Properties

Label 1080.4.a.e.1.2
Level $1080$
Weight $4$
Character 1080.1
Self dual yes
Analytic conductor $63.722$
Analytic rank $1$
Dimension $3$
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] = [1080,4,Mod(1,1080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1080.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1080.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: \(63.7220628062\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.4281.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 12x + 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.26757\) of defining polynomial
Character \(\chi\) \(=\) 1080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.00000 q^{5} +10.2514 q^{7} +O(q^{10})\) \(q-5.00000 q^{5} +10.2514 q^{7} +44.3754 q^{11} -50.8782 q^{13} -25.2514 q^{17} -31.3754 q^{19} -76.1296 q^{23} +25.0000 q^{25} -156.008 q^{29} +134.505 q^{31} -51.2571 q^{35} +81.2423 q^{37} -326.256 q^{41} +422.526 q^{43} +452.635 q^{47} -237.909 q^{49} -98.1105 q^{53} -221.877 q^{55} -540.118 q^{59} +522.292 q^{61} +254.391 q^{65} -129.272 q^{67} -26.6034 q^{71} -147.873 q^{73} +454.910 q^{77} -1088.91 q^{79} -594.611 q^{83} +126.257 q^{85} -592.229 q^{89} -521.573 q^{91} +156.877 q^{95} -666.786 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 15 q^{5} - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 15 q^{5} - 8 q^{7} + 10 q^{11} + 48 q^{13} - 37 q^{17} + 29 q^{19} + 11 q^{23} + 75 q^{25} - 28 q^{29} + 41 q^{31} + 40 q^{35} + 230 q^{37} - 370 q^{41} - 130 q^{43} + 56 q^{47} + 547 q^{49} - 805 q^{53} - 50 q^{55} - 576 q^{59} - 257 q^{61} - 240 q^{65} - 14 q^{67} - 1238 q^{71} - 398 q^{73} - 1296 q^{77} - 321 q^{79} - 687 q^{83} + 185 q^{85} - 2358 q^{89} - 1968 q^{91} - 145 q^{95} + 576 q^{97}+O(q^{100}) \) 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
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 10.2514 0.553524 0.276762 0.960938i \(-0.410739\pi\)
0.276762 + 0.960938i \(0.410739\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 44.3754 1.21633 0.608167 0.793809i \(-0.291905\pi\)
0.608167 + 0.793809i \(0.291905\pi\)
\(12\) 0 0
\(13\) −50.8782 −1.08547 −0.542733 0.839905i \(-0.682611\pi\)
−0.542733 + 0.839905i \(0.682611\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −25.2514 −0.360257 −0.180128 0.983643i \(-0.557651\pi\)
−0.180128 + 0.983643i \(0.557651\pi\)
\(18\) 0 0
\(19\) −31.3754 −0.378842 −0.189421 0.981896i \(-0.560661\pi\)
−0.189421 + 0.981896i \(0.560661\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −76.1296 −0.690179 −0.345090 0.938570i \(-0.612151\pi\)
−0.345090 + 0.938570i \(0.612151\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −156.008 −0.998963 −0.499481 0.866325i \(-0.666476\pi\)
−0.499481 + 0.866325i \(0.666476\pi\)
\(30\) 0 0
\(31\) 134.505 0.779284 0.389642 0.920966i \(-0.372599\pi\)
0.389642 + 0.920966i \(0.372599\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −51.2571 −0.247544
\(36\) 0 0
\(37\) 81.2423 0.360977 0.180488 0.983577i \(-0.442232\pi\)
0.180488 + 0.983577i \(0.442232\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −326.256 −1.24275 −0.621373 0.783515i \(-0.713425\pi\)
−0.621373 + 0.783515i \(0.713425\pi\)
\(42\) 0 0
\(43\) 422.526 1.49848 0.749240 0.662299i \(-0.230419\pi\)
0.749240 + 0.662299i \(0.230419\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 452.635 1.40476 0.702379 0.711803i \(-0.252121\pi\)
0.702379 + 0.711803i \(0.252121\pi\)
\(48\) 0 0
\(49\) −237.909 −0.693611
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −98.1105 −0.254274 −0.127137 0.991885i \(-0.540579\pi\)
−0.127137 + 0.991885i \(0.540579\pi\)
\(54\) 0 0
\(55\) −221.877 −0.543961
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −540.118 −1.19182 −0.595910 0.803051i \(-0.703208\pi\)
−0.595910 + 0.803051i \(0.703208\pi\)
\(60\) 0 0
\(61\) 522.292 1.09627 0.548136 0.836389i \(-0.315338\pi\)
0.548136 + 0.836389i \(0.315338\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 254.391 0.485436
\(66\) 0 0
