Properties

Label 1296.4.a.i.1.2
Level $1296$
Weight $4$
Character 1296.1
Self dual yes
Analytic conductor $76.466$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1296,4,Mod(1,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1296.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1296.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: \(76.4664753674\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 9)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 1296.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.62772 q^{5} -12.1168 q^{7} +O(q^{10})\) \(q-4.62772 q^{5} -12.1168 q^{7} +10.0217 q^{11} -48.5842 q^{13} -75.3505 q^{17} +116.052 q^{19} -38.0733 q^{23} -103.584 q^{25} +22.6277 q^{29} -30.1168 q^{31} +56.0733 q^{35} +130.103 q^{37} -347.484 q^{41} -26.7663 q^{43} +460.878 q^{47} -196.182 q^{49} +438.310 q^{53} -46.3778 q^{55} +8.36974 q^{59} -82.0895 q^{61} +224.834 q^{65} -683.571 q^{67} +1097.61 q^{71} +470.464 q^{73} -121.432 q^{77} -486.035 q^{79} +99.1659 q^{83} +348.701 q^{85} -8.80426 q^{89} +588.687 q^{91} -537.054 q^{95} +660.973 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 15 q^{5} - 7 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 15 q^{5} - 7 q^{7} + 66 q^{11} - 11 q^{13} - 99 q^{17} + 77 q^{19} + 33 q^{23} - 121 q^{25} + 51 q^{29} - 43 q^{31} + 3 q^{35} - 50 q^{37} - 132 q^{41} - 88 q^{43} + 399 q^{47} - 513 q^{49} - 54 q^{53} - 627 q^{55} + 798 q^{59} + 439 q^{61} - 165 q^{65} - 988 q^{67} + 1368 q^{71} - 455 q^{73} + 165 q^{77} + 803 q^{79} + 813 q^{83} + 594 q^{85} + 396 q^{89} + 781 q^{91} - 132 q^{95} + 736 q^{97}+O(q^{100}) \) 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
\(5\) −4.62772 −0.413916 −0.206958 0.978350i \(-0.566356\pi\)
−0.206958 + 0.978350i \(0.566356\pi\)
\(6\) 0 0
\(7\) −12.1168 −0.654248 −0.327124 0.944981i \(-0.606079\pi\)
−0.327124 + 0.944981i \(0.606079\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 10.0217 0.274697 0.137349 0.990523i \(-0.456142\pi\)
0.137349 + 0.990523i \(0.456142\pi\)
\(12\) 0 0
\(13\) −48.5842 −1.03653 −0.518263 0.855221i \(-0.673421\pi\)
−0.518263 + 0.855221i \(0.673421\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −75.3505 −1.07501 −0.537506 0.843260i \(-0.680633\pi\)
−0.537506 + 0.843260i \(0.680633\pi\)
\(18\) 0 0
\(19\) 116.052 1.40127 0.700633 0.713522i \(-0.252901\pi\)
0.700633 + 0.713522i \(0.252901\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −38.0733 −0.345167 −0.172584 0.984995i \(-0.555211\pi\)
−0.172584 + 0.984995i \(0.555211\pi\)
\(24\) 0 0
\(25\) −103.584 −0.828674
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 22.6277 0.144892 0.0724459 0.997372i \(-0.476920\pi\)
0.0724459 + 0.997372i \(0.476920\pi\)
\(30\) 0 0
\(31\) −30.1168 −0.174489 −0.0872443 0.996187i \(-0.527806\pi\)
−0.0872443 + 0.996187i \(0.527806\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 56.0733 0.270804
\(36\) 0 0
\(37\) 130.103 0.578077 0.289038 0.957318i \(-0.406665\pi\)
0.289038 + 0.957318i \(0.406665\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −347.484 −1.32361 −0.661803 0.749678i \(-0.730208\pi\)
−0.661803 + 0.749678i \(0.730208\pi\)
\(42\) 0 0
\(43\) −26.7663 −0.0949261 −0.0474631 0.998873i \(-0.515114\pi\)
−0.0474631 + 0.998873i \(0.515114\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 460.878 1.43034 0.715169 0.698951i \(-0.246350\pi\)
0.715169 + 0.698951i \(0.246350\pi\)
\(48\) 0 0
\(49\) −196.182 −0.571959
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 438.310 1.13597 0.567985 0.823039i \(-0.307723\pi\)
0.567985 + 0.823039i \(0.307723\pi\)
\(54\) 0 0
\(55\) −46.3778 −0.113702
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 8.36974 0.0184686 0.00923430 0.999957i \(-0.497061\pi\)
0.00923430 + 0.999957i \(0.497061\pi\)
\(60\) 0 0
\(61\) −82.0895 −0.172303 −0.0861515 0.996282i \(-0.527457\pi\)
−0.0861515 + 0.996282i \(0.527457\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 224.834 0.429034
\(66\) 0 0
\(67\) −683.571 −1.24644 −0.623220 0.782047i \(-0.714176\pi\)
−0.623220 + 0.782047i \(0.714176\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1097.61 1.83468 0.917339 0.398107i \(-0.130333\pi\)
