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Properties

Label 400.4.a.r.1.1
Level $400$
Weight $4$
Character 400.1
Self dual yes
Analytic conductor $23.601$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,4,Mod(1,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, 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("400.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 400.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: \(23.6007640023\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 50)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.00000 q^{3} -34.0000 q^{7} +22.0000 q^{9} -27.0000 q^{11} +28.0000 q^{13} -21.0000 q^{17} -35.0000 q^{19} -238.000 q^{21} -78.0000 q^{23} -35.0000 q^{27} -120.000 q^{29} -182.000 q^{31} -189.000 q^{33} -146.000 q^{37} +196.000 q^{39} +357.000 q^{41} -148.000 q^{43} -84.0000 q^{47} +813.000 q^{49} -147.000 q^{51} -702.000 q^{53} -245.000 q^{57} +840.000 q^{59} -238.000 q^{61} -748.000 q^{63} +461.000 q^{67} -546.000 q^{69} +708.000 q^{71} +133.000 q^{73} +918.000 q^{77} -650.000 q^{79} -839.000 q^{81} -903.000 q^{83} -840.000 q^{87} +735.000 q^{89} -952.000 q^{91} -1274.00 q^{93} -1106.00 q^{97} -594.000 q^{99} +O(q^{100})\)

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
\(3\) 7.00000 1.34715 0.673575 − 0.739119i \(-0.264758\pi\)
0.673575 + 0.739119i \(0.264758\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −34.0000 −1.83583 −0.917914 − 0.396780i \(-0.870128\pi\)
−0.917914 + 0.396780i \(0.870128\pi\)
\(8\) 0 0
\(9\) 22.0000 0.814815
\(10\) 0 0
\(11\) −27.0000 −0.740073 −0.370037 − 0.929017i \(-0.620655\pi\)
−0.370037 + 0.929017i \(0.620655\pi\)
\(12\) 0 0
\(13\) 28.0000 0.597369 0.298685 − 0.954352i \(-0.403452\pi\)
0.298685 + 0.954352i \(0.403452\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −21.0000 −0.299603 −0.149801 − 0.988716i \(-0.547863\pi\)
−0.149801 + 0.988716i \(0.547863\pi\)
\(18\) 0 0
\(19\) −35.0000 −0.422608 −0.211304 − 0.977420i \(-0.567771\pi\)
−0.211304 + 0.977420i \(0.567771\pi\)
\(20\) 0 0
\(21\) −238.000 −2.47314
\(22\) 0 0
\(23\) −78.0000 −0.707136 −0.353568 − 0.935409i \(-0.615032\pi\)
−0.353568 + 0.935409i \(0.615032\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −35.0000 −0.249472
\(28\) 0 0
\(29\) −120.000 −0.768395 −0.384197 − 0.923251i \(-0.625522\pi\)
−0.384197 + 0.923251i \(0.625522\pi\)
\(30\) 0 0
\(31\) −182.000 −1.05446 −0.527228 − 0.849724i \(-0.676769\pi\)
−0.527228 + 0.849724i \(0.676769\pi\)
\(32\) 0 0
\(33\) −189.000 −0.996990
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −146.000 −0.648710 −0.324355 − 0.945936i \(-0.605147\pi\)
−0.324355 + 0.945936i \(0.605147\pi\)
\(38\) 0 0
\(39\) 196.000 0.804747
\(40\) 0 0
\(41\) 357.000 1.35985 0.679927 − 0.733280i \(-0.262011\pi\)
0.679927 + 0.733280i \(0.262011\pi\)