\(67\) −129.272 −0.235717 −0.117859 0.993030i \(-0.537603\pi\)
−0.117859 + 0.993030i \(0.537603\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −26.6034 −0.0444682 −0.0222341 0.999753i \(-0.507078\pi\)
−0.0222341 + 0.999753i \(0.507078\pi\)
\(72\) 0 0
\(73\) −147.873 −0.237085 −0.118542 0.992949i \(-0.537822\pi\)
−0.118542 + 0.992949i \(0.537822\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 454.910 0.673270
\(78\) 0 0
\(79\) −1088.91 −1.55079 −0.775394 0.631478i \(-0.782449\pi\)
−0.775394 + 0.631478i \(0.782449\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −594.611 −0.786350 −0.393175 0.919464i \(-0.628623\pi\)
−0.393175 + 0.919464i \(0.628623\pi\)
\(84\) 0 0
\(85\) 126.257 0.161112
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −592.229 −0.705350 −0.352675 0.935746i \(-0.614728\pi\)
−0.352675 + 0.935746i \(0.614728\pi\)
\(90\) 0 0
\(91\) −521.573 −0.600832
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 156.877 0.169423
\(96\) 0 0
\(97\) −666.786 −0.697958 −0.348979 0.937131i \(-0.613471\pi\)
−0.348979 + 0.937131i \(0.613471\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1218.56 −1.20051 −0.600254 0.799809i \(-0.704934\pi\)
−0.600254 + 0.799809i \(0.704934\pi\)
\(102\) 0 0
\(103\) −231.789 −0.221736 −0.110868 0.993835i \(-0.535363\pi\)
−0.110868 + 0.993835i \(0.535363\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1979.96 −1.78888 −0.894439 0.447189i \(-0.852425\pi\)
−0.894439 + 0.447189i \(0.852425\pi\)
\(108\) 0 0
\(109\) 1075.41 0.945005 0.472502 0.881329i \(-0.343351\pi\)
0.472502 + 0.881329i \(0.343351\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −693.410 −0.577261 −0.288631 0.957441i \(-0.593200\pi\)
−0.288631 + 0.957441i \(0.593200\pi\)
\(114\) 0 0
\(115\) 380.648 0.308657
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −258.863 −0.199411
\(120\) 0 0
\(121\) 638.173 0.479469
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −833.579 −0.582427 −0.291213 0.956658i \(-0.594059\pi\)
−0.291213 + 0.956658i \(0.594059\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −353.721 −0.235914 −0.117957 0.993019i \(-0.537635\pi\)
−0.117957 + 0.993019i \(0.537635\pi\)
\(132\) 0 0
\(133\) −321.642 −0.209698
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1509.90 −0.941601 −0.470800 0.882240i \(-0.656035\pi\)
−0.470800 + 0.882240i \(0.656035\pi\)
\(138\) 0 0
\(139\) −618.485 −0.377404 −0.188702 0.982034i \(-0.560428\pi\)
−0.188702 + 0.982034i \(0.560428\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −2257.74 −1.32029
\(144\) 0 0
\(145\) 780.039 0.446750
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2405.41 −1.32254 −0.661271 0.750147i \(-0.729983\pi\)
−0.661271 + 0.750147i \(0.729983\pi\)
\(150\) 0 0
\(151\) 1601.13 0.862900 0.431450 0.902137i \(-0.358002\pi\)
0.431450 + 0.902137i \(0.358002\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −672.525 −0.348506
\(156\) 0 0
\(157\) 3208.39 1.63094 0.815470 0.578799i \(-0.196479\pi\)
0.815470 + 0.578799i \(0.196479\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −780.436 −0.382031
\(162\) 0 0
\(163\) −392.481 −0.188598 −0.0942991 0.995544i \(-0.530061\pi\)
−0.0942991 + 0.995544i \(0.530061\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 260.818 0.120854 0.0604272 0.998173i \(-0.480754\pi\)
0.0604272 + 0.998173i \(0.480754\pi\)
\(168\) 0 0
\(169\) 391.590 0.178239
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 204.870 0.0900346 0.0450173 0.998986i \(-0.485666\pi\)
0.0450173 + 0.998986i \(0.485666\pi\)
\(174\) 0 0
\(175\) 256.285 0.110705
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1836.92 0.767029 0.383514 0.923535i \(-0.374714\pi\)
0.383514 + 0.923535i \(0.374714\pi\)
\(180\) 0 0
\(181\) −2880.67 −1.18298 −0.591488 0.806314i \(-0.701459\pi\)
−0.591488 + 0.806314i \(0.701459\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −406.211 −0.161434
\(186\) 0 0
\(187\) −1120.54 −0.438193
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −748.242 −0.283460 −0.141730 0.989905i \(-0.545267\pi\)
