Properties

Label 2200.4.a.b.1.1
Level $2200$
Weight $4$
Character 2200.1
Self dual yes
Analytic conductor $129.804$
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] = [2200,4,Mod(1,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, 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("2200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2200.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: \(129.804202013\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 440)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.00000 q^{3} +32.0000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-6.00000 q^{3} +32.0000 q^{7} +9.00000 q^{9} +11.0000 q^{11} +48.0000 q^{13} +36.0000 q^{17} -44.0000 q^{19} -192.000 q^{21} -58.0000 q^{23} +108.000 q^{27} -278.000 q^{29} -112.000 q^{31} -66.0000 q^{33} -194.000 q^{37} -288.000 q^{39} -314.000 q^{41} -396.000 q^{43} +410.000 q^{47} +681.000 q^{49} -216.000 q^{51} -170.000 q^{53} +264.000 q^{57} +404.000 q^{59} +250.000 q^{61} +288.000 q^{63} +26.0000 q^{67} +348.000 q^{69} -468.000 q^{71} +164.000 q^{73} +352.000 q^{77} -664.000 q^{79} -891.000 q^{81} -1348.00 q^{83} +1668.00 q^{87} +534.000 q^{89} +1536.00 q^{91} +672.000 q^{93} +1498.00 q^{97} +99.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −6.00000 −1.15470 −0.577350 0.816497i \(-0.695913\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 32.0000 1.72784 0.863919 0.503631i \(-0.168003\pi\)
0.863919 + 0.503631i \(0.168003\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 0 0
\(13\) 48.0000 1.02406 0.512031 0.858967i \(-0.328893\pi\)
0.512031 + 0.858967i \(0.328893\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 36.0000 0.513605 0.256802 0.966464i \(-0.417331\pi\)
0.256802 + 0.966464i \(0.417331\pi\)
\(18\) 0 0
\(19\) −44.0000 −0.531279 −0.265639 0.964072i \(-0.585583\pi\)
−0.265639 + 0.964072i \(0.585583\pi\)
\(20\) 0 0
\(21\) −192.000 −1.99513
\(22\) 0 0
\(23\) −58.0000 −0.525819 −0.262909 0.964821i \(-0.584682\pi\)
−0.262909 + 0.964821i \(0.584682\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 108.000 0.769800
\(28\) 0 0
\(29\) −278.000 −1.78011 −0.890057 0.455849i \(-0.849336\pi\)
−0.890057 + 0.455849i \(0.849336\pi\)
\(30\) 0 0
\(31\) −112.000 −0.648897 −0.324448 0.945903i \(-0.605179\pi\)
−0.324448 + 0.945903i \(0.605179\pi\)
\(32\) 0 0
\(33\) −66.0000 −0.348155
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −194.000 −0.861984 −0.430992 0.902356i \(-0.641836\pi\)
−0.430992 + 0.902356i \(0.641836\pi\)
\(38\) 0 0
\(39\) −288.000 −1.18248
\(40\) 0 0
\(41\) −314.000 −1.19606 −0.598031 0.801473i \(-0.704050\pi\)
−0.598031 + 0.801473i \(0.704050\pi\)
\(42\) 0 0
\(43\) −396.000 −1.40441 −0.702203 0.711977i \(-0.747800\pi\)
−0.702203 + 0.711977i \(0.747800\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 410.000 1.27244 0.636220 0.771508i \(-0.280497\pi\)
0.636220 + 0.771508i \(0.280497\pi\)
\(48\) 0 0
\(49\) 681.000 1.98542
\(50\) 0 0
\(51\) −216.000 −0.593060
\(52\) 0 0
\(53\) −170.000 −0.440590 −0.220295 0.975433i \(-0.570702\pi\)
−0.220295 + 0.975433i \(0.570702\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 264.000 0.613468
\(58\) 0 0
\(59\) 404.000 0.891463 0.445732 0.895167i \(-0.352944\pi\)
0.445732 + 0.895167i \(0.352944\pi\)
\(60\) 0 0
\(61\) 250.000 0.524741 0.262371 0.964967i \(-0.415496\pi\)
0.262371 + 0.964967i \(0.415496\pi\)
\(62\) 0 0
\(63\) 288.000 0.575946
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 26.0000 0.0474090 0.0237045 0.999719i \(-0.492454\pi\)
0.0237045 + 0.999719i \(0.492454\pi\)
\(68\) 0 0
\(69\) 348.000 0.607163
\(70\) 0 0
\(71\) −468.000 −0.782273 −0.391136 0.920333i \(-0.627918\pi\)
−0.391136 + 0.920333i \(0.627918\pi\)
\(72\) 0 0
\(73\) 164.000 0.262942 0.131471 0.991320i \(-0.458030\pi\)
0.131471 + 0.991320i \(0.458030\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 352.000 0.520963
\(78\) 0 0
\(79\) −664.000 −0.945644 −0.472822 0.881158i \(-0.656765\pi\)
−0.472822 + 0.881158i \(0.656765\pi\)