\(42\) 0 0
\(43\) −148.000 −0.524879 −0.262439 − 0.964948i \(-0.584527\pi\)
−0.262439 + 0.964948i \(0.584527\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −84.0000 −0.260695 −0.130347 − 0.991468i \(-0.541609\pi\)
−0.130347 + 0.991468i \(0.541609\pi\)
\(48\) 0 0
\(49\) 813.000 2.37026
\(50\) 0 0
\(51\) −147.000 −0.403610
\(52\) 0 0
\(53\) −702.000 −1.81938 −0.909690 − 0.415288i \(-0.863681\pi\)
−0.909690 + 0.415288i \(0.863681\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −245.000 −0.569317
\(58\) 0 0
\(59\) 840.000 1.85354 0.926769 − 0.375633i \(-0.122575\pi\)
0.926769 + 0.375633i \(0.122575\pi\)
\(60\) 0 0
\(61\) −238.000 −0.499554 −0.249777 − 0.968303i \(-0.580357\pi\)
−0.249777 + 0.968303i \(0.580357\pi\)
\(62\) 0 0
\(63\) −748.000 −1.49586
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 461.000 0.840599 0.420299 − 0.907386i \(-0.361925\pi\)
0.420299 + 0.907386i \(0.361925\pi\)
\(68\) 0 0
\(69\) −546.000 −0.952618
\(70\) 0 0
\(71\) 708.000 1.18344 0.591719 − 0.806144i \(-0.298449\pi\)
0.591719 + 0.806144i \(0.298449\pi\)
\(72\) 0 0
\(73\) 133.000 0.213239 0.106620 − 0.994300i \(-0.465997\pi\)
0.106620 + 0.994300i \(0.465997\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 918.000 1.35865
\(78\) 0 0
\(79\) −650.000 −0.925705 −0.462853 − 0.886435i \(-0.653174\pi\)
−0.462853 + 0.886435i \(0.653174\pi\)
\(80\) 0 0
\(81\) −839.000 −1.15089
\(82\) 0 0
\(83\) −903.000 −1.19418 −0.597091 − 0.802173i \(-0.703677\pi\)
−0.597091 + 0.802173i \(0.703677\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −840.000 −1.03514
\(88\) 0 0
\(89\) 735.000 0.875392 0.437696 − 0.899123i \(-0.355795\pi\)
0.437696 + 0.899123i \(0.355795\pi\)
\(90\) 0 0
\(91\) −952.000 −1.09667
\(92\) 0 0
\(93\) −1274.00 −1.42051
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1106.00 −1.15770 −0.578852 − 0.815433i \(-0.696499\pi\)
−0.578852 + 0.815433i \(0.696499\pi\)
\(98\) 0 0
\(99\) −594.000 −0.603023
\(100\) 0 0
\(101\) 462.000 0.455156 0.227578 − 0.973760i \(-0.426919\pi\)
0.227578 + 0.973760i \(0.426919\pi\)
\(102\) 0 0
\(103\) 812.000 0.776784 0.388392 − 0.921494i \(-0.373031\pi\)
0.388392 + 0.921494i \(0.373031\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −789.000 −0.712855 −0.356428 − 0.934323i \(-0.616005\pi\)
−0.356428 + 0.934323i \(0.616005\pi\)
\(108\) 0 0
\(109\) 230.000 0.202110 0.101055 − 0.994881i \(-0.467778\pi\)
0.101055 + 0.994881i \(0.467778\pi\)
\(110\) 0 0
\(111\) −1022.00 −0.873909
\(112\) 0 0
\(113\) 2073.00 1.72576 0.862882 − 0.505405i \(-0.168657\pi\)
0.862882 + 0.505405i \(0.168657\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 616.000 0.486745
\(118\) 0 0
\(119\) 714.000 0.550019
\(120\) 0 0
\(121\) −602.000 −0.452292
\(122\) 0 0
\(123\) 2499.00 1.83193