0.917339 + 0.398107i \(0.130333\pi\)
\(72\) 0 0
\(73\) 470.464 0.754297 0.377149 0.926153i \(-0.376905\pi\)
0.377149 + 0.926153i \(0.376905\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −121.432 −0.179720
\(78\) 0 0
\(79\) −486.035 −0.692192 −0.346096 0.938199i \(-0.612493\pi\)
−0.346096 + 0.938199i \(0.612493\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 99.1659 0.131143 0.0655715 0.997848i \(-0.479113\pi\)
0.0655715 + 0.997848i \(0.479113\pi\)
\(84\) 0 0
\(85\) 348.701 0.444964
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −8.80426 −0.0104859 −0.00524297 0.999986i \(-0.501669\pi\)
−0.00524297 + 0.999986i \(0.501669\pi\)
\(90\) 0 0
\(91\) 588.687 0.678145
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −537.054 −0.580006
\(96\) 0 0
\(97\) 660.973 0.691872 0.345936 0.938258i \(-0.387561\pi\)
0.345936 + 0.938258i \(0.387561\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 565.122 0.556750 0.278375 0.960472i \(-0.410204\pi\)
0.278375 + 0.960472i \(0.410204\pi\)
\(102\) 0 0
\(103\) 971.182 0.929062 0.464531 0.885557i \(-0.346223\pi\)
0.464531 + 0.885557i \(0.346223\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −563.845 −0.509430 −0.254715 0.967016i \(-0.581982\pi\)
−0.254715 + 0.967016i \(0.581982\pi\)
\(108\) 0 0
\(109\) 225.484 0.198142 0.0990709 0.995080i \(-0.468413\pi\)
0.0990709 + 0.995080i \(0.468413\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 691.063 0.575307 0.287654 0.957735i \(-0.407125\pi\)
0.287654 + 0.957735i \(0.407125\pi\)
\(114\) 0 0
\(115\) 176.193 0.142870
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 913.011 0.703324
\(120\) 0 0
\(121\) −1230.56 −0.924541
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1057.82 0.756917
\(126\) 0 0
\(127\) 895.897 0.625968 0.312984 0.949758i \(-0.398671\pi\)
0.312984 + 0.949758i \(0.398671\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1654.86 1.10371 0.551853 0.833942i \(-0.313921\pi\)
0.551853 + 0.833942i \(0.313921\pi\)
\(132\) 0 0
\(133\) −1406.18 −0.916776
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2651.09 1.65327 0.826635 0.562738i \(-0.190252\pi\)
0.826635 + 0.562738i \(0.190252\pi\)
\(138\) 0 0
\(139\) −634.168 −0.386975 −0.193487 0.981103i \(-0.561980\pi\)
−0.193487 + 0.981103i \(0.561980\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −486.899 −0.284731
\(144\) 0 0
\(145\) −104.715 −0.0599730
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3406.32 1.87286 0.936432 0.350849i \(-0.114107\pi\)
0.936432 + 0.350849i \(0.114107\pi\)
\(150\) 0 0
\(151\) 1750.32 0.943303 0.471652 0.881785i \(-0.343658\pi\)
0.471652 + 0.881785i \(0.343658\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 139.372 0.0722236
\(156\) 0 0
\(157\) 2178.57 1.10745 0.553723 0.832701i \(-0.313207\pi\)
0.553723 + 0.832701i \(0.313207\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 461.329 0.225825
\(162\) 0 0
\(163\) −2188.41 −1.05159 −0.525797 0.850610i \(-0.676233\pi\)
−0.525797 + 0.850610i \(0.676233\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1921.04 0.890147 0.445074 0.895494i \(-0.353178\pi\)
0.445074 + 0.895494i \(0.353178\pi\)
\(168\) 0 0
\(169\) 163.426 0.0743862
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −3169.83 −1.39305 −0.696525 0.717533i \(-0.745271\pi\)
−0.696525 + 0.717533i \(0.745271\pi\)
\(174\) 0 0
\(175\) 1255.11 0.542158
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1368.78 0.571551 0.285776 0.958297i \(-0.407749\pi\)
0.285776 + 0.958297i \(0.407749\pi\)
\(180\) 0 0
\(181\) −3951.44 −1.62270 −0.811350 0.584561i \(-0.801267\pi\)
−0.811350 + 0.584561i \(0.801267\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −602.081 −0.239275
\(186\) 0 0
\(187\) −755.144 −0.295303
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2402.59 −0.910184 −0.455092 0.890444i \(-0.650394\pi\)
−0.455092 + 0.890444i \(0.650394\pi\)
\(192\) 0 0
\(193\) −1335.07 −0.497930 −0.248965 0.968513i \(-0.580090\pi\)
−0.248965 + 0.968513i \(0.580090\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2630.89 0.951487 0.475743 0.879584i \(-0.342179\pi\)