\(80\) 0 0
\(81\) −891.000 −1.22222
\(82\) 0 0
\(83\) −1348.00 −1.78268 −0.891339 0.453338i \(-0.850233\pi\)
−0.891339 + 0.453338i \(0.850233\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1668.00 2.05550
\(88\) 0 0
\(89\) 534.000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 1536.00 1.76941
\(92\) 0 0
\(93\) 672.000 0.749281
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1498.00 1.56803 0.784015 0.620742i \(-0.213169\pi\)
0.784015 + 0.620742i \(0.213169\pi\)
\(98\) 0 0
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) −1930.00 −1.90141 −0.950704 0.310100i \(-0.899637\pi\)
−0.950704 + 0.310100i \(0.899637\pi\)
\(102\) 0 0
\(103\) −566.000 −0.541453 −0.270726 0.962656i \(-0.587264\pi\)
−0.270726 + 0.962656i \(0.587264\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 136.000 0.122875 0.0614375 0.998111i \(-0.480432\pi\)
0.0614375 + 0.998111i \(0.480432\pi\)
\(108\) 0 0
\(109\) −506.000 −0.444642 −0.222321 0.974973i \(-0.571363\pi\)
−0.222321 + 0.974973i \(0.571363\pi\)
\(110\) 0 0
\(111\) 1164.00 0.995333
\(112\) 0 0
\(113\) 2098.00 1.74658 0.873289 0.487203i \(-0.161983\pi\)
0.873289 + 0.487203i \(0.161983\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 432.000 0.341354
\(118\) 0 0
\(119\) 1152.00 0.887426
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 0 0
\(123\) 1884.00 1.38109
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2476.00 −1.73000 −0.864998 0.501775i \(-0.832680\pi\)
−0.864998 + 0.501775i \(0.832680\pi\)
\(128\) 0 0
\(129\) 2376.00 1.62167
\(130\) 0 0
\(131\) 2620.00 1.74741 0.873704 0.486458i \(-0.161711\pi\)
0.873704 + 0.486458i \(0.161711\pi\)
\(132\) 0 0
\(133\) −1408.00 −0.917963
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1854.00 −1.15619 −0.578095 0.815970i \(-0.696204\pi\)
−0.578095 + 0.815970i \(0.696204\pi\)
\(138\) 0 0
\(139\) 740.000 0.451554 0.225777 0.974179i \(-0.427508\pi\)
0.225777 + 0.974179i \(0.427508\pi\)
\(140\) 0 0
\(141\) −2460.00 −1.46929
\(142\) 0 0
\(143\) 528.000 0.308766
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −4086.00 −2.29257
\(148\) 0 0
\(149\) −786.000 −0.432159 −0.216079 0.976376i \(-0.569327\pi\)
−0.216079 + 0.976376i \(0.569327\pi\)
\(150\) 0 0
\(151\) −1968.00 −1.06062 −0.530310 0.847804i \(-0.677924\pi\)
−0.530310 + 0.847804i \(0.677924\pi\)
\(152\) 0 0
\(153\) 324.000 0.171202
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1970.00 1.00142 0.500711 0.865615i \(-0.333072\pi\)
0.500711 + 0.865615i \(0.333072\pi\)
\(158\) 0 0
\(159\) 1020.00 0.508750
\(160\) 0 0
\(161\) −1856.00 −0.908530
\(162\) 0 0
\(163\) 10.0000 0.00480528 0.00240264 0.999997i \(-0.499235\pi\)
0.00240264 + 0.999997i \(0.499235\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 980.000 0.454100 0.227050 0.973883i \(-0.427092\pi\)
0.227050 + 0.973883i \(0.427092\pi\)
\(168\) 0 0
\(169\) 107.000 0.0487028
\(170\) 0 0
\(171\) −396.000 −0.177093
\(172\) 0 0
\(173\) −1116.00 −0.490450 −0.245225 0.969466i \(-0.578862\pi\)
−0.245225 + 0.969466i \(0.578862\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −2424.00 −1.02937
\(178\) 0 0
\(179\) −1664.00 −0.694822 −0.347411 0.937713i \(-0.612939\pi\)
−0.347411 + 0.937713i \(0.612939\pi\)
\(180\) 0 0
\(181\) 2338.00 0.960122 0.480061 0.877235i \(-0.340614\pi\)
0.480061 + 0.877235i \(0.340614\pi\)
\(182\) 0 0
\(183\) −1500.00 −0.605919
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 396.000 0.154858
\(188\) 0 0
\(189\) 3456.00 1.33009
\(190\) 0 0
\(191\) 1680.00 0.636443 0.318221 0.948016i \(-0.396914\pi\)
0.318221 + 0.948016i \(0.396914\pi\)
\(192\) 0 0
\(193\) 3932.00 1.46648 0.733242 0.679967i \(-0.238006\pi\)
0.733242 + 0.679967i \(0.238006\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1452.00 −0.525131 −0.262565 0.964914i \(-0.584569\pi\)
−0.262565 + 0.964914i \(0.584569\pi\)
\(198\) 0 0
\(199\) 2336.00 0.832134 0.416067 0.909334i \(-0.363408\pi\)
0.416067 + 0.909334i \(0.363408\pi\)
\(200\) 0 0
\(201\) −156.000 −0.0547432
\(202\) 0 0