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −1114.00 −0.778358 −0.389179 − 0.921162i \(-0.627241\pi\)
−0.389179 + 0.921162i \(0.627241\pi\)
\(128\) 0 0
\(129\) −1036.00 −0.707091
\(130\) 0 0
\(131\) −252.000 −0.168071 −0.0840357 − 0.996463i \(-0.526781\pi\)
−0.0840357 + 0.996463i \(0.526781\pi\)
\(132\) 0 0
\(133\) 1190.00 0.775835
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1941.00 −1.21044 −0.605222 − 0.796057i \(-0.706916\pi\)
−0.605222 + 0.796057i \(0.706916\pi\)
\(138\) 0 0
\(139\) 1645.00 1.00379 0.501896 − 0.864928i \(-0.332636\pi\)
0.501896 + 0.864928i \(0.332636\pi\)
\(140\) 0 0
\(141\) −588.000 −0.351195
\(142\) 0 0
\(143\) −756.000 −0.442097
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5691.00 3.19310
\(148\) 0 0
\(149\) −1800.00 −0.989676 −0.494838 − 0.868985i \(-0.664773\pi\)
−0.494838 + 0.868985i \(0.664773\pi\)
\(150\) 0 0
\(151\) 3298.00 1.77740 0.888700 − 0.458489i \(-0.151609\pi\)
0.888700 + 0.458489i \(0.151609\pi\)
\(152\) 0 0
\(153\) −462.000 −0.244121
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −266.000 −0.135217 −0.0676086 − 0.997712i \(-0.521537\pi\)
−0.0676086 + 0.997712i \(0.521537\pi\)
\(158\) 0 0
\(159\) −4914.00 −2.45098
\(160\) 0 0
\(161\) 2652.00 1.29818
\(162\) 0 0
\(163\) 1157.00 0.555971 0.277985 − 0.960585i \(-0.410333\pi\)
0.277985 + 0.960585i \(0.410333\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1764.00 −0.817380 −0.408690 − 0.912673i \(-0.634014\pi\)
−0.408690 + 0.912673i \(0.634014\pi\)
\(168\) 0 0
\(169\) −1413.00 −0.643150
\(170\) 0 0
\(171\) −770.000 −0.344347
\(172\) 0 0
\(173\) 1848.00 0.812144 0.406072 − 0.913841i \(-0.366898\pi\)
0.406072 + 0.913841i \(0.366898\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5880.00 2.49699
\(178\) 0 0
\(179\) −135.000 −0.0563708 −0.0281854 − 0.999603i \(-0.508973\pi\)
−0.0281854 + 0.999603i \(0.508973\pi\)
\(180\) 0 0
\(181\) 2282.00 0.937126 0.468563 − 0.883430i \(-0.344772\pi\)
0.468563 + 0.883430i \(0.344772\pi\)
\(182\) 0 0
\(183\) −1666.00 −0.672974
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 567.000 0.221728
\(188\) 0 0
\(189\) 1190.00 0.457988
\(190\) 0 0
\(191\) 1398.00 0.529611 0.264806 − 0.964302i \(-0.414692\pi\)
0.264806 + 0.964302i \(0.414692\pi\)
\(192\) 0 0
\(193\) −3317.00 −1.23711 −0.618557 − 0.785740i \(-0.712282\pi\)
−0.618557 + 0.785740i \(0.712282\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1686.00 −0.609759 −0.304880 − 0.952391i \(-0.598616\pi\)
−0.304880 + 0.952391i \(0.598616\pi\)
\(198\) 0 0
\(199\) 1540.00 0.548581 0.274291 − 0.961647i \(-0.411557\pi\)
0.274291 + 0.961647i \(0.411557\pi\)
\(200\) 0 0
\(201\) 3227.00 1.13241
\(202\) 0 0
\(203\) 4080.00 1.41064
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −1716.00 −0.576185
\(208\) 0 0
\(209\) 945.000 0.312761
\(210\) 0 0
\(211\) 3043.00 0.992838 0.496419 − 0.868083i \(-0.334648\pi\)