0.475743 + 0.879584i \(0.342179\pi\)
\(198\) 0 0
\(199\) −2477.34 −0.882483 −0.441241 0.897388i \(-0.645462\pi\)
−0.441241 + 0.897388i \(0.645462\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −274.177 −0.0947952
\(204\) 0 0
\(205\) 1608.06 0.547861
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1163.04 0.384924
\(210\) 0 0
\(211\) 2784.73 0.908572 0.454286 0.890856i \(-0.349894\pi\)
0.454286 + 0.890856i \(0.349894\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 123.867 0.0392914
\(216\) 0 0
\(217\) 364.921 0.114159
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3660.85 1.11428
\(222\) 0 0
\(223\) 43.0576 0.0129298 0.00646491 0.999979i \(-0.497942\pi\)
0.00646491 + 0.999979i \(0.497942\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 683.260 0.199778 0.0998889 0.994999i \(-0.468151\pi\)
0.0998889 + 0.994999i \(0.468151\pi\)
\(228\) 0 0
\(229\) 4294.31 1.23920 0.619598 0.784920i \(-0.287296\pi\)
0.619598 + 0.784920i \(0.287296\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −3466.34 −0.974625 −0.487313 0.873228i \(-0.662023\pi\)
−0.487313 + 0.873228i \(0.662023\pi\)
\(234\) 0 0
\(235\) −2132.81 −0.592040
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5643.37 1.52736 0.763681 0.645594i \(-0.223390\pi\)
0.763681 + 0.645594i \(0.223390\pi\)
\(240\) 0 0
\(241\) 6589.43 1.76125 0.880627 0.473810i \(-0.157121\pi\)
0.880627 + 0.473810i \(0.157121\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 907.876 0.236743
\(246\) 0 0
\(247\) −5638.28 −1.45245
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4135.47 1.03996 0.519978 0.854180i \(-0.325940\pi\)
0.519978 + 0.854180i \(0.325940\pi\)
\(252\) 0 0
\(253\) −381.562 −0.0948165
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3344.62 0.811796 0.405898 0.913918i \(-0.366959\pi\)
0.405898 + 0.913918i \(0.366959\pi\)
\(258\) 0 0
\(259\) −1576.44 −0.378205
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6521.50 1.52902 0.764511 0.644611i \(-0.222981\pi\)
0.764511 + 0.644611i \(0.222981\pi\)
\(264\) 0 0
\(265\) −2028.37 −0.470196
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2904.99 0.658441 0.329220 0.944253i \(-0.393214\pi\)
0.329220 + 0.944253i \(0.393214\pi\)
\(270\) 0 0
\(271\) 1335.38 0.299331 0.149665 0.988737i \(-0.452180\pi\)
0.149665 + 0.988737i \(0.452180\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1038.10 −0.227635
\(276\) 0 0
\(277\) −8375.64 −1.81676 −0.908381 0.418143i \(-0.862681\pi\)
−0.908381 + 0.418143i \(0.862681\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5179.33 1.09955 0.549774 0.835313i \(-0.314714\pi\)
0.549774 + 0.835313i \(0.314714\pi\)
\(282\) 0 0
\(283\) 3080.07 0.646965 0.323482 0.946234i \(-0.395146\pi\)
0.323482 + 0.946234i \(0.395146\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4210.40 0.865966
\(288\) 0 0
\(289\) 764.703 0.155649
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −3648.89 −0.727545 −0.363773 0.931488i \(-0.618511\pi\)
−0.363773 + 0.931488i \(0.618511\pi\)
\(294\) 0 0
\(295\) −38.7328 −0.00764444
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1849.76 0.357775
\(300\) 0 0
\(301\) 324.323 0.0621052
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 379.887 0.0713190
\(306\) 0 0
\(307\) 3439.25 0.639376 0.319688 0.947523i \(-0.396422\pi\)
0.319688 + 0.947523i \(0.396422\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7175.33 1.30828 0.654141 0.756373i \(-0.273030\pi\)
0.654141 + 0.756373i \(0.273030\pi\)
\(312\) 0 0
\(313\) −4857.30 −0.877159 −0.438579 0.898692i \(-0.644518\pi\)
−0.438579 + 0.898692i \(0.644518\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7547.93 −1.33733 −0.668666 0.743563i \(-0.733134\pi\)
−0.668666 + 0.743563i \(0.733134\pi\)
\(318\) 0 0
\(319\) 226.769 0.0398014
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −8744.55 −1.50638
\(324\) 0 0
\(325\) 5032.56 0.858942
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −5584.38 −0.935796
\(330\) 0 0
\(331\) 3129.72 0.519712 0.259856 0.965647i \(-0.416325\pi\)
0.259856 + 0.965647i \(0.416325\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3163.37 0.515921
\(336\) 0 0