\(203\) −8896.00 −3.07575
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −522.000 −0.175273
\(208\) 0 0
\(209\) −484.000 −0.160187
\(210\) 0 0
\(211\) −1996.00 −0.651234 −0.325617 0.945502i \(-0.605572\pi\)
−0.325617 + 0.945502i \(0.605572\pi\)
\(212\) 0 0
\(213\) 2808.00 0.903291
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −3584.00 −1.12119
\(218\) 0 0
\(219\) −984.000 −0.303619
\(220\) 0 0
\(221\) 1728.00 0.525963
\(222\) 0 0
\(223\) −2210.00 −0.663644 −0.331822 0.943342i \(-0.607663\pi\)
−0.331822 + 0.943342i \(0.607663\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5484.00 −1.60346 −0.801731 0.597685i \(-0.796087\pi\)
−0.801731 + 0.597685i \(0.796087\pi\)
\(228\) 0 0
\(229\) 4566.00 1.31760 0.658799 0.752319i \(-0.271065\pi\)
0.658799 + 0.752319i \(0.271065\pi\)
\(230\) 0 0
\(231\) −2112.00 −0.601556
\(232\) 0 0
\(233\) −4712.00 −1.32486 −0.662432 0.749122i \(-0.730476\pi\)
−0.662432 + 0.749122i \(0.730476\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 3984.00 1.09194
\(238\) 0 0
\(239\) 496.000 0.134241 0.0671204 0.997745i \(-0.478619\pi\)
0.0671204 + 0.997745i \(0.478619\pi\)
\(240\) 0 0
\(241\) 4806.00 1.28457 0.642286 0.766465i \(-0.277986\pi\)
0.642286 + 0.766465i \(0.277986\pi\)
\(242\) 0 0
\(243\) 2430.00 0.641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2112.00 −0.544062
\(248\) 0 0
\(249\) 8088.00 2.05846
\(250\) 0 0
\(251\) −1288.00 −0.323896 −0.161948 0.986799i \(-0.551778\pi\)
−0.161948 + 0.986799i \(0.551778\pi\)
\(252\) 0 0
\(253\) −638.000 −0.158540
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 238.000 0.0577667 0.0288833 0.999583i \(-0.490805\pi\)
0.0288833 + 0.999583i \(0.490805\pi\)
\(258\) 0 0
\(259\) −6208.00 −1.48937
\(260\) 0 0
\(261\) −2502.00 −0.593371
\(262\) 0 0
\(263\) −428.000 −0.100348 −0.0501742 0.998740i \(-0.515978\pi\)
−0.0501742 + 0.998740i \(0.515978\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −3204.00 −0.734388
\(268\) 0 0
\(269\) −5686.00 −1.28878 −0.644389 0.764697i \(-0.722888\pi\)
−0.644389 + 0.764697i \(0.722888\pi\)
\(270\) 0 0
\(271\) −8184.00 −1.83447 −0.917237 0.398341i \(-0.869586\pi\)
−0.917237 + 0.398341i \(0.869586\pi\)
\(272\) 0 0
\(273\) −9216.00 −2.04314
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2648.00 0.574379 0.287189 0.957874i \(-0.407279\pi\)
0.287189 + 0.957874i \(0.407279\pi\)
\(278\) 0 0
\(279\) −1008.00 −0.216299
\(280\) 0 0
\(281\) −6018.00 −1.27759 −0.638797 0.769376i \(-0.720568\pi\)
−0.638797 + 0.769376i \(0.720568\pi\)
\(282\) 0 0
\(283\) −4444.00 −0.933457 −0.466729 0.884401i \(-0.654568\pi\)
−0.466729 + 0.884401i \(0.654568\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −10048.0 −2.06660
\(288\) 0 0
\(289\) −3617.00 −0.736210
\(290\) 0 0
\(291\) −8988.00 −1.81060
\(292\) 0 0
\(293\) 3312.00 0.660372 0.330186 0.943916i \(-0.392888\pi\)
0.330186 + 0.943916i \(0.392888\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1188.00 0.232104
\(298\) 0 0
\(299\) −2784.00 −0.538471
\(300\) 0 0
\(301\) −12672.0 −2.42658
\(302\) 0 0
\(303\) 11580.0 2.19556
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4496.00 −0.835832 −0.417916 0.908486i \(-0.637239\pi\)
−0.417916 + 0.908486i \(0.637239\pi\)
\(308\) 0 0
\(309\) 3396.00 0.625216
\(310\) 0 0
\(311\) −5028.00 −0.916758 −0.458379 0.888757i \(-0.651570\pi\)
−0.458379 + 0.888757i \(0.651570\pi\)
\(312\) 0 0
\(313\) −2250.00 −0.406318 −0.203159 0.979146i \(-0.565121\pi\)
−0.203159 + 0.979146i \(0.565121\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 834.000 0.147767 0.0738834 0.997267i \(-0.476461\pi\)
0.0738834 + 0.997267i \(0.476461\pi\)
\(318\) 0 0
\(319\) −3058.00 −0.536725
\(320\) 0 0
\(321\) −816.000 −0.141884
\(322\) 0 0
\(323\) −1584.00 −0.272867
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 3036.00 0.513429
\(328\) 0 0
\(329\) 13120.0 2.19857
\(330\) 0 0
\(331\) −7244.00 −1.20292 −0.601460 0.798903i \(-0.705414\pi\)
−0.601460 + 0.798903i \(0.705414\pi\)
\(332\) 0 0
\(333\) −1746.00 −0.287328