0.496419 + 0.868083i \(0.334648\pi\)
\(212\) 0 0
\(213\) 4956.00 1.59427
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6188.00 1.93580
\(218\) 0 0
\(219\) 931.000 0.287266
\(220\) 0 0
\(221\) −588.000 −0.178974
\(222\) 0 0
\(223\) 3332.00 1.00057 0.500285 − 0.865861i \(-0.333228\pi\)
0.500285 + 0.865861i \(0.333228\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1596.00 0.466653 0.233327 − 0.972398i \(-0.425039\pi\)
0.233327 + 0.972398i \(0.425039\pi\)
\(228\) 0 0
\(229\) 4340.00 1.25238 0.626191 − 0.779670i \(-0.284613\pi\)
0.626191 + 0.779670i \(0.284613\pi\)
\(230\) 0 0
\(231\) 6426.00 1.83030
\(232\) 0 0
\(233\) 3018.00 0.848565 0.424283 − 0.905530i \(-0.360526\pi\)
0.424283 + 0.905530i \(0.360526\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4550.00 −1.24706
\(238\) 0 0
\(239\) −4440.00 −1.20167 −0.600836 − 0.799372i \(-0.705166\pi\)
−0.600836 + 0.799372i \(0.705166\pi\)
\(240\) 0 0
\(241\) −3703.00 −0.989756 −0.494878 − 0.868962i \(-0.664787\pi\)
−0.494878 + 0.868962i \(0.664787\pi\)
\(242\) 0 0
\(243\) −4928.00 −1.30095
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −980.000 −0.252453
\(248\) 0 0
\(249\) −6321.00 −1.60874
\(250\) 0 0
\(251\) −7077.00 −1.77967 −0.889833 − 0.456286i \(-0.849179\pi\)
−0.889833 + 0.456286i \(0.849179\pi\)
\(252\) 0 0
\(253\) 2106.00 0.523332
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −6846.00 −1.66164 −0.830821 − 0.556540i \(-0.812128\pi\)
−0.830821 + 0.556540i \(0.812128\pi\)
\(258\) 0 0
\(259\) 4964.00 1.19092
\(260\) 0 0
\(261\) −2640.00 −0.626099
\(262\) 0 0
\(263\) −3438.00 −0.806069 −0.403035 − 0.915185i \(-0.632045\pi\)
−0.403035 + 0.915185i \(0.632045\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 5145.00 1.17928
\(268\) 0 0
\(269\) −1680.00 −0.380786 −0.190393 − 0.981708i \(-0.560976\pi\)
−0.190393 + 0.981708i \(0.560976\pi\)
\(270\) 0 0
\(271\) −5222.00 −1.17053 −0.585266 − 0.810842i \(-0.699010\pi\)
−0.585266 + 0.810842i \(0.699010\pi\)
\(272\) 0 0
\(273\) −6664.00 −1.47738
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1384.00 0.300204 0.150102 − 0.988671i \(-0.452040\pi\)
0.150102 + 0.988671i \(0.452040\pi\)
\(278\) 0 0
\(279\) −4004.00 −0.859187
\(280\) 0 0
\(281\) −3858.00 −0.819036 −0.409518 − 0.912302i \(-0.634303\pi\)
−0.409518 + 0.912302i \(0.634303\pi\)
\(282\) 0 0
\(283\) 4277.00 0.898379 0.449190 − 0.893437i \(-0.351713\pi\)
0.449190 + 0.893437i \(0.351713\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −12138.0 −2.49646
\(288\) 0 0
\(289\) −4472.00 −0.910238
\(290\) 0 0
\(291\) −7742.00 −1.55960
\(292\) 0 0
\(293\) −6342.00 −1.26452 −0.632259 − 0.774757i \(-0.717872\pi\)
−0.632259 + 0.774757i \(0.717872\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 945.000 0.184628
\(298\) 0 0
\(299\) −2184.00 −0.422421