\(337\) −9229.98 −1.49196 −0.745978 0.665971i \(-0.768017\pi\)
−0.745978 + 0.665971i \(0.768017\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −301.823 −0.0479315
\(342\) 0 0
\(343\) 6533.19 1.02845
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8104.40 −1.25379 −0.626897 0.779102i \(-0.715675\pi\)
−0.626897 + 0.779102i \(0.715675\pi\)
\(348\) 0 0
\(349\) 3076.42 0.471854 0.235927 0.971771i \(-0.424187\pi\)
0.235927 + 0.971771i \(0.424187\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 6150.08 0.927297 0.463649 0.886019i \(-0.346540\pi\)
0.463649 + 0.886019i \(0.346540\pi\)
\(354\) 0 0
\(355\) −5079.42 −0.759402
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3307.94 0.486313 0.243156 0.969987i \(-0.421817\pi\)
0.243156 + 0.969987i \(0.421817\pi\)
\(360\) 0 0
\(361\) 6608.97 0.963548
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2177.18 −0.312215
\(366\) 0 0
\(367\) −2949.29 −0.419488 −0.209744 0.977756i \(-0.567263\pi\)
−0.209744 + 0.977756i \(0.567263\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5310.93 −0.743207
\(372\) 0 0
\(373\) 1163.58 0.161522 0.0807612 0.996733i \(-0.474265\pi\)
0.0807612 + 0.996733i \(0.474265\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1099.35 −0.150184
\(378\) 0 0
\(379\) 4016.67 0.544387 0.272193 0.962243i \(-0.412251\pi\)
0.272193 + 0.962243i \(0.412251\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −10803.3 −1.44131 −0.720656 0.693293i \(-0.756159\pi\)
−0.720656 + 0.693293i \(0.756159\pi\)
\(384\) 0 0
\(385\) 561.953 0.0743890
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2064.59 −0.269097 −0.134549 0.990907i \(-0.542958\pi\)
−0.134549 + 0.990907i \(0.542958\pi\)
\(390\) 0 0
\(391\) 2868.85 0.371058
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 2249.23 0.286509
\(396\) 0 0
\(397\) −7937.61 −1.00347 −0.501735 0.865022i \(-0.667305\pi\)
−0.501735 + 0.865022i \(0.667305\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2578.21 −0.321071 −0.160536 0.987030i \(-0.551322\pi\)
−0.160536 + 0.987030i \(0.551322\pi\)
\(402\) 0 0
\(403\) 1463.20 0.180862
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1303.86 0.158796
\(408\) 0 0
\(409\) 5845.76 0.706734 0.353367 0.935485i \(-0.385037\pi\)
0.353367 + 0.935485i \(0.385037\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −101.415 −0.0120830
\(414\) 0 0
\(415\) −458.912 −0.0542822
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −8741.31 −1.01919 −0.509596 0.860414i \(-0.670205\pi\)
−0.509596 + 0.860414i \(0.670205\pi\)
\(420\) 0 0
\(421\) 1056.51 0.122307 0.0611533 0.998128i \(-0.480522\pi\)
0.0611533 + 0.998128i \(0.480522\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 7805.13 0.890833
\(426\) 0 0
\(427\) 994.666 0.112729
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9868.64 1.10291 0.551457 0.834203i \(-0.314072\pi\)
0.551457 + 0.834203i \(0.314072\pi\)
\(432\) 0 0
\(433\) 477.948 0.0530456 0.0265228 0.999648i \(-0.491557\pi\)
0.0265228 + 0.999648i \(0.491557\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −4418.47 −0.483671
\(438\) 0 0
\(439\) 1052.48 0.114424 0.0572119 0.998362i \(-0.481779\pi\)
0.0572119 + 0.998362i \(0.481779\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −10498.7 −1.12598 −0.562989 0.826464i \(-0.690349\pi\)
−0.562989 + 0.826464i \(0.690349\pi\)
\(444\) 0 0
\(445\) 40.7436 0.00434030
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7329.40 0.770369 0.385184 0.922840i \(-0.374138\pi\)
0.385184 + 0.922840i \(0.374138\pi\)
\(450\) 0 0
\(451\) −3482.39 −0.363591
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2724.28 −0.280695
\(456\) 0 0
\(457\) 7144.40 0.731293 0.365646 0.930754i \(-0.380848\pi\)
0.365646 + 0.930754i \(0.380848\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −5718.69 −0.577757 −0.288878 0.957366i \(-0.593282\pi\)
−0.288878 + 0.957366i \(0.593282\pi\)
\(462\) 0 0
\(463\) −788.466 −0.0791428 −0.0395714 0.999217i \(-0.512599\pi\)
−0.0395714 + 0.999217i \(0.512599\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −17068.0 −1.69125 −0.845626 0.533776i \(-0.820772\pi\)
−0.845626 + 0.533776i \(0.820772\pi\)
\(468\) 0 0