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4516.00 0.729977 0.364988 0.931012i \(-0.381073\pi\)
0.364988 + 0.931012i \(0.381073\pi\)
\(338\) 0 0
\(339\) −12588.0 −2.01677
\(340\) 0 0
\(341\) −1232.00 −0.195650
\(342\) 0 0
\(343\) 10816.0 1.70265
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −10720.0 −1.65844 −0.829222 0.558920i \(-0.811216\pi\)
−0.829222 + 0.558920i \(0.811216\pi\)
\(348\) 0 0
\(349\) −3062.00 −0.469642 −0.234821 0.972039i \(-0.575450\pi\)
−0.234821 + 0.972039i \(0.575450\pi\)
\(350\) 0 0
\(351\) 5184.00 0.788323
\(352\) 0 0
\(353\) 11166.0 1.68359 0.841794 0.539800i \(-0.181500\pi\)
0.841794 + 0.539800i \(0.181500\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −6912.00 −1.02471
\(358\) 0 0
\(359\) −11184.0 −1.64420 −0.822102 0.569341i \(-0.807198\pi\)
−0.822102 + 0.569341i \(0.807198\pi\)
\(360\) 0 0
\(361\) −4923.00 −0.717743
\(362\) 0 0
\(363\) −726.000 −0.104973
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1350.00 −0.192015 −0.0960074 0.995381i \(-0.530607\pi\)
−0.0960074 + 0.995381i \(0.530607\pi\)
\(368\) 0 0
\(369\) −2826.00 −0.398687
\(370\) 0 0
\(371\) −5440.00 −0.761269
\(372\) 0 0
\(373\) 8068.00 1.11996 0.559980 0.828506i \(-0.310809\pi\)
0.559980 + 0.828506i \(0.310809\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −13344.0 −1.82295
\(378\) 0 0
\(379\) −380.000 −0.0515021 −0.0257510 0.999668i \(-0.508198\pi\)
−0.0257510 + 0.999668i \(0.508198\pi\)
\(380\) 0 0
\(381\) 14856.0 1.99763
\(382\) 0 0
\(383\) −2230.00 −0.297514 −0.148757 0.988874i \(-0.547527\pi\)
−0.148757 + 0.988874i \(0.547527\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3564.00 −0.468135
\(388\) 0 0
\(389\) −3974.00 −0.517969 −0.258984 0.965882i \(-0.583388\pi\)
−0.258984 + 0.965882i \(0.583388\pi\)
\(390\) 0 0
\(391\) −2088.00 −0.270063
\(392\) 0 0
\(393\) −15720.0 −2.01773
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −5470.00 −0.691515 −0.345757 0.938324i \(-0.612378\pi\)
−0.345757 + 0.938324i \(0.612378\pi\)
\(398\) 0 0
\(399\) 8448.00 1.05997
\(400\) 0 0
\(401\) −834.000 −0.103860 −0.0519301 0.998651i \(-0.516537\pi\)
−0.0519301 + 0.998651i \(0.516537\pi\)
\(402\) 0 0
\(403\) −5376.00 −0.664510
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2134.00 −0.259898
\(408\) 0 0
\(409\) 3610.00 0.436438 0.218219 0.975900i \(-0.429975\pi\)
0.218219 + 0.975900i \(0.429975\pi\)
\(410\) 0 0
\(411\) 11124.0 1.33505
\(412\) 0 0
\(413\) 12928.0 1.54030
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −4440.00 −0.521409
\(418\) 0 0
\(419\) −5608.00 −0.653863 −0.326932 0.945048i \(-0.606015\pi\)
−0.326932 + 0.945048i \(0.606015\pi\)
\(420\) 0 0
\(421\) −6450.00 −0.746684 −0.373342 0.927694i \(-0.621788\pi\)
−0.373342 + 0.927694i \(0.621788\pi\)
\(422\) 0 0
\(423\) 3690.00 0.424146
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8000.00 0.906668
\(428\) 0 0
\(429\) −3168.00 −0.356533
\(430\) 0 0
\(431\) 112.000 0.0125171 0.00625853 0.999980i \(-0.498008\pi\)
0.00625853 + 0.999980i \(0.498008\pi\)
\(432\) 0 0
\(433\) 3194.00 0.354489 0.177245 0.984167i \(-0.443282\pi\)
0.177245 + 0.984167i \(0.443282\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2552.00 0.279356
\(438\) 0 0
\(439\) 16816.0 1.82821 0.914105 0.405478i \(-0.132895\pi\)
0.914105 + 0.405478i \(0.132895\pi\)
\(440\) 0 0
\(441\) 6129.00 0.661808
\(442\) 0 0
\(443\) −198.000 −0.0212354 −0.0106177 0.999944i \(-0.503380\pi\)
−0.0106177 + 0.999944i \(0.503380\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 4716.00 0.499014
\(448\) 0 0
\(449\) 6498.00 0.682983 0.341492 0.939885i \(-0.389068\pi\)
0.341492 + 0.939885i \(0.389068\pi\)
\(450\) 0 0
\(451\) −3454.00 −0.360626
\(452\) 0 0
\(453\) 11808.0 1.22470
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 9720.00 0.994929 0.497464 0.867484i \(-0.334265\pi\)
0.497464 + 0.867484i \(0.334265\pi\)
\(458\) 0 0
\(459\) 3888.00 0.395373
\(460\) 0 0
\(461\) −4666.00 −0.471404 −0.235702 0.971825i \(-0.575739\pi\)