\(300\) 0 0
\(301\) 5032.00 0.963587
\(302\) 0 0
\(303\) 3234.00 0.613163
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5831.00 1.08402 0.542008 − 0.840373i \(-0.317664\pi\)
0.542008 + 0.840373i \(0.317664\pi\)
\(308\) 0 0
\(309\) 5684.00 1.04644
\(310\) 0 0
\(311\) 2478.00 0.451815 0.225908 − 0.974149i \(-0.427465\pi\)
0.225908 + 0.974149i \(0.427465\pi\)
\(312\) 0 0
\(313\) 2758.00 0.498056 0.249028 − 0.968496i \(-0.419889\pi\)
0.249028 + 0.968496i \(0.419889\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −636.000 −0.112686 −0.0563428 − 0.998411i \(-0.517944\pi\)
−0.0563428 + 0.998411i \(0.517944\pi\)
\(318\) 0 0
\(319\) 3240.00 0.568668
\(320\) 0 0
\(321\) −5523.00 −0.960323
\(322\) 0 0
\(323\) 735.000 0.126615
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1610.00 0.272273
\(328\) 0 0
\(329\) 2856.00 0.478591
\(330\) 0 0
\(331\) −6887.00 −1.14364 −0.571818 − 0.820380i \(-0.693762\pi\)
−0.571818 + 0.820380i \(0.693762\pi\)
\(332\) 0 0
\(333\) −3212.00 −0.528578
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7331.00 −1.18500 −0.592500 − 0.805570i \(-0.701859\pi\)
−0.592500 + 0.805570i \(0.701859\pi\)
\(338\) 0 0
\(339\) 14511.0 2.32487
\(340\) 0 0
\(341\) 4914.00 0.780375
\(342\) 0 0
\(343\) −15980.0 −2.51557
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −5349.00 −0.827520 −0.413760 − 0.910386i \(-0.635785\pi\)
−0.413760 + 0.910386i \(0.635785\pi\)
\(348\) 0 0
\(349\) 11270.0 1.72857 0.864283 − 0.503007i \(-0.167773\pi\)
0.864283 + 0.503007i \(0.167773\pi\)
\(350\) 0 0
\(351\) −980.000 −0.149027
\(352\) 0 0
\(353\) −1302.00 −0.196313 −0.0981565 − 0.995171i \(-0.531295\pi\)
−0.0981565 + 0.995171i \(0.531295\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4998.00 0.740959
\(358\) 0 0
\(359\) 750.000 0.110260 0.0551302 − 0.998479i \(-0.482443\pi\)
0.0551302 + 0.998479i \(0.482443\pi\)
\(360\) 0 0
\(361\) −5634.00 −0.821403
\(362\) 0 0
\(363\) −4214.00 −0.609305
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1204.00 −0.171249 −0.0856244 − 0.996327i \(-0.527288\pi\)
−0.0856244 + 0.996327i \(0.527288\pi\)
\(368\) 0 0
\(369\) 7854.00 1.10803
\(370\) 0 0
\(371\) 23868.0 3.34007
\(372\) 0 0
\(373\) 1198.00 0.166301 0.0831503 − 0.996537i \(-0.473502\pi\)
0.0831503 + 0.996537i \(0.473502\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3360.00 −0.459015
\(378\) 0 0
\(379\) −8105.00 −1.09849 −0.549243 − 0.835663i \(-0.685084\pi\)
−0.549243 + 0.835663i \(0.685084\pi\)
\(380\) 0 0
\(381\) −7798.00 −1.04857
\(382\) 0 0
\(383\) −3318.00 −0.442668 −0.221334 − 0.975198i \(-0.571041\pi\)
−0.221334 + 0.975198i \(0.571041\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3256.00 −0.427679
\(388\) 0 0
\(389\) −13770.0 −1.79477 −0.897387 − 0.441245i \(-0.854537\pi\)
−0.897387 + 0.441245i \(0.854537\pi\)
\(390\) 0 0