\(469\) 8282.72 0.815481
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −268.245 −0.0260760
\(474\) 0 0
\(475\) −12021.1 −1.16119
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 1517.99 0.144799 0.0723994 0.997376i \(-0.476934\pi\)
0.0723994 + 0.997376i \(0.476934\pi\)
\(480\) 0 0
\(481\) −6320.96 −0.599191
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3058.80 −0.286377
\(486\) 0 0
\(487\) −12737.3 −1.18518 −0.592591 0.805503i \(-0.701895\pi\)
−0.592591 + 0.805503i \(0.701895\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5647.70 0.519098 0.259549 0.965730i \(-0.416426\pi\)
0.259549 + 0.965730i \(0.416426\pi\)
\(492\) 0 0
\(493\) −1705.01 −0.155760
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −13299.6 −1.20033
\(498\) 0 0
\(499\) −10846.3 −0.973038 −0.486519 0.873670i \(-0.661734\pi\)
−0.486519 + 0.873670i \(0.661734\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −12345.7 −1.09437 −0.547186 0.837011i \(-0.684301\pi\)
−0.547186 + 0.837011i \(0.684301\pi\)
\(504\) 0 0
\(505\) −2615.23 −0.230448
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5895.75 0.513408 0.256704 0.966490i \(-0.417364\pi\)
0.256704 + 0.966490i \(0.417364\pi\)
\(510\) 0 0
\(511\) −5700.54 −0.493497
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4494.36 −0.384554
\(516\) 0 0
\(517\) 4618.80 0.392910
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5211.51 −0.438235 −0.219118 0.975698i \(-0.570318\pi\)
−0.219118 + 0.975698i \(0.570318\pi\)
\(522\) 0 0
\(523\) 9809.86 0.820182 0.410091 0.912045i \(-0.365497\pi\)
0.410091 + 0.912045i \(0.365497\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2269.32 0.187577
\(528\) 0 0
\(529\) −10717.4 −0.880860
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 16882.2 1.37195
\(534\) 0 0
\(535\) 2609.32 0.210861
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1966.09 −0.157116
\(540\) 0 0
\(541\) 8084.25 0.642456 0.321228 0.947002i \(-0.395904\pi\)
0.321228 + 0.947002i \(0.395904\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1043.48 −0.0820140
\(546\) 0 0
\(547\) 24067.1 1.88123 0.940617 0.339470i \(-0.110248\pi\)
0.940617 + 0.339470i \(0.110248\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2625.98 0.203032
\(552\) 0 0
\(553\) 5889.21 0.452866
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4582.37 −0.348584 −0.174292 0.984694i \(-0.555764\pi\)
−0.174292 + 0.984694i \(0.555764\pi\)
\(558\) 0 0
\(559\) 1300.42 0.0983934
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3095.31 −0.231709 −0.115854 0.993266i \(-0.536961\pi\)
−0.115854 + 0.993266i \(0.536961\pi\)
\(564\) 0 0
\(565\) −3198.04 −0.238129
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −20565.1 −1.51517 −0.757586 0.652735i \(-0.773621\pi\)
−0.757586 + 0.652735i \(0.773621\pi\)
\(570\) 0 0
\(571\) 1169.98 0.0857483 0.0428742 0.999080i \(-0.486349\pi\)
0.0428742 + 0.999080i \(0.486349\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3943.80 0.286031
\(576\) 0 0
\(577\) −13073.0 −0.943214 −0.471607 0.881809i \(-0.656326\pi\)
−0.471607 + 0.881809i \(0.656326\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −1201.58 −0.0858001
\(582\) 0 0
\(583\) 4392.63 0.312048
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −14547.8 −1.02292 −0.511459 0.859308i \(-0.670895\pi\)
−0.511459 + 0.859308i \(0.670895\pi\)
\(588\) 0 0
\(589\) −3495.11 −0.244505
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −19018.0 −1.31699 −0.658494 0.752586i \(-0.728806\pi\)
−0.658494 + 0.752586i \(0.728806\pi\)
\(594\) 0 0
\(595\) −4225.16 −0.291117
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11174.4 0.762227 0.381113 0.924528i \(-0.375541\pi\)
0.381113 + 0.924528i \(0.375541\pi\)
\(600\) 0 0
\(601\) 4588.54 0.311432 0.155716 0.987802i \(-0.450232\pi\)
0.155716 + 0.987802i \(0.450232\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5694.71 0.382682
\(606\) 0 0
\(607\) 11490.0 0.768309 0.384155 0.923269i \(-0.374493\pi\)
0.384155 + 0.923269i \(0.374493\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −22391.4 −1.48258
\(612\) 0 0
\(613\) −22966.9 −1.51326 −0.756628 0.653846i \(-0.773154\pi\)