−0.235702 + 0.971825i \(0.575739\pi\)
\(462\) 0 0
\(463\) 15942.0 1.60019 0.800095 0.599874i \(-0.204783\pi\)
0.800095 + 0.599874i \(0.204783\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −16254.0 −1.61059 −0.805295 0.592874i \(-0.797993\pi\)
−0.805295 + 0.592874i \(0.797993\pi\)
\(468\) 0 0
\(469\) 832.000 0.0819151
\(470\) 0 0
\(471\) −11820.0 −1.15634
\(472\) 0 0
\(473\) −4356.00 −0.423444
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1530.00 −0.146863
\(478\) 0 0
\(479\) −15424.0 −1.47127 −0.735637 0.677376i \(-0.763117\pi\)
−0.735637 + 0.677376i \(0.763117\pi\)
\(480\) 0 0
\(481\) −9312.00 −0.882725
\(482\) 0 0
\(483\) 11136.0 1.04908
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 8322.00 0.774345 0.387172 0.922007i \(-0.373452\pi\)
0.387172 + 0.922007i \(0.373452\pi\)
\(488\) 0 0
\(489\) −60.0000 −0.00554866
\(490\) 0 0
\(491\) −13012.0 −1.19597 −0.597987 0.801506i \(-0.704033\pi\)
−0.597987 + 0.801506i \(0.704033\pi\)
\(492\) 0 0
\(493\) −10008.0 −0.914275
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −14976.0 −1.35164
\(498\) 0 0
\(499\) 13972.0 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) −5880.00 −0.524349
\(502\) 0 0
\(503\) −4896.00 −0.434000 −0.217000 0.976172i \(-0.569627\pi\)
−0.217000 + 0.976172i \(0.569627\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −642.000 −0.0562371
\(508\) 0 0
\(509\) 16574.0 1.44328 0.721640 0.692268i \(-0.243389\pi\)
0.721640 + 0.692268i \(0.243389\pi\)
\(510\) 0 0
\(511\) 5248.00 0.454321
\(512\) 0 0
\(513\) −4752.00 −0.408978
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 4510.00 0.383655
\(518\) 0 0
\(519\) 6696.00 0.566323
\(520\) 0 0
\(521\) 18070.0 1.51950 0.759752 0.650214i \(-0.225321\pi\)
0.759752 + 0.650214i \(0.225321\pi\)
\(522\) 0 0
\(523\) 5128.00 0.428741 0.214371 0.976752i \(-0.431230\pi\)
0.214371 + 0.976752i \(0.431230\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −4032.00 −0.333276
\(528\) 0 0
\(529\) −8803.00 −0.723514
\(530\) 0 0
\(531\) 3636.00 0.297154
\(532\) 0 0
\(533\) −15072.0 −1.22484
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 9984.00 0.802312
\(538\) 0 0
\(539\) 7491.00 0.598627
\(540\) 0 0
\(541\) 14670.0 1.16583 0.582914 0.812534i \(-0.301913\pi\)
0.582914 + 0.812534i \(0.301913\pi\)
\(542\) 0 0
\(543\) −14028.0 −1.10865
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −7956.00 −0.621890 −0.310945 0.950428i \(-0.600646\pi\)
−0.310945 + 0.950428i \(0.600646\pi\)
\(548\) 0 0
\(549\) 2250.00 0.174914
\(550\) 0 0
\(551\) 12232.0 0.945736
\(552\) 0 0
\(553\) −21248.0 −1.63392
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11244.0 0.855339 0.427669 0.903935i \(-0.359335\pi\)
0.427669 + 0.903935i \(0.359335\pi\)
\(558\) 0 0
\(559\) −19008.0 −1.43820
\(560\) 0 0
\(561\) −2376.00 −0.178814
\(562\) 0 0
\(563\) 17688.0 1.32409 0.662043 0.749466i \(-0.269690\pi\)
0.662043 + 0.749466i \(0.269690\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −28512.0 −2.11180
\(568\) 0 0
\(569\) −6750.00 −0.497319 −0.248660 0.968591i \(-0.579990\pi\)
−0.248660 + 0.968591i \(0.579990\pi\)
\(570\) 0 0
\(571\) −26812.0 −1.96506 −0.982528 0.186113i \(-0.940411\pi\)
−0.982528 + 0.186113i \(0.940411\pi\)
\(572\) 0 0
\(573\) −10080.0 −0.734901
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 20762.0 1.49798 0.748989 0.662582i \(-0.230540\pi\)
0.748989 + 0.662582i \(0.230540\pi\)
\(578\) 0 0
\(579\) −23592.0 −1.69335
\(580\) 0 0
\(581\) −43136.0 −3.08018
\(582\) 0 0
\(583\) −1870.00 −0.132843
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22314.0 1.56899 0.784495 0.620135i \(-0.212922\pi\)
0.784495 + 0.620135i \(0.212922\pi\)
\(588\) 0 0
\(589\) 4928.00 0.344745
\(590\) 0 0
\(591\) 8712.00 0.606369
\(592\) 0 0
\(593\) −16676.0 −1.15481 −0.577404 0.816459i \(-0.695934\pi\)
−0.577404 + 0.816459i \(0.695934\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −14016.0 −0.960865
\(598\) 0 0
\(599\) 10380.0 0.708039 0.354019 0.935238i \(-0.384815\pi\)
0.354019 + 0.935238i \(0.384815\pi\)