\(391\) 1638.00 0.211860
\(392\) 0 0
\(393\) −1764.00 −0.226417
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3724.00 0.470786 0.235393 − 0.971900i \(-0.424362\pi\)
0.235393 + 0.971900i \(0.424362\pi\)
\(398\) 0 0
\(399\) 8330.00 1.04517
\(400\) 0 0
\(401\) 6117.00 0.761767 0.380883 − 0.924623i \(-0.375620\pi\)
0.380883 + 0.924623i \(0.375620\pi\)
\(402\) 0 0
\(403\) −5096.00 −0.629900
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 3942.00 0.480093
\(408\) 0 0
\(409\) 6125.00 0.740493 0.370247 − 0.928933i \(-0.379273\pi\)
0.370247 + 0.928933i \(0.379273\pi\)
\(410\) 0 0
\(411\) −13587.0 −1.63065
\(412\) 0 0
\(413\) −28560.0 −3.40277
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 11515.0 1.35226
\(418\) 0 0
\(419\) −1575.00 −0.183637 −0.0918184 − 0.995776i \(-0.529268\pi\)
−0.0918184 + 0.995776i \(0.529268\pi\)
\(420\) 0 0
\(421\) −988.000 −0.114376 −0.0571879 − 0.998363i \(-0.518213\pi\)
−0.0571879 + 0.998363i \(0.518213\pi\)
\(422\) 0 0
\(423\) −1848.00 −0.212418
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8092.00 0.917094
\(428\) 0 0
\(429\) −5292.00 −0.595571
\(430\) 0 0
\(431\) 558.000 0.0623617 0.0311809 − 0.999514i \(-0.490073\pi\)
0.0311809 + 0.999514i \(0.490073\pi\)
\(432\) 0 0
\(433\) 2443.00 0.271139 0.135569 − 0.990768i \(-0.456714\pi\)
0.135569 + 0.990768i \(0.456714\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2730.00 0.298841
\(438\) 0 0
\(439\) −12320.0 −1.33941 −0.669706 − 0.742627i \(-0.733580\pi\)
−0.669706 + 0.742627i \(0.733580\pi\)
\(440\) 0 0
\(441\) 17886.0 1.93132
\(442\) 0 0
\(443\) −2343.00 −0.251285 −0.125643 − 0.992076i \(-0.540099\pi\)
−0.125643 + 0.992076i \(0.540099\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −12600.0 −1.33324
\(448\) 0 0
\(449\) 10905.0 1.14619 0.573094 − 0.819489i \(-0.305743\pi\)
0.573094 + 0.819489i \(0.305743\pi\)
\(450\) 0 0
\(451\) −9639.00 −1.00639
\(452\) 0 0
\(453\) 23086.0 2.39443
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3319.00 0.339729 0.169865 − 0.985467i \(-0.445667\pi\)
0.169865 + 0.985467i \(0.445667\pi\)
\(458\) 0 0
\(459\) 735.000 0.0747426
\(460\) 0 0
\(461\) −6468.00 −0.653459 −0.326730 − 0.945118i \(-0.605947\pi\)
−0.326730 + 0.945118i \(0.605947\pi\)
\(462\) 0 0
\(463\) 11972.0 1.20170 0.600849 − 0.799363i \(-0.294829\pi\)
0.600849 + 0.799363i \(0.294829\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6636.00 0.657553 0.328777 − 0.944408i \(-0.393364\pi\)
0.328777 + 0.944408i \(0.393364\pi\)
\(468\) 0 0
\(469\) −15674.0 −1.54319
\(470\) 0 0
\(471\) −1862.00 −0.182158
\(472\) 0 0
\(473\) 3996.00 0.388449
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −15444.0 −1.48246
\(478\) 0 0
\(479\) 630.000 0.0600949 0.0300474 − 0.999548i \(-0.490434\pi\)
0.0300474 + 0.999548i \(0.490434\pi\)
\(480\) 0 0