−0.141730 + 0.989905i \(0.545267\pi\)
\(192\) 0 0
\(193\) 621.620 0.231840 0.115920 0.993259i \(-0.463018\pi\)
0.115920 + 0.993259i \(0.463018\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −588.243 −0.212744 −0.106372 0.994326i \(-0.533923\pi\)
−0.106372 + 0.994326i \(0.533923\pi\)
\(198\) 0 0
\(199\) −4015.31 −1.43034 −0.715171 0.698950i \(-0.753651\pi\)
−0.715171 + 0.698950i \(0.753651\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −1599.30 −0.552950
\(204\) 0 0
\(205\) 1631.28 0.555773
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1392.29 −0.460799
\(210\) 0 0
\(211\) −640.632 −0.209019 −0.104509 0.994524i \(-0.533327\pi\)
−0.104509 + 0.994524i \(0.533327\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −2112.63 −0.670141
\(216\) 0 0
\(217\) 1378.87 0.431353
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1284.75 0.391047
\(222\) 0 0
\(223\) −4147.71 −1.24552 −0.622761 0.782412i \(-0.713989\pi\)
−0.622761 + 0.782412i \(0.713989\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5012.97 −1.46574 −0.732869 0.680370i \(-0.761819\pi\)
−0.732869 + 0.680370i \(0.761819\pi\)
\(228\) 0 0
\(229\) 6763.07 1.95160 0.975800 0.218664i \(-0.0701700\pi\)
0.975800 + 0.218664i \(0.0701700\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1995.20 0.560987 0.280493 0.959856i \(-0.409502\pi\)
0.280493 + 0.959856i \(0.409502\pi\)
\(234\) 0 0
\(235\) −2263.18 −0.628227
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1681.95 −0.455213 −0.227607 0.973753i \(-0.573090\pi\)
−0.227607 + 0.973753i \(0.573090\pi\)
\(240\) 0 0
\(241\) 3573.58 0.955164 0.477582 0.878587i \(-0.341513\pi\)
0.477582 + 0.878587i \(0.341513\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 1189.54 0.310192
\(246\) 0 0
\(247\) 1596.32 0.411221
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −808.137 −0.203224 −0.101612 0.994824i \(-0.532400\pi\)
−0.101612 + 0.994824i \(0.532400\pi\)
\(252\) 0 0
\(253\) −3378.28 −0.839488
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4824.76 −1.17105 −0.585526 0.810654i \(-0.699112\pi\)
−0.585526 + 0.810654i \(0.699112\pi\)
\(258\) 0 0
\(259\) 832.848 0.199809
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4252.92 0.997134 0.498567 0.866851i \(-0.333860\pi\)
0.498567 + 0.866851i \(0.333860\pi\)
\(264\) 0 0
\(265\) 490.553 0.113715
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2198.92 0.498403 0.249202 0.968452i \(-0.419832\pi\)
0.249202 + 0.968452i \(0.419832\pi\)
\(270\) 0 0
\(271\) −4904.13 −1.09928 −0.549639 0.835402i \(-0.685235\pi\)
−0.549639 + 0.835402i \(0.685235\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1109.38 0.243267
\(276\) 0 0
\(277\) 933.019 0.202382 0.101191 0.994867i \(-0.467735\pi\)
0.101191 + 0.994867i \(0.467735\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −4412.79 −0.936814 −0.468407 0.883513i \(-0.655172\pi\)
−0.468407 + 0.883513i \(0.655172\pi\)
\(282\) 0 0
\(283\) 748.931 0.157312 0.0786561 0.996902i \(-0.474937\pi\)
0.0786561 + 0.996902i \(0.474937\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3344.58 −0.687890
\(288\) 0 0
\(289\) −4275.37 −0.870215
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1706.45 −0.340246 −0.170123 0.985423i \(-0.554416\pi\)
−0.170123 + 0.985423i \(0.554416\pi\)
\(294\) 0 0
\(295\) 2700.59 0.532998
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3873.34 0.749167
\(300\) 0 0
\(301\) 4331.49 0.829445
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2611.46 −0.490268
\(306\) 0 0
\(307\) 5773.97 1.07341 0.536706 0.843769i \(-0.319668\pi\)
0.536706 + 0.843769i \(0.319668\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3777.25 −0.688709 −0.344354 0.938840i \(-0.611902\pi\)
−0.344354 + 0.938840i \(0.611902\pi\)
\(312\) 0 0
\(313\) 6966.88 1.25812 0.629060 0.777357i \(-0.283440\pi\)
0.629060 + 0.777357i \(0.283440\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8482.29 1.50288 0.751440 0.659801i \(-0.229360\pi\)
0.751440 + 0.659801i \(0.229360\pi\)
\(318\) 0 0