\(600\) 0 0
\(601\) 26390.0 1.79113 0.895566 0.444928i \(-0.146771\pi\)
0.895566 + 0.444928i \(0.146771\pi\)
\(602\) 0 0
\(603\) 234.000 0.0158030
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 16284.0 1.08888 0.544438 0.838801i \(-0.316743\pi\)
0.544438 + 0.838801i \(0.316743\pi\)
\(608\) 0 0
\(609\) 53376.0 3.55157
\(610\) 0 0
\(611\) 19680.0 1.30306
\(612\) 0 0
\(613\) −25772.0 −1.69808 −0.849039 0.528331i \(-0.822818\pi\)
−0.849039 + 0.528331i \(0.822818\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3462.00 −0.225891 −0.112946 0.993601i \(-0.536029\pi\)
−0.112946 + 0.993601i \(0.536029\pi\)
\(618\) 0 0
\(619\) −10104.0 −0.656081 −0.328040 0.944664i \(-0.606388\pi\)
−0.328040 + 0.944664i \(0.606388\pi\)
\(620\) 0 0
\(621\) −6264.00 −0.404776
\(622\) 0 0
\(623\) 17088.0 1.09890
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 2904.00 0.184967
\(628\) 0 0
\(629\) −6984.00 −0.442719
\(630\) 0 0
\(631\) 21948.0 1.38468 0.692342 0.721569i \(-0.256579\pi\)
0.692342 + 0.721569i \(0.256579\pi\)
\(632\) 0 0
\(633\) 11976.0 0.751980
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 32688.0 2.03320
\(638\) 0 0
\(639\) −4212.00 −0.260758
\(640\) 0 0
\(641\) 11538.0 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) 7706.00 0.472620 0.236310 0.971678i \(-0.424062\pi\)
0.236310 + 0.971678i \(0.424062\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22566.0 1.37119 0.685596 0.727982i \(-0.259542\pi\)
0.685596 + 0.727982i \(0.259542\pi\)
\(648\) 0 0
\(649\) 4444.00 0.268786
\(650\) 0 0
\(651\) 21504.0 1.29464
\(652\) 0 0
\(653\) −9938.00 −0.595565 −0.297783 0.954634i \(-0.596247\pi\)
−0.297783 + 0.954634i \(0.596247\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1476.00 0.0876473
\(658\) 0 0
\(659\) −8444.00 −0.499137 −0.249569 0.968357i \(-0.580289\pi\)
−0.249569 + 0.968357i \(0.580289\pi\)
\(660\) 0 0
\(661\) −18478.0 −1.08731 −0.543654 0.839309i \(-0.682960\pi\)
−0.543654 + 0.839309i \(0.682960\pi\)
\(662\) 0 0
\(663\) −10368.0 −0.607330
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 16124.0 0.936018
\(668\) 0 0
\(669\) 13260.0 0.766310
\(670\) 0 0
\(671\) 2750.00 0.158215
\(672\) 0 0
\(673\) 3772.00 0.216047 0.108024 0.994148i \(-0.465548\pi\)
0.108024 + 0.994148i \(0.465548\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 22296.0 1.26574 0.632869 0.774259i \(-0.281877\pi\)
0.632869 + 0.774259i \(0.281877\pi\)
\(678\) 0 0
\(679\) 47936.0 2.70930
\(680\) 0 0
\(681\) 32904.0 1.85152
\(682\) 0 0
\(683\) −14930.0 −0.836428 −0.418214 0.908348i \(-0.637344\pi\)
−0.418214 + 0.908348i \(0.637344\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −27396.0 −1.52143
\(688\) 0 0
\(689\) −8160.00 −0.451192
\(690\) 0 0
\(691\) −33656.0 −1.85287 −0.926436 0.376452i \(-0.877144\pi\)
−0.926436 + 0.376452i \(0.877144\pi\)
\(692\) 0 0
\(693\) 3168.00 0.173654
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −11304.0 −0.614303
\(698\) 0 0
\(699\) 28272.0 1.52982
\(700\) 0 0
\(701\) −6742.00 −0.363255 −0.181628 0.983367i \(-0.558137\pi\)
−0.181628 + 0.983367i \(0.558137\pi\)
\(702\) 0 0
\(703\) 8536.00 0.457954
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −61760.0 −3.28532
\(708\) 0 0
\(709\) −25598.0 −1.35593 −0.677964 0.735095i \(-0.737138\pi\)
−0.677964 + 0.735095i \(0.737138\pi\)
\(710\) 0 0
\(711\) −5976.00 −0.315215
\(712\) 0 0
\(713\) 6496.00 0.341202
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −2976.00 −0.155008
\(718\) 0 0
\(719\) −15668.0 −0.812681 −0.406341 0.913722i \(-0.633195\pi\)
−0.406341 + 0.913722i \(0.633195\pi\)
\(720\) 0 0
\(721\) −18112.0 −0.935542
\(722\) 0 0
\(723\) −28836.0 −1.48330
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −7810.00 −0.398428 −0.199214 0.979956i \(-0.563839\pi\)
−0.199214 + 0.979956i \(0.563839\pi\)
\(728\) 0 0
\(729\) 9477.00 0.481481
\(730\) 0 0
\(731\) −14256.0 −0.721309
\(732\) 0 0
\(733\) −19708.0 −0.993085 −0.496543 0.868012i \(-0.665397\pi\)
