i \(0.398159\pi\)
\(648\) 0 0
\(649\) −40.0000 −1.57014
\(650\) 0 0
\(651\) 0 0
\(652\) −2.00000 −0.0783260
\(653\) −10.0000 −0.391330 −0.195665 − 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) −14.0000 −0.547025
\(656\) −6.00000 −0.234261
\(657\) 0 0
\(658\) −8.00000 −0.311872
\(659\) 42.0000 1.63609 0.818044 − 0.575156i \(-0.195059\pi\)
0.818044 + 0.575156i \(0.195059\pi\)
\(660\) 0 0
\(661\) −2.00000 −0.0777910 −0.0388955 − 0.999243i \(-0.512384\pi\)
−0.0388955 + 0.999243i \(0.512384\pi\)
\(662\) 4.00000 0.155464
\(663\) 0 0
\(664\) −12.0000 −0.465690
\(665\) −16.0000 −0.620453
\(666\) 0 0
\(667\) 0 0
\(668\) 16.0000 0.619059
\(669\) 0 0
\(670\) 2.00000 0.0772667
\(671\) −56.0000 −2.16186
\(672\) 0 0
\(673\) −44.0000 −1.69608 −0.848038 − 0.529936i \(-0.822216\pi\)
−0.848038 + 0.529936i \(0.822216\pi\)
\(674\) −28.0000 −1.07852
\(675\) 0 0
\(676\) 13.0000 0.500000
\(677\) 26.0000 0.999261 0.499631 − 0.866239i \(-0.333469\pi\)
0.499631 + 0.866239i \(0.333469\pi\)
\(678\) 0 0
\(679\) −12.0000 −0.460518
\(680\) 6.00000 0.230089
\(681\) 0 0
\(682\) −4.00000 −0.153168
\(683\) 4.00000 0.153056 0.0765279 − 0.997067i \(-0.475617\pi\)
0.0765279 + 0.997067i \(0.475617\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) −20.0000 −0.763604
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 12.0000 0.456502 0.228251 − 0.973602i \(-0.426699\pi\)
0.228251 + 0.973602i \(0.426699\pi\)
\(692\) 6.00000 0.228086
\(693\) 0 0
\(694\) −12.0000 −0.455514
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 14.0000 0.529908
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) −64.0000 −2.41381
\(704\) 28.0000 1.05529
\(705\) 0 0
\(706\) −14.0000 −0.526897
\(707\) 28.0000 1.05305
\(708\) 0 0
\(709\) −42.0000 −1.57734 −0.788672 − 0.614815i \(-0.789231\pi\)
−0.788672 + 0.614815i \(0.789231\pi\)
\(710\) −6.00000 −0.225176
\(711\) 0 0
\(712\) −12.0000 −0.449719
\(713\) 8.00000 0.299602
\(714\) 0 0
\(715\) 0 0
\(716\) −4.00000 −0.149487
\(717\) 0 0
\(718\) 26.0000 0.970311
\(719\) 8.00000 0.298350 0.149175 − 0.988811i \(-0.452338\pi\)
0.149175 + 0.988811i \(0.452338\pi\)
\(720\) 0 0
\(721\) −4.00000 −0.148968
\(722\) −45.0000 −1.67473
\(723\) 0 0
\(724\) 6.00000 0.222988
\(725\) 0 0
\(726\) 0 0
\(727\) 30.0000 1.11264 0.556319 − 0.830969i \(-0.312213\pi\)
0.556319 + 0.830969i \(0.312213\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −16.0000 −0.592187
\(731\) 0 0
\(732\) 0 0
\(733\) −26.0000 −0.960332 −0.480166 − 0.877178i \(-0.659424\pi\)
−0.480166 + 0.877178i \(0.659424\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) −40.0000 −1.47442
\(737\) 8.00000 0.294684
\(738\) 0 0
\(739\) 28.0000 1.03000 0.514998 − 0.857191i \(-0.327793\pi\)
0.514998 + 0.857191i \(0.327793\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 22.0000 0.805477
\(747\) 0 0
\(748\) 8.00000 0.292509
\(749\) 32.0000 1.16925
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 − 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) 4.00000 0.145865
\(753\) 0 0
\(754\) 0 0
\(755\) 24.0000 0.873449
\(756\) 0 0
\(757\) −8.00000 −0.290765 −0.145382 − 0.989376i \(-0.546441\pi\)
−0.145382 + 0.989376i \(0.546441\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 24.0000 0.870572
\(761\) 24.0000 0.869999 0.435000 − 0.900431i \(-0.356748\pi\)
0.435000 + 0.900431i \(0.356748\pi\)
\(762\) 0 0
\(763\) 28.0000 1.01367
\(764\) −14.0000 −0.506502
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 46.0000 1.65880 0.829401 − 0.558653i \(-0.188682\pi\)
0.829401 + 0.558653i \(0.188682\pi\)
\(770\) −8.00000 −0.288300
\(771\) 0 0
\(772\) −6.00000 −0.215945