\(319\) −6922.90 −1.21507
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 792.272 0.136481
\(324\) 0 0
\(325\) −1271.95 −0.217093
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4640.15 0.777568
\(330\) 0 0
\(331\) −7673.68 −1.27427 −0.637136 0.770752i \(-0.719881\pi\)
−0.637136 + 0.770752i \(0.719881\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 646.359 0.105416
\(336\) 0 0
\(337\) 7135.56 1.15341 0.576704 0.816953i \(-0.304339\pi\)
0.576704 + 0.816953i \(0.304339\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5968.71 0.947870
\(342\) 0 0
\(343\) −5955.13 −0.937455
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6809.59 1.05348 0.526741 0.850026i \(-0.323414\pi\)
0.526741 + 0.850026i \(0.323414\pi\)
\(348\) 0 0
\(349\) −10666.0 −1.63593 −0.817963 0.575271i \(-0.804897\pi\)
−0.817963 + 0.575271i \(0.804897\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −6399.50 −0.964904 −0.482452 0.875922i \(-0.660254\pi\)
−0.482452 + 0.875922i \(0.660254\pi\)
\(354\) 0 0
\(355\) 133.017 0.0198868
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1340.16 0.197022 0.0985112 0.995136i \(-0.468592\pi\)
0.0985112 + 0.995136i \(0.468592\pi\)
\(360\) 0 0
\(361\) −5874.59 −0.856479
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 739.363 0.106027
\(366\) 0 0
\(367\) 2856.96 0.406354 0.203177 0.979142i \(-0.434873\pi\)
0.203177 + 0.979142i \(0.434873\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1005.77 −0.140747
\(372\) 0 0
\(373\) 9057.97 1.25738 0.628691 0.777655i \(-0.283591\pi\)
0.628691 + 0.777655i \(0.283591\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 7937.39 1.08434
\(378\) 0 0
\(379\) −10770.0 −1.45968 −0.729839 0.683619i \(-0.760405\pi\)
−0.729839 + 0.683619i \(0.760405\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1249.75 −0.166734 −0.0833671 0.996519i \(-0.526567\pi\)
−0.0833671 + 0.996519i \(0.526567\pi\)
\(384\) 0 0
\(385\) −2274.55 −0.301096
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −14409.5 −1.87812 −0.939060 0.343753i \(-0.888302\pi\)
−0.939060 + 0.343753i \(0.888302\pi\)
\(390\) 0 0
\(391\) 1922.38 0.248642
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 5444.56 0.693533
\(396\) 0 0
\(397\) 8504.39 1.07512 0.537561 0.843225i \(-0.319346\pi\)
0.537561 + 0.843225i \(0.319346\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11812.6 1.47106 0.735528 0.677494i \(-0.236934\pi\)
0.735528 + 0.677494i \(0.236934\pi\)
\(402\) 0 0
\(403\) −6843.37 −0.845887
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3605.16 0.439069
\(408\) 0 0
\(409\) −3009.71 −0.363864 −0.181932 0.983311i \(-0.558235\pi\)
−0.181932 + 0.983311i \(0.558235\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5536.97 −0.659701
\(414\) 0 0
\(415\) 2973.06 0.351667
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1354.28 0.157902 0.0789510 0.996878i \(-0.474843\pi\)
0.0789510 + 0.996878i \(0.474843\pi\)
\(420\) 0 0
\(421\) 14411.5 1.66835 0.834175 0.551500i \(-0.185944\pi\)
0.834175 + 0.551500i \(0.185944\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −631.285 −0.0720514
\(426\) 0 0
\(427\) 5354.23 0.606813
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6762.24 0.755744 0.377872 0.925858i \(-0.376656\pi\)
0.377872 + 0.925858i \(0.376656\pi\)
\(432\) 0 0
\(433\) −3323.67 −0.368881 −0.184440 0.982844i \(-0.559047\pi\)
−0.184440 + 0.982844i \(0.559047\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2388.59 0.261469
\(438\) 0 0
\(439\) 4649.90 0.505530 0.252765 0.967528i \(-0.418660\pi\)
0.252765 + 0.967528i \(0.418660\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 14434.0 1.54804 0.774019 0.633162i \(-0.218243\pi\)
0.774019 + 0.633162i \(0.218243\pi\)
\(444\) 0 0
\(445\) 2961.14 0.315442
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −3428.29 −0.360336 −0.180168 0.983636i \(-0.557664\pi\)
−0.180168 + 0.983636i \(0.557664\pi\)
\(450\) 0 0
\(451\) −14477.7 −1.51159
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 2607.87 0.268700
\(456\) 0 0