−0.496543 + 0.868012i \(0.665397\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 286.000 0.0142944
\(738\) 0 0
\(739\) −11588.0 −0.576822 −0.288411 0.957507i \(-0.593127\pi\)
−0.288411 + 0.957507i \(0.593127\pi\)
\(740\) 0 0
\(741\) 12672.0 0.628229
\(742\) 0 0
\(743\) −10812.0 −0.533854 −0.266927 0.963717i \(-0.586008\pi\)
−0.266927 + 0.963717i \(0.586008\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −12132.0 −0.594226
\(748\) 0 0
\(749\) 4352.00 0.212308
\(750\) 0 0
\(751\) 5972.00 0.290175 0.145087 0.989419i \(-0.453654\pi\)
0.145087 + 0.989419i \(0.453654\pi\)
\(752\) 0 0
\(753\) 7728.00 0.374003
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 16034.0 0.769836 0.384918 0.922951i \(-0.374230\pi\)
0.384918 + 0.922951i \(0.374230\pi\)
\(758\) 0 0
\(759\) 3828.00 0.183067
\(760\) 0 0
\(761\) −5862.00 −0.279234 −0.139617 0.990206i \(-0.544587\pi\)
−0.139617 + 0.990206i \(0.544587\pi\)
\(762\) 0 0
\(763\) −16192.0 −0.768270
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 19392.0 0.912913
\(768\) 0 0
\(769\) 2926.00 0.137210 0.0686048 0.997644i \(-0.478145\pi\)
0.0686048 + 0.997644i \(0.478145\pi\)
\(770\) 0 0
\(771\) −1428.00 −0.0667032
\(772\) 0 0
\(773\) −39066.0 −1.81773 −0.908866 0.417089i \(-0.863050\pi\)
−0.908866 + 0.417089i \(0.863050\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 37248.0 1.71977
\(778\) 0 0
\(779\) 13816.0 0.635442
\(780\) 0 0
\(781\) −5148.00 −0.235864
\(782\) 0 0
\(783\) −30024.0 −1.37033
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 34752.0 1.57405 0.787024 0.616923i \(-0.211621\pi\)
0.787024 + 0.616923i \(0.211621\pi\)
\(788\) 0 0
\(789\) 2568.00 0.115872
\(790\) 0 0
\(791\) 67136.0 3.01780
\(792\) 0 0
\(793\) 12000.0 0.537368
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 4034.00 0.179287 0.0896434 0.995974i \(-0.471427\pi\)
0.0896434 + 0.995974i \(0.471427\pi\)
\(798\) 0 0
\(799\) 14760.0 0.653531
\(800\) 0 0
\(801\) 4806.00 0.212000
\(802\) 0 0
\(803\) 1804.00 0.0792799
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 34116.0 1.48815
\(808\) 0 0
\(809\) −40890.0 −1.77703 −0.888514 0.458849i \(-0.848262\pi\)
−0.888514 + 0.458849i \(0.848262\pi\)
\(810\) 0 0
\(811\) −20060.0 −0.868560 −0.434280 0.900778i \(-0.642997\pi\)
−0.434280 + 0.900778i \(0.642997\pi\)
\(812\) 0 0
\(813\) 49104.0 2.11827
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 17424.0 0.746130
\(818\) 0 0
\(819\) 13824.0 0.589804
\(820\) 0 0
\(821\) −33238.0 −1.41293 −0.706464 0.707749i \(-0.749711\pi\)
−0.706464 + 0.707749i \(0.749711\pi\)
\(822\) 0 0
\(823\) 31114.0 1.31782 0.658910 0.752222i \(-0.271018\pi\)
0.658910 + 0.752222i \(0.271018\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −42296.0 −1.77845 −0.889224 0.457473i \(-0.848755\pi\)
−0.889224 + 0.457473i \(0.848755\pi\)
\(828\) 0 0
\(829\) −11834.0 −0.495792 −0.247896 0.968787i \(-0.579739\pi\)
−0.247896 + 0.968787i \(0.579739\pi\)
\(830\) 0 0
\(831\) −15888.0 −0.663235
\(832\) 0 0
\(833\) 24516.0 1.01972
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −12096.0 −0.499521
\(838\) 0 0
\(839\) −7032.00 −0.289358 −0.144679 0.989479i \(-0.546215\pi\)
−0.144679 + 0.989479i \(0.546215\pi\)
\(840\) 0 0
\(841\) 52895.0 2.16881
\(842\) 0 0
\(843\) 36108.0 1.47524
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3872.00 0.157076
\(848\) 0 0
\(849\) 26664.0 1.07786
\(850\) 0 0
\(851\) 11252.0 0.453247
\(852\) 0 0
\(853\) 16268.0 0.652996 0.326498 0.945198i \(-0.394131\pi\)
0.326498 + 0.945198i \(0.394131\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 31980.0 1.27470 0.637349 0.770575i \(-0.280031\pi\)
0.637349 + 0.770575i \(0.280031\pi\)
\(858\) 0 0
\(859\) −30160.0 −1.19796 −0.598979 0.800765i \(-0.704427\pi\)
−0.598979 + 0.800765i \(0.704427\pi\)
\(860\) 0 0
\(861\) 60288.0 2.38631
\(862\) 0 0
\(863\) −21518.0 −0.848762 −0.424381 0.905484i \(-0.639508\pi\)
−0.424381 + 0.905484i \(0.639508\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 21702.0 0.850102
\(868\) 0 0
\(869\) −7304.00 −0.285122