\(773\) −30.0000 −1.07903 −0.539513 − 0.841978i \(-0.681391\pi\)
−0.539513 + 0.841978i \(0.681391\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 18.0000 0.646162
\(777\) 0 0
\(778\) 16.0000 0.573628
\(779\) −48.0000 −1.71978
\(780\) 0 0
\(781\) −24.0000 −0.858788
\(782\) 16.0000 0.572159
\(783\) 0 0
\(784\) 3.00000 0.107143
\(785\) −2.00000 −0.0713831
\(786\) 0 0
\(787\) 52.0000 1.85360 0.926800 − 0.375555i \(-0.122548\pi\)
0.926800 + 0.375555i \(0.122548\pi\)
\(788\) 2.00000 0.0712470
\(789\) 0 0
\(790\) 0 0
\(791\) −12.0000 −0.426671
\(792\) 0 0
\(793\) 0 0
\(794\) −14.0000 −0.496841
\(795\) 0 0
\(796\) 8.00000 0.283552
\(797\) −42.0000 −1.48772 −0.743858 − 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) 8.00000 0.283020
\(800\) −5.00000 −0.176777
\(801\) 0 0
\(802\) 16.0000 0.564980
\(803\) −64.0000 −2.25851
\(804\) 0 0
\(805\) 16.0000 0.563926
\(806\) 0 0
\(807\) 0 0
\(808\) −42.0000 −1.47755
\(809\) −24.0000 −0.843795 −0.421898 − 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 0 0
\(811\) 32.0000 1.12367 0.561836 − 0.827249i \(-0.310095\pi\)
0.561836 + 0.827249i \(0.310095\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −32.0000 −1.12160
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) 0 0
\(818\) 18.0000 0.629355
\(819\) 0 0
\(820\) 6.00000 0.209529
\(821\) −24.0000 −0.837606 −0.418803 − 0.908077i \(-0.637550\pi\)
−0.418803 + 0.908077i \(0.637550\pi\)
\(822\) 0 0
\(823\) −44.0000 −1.53374 −0.766872 − 0.641800i \(-0.778188\pi\)
−0.766872 + 0.641800i \(0.778188\pi\)
\(824\) 6.00000 0.209020
\(825\) 0 0
\(826\) −20.0000 −0.695889
\(827\) 12.0000 0.417281 0.208640 − 0.977992i \(-0.433096\pi\)
0.208640 + 0.977992i \(0.433096\pi\)
\(828\) 0 0
\(829\) −22.0000 −0.764092 −0.382046 − 0.924143i \(-0.624780\pi\)
−0.382046 + 0.924143i \(0.624780\pi\)
\(830\) −4.00000 −0.138842
\(831\) 0 0
\(832\) 0 0
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 16.0000 0.553703
\(836\) 32.0000 1.10674
\(837\) 0 0
\(838\) 14.0000 0.483622
\(839\) 6.00000 0.207143 0.103572 − 0.994622i \(-0.466973\pi\)
0.103572 + 0.994622i \(0.466973\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −18.0000 −0.620321
\(843\) 0 0
\(844\) −8.00000 −0.275371
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) −10.0000 −0.343604
\(848\) 6.00000 0.206041
\(849\) 0 0
\(850\) 2.00000 0.0685994
\(851\) 64.0000 2.19389
\(852\) 0 0
\(853\) −2.00000 −0.0684787 −0.0342393 − 0.999414i \(-0.510901\pi\)
−0.0342393 + 0.999414i \(0.510901\pi\)
\(854\) −28.0000 −0.958140
\(855\) 0 0
\(856\) −48.0000 −1.64061
\(857\) 6.00000 0.204956 0.102478 − 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 28.0000 0.955348 0.477674 − 0.878537i \(-0.341480\pi\)
0.477674 + 0.878537i \(0.341480\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 2.00000 0.0681203
\(863\) −8.00000 −0.272323 −0.136162 − 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(864\) 0 0
\(865\) 6.00000 0.204006
\(866\) −24.0000 −0.815553
\(867\) 0 0
\(868\) 2.00000 0.0678844
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −42.0000 −1.42230
\(873\) 0 0
\(874\) 64.0000 2.16483
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) −18.0000 −0.607817 −0.303908 − 0.952701i \(-0.598292\pi\)
−0.303908 + 0.952701i \(0.598292\pi\)
\(878\) −24.0000 −0.809961
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) 28.0000 0.943344 0.471672 − 0.881774i \(-0.343651\pi\)
0.471672 + 0.881774i \(0.343651\pi\)
\(882\) 0 0
\(883\) 28.0000 0.942275 0.471138 − 0.882060i \(-0.343844\pi\)
0.471138 + 0.882060i \(0.343844\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −20.0000 −0.671913