−0.756628 + 0.653846i \(0.773154\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8496.63 0.554394 0.277197 0.960813i \(-0.410594\pi\)
0.277197 + 0.960813i \(0.410594\pi\)
\(618\) 0 0
\(619\) −14339.9 −0.931129 −0.465564 0.885014i \(-0.654149\pi\)
−0.465564 + 0.885014i \(0.654149\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 106.680 0.00686041
\(624\) 0 0
\(625\) 8052.72 0.515374
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −9803.34 −0.621439
\(630\) 0 0
\(631\) −17834.3 −1.12516 −0.562578 0.826744i \(-0.690190\pi\)
−0.562578 + 0.826744i \(0.690190\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4145.96 −0.259098
\(636\) 0 0
\(637\) 9531.35 0.592851
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −26346.0 −1.62341 −0.811705 0.584068i \(-0.801460\pi\)
−0.811705 + 0.584068i \(0.801460\pi\)
\(642\) 0 0
\(643\) 21565.1 1.32262 0.661309 0.750114i \(-0.270001\pi\)
0.661309 + 0.750114i \(0.270001\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4186.32 0.254376 0.127188 0.991879i \(-0.459405\pi\)
0.127188 + 0.991879i \(0.459405\pi\)
\(648\) 0 0
\(649\) 83.8794 0.00507328
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2805.14 −0.168107 −0.0840534 0.996461i \(-0.526787\pi\)
−0.0840534 + 0.996461i \(0.526787\pi\)
\(654\) 0 0
\(655\) −7658.21 −0.456841
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −19072.3 −1.12739 −0.563696 0.825982i \(-0.690621\pi\)
−0.563696 + 0.825982i \(0.690621\pi\)
\(660\) 0 0
\(661\) −24513.1 −1.44243 −0.721216 0.692710i \(-0.756417\pi\)
−0.721216 + 0.692710i \(0.756417\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6507.40 0.379468
\(666\) 0 0
\(667\) −861.513 −0.0500119
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −822.681 −0.0473312
\(672\) 0 0
\(673\) −3546.65 −0.203140 −0.101570 0.994828i \(-0.532387\pi\)
−0.101570 + 0.994828i \(0.532387\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −17875.0 −1.01476 −0.507380 0.861723i \(-0.669386\pi\)
−0.507380 + 0.861723i \(0.669386\pi\)
\(678\) 0 0
\(679\) −8008.90 −0.452656
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2857.23 0.160072 0.0800358 0.996792i \(-0.474497\pi\)
0.0800358 + 0.996792i \(0.474497\pi\)
\(684\) 0 0
\(685\) −12268.5 −0.684315
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −21294.9 −1.17746
\(690\) 0 0
\(691\) 15627.6 0.860349 0.430175 0.902746i \(-0.358452\pi\)
0.430175 + 0.902746i \(0.358452\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2934.75 0.160175
\(696\) 0 0
\(697\) 26183.1 1.42289
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −17562.6 −0.946264 −0.473132 0.880992i \(-0.656877\pi\)
−0.473132 + 0.880992i \(0.656877\pi\)
\(702\) 0 0
\(703\) 15098.7 0.810039
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −6847.50 −0.364253
\(708\) 0 0
\(709\) 20003.6 1.05959 0.529795 0.848126i \(-0.322269\pi\)
0.529795 + 0.848126i \(0.322269\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1146.65 0.0602277
\(714\) 0 0
\(715\) 2253.23 0.117855
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 25504.1 1.32287 0.661435 0.750002i \(-0.269948\pi\)
0.661435 + 0.750002i \(0.269948\pi\)
\(720\) 0 0
\(721\) −11767.7 −0.607837
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2343.87 −0.120068
\(726\) 0 0
\(727\) 23909.8 1.21976 0.609879 0.792494i \(-0.291218\pi\)
0.609879 + 0.792494i \(0.291218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2016.86 0.102047
\(732\) 0 0
\(733\) −8505.54 −0.428594 −0.214297 0.976769i \(-0.568746\pi\)
−0.214297 + 0.976769i \(0.568746\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6850.57 −0.342394
\(738\) 0 0
\(739\) −25802.5 −1.28438 −0.642192 0.766544i \(-0.721975\pi\)
−0.642192 + 0.766544i \(0.721975\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 27337.6 1.34982 0.674911 0.737899i \(-0.264182\pi\)
0.674911 + 0.737899i \(0.264182\pi\)
\(744\) 0 0
\(745\) −15763.5 −0.775208
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 6832.02 0.333293
\(750\) 0 0
\(751\) −3336.64 −0.162125 −0.0810623 0.996709i \(-0.525831\pi\)
−0.0810623 + 0.996709i \(0.525831\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8099.98 −0.390448