\(870\) 0 0
\(871\) 1248.00 0.0485498
\(872\) 0 0
\(873\) 13482.0 0.522676
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 48124.0 1.85294 0.926472 0.376364i \(-0.122826\pi\)
0.926472 + 0.376364i \(0.122826\pi\)
\(878\) 0 0
\(879\) −19872.0 −0.762532
\(880\) 0 0
\(881\) −8146.00 −0.311516 −0.155758 0.987795i \(-0.549782\pi\)
−0.155758 + 0.987795i \(0.549782\pi\)
\(882\) 0 0
\(883\) 47218.0 1.79956 0.899780 0.436343i \(-0.143727\pi\)
0.899780 + 0.436343i \(0.143727\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −28796.0 −1.09005 −0.545025 0.838420i \(-0.683480\pi\)
−0.545025 + 0.838420i \(0.683480\pi\)
\(888\) 0 0
\(889\) −79232.0 −2.98915
\(890\) 0 0
\(891\) −9801.00 −0.368514
\(892\) 0 0
\(893\) −18040.0 −0.676020
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 16704.0 0.621773
\(898\) 0 0
\(899\) 31136.0 1.15511
\(900\) 0 0
\(901\) −6120.00 −0.226289
\(902\) 0 0
\(903\) 76032.0 2.80198
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −50374.0 −1.84415 −0.922073 0.387015i \(-0.873506\pi\)
−0.922073 + 0.387015i \(0.873506\pi\)
\(908\) 0 0
\(909\) −17370.0 −0.633803
\(910\) 0 0
\(911\) 41328.0 1.50303 0.751514 0.659718i \(-0.229324\pi\)
0.751514 + 0.659718i \(0.229324\pi\)
\(912\) 0 0
\(913\) −14828.0 −0.537497
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 83840.0 3.01924
\(918\) 0 0
\(919\) −39320.0 −1.41137 −0.705684 0.708527i \(-0.749360\pi\)
−0.705684 + 0.708527i \(0.749360\pi\)
\(920\) 0 0
\(921\) 26976.0 0.965135
\(922\) 0 0
\(923\) −22464.0 −0.801096
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −5094.00 −0.180484
\(928\) 0 0
\(929\) −27682.0 −0.977629 −0.488814 0.872388i \(-0.662570\pi\)
−0.488814 + 0.872388i \(0.662570\pi\)
\(930\) 0 0
\(931\) −29964.0 −1.05481
\(932\) 0 0
\(933\) 30168.0 1.05858
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −57208.0 −1.99456 −0.997281 0.0736977i \(-0.976520\pi\)
−0.997281 + 0.0736977i \(0.976520\pi\)
\(938\) 0 0
\(939\) 13500.0 0.469176
\(940\) 0 0
\(941\) 41350.0 1.43249 0.716244 0.697850i \(-0.245860\pi\)
0.716244 + 0.697850i \(0.245860\pi\)
\(942\) 0 0
\(943\) 18212.0 0.628912
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 38506.0 1.32131 0.660653 0.750691i \(-0.270279\pi\)
0.660653 + 0.750691i \(0.270279\pi\)
\(948\) 0 0
\(949\) 7872.00 0.269269
\(950\) 0 0
\(951\) −5004.00 −0.170627
\(952\) 0 0
\(953\) 7816.00 0.265672 0.132836 0.991138i \(-0.457592\pi\)
0.132836 + 0.991138i \(0.457592\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 18348.0 0.619756
\(958\) 0 0
\(959\) −59328.0 −1.99771
\(960\) 0 0
\(961\) −17247.0 −0.578933
\(962\) 0 0
\(963\) 1224.00 0.0409583
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 24596.0 0.817946 0.408973 0.912546i \(-0.365887\pi\)
0.408973 + 0.912546i \(0.365887\pi\)
\(968\) 0 0
\(969\) 9504.00 0.315080
\(970\) 0 0
\(971\) 48236.0 1.59420 0.797099 0.603848i \(-0.206367\pi\)
0.797099 + 0.603848i \(0.206367\pi\)
\(972\) 0 0
\(973\) 23680.0 0.780212
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −32230.0 −1.05540 −0.527702 0.849430i \(-0.676946\pi\)
−0.527702 + 0.849430i \(0.676946\pi\)
\(978\) 0 0
\(979\) 5874.00 0.191761
\(980\) 0 0
\(981\) −4554.00 −0.148214
\(982\) 0 0
\(983\) 1874.00 0.0608050 0.0304025 0.999538i \(-0.490321\pi\)
0.0304025 + 0.999538i \(0.490321\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −78720.0 −2.53869
\(988\) 0 0
\(989\) 22968.0 0.738463
\(990\) 0 0
\(991\) 7144.00 0.228998 0.114499 0.993423i \(-0.463474\pi\)
0.114499 + 0.993423i \(0.463474\pi\)
\(992\) 0 0
\(993\) 43464.0 1.38901
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −31892.0 −1.01307 −0.506534 0.862220i \(-0.669074\pi\)
−0.506534 + 0.862220i \(0.669074\pi\)
\(998\) 0 0
\(999\) −20952.0 −0.663555
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 2200.4.a.b.1.1 1
5.4 even 2 440.4.a.d.1.1 1
20.19 odd 2 880.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
440.4.a.d.1.1 1 5.4 even 2
880.4.a.d.1.1 1 20.19 odd 2
2200.4.a.b.1.1 1 1.1 even 1 trivial