\(481\) −4088.00 −0.387519
\(482\) 0 0
\(483\) 18564.0 1.74884
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 10646.0 0.990588 0.495294 − 0.868725i \(-0.335060\pi\)
0.495294 + 0.868725i \(0.335060\pi\)
\(488\) 0 0
\(489\) 8099.00 0.748976
\(490\) 0 0
\(491\) 2388.00 0.219489 0.109744 − 0.993960i \(-0.464997\pi\)
0.109744 + 0.993960i \(0.464997\pi\)
\(492\) 0 0
\(493\) 2520.00 0.230213
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −24072.0 −2.17259
\(498\) 0 0
\(499\) 10540.0 0.945562 0.472781 − 0.881180i \(-0.343250\pi\)
0.472781 + 0.881180i \(0.343250\pi\)
\(500\) 0 0
\(501\) −12348.0 −1.10113
\(502\) 0 0
\(503\) 6972.00 0.618024 0.309012 − 0.951058i \(-0.400002\pi\)
0.309012 + 0.951058i \(0.400002\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −9891.00 −0.866420
\(508\) 0 0
\(509\) −9030.00 −0.786341 −0.393171 − 0.919466i \(-0.628622\pi\)
−0.393171 + 0.919466i \(0.628622\pi\)
\(510\) 0 0
\(511\) −4522.00 −0.391471
\(512\) 0 0
\(513\) 1225.00 0.105429
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 2268.00 0.192933
\(518\) 0 0
\(519\) 12936.0 1.09408
\(520\) 0 0
\(521\) 10437.0 0.877645 0.438823 − 0.898574i \(-0.355396\pi\)
0.438823 + 0.898574i \(0.355396\pi\)
\(522\) 0 0
\(523\) −8113.00 −0.678311 −0.339156 − 0.940730i \(-0.610141\pi\)
−0.339156 + 0.940730i \(0.610141\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3822.00 0.315918
\(528\) 0 0
\(529\) −6083.00 −0.499959
\(530\) 0 0
\(531\) 18480.0 1.51029
\(532\) 0 0
\(533\) 9996.00 0.812336
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −945.000 −0.0759400
\(538\) 0 0
\(539\) −21951.0 −1.75417
\(540\) 0 0
\(541\) −14848.0 −1.17997 −0.589986 − 0.807413i \(-0.700867\pi\)
−0.589986 + 0.807413i \(0.700867\pi\)
\(542\) 0 0
\(543\) 15974.0 1.26245
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −20329.0 −1.58904 −0.794520 − 0.607238i \(-0.792278\pi\)
−0.794520 + 0.607238i \(0.792278\pi\)
\(548\) 0 0
\(549\) −5236.00 −0.407044
\(550\) 0 0
\(551\) 4200.00 0.324730
\(552\) 0 0
\(553\) 22100.0 1.69944
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15324.0 1.16571 0.582853 − 0.812577i \(-0.301936\pi\)
0.582853 + 0.812577i \(0.301936\pi\)
\(558\) 0 0
\(559\) −4144.00 −0.313547
\(560\) 0 0
\(561\) 3969.00 0.298701
\(562\) 0 0
\(563\) −9408.00 −0.704263 −0.352131 − 0.935951i \(-0.614543\pi\)
−0.352131 + 0.935951i \(0.614543\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 28526.0 2.11284
\(568\) 0 0
\(569\) −24375.0 −1.79588 −0.897938 − 0.440122i \(-0.854935\pi\)
−0.897938 + 0.440122i \(0.854935\pi\)
\(570\) 0 0
\(571\) 988.000 0.0724107 0.0362054 − 0.999344i \(-0.488473\pi\)
0.0362054 + 0.999344i \(0.488473\pi\)
\(572\) 0 0
\(573\) 9786.00 0.713466
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 16429.0 1.18535 0.592676 − 0.805441i \(-0.298071\pi\)