\(457\) 18647.8 1.90877 0.954384 0.298583i \(-0.0965141\pi\)
0.954384 + 0.298583i \(0.0965141\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −144.345 −0.0145831 −0.00729157 0.999973i \(-0.502321\pi\)
−0.00729157 + 0.999973i \(0.502321\pi\)
\(462\) 0 0
\(463\) 3418.69 0.343154 0.171577 0.985171i \(-0.445114\pi\)
0.171577 + 0.985171i \(0.445114\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9571.86 0.948465 0.474232 0.880400i \(-0.342726\pi\)
0.474232 + 0.880400i \(0.342726\pi\)
\(468\) 0 0
\(469\) −1325.22 −0.130475
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 18749.8 1.82265
\(474\) 0 0
\(475\) −784.384 −0.0757685
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 17144.0 1.63534 0.817670 0.575687i \(-0.195265\pi\)
0.817670 + 0.575687i \(0.195265\pi\)
\(480\) 0 0
\(481\) −4133.46 −0.391829
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3333.93 0.312136
\(486\) 0 0
\(487\) −16588.4 −1.54351 −0.771757 0.635917i \(-0.780622\pi\)
−0.771757 + 0.635917i \(0.780622\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5835.61 0.536369 0.268185 0.963368i \(-0.413576\pi\)
0.268185 + 0.963368i \(0.413576\pi\)
\(492\) 0 0
\(493\) 3939.42 0.359883
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −272.722 −0.0246142
\(498\) 0 0
\(499\) 9028.41 0.809954 0.404977 0.914327i \(-0.367280\pi\)
0.404977 + 0.914327i \(0.367280\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −8169.36 −0.724163 −0.362081 0.932147i \(-0.617934\pi\)
−0.362081 + 0.932147i \(0.617934\pi\)
\(504\) 0 0
\(505\) 6092.80 0.536884
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7955.70 0.692791 0.346395 0.938089i \(-0.387406\pi\)
0.346395 + 0.938089i \(0.387406\pi\)
\(510\) 0 0
\(511\) −1515.90 −0.131232
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1158.94 0.0991634
\(516\) 0 0
\(517\) 20085.9 1.70866
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 5276.52 0.443701 0.221851 0.975081i \(-0.428790\pi\)
0.221851 + 0.975081i \(0.428790\pi\)
\(522\) 0 0
\(523\) −17227.2 −1.44033 −0.720167 0.693801i \(-0.755935\pi\)
−0.720167 + 0.693801i \(0.755935\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3396.44 −0.280742
\(528\) 0 0
\(529\) −6371.28 −0.523653
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 16599.3 1.34896
\(534\) 0 0
\(535\) 9899.80 0.800011
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −10557.3 −0.843663
\(540\) 0 0
\(541\) 4830.69 0.383895 0.191948 0.981405i \(-0.438520\pi\)
0.191948 + 0.981405i \(0.438520\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5377.04 −0.422619
\(546\) 0 0
\(547\) 16766.0 1.31053 0.655267 0.755398i \(-0.272556\pi\)
0.655267 + 0.755398i \(0.272556\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4894.80 0.378449
\(552\) 0 0
\(553\) −11162.9 −0.858398
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17819.2 1.35552 0.677758 0.735285i \(-0.262952\pi\)
0.677758 + 0.735285i \(0.262952\pi\)
\(558\) 0 0
\(559\) −21497.4 −1.62655
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −13031.8 −0.975534 −0.487767 0.872974i \(-0.662189\pi\)
−0.487767 + 0.872974i \(0.662189\pi\)
\(564\) 0 0
\(565\) 3467.05 0.258159
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7591.83 0.559342 0.279671 0.960096i \(-0.409775\pi\)
0.279671 + 0.960096i \(0.409775\pi\)
\(570\) 0 0
\(571\) 9344.83 0.684884 0.342442 0.939539i \(-0.388746\pi\)
0.342442 + 0.939539i \(0.388746\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1903.24 −0.138036
\(576\) 0 0
\(577\) −20553.5 −1.48294 −0.741469 0.670987i \(-0.765870\pi\)
−0.741469 + 0.670987i \(0.765870\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6095.61 −0.435264
\(582\) 0 0
\(583\) −4353.69 −0.309282
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 4842.95 0.340528 0.170264 0.985398i \(-0.445538\pi\)
0.170264 + 0.985398i \(0.445538\pi\)
\(588\) 0 0
\(589\) −4220.14 −0.295226
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 15403.8 1.06671 0.533355 0.845891i \(-0.320931\pi\)
0.533355 + 0.845891i \(0.320931\pi\)