\(756\) 0 0
\(757\) 33149.5 1.59160 0.795798 0.605562i \(-0.207052\pi\)
0.795798 + 0.605562i \(0.207052\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 10996.8 0.523831 0.261915 0.965091i \(-0.415646\pi\)
0.261915 + 0.965091i \(0.415646\pi\)
\(762\) 0 0
\(763\) −2732.15 −0.129634
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −406.637 −0.0191432
\(768\) 0 0
\(769\) 33285.0 1.56084 0.780420 0.625255i \(-0.215005\pi\)
0.780420 + 0.625255i \(0.215005\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 7242.46 0.336990 0.168495 0.985703i \(-0.446109\pi\)
0.168495 + 0.985703i \(0.446109\pi\)
\(774\) 0 0
\(775\) 3119.63 0.144594
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −40326.0 −1.85472
\(780\) 0 0
\(781\) 11000.0 0.503981
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −10081.8 −0.458389
\(786\) 0 0
\(787\) −17414.3 −0.788758 −0.394379 0.918948i \(-0.629040\pi\)
−0.394379 + 0.918948i \(0.629040\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8373.50 −0.376394
\(792\) 0 0
\(793\) 3988.26 0.178597
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29132.9 −1.29478 −0.647391 0.762158i \(-0.724140\pi\)
−0.647391 + 0.762158i \(0.724140\pi\)
\(798\) 0 0
\(799\) −34727.4 −1.53763
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4714.88 0.207203
\(804\) 0 0
\(805\) −2134.90 −0.0934725
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 36440.1 1.58364 0.791820 0.610754i \(-0.209134\pi\)
0.791820 + 0.610754i \(0.209134\pi\)
\(810\) 0 0
\(811\) 18922.0 0.819286 0.409643 0.912246i \(-0.365653\pi\)
0.409643 + 0.912246i \(0.365653\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 10127.4 0.435271
\(816\) 0 0
\(817\) −3106.27 −0.133017
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 17910.7 0.761374 0.380687 0.924704i \(-0.375688\pi\)
0.380687 + 0.924704i \(0.375688\pi\)
\(822\) 0 0
\(823\) −17524.4 −0.742240 −0.371120 0.928585i \(-0.621026\pi\)
−0.371120 + 0.928585i \(0.621026\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −17643.9 −0.741885 −0.370943 0.928656i \(-0.620965\pi\)
−0.370943 + 0.928656i \(0.620965\pi\)
\(828\) 0 0
\(829\) 45178.6 1.89278 0.946391 0.323023i \(-0.104699\pi\)
0.946391 + 0.323023i \(0.104699\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 14782.4 0.614863
\(834\) 0 0
\(835\) −8890.03 −0.368446
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 26776.0 1.10180 0.550901 0.834571i \(-0.314284\pi\)
0.550901 + 0.834571i \(0.314284\pi\)
\(840\) 0 0
\(841\) −23877.0 −0.979006
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −756.291 −0.0307896
\(846\) 0 0
\(847\) 14910.6 0.604879
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4953.46 −0.199533
\(852\) 0 0
\(853\) −7911.75 −0.317577 −0.158789 0.987313i \(-0.550759\pi\)
−0.158789 + 0.987313i \(0.550759\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1848.30 −0.0736717 −0.0368359 0.999321i \(-0.511728\pi\)
−0.0368359 + 0.999321i \(0.511728\pi\)
\(858\) 0 0
\(859\) 18854.2 0.748890 0.374445 0.927249i \(-0.377833\pi\)
0.374445 + 0.927249i \(0.377833\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2086.03 0.0822821 0.0411410 0.999153i \(-0.486901\pi\)
0.0411410 + 0.999153i \(0.486901\pi\)
\(864\) 0 0
\(865\) 14669.1 0.576605
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −4870.92 −0.190143
\(870\) 0 0
\(871\) 33210.7 1.29197
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −12817.5 −0.495211
\(876\) 0 0
\(877\) −24777.2 −0.954010 −0.477005 0.878901i \(-0.658278\pi\)
−0.477005 + 0.878901i \(0.658278\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −3741.26 −0.143072 −0.0715359 0.997438i \(-0.522790\pi\)
−0.0715359 + 0.997438i \(0.522790\pi\)
\(882\) 0 0
\(883\) −14131.6 −0.538580 −0.269290 0.963059i \(-0.586789\pi\)
−0.269290 + 0.963059i \(0.586789\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 26622.5 1.00778 0.503888 0.863769i \(-0.331903\pi\)
0.503888 + 0.863769i \(0.331903\pi\)
\(888\) 0 0
\(889\) −10855.4 −0.409539
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 53485.6 2.00429
\(894\) 0 0
\(895\) −6334.34 −0.236574