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 8.00000 0.268311
\(890\) −4.00000 −0.134080
\(891\) 0 0
\(892\) −16.0000 −0.535720
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) −6.00000 −0.200446
\(897\) 0 0
\(898\) −8.00000 −0.266963
\(899\) 0 0
\(900\) 0 0
\(901\) 12.0000 0.399778
\(902\) −24.0000 −0.799113
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) 6.00000 0.199447
\(906\) 0 0
\(907\) −18.0000 −0.597680 −0.298840 − 0.954303i \(-0.596600\pi\)
−0.298840 + 0.954303i \(0.596600\pi\)
\(908\) −20.0000 −0.663723
\(909\) 0 0
\(910\) 0 0
\(911\) 8.00000 0.265052 0.132526 − 0.991180i \(-0.457691\pi\)
0.132526 + 0.991180i \(0.457691\pi\)
\(912\) 0 0
\(913\) −16.0000 −0.529523
\(914\) 8.00000 0.264616
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) −28.0000 −0.924641
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 − 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) −24.0000 −0.791257
\(921\) 0 0
\(922\) 12.0000 0.395199
\(923\) 0 0
\(924\) 0 0
\(925\) 8.00000 0.263038
\(926\) −20.0000 −0.657241
\(927\) 0 0
\(928\) 0 0
\(929\) 44.0000 1.44359 0.721797 − 0.692105i \(-0.243317\pi\)
0.721797 + 0.692105i \(0.243317\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 40.0000 1.30884
\(935\) 8.00000 0.261628
\(936\) 0 0
\(937\) 6.00000 0.196011 0.0980057 − 0.995186i \(-0.468754\pi\)
0.0980057 + 0.995186i \(0.468754\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) −4.00000 −0.130466
\(941\) −24.0000 −0.782378 −0.391189 − 0.920310i \(-0.627936\pi\)
−0.391189 + 0.920310i \(0.627936\pi\)
\(942\) 0 0
\(943\) 48.0000 1.56310
\(944\) 10.0000 0.325472
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 8.00000 0.259554
\(951\) 0 0
\(952\) 12.0000 0.388922
\(953\) −30.0000 −0.971795 −0.485898 − 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(954\) 0 0
\(955\) −14.0000 −0.453029
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) −10.0000 −0.323085
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 0 0
\(964\) 22.0000 0.708572
\(965\) −6.00000 −0.193147
\(966\) 0 0
\(967\) 24.0000 0.771788 0.385894 − 0.922543i \(-0.373893\pi\)
0.385894 + 0.922543i \(0.373893\pi\)
\(968\) 15.0000 0.482118
\(969\) 0 0
\(970\) 6.00000 0.192648
\(971\) 42.0000 1.34784 0.673922 − 0.738802i \(-0.264608\pi\)
0.673922 + 0.738802i \(0.264608\pi\)
\(972\) 0 0
\(973\) 24.0000 0.769405
\(974\) 4.00000 0.128168
\(975\) 0 0
\(976\) 14.0000 0.448129
\(977\) 30.0000 0.959785 0.479893 − 0.877327i \(-0.340676\pi\)
0.479893 + 0.877327i \(0.340676\pi\)
\(978\) 0 0
\(979\) −16.0000 −0.511362
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) −24.0000 −0.765871
\(983\) 8.00000 0.255160 0.127580 − 0.991828i \(-0.459279\pi\)
0.127580 + 0.991828i \(0.459279\pi\)
\(984\) 0 0
\(985\) 2.00000 0.0637253
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −56.0000 −1.77890 −0.889449 − 0.457034i \(-0.848912\pi\)
−0.889449 + 0.457034i \(0.848912\pi\)
\(992\) −5.00000 −0.158750
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) 42.0000 1.33015 0.665077 − 0.746775i \(-0.268399\pi\)
0.665077 + 0.746775i \(0.268399\pi\)
\(998\) 20.0000 0.633089
\(999\) 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 1395.2.a.a.1.1 1
3.2 odd 2 465.2.a.b.1.1 ✓ 1
5.4 even 2 6975.2.a.q.1.1 1
12.11 even 2 7440.2.a.ba.1.1 1
15.2 even 4 2325.2.c.d.1024.2 2
15.8 even 4 2325.2.c.d.1024.1 2
15.14 odd 2 2325.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
465.2.a.b.1.1 ✓ 1 3.2 odd 2
1395.2.a.a.1.1 1 1.1 even 1 trivial
2325.2.a.d.1.1 1 15.14 odd 2
2325.2.c.d.1024.1 2 15.8 even 4
2325.2.c.d.1024.2 2 15.2 even 4
6975.2.a.q.1.1 1 5.4 even 2
7440.2.a.ba.1.1 1 12.11 even 2