\(594\) 0 0
\(595\) 1294.31 0.0891793
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3971.66 −0.270914 −0.135457 0.990783i \(-0.543250\pi\)
−0.135457 + 0.990783i \(0.543250\pi\)
\(600\) 0 0
\(601\) −20636.8 −1.40065 −0.700326 0.713824i \(-0.746962\pi\)
−0.700326 + 0.713824i \(0.746962\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3190.87 −0.214425
\(606\) 0 0
\(607\) 18039.1 1.20624 0.603118 0.797652i \(-0.293925\pi\)
0.603118 + 0.797652i \(0.293925\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −23029.3 −1.52482
\(612\) 0 0
\(613\) −14654.4 −0.965559 −0.482779 0.875742i \(-0.660373\pi\)
−0.482779 + 0.875742i \(0.660373\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −19933.8 −1.30066 −0.650329 0.759653i \(-0.725369\pi\)
−0.650329 + 0.759653i \(0.725369\pi\)
\(618\) 0 0
\(619\) −10548.9 −0.684969 −0.342485 0.939523i \(-0.611268\pi\)
−0.342485 + 0.939523i \(0.611268\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −6071.18 −0.390428
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2051.48 −0.130044
\(630\) 0 0
\(631\) 23088.8 1.45666 0.728330 0.685227i \(-0.240297\pi\)
0.728330 + 0.685227i \(0.240297\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4167.89 0.260469
\(636\) 0 0
\(637\) 12104.4 0.752892
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −711.767 −0.0438582 −0.0219291 0.999760i \(-0.506981\pi\)
−0.0219291 + 0.999760i \(0.506981\pi\)
\(642\) 0 0
\(643\) −6867.64 −0.421203 −0.210601 0.977572i \(-0.567542\pi\)
−0.210601 + 0.977572i \(0.567542\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1122.52 −0.0682086 −0.0341043 0.999418i \(-0.510858\pi\)
−0.0341043 + 0.999418i \(0.510858\pi\)
\(648\) 0 0
\(649\) −23967.9 −1.44965
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5153.57 0.308843 0.154422 0.988005i \(-0.450649\pi\)
0.154422 + 0.988005i \(0.450649\pi\)
\(654\) 0 0
\(655\) 1768.60 0.105504
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 28258.6 1.67041 0.835205 0.549939i \(-0.185349\pi\)
0.835205 + 0.549939i \(0.185349\pi\)
\(660\) 0 0
\(661\) 2401.89 0.141335 0.0706676 0.997500i \(-0.477487\pi\)
0.0706676 + 0.997500i \(0.477487\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1608.21 0.0937800
\(666\) 0 0
\(667\) 11876.8 0.689463
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 23176.9 1.33343
\(672\) 0 0
\(673\) −21090.9 −1.20801 −0.604007 0.796979i \(-0.706430\pi\)
−0.604007 + 0.796979i \(0.706430\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −9098.10 −0.516497 −0.258249 0.966079i \(-0.583145\pi\)
−0.258249 + 0.966079i \(0.583145\pi\)
\(678\) 0 0
\(679\) −6835.50 −0.386336
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −9488.98 −0.531604 −0.265802 0.964028i \(-0.585637\pi\)
−0.265802 + 0.964028i \(0.585637\pi\)
\(684\) 0 0
\(685\) 7549.49 0.421097
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4991.69 0.276006
\(690\) 0 0
\(691\) −21798.7 −1.20009 −0.600043 0.799967i \(-0.704850\pi\)
−0.600043 + 0.799967i \(0.704850\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3092.42 0.168780
\(696\) 0 0
\(697\) 8238.42 0.447708
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 13799.3 0.743499 0.371750 0.928333i \(-0.378758\pi\)
0.371750 + 0.928333i \(0.378758\pi\)
\(702\) 0 0
\(703\) −2549.01 −0.136753
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −12492.0 −0.664510
\(708\) 0 0
\(709\) 1561.32 0.0827030 0.0413515 0.999145i \(-0.486834\pi\)
0.0413515 + 0.999145i \(0.486834\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −10239.8 −0.537846
\(714\) 0 0
\(715\) 11288.7 0.590452
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 14008.2 0.726590 0.363295 0.931674i \(-0.381652\pi\)
0.363295 + 0.931674i \(0.381652\pi\)
\(720\) 0 0
\(721\) −2376.16 −0.122736
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3900.19 −0.199793
\(726\) 0 0
\(727\) 20977.4 1.07016 0.535081 0.844801i \(-0.320281\pi\)
0.535081 + 0.844801i \(0.320281\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −10669.4 −0.539838
\(732\) 0 0