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −681.475 −0.0252820
\(900\) 0 0
\(901\) −33026.9 −1.22118
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18286.2 0.671661
\(906\) 0 0
\(907\) −52619.4 −1.92635 −0.963174 0.268878i \(-0.913347\pi\)
−0.963174 + 0.268878i \(0.913347\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 10180.4 0.370245 0.185122 0.982715i \(-0.440732\pi\)
0.185122 + 0.982715i \(0.440732\pi\)
\(912\) 0 0
\(913\) 993.816 0.0360246
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −20051.6 −0.722097
\(918\) 0 0
\(919\) 45618.2 1.63744 0.818718 0.574195i \(-0.194685\pi\)
0.818718 + 0.574195i \(0.194685\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −53326.5 −1.90169
\(924\) 0 0
\(925\) −13476.6 −0.479037
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 13200.4 0.466190 0.233095 0.972454i \(-0.425115\pi\)
0.233095 + 0.972454i \(0.425115\pi\)
\(930\) 0 0
\(931\) −22767.2 −0.801468
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3494.59 0.122230
\(936\) 0 0
\(937\) −13468.1 −0.469565 −0.234783 0.972048i \(-0.575438\pi\)
−0.234783 + 0.972048i \(0.575438\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −13884.1 −0.480987 −0.240493 0.970651i \(-0.577309\pi\)
−0.240493 + 0.970651i \(0.577309\pi\)
\(942\) 0 0
\(943\) 13229.9 0.456865
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5276.64 0.181064 0.0905320 0.995894i \(-0.471143\pi\)
0.0905320 + 0.995894i \(0.471143\pi\)
\(948\) 0 0
\(949\) −22857.1 −0.781849
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 26131.4 0.888225 0.444112 0.895971i \(-0.353519\pi\)
0.444112 + 0.895971i \(0.353519\pi\)
\(954\) 0 0
\(955\) 11118.5 0.376740
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −32122.9 −1.08165
\(960\) 0 0
\(961\) −28884.0 −0.969554
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 6178.33 0.206101
\(966\) 0 0
\(967\) 3997.53 0.132939 0.0664695 0.997788i \(-0.478827\pi\)
0.0664695 + 0.997788i \(0.478827\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 41785.1 1.38100 0.690499 0.723334i \(-0.257391\pi\)
0.690499 + 0.723334i \(0.257391\pi\)
\(972\) 0 0
\(973\) 7684.12 0.253177
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12245.1 −0.400978 −0.200489 0.979696i \(-0.564253\pi\)
−0.200489 + 0.979696i \(0.564253\pi\)
\(978\) 0 0
\(979\) −88.2340 −0.00288046
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 43783.1 1.42061 0.710307 0.703892i \(-0.248556\pi\)
0.710307 + 0.703892i \(0.248556\pi\)
\(984\) 0 0
\(985\) −12175.0 −0.393835
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1019.08 0.0327654
\(990\) 0 0
\(991\) −5178.38 −0.165991 −0.0829953 0.996550i \(-0.526449\pi\)
−0.0829953 + 0.996550i \(0.526449\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 11464.4 0.365274
\(996\) 0 0
\(997\) 25720.9 0.817041 0.408520 0.912749i \(-0.366045\pi\)
0.408520 + 0.912749i \(0.366045\pi\)
\(998\) 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 1296.4.a.i.1.2 2
3.2 odd 2 1296.4.a.u.1.1 2
4.3 odd 2 81.4.a.a.1.1 2
9.2 odd 6 144.4.i.c.49.1 4
9.4 even 3 432.4.i.c.289.1 4
9.5 odd 6 144.4.i.c.97.1 4
9.7 even 3 432.4.i.c.145.1 4
12.11 even 2 81.4.a.d.1.2 2
20.19 odd 2 2025.4.a.n.1.2 2
36.7 odd 6 27.4.c.a.10.2 4
36.11 even 6 9.4.c.a.4.1 4
36.23 even 6 9.4.c.a.7.1 yes 4
36.31 odd 6 27.4.c.a.19.2 4
60.59 even 2 2025.4.a.g.1.1 2
180.23 odd 12 225.4.k.b.124.4 8
180.47 odd 12 225.4.k.b.49.4 8
180.59 even 6 225.4.e.b.151.2 4
180.83 odd 12 225.4.k.b.49.1 8
180.119 even 6 225.4.e.b.76.2 4
180.167 odd 12 225.4.k.b.124.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9.4.c.a.4.1 4 36.11 even 6
9.4.c.a.7.1 yes 4 36.23 even 6
27.4.c.a.10.2 4 36.7 odd 6
27.4.c.a.19.2 4 36.31 odd 6
81.4.a.a.1.1 2 4.3 odd 2
81.4.a.d.1.2 2 12.11 even 2
144.4.i.c.49.1 4 9.2 odd 6
144.4.i.c.97.1 4 9.5 odd 6
225.4.e.b.76.2 4 180.119 even 6
225.4.e.b.151.2 4 180.59 even 6
225.4.k.b.49.1 8 180.83 odd 12
225.4.k.b.49.4 8 180.47 odd 12
225.4.k.b.124.1 8 180.167 odd 12
225.4.k.b.124.4 8 180.23 odd 12
432.4.i.c.145.1 4 9.7 even 3
432.4.i.c.289.1 4 9.4 even 3
1296.4.a.i.1.2 2 1.1 even 1 trivial
1296.4.a.u.1.1 2 3.2 odd 2
2025.4.a.g.1.1 2 60.59 even 2
2025.4.a.n.1.2 2 20.19 odd 2