\(733\) 20802.8 1.04825 0.524126 0.851641i \(-0.324392\pi\)
0.524126 + 0.851641i \(0.324392\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5736.49 −0.286711
\(738\) 0 0
\(739\) 3111.47 0.154881 0.0774407 0.996997i \(-0.475325\pi\)
0.0774407 + 0.996997i \(0.475325\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5449.19 −0.269060 −0.134530 0.990910i \(-0.542952\pi\)
−0.134530 + 0.990910i \(0.542952\pi\)
\(744\) 0 0
\(745\) 12027.0 0.591459
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −20297.4 −0.990188
\(750\) 0 0
\(751\) 28975.5 1.40790 0.703949 0.710251i \(-0.251419\pi\)
0.703949 + 0.710251i \(0.251419\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8005.64 −0.385901
\(756\) 0 0
\(757\) 20654.2 0.991666 0.495833 0.868418i \(-0.334863\pi\)
0.495833 + 0.868418i \(0.334863\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13619.4 0.648757 0.324378 0.945927i \(-0.394845\pi\)
0.324378 + 0.945927i \(0.394845\pi\)
\(762\) 0 0
\(763\) 11024.5 0.523083
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 27480.2 1.29368
\(768\) 0 0
\(769\) −15267.8 −0.715959 −0.357980 0.933729i \(-0.616534\pi\)
−0.357980 + 0.933729i \(0.616534\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 16108.3 0.749514 0.374757 0.927123i \(-0.377726\pi\)
0.374757 + 0.927123i \(0.377726\pi\)
\(774\) 0 0
\(775\) 3362.62 0.155857
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 10236.4 0.470805
\(780\) 0 0
\(781\) −1180.54 −0.0540882
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −16042.0 −0.729379
\(786\) 0 0
\(787\) 25511.0 1.15549 0.577743 0.816219i \(-0.303933\pi\)
0.577743 + 0.816219i \(0.303933\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −7108.43 −0.319528
\(792\) 0 0
\(793\) −26573.3 −1.18997
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −26626.1 −1.18337 −0.591685 0.806169i \(-0.701537\pi\)
−0.591685 + 0.806169i \(0.701537\pi\)
\(798\) 0 0
\(799\) −11429.7 −0.506074
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6561.90 −0.288374
\(804\) 0 0
\(805\) 3902.18 0.170849
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 18935.5 0.822911 0.411456 0.911430i \(-0.365020\pi\)
0.411456 + 0.911430i \(0.365020\pi\)
\(810\) 0 0
\(811\) −18366.2 −0.795220 −0.397610 0.917554i \(-0.630160\pi\)
−0.397610 + 0.917554i \(0.630160\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1962.41 0.0843437
\(816\) 0 0
\(817\) −13256.9 −0.567688
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −32495.2 −1.38135 −0.690675 0.723165i \(-0.742687\pi\)
−0.690675 + 0.723165i \(0.742687\pi\)
\(822\) 0 0
\(823\) −11442.4 −0.484637 −0.242318 0.970197i \(-0.577908\pi\)
−0.242318 + 0.970197i \(0.577908\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −32378.4 −1.36143 −0.680717 0.732546i \(-0.738332\pi\)
−0.680717 + 0.732546i \(0.738332\pi\)
\(828\) 0 0
\(829\) −35365.1 −1.48164 −0.740821 0.671702i \(-0.765563\pi\)
−0.740821 + 0.671702i \(0.765563\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 6007.53 0.249878
\(834\) 0 0
\(835\) −1304.09 −0.0540477
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −6335.06 −0.260680 −0.130340 0.991469i \(-0.541607\pi\)
−0.130340 + 0.991469i \(0.541607\pi\)
\(840\) 0 0
\(841\) −50.5692 −0.00207344
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1957.95 −0.0797107
\(846\) 0 0
\(847\) 6542.18 0.265398
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −6184.94 −0.249139
\(852\) 0 0
\(853\) 4491.51 0.180289 0.0901445 0.995929i \(-0.471267\pi\)
0.0901445 + 0.995929i \(0.471267\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −10823.8 −0.431430 −0.215715 0.976456i \(-0.569208\pi\)
−0.215715 + 0.976456i \(0.569208\pi\)
\(858\) 0 0
\(859\) −45625.9 −1.81226 −0.906132 0.422995i \(-0.860979\pi\)
−0.906132 + 0.422995i \(0.860979\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −29567.9 −1.16629 −0.583143 0.812370i \(-0.698177\pi\)
−0.583143 + 0.812370i \(0.698177\pi\)
\(864\) 0 0
\(865\) −1024.35 −0.0402647
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −48320.9 −1.88628
\(870\) 0 0
\(871\) 6577.12 0.255864
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1281.43 −0.0495087
\(876\) 0 0
\(877\) −1813.37 −0.0698212 −0.0349106 0.999390i \(-0.511115\pi\)
−0.0349106 + 0.999390i \(0.511115\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −29307.8 −1.12078 −0.560390 0.828229i \(-0.689349\pi\)
−0.560390 + 0.828229i \(0.689349\pi\)
\(882\) 0 0
\(883\) 48560.6 1.85073 0.925365 0.379077i \(-0.123759\pi\)
0.925365 + 0.379077i \(0.123759\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −10983.5 −0.415770 −0.207885 0.978153i \(-0.566658\pi\)
−0.207885 + 0.978153i \(0.566658\pi\)
\(888\) 0 0
\(889\) −8545.36 −0.322387
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −14201.6 −0.532182
\(894\) 0 0
\(895\) −9184.62 −0.343026
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −20983.8 −0.778476
\(900\) 0 0
\(901\) 2477.43 0.0916039
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 14403.4 0.529043
\(906\) 0 0
\(907\) 41652.9 1.52488 0.762438 0.647061i \(-0.224002\pi\)
0.762438 + 0.647061i \(0.224002\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −7771.00 −0.282618 −0.141309 0.989966i \(-0.545131\pi\)
−0.141309 + 0.989966i \(0.545131\pi\)
\(912\) 0 0
\(913\) −26386.1 −0.956465
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3626.14 −0.130584
\(918\) 0 0
\(919\) −11284.4 −0.405045 −0.202523 0.979278i \(-0.564914\pi\)
−0.202523 + 0.979278i \(0.564914\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1353.53 0.0482688
\(924\) 0 0
\(925\) 2031.06 0.0721954
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −50206.4 −1.77311 −0.886555 0.462623i \(-0.846908\pi\)
−0.886555 + 0.462623i \(0.846908\pi\)
\(930\) 0 0
\(931\) 7464.47 0.262769
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 5602.70 0.195966
\(936\) 0 0
\(937\) 16233.7 0.565988 0.282994 0.959122i \(-0.408672\pi\)
0.282994 + 0.959122i \(0.408672\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 33113.6 1.14715 0.573577 0.819152i \(-0.305556\pi\)
0.573577 + 0.819152i \(0.305556\pi\)
\(942\) 0 0
\(943\) 24837.7 0.857717
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 19025.7 0.652855 0.326427 0.945222i \(-0.394155\pi\)
0.326427 + 0.945222i \(0.394155\pi\)
\(948\) 0 0
\(949\) 7523.49 0.257347
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 46547.9 1.58220 0.791098 0.611689i \(-0.209510\pi\)
0.791098 + 0.611689i \(0.209510\pi\)
\(954\) 0 0
\(955\) 3741.21 0.126767
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −15478.6 −0.521199
\(960\) 0 0
\(961\) −11699.4 −0.392716
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −3108.10 −0.103682
\(966\) 0 0
\(967\) −1793.49 −0.0596430 −0.0298215 0.999555i \(-0.509494\pi\)
−0.0298215 + 0.999555i \(0.509494\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −34899.8 −1.15344 −0.576719 0.816942i \(-0.695667\pi\)
−0.576719 + 0.816942i \(0.695667\pi\)
\(972\) 0 0
\(973\) −6340.34 −0.208902
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 31360.4 1.02693 0.513464 0.858111i \(-0.328362\pi\)
0.513464 + 0.858111i \(0.328362\pi\)
\(978\) 0 0
\(979\) −26280.4 −0.857941
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −56392.0 −1.82973 −0.914866 0.403758i \(-0.867704\pi\)
−0.914866 + 0.403758i \(0.867704\pi\)
\(984\) 0 0
\(985\) 2941.22 0.0951421
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −32166.8 −1.03422
\(990\) 0 0
\(991\) 10086.0 0.323303 0.161652 0.986848i \(-0.448318\pi\)
0.161652 + 0.986848i \(0.448318\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 20076.6 0.639668
\(996\) 0 0
\(997\) 33875.2 1.07607 0.538033 0.842924i \(-0.319168\pi\)
0.538033 + 0.842924i \(0.319168\pi\)
\(998\) 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 1080.4.a.e.1.2 3
3.2 odd 2 1080.4.a.k.1.2 yes 3
4.3 odd 2 2160.4.a.bj.1.2 3
12.11 even 2 2160.4.a.br.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1080.4.a.e.1.2 3 1.1 even 1 trivial
1080.4.a.k.1.2 yes 3 3.2 odd 2
2160.4.a.bj.1.2 3 4.3 odd 2
2160.4.a.br.1.2 3 12.11 even 2