0.592676 + 0.805441i \(0.298071\pi\)
\(578\) 0 0
\(579\) −23219.0 −1.66658
\(580\) 0 0
\(581\) 30702.0 2.19231
\(582\) 0 0
\(583\) 18954.0 1.34647
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13839.0 −0.973078 −0.486539 − 0.873659i \(-0.661741\pi\)
−0.486539 + 0.873659i \(0.661741\pi\)
\(588\) 0 0
\(589\) 6370.00 0.445622
\(590\) 0 0
\(591\) −11802.0 −0.821437
\(592\) 0 0
\(593\) −14007.0 −0.969981 −0.484990 − 0.874520i \(-0.661177\pi\)
−0.484990 + 0.874520i \(0.661177\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10780.0 0.739022
\(598\) 0 0
\(599\) −21090.0 −1.43859 −0.719294 − 0.694706i \(-0.755535\pi\)
−0.719294 + 0.694706i \(0.755535\pi\)
\(600\) 0 0
\(601\) 5747.00 0.390058 0.195029 − 0.980797i \(-0.437520\pi\)
0.195029 + 0.980797i \(0.437520\pi\)
\(602\) 0 0
\(603\) 10142.0 0.684932
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 17696.0 1.18329 0.591646 − 0.806198i \(-0.298478\pi\)
0.591646 + 0.806198i \(0.298478\pi\)
\(608\) 0 0
\(609\) 28560.0 1.90034
\(610\) 0 0
\(611\) −2352.00 −0.155731
\(612\) 0 0
\(613\) −26102.0 −1.71982 −0.859910 − 0.510445i \(-0.829481\pi\)
−0.859910 + 0.510445i \(0.829481\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4194.00 0.273653 0.136827 − 0.990595i \(-0.456310\pi\)
0.136827 + 0.990595i \(0.456310\pi\)
\(618\) 0 0
\(619\) 7420.00 0.481801 0.240901 − 0.970550i \(-0.422557\pi\)
0.240901 + 0.970550i \(0.422557\pi\)
\(620\) 0 0
\(621\) 2730.00 0.176411
\(622\) 0 0
\(623\) −24990.0 −1.60707
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 6615.00 0.421336
\(628\) 0 0
\(629\) 3066.00 0.194355
\(630\) 0 0
\(631\) 5818.00 0.367054 0.183527 − 0.983015i \(-0.441249\pi\)
0.183527 + 0.983015i \(0.441249\pi\)
\(632\) 0 0
\(633\) 21301.0 1.33750
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 22764.0 1.41592
\(638\) 0 0
\(639\) 15576.0 0.964283
\(640\) 0 0
\(641\) −29478.0 −1.81640 −0.908199 − 0.418539i \(-0.862542\pi\)
−0.908199 + 0.418539i \(0.862542\pi\)
\(642\) 0 0
\(643\) 5852.00 0.358912 0.179456 − 0.983766i \(-0.442566\pi\)
0.179456 + 0.983766i \(0.442566\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −29484.0 −1.79155 −0.895777 − 0.444503i \(-0.853380\pi\)
−0.895777 + 0.444503i \(0.853380\pi\)
\(648\) 0 0
\(649\) −22680.0 −1.37175
\(650\) 0 0
\(651\) 43316.0 2.60782
\(652\) 0 0
\(653\) 3498.00 0.209628 0.104814 − 0.994492i \(-0.466575\pi\)
0.104814 + 0.994492i \(0.466575\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 2926.00 0.173751
\(658\) 0 0
\(659\) −7905.00 −0.467276 −0.233638 − 0.972324i \(-0.575063\pi\)
−0.233638 + 0.972324i \(0.575063\pi\)
\(660\) 0 0
\(661\) 27272.0 1.60478 0.802389 − 0.596802i \(-0.203562\pi\)
0.802389 + 0.596802i \(0.203562\pi\)
\(662\) 0 0
\(663\) −4116.00 −0.241104
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 9360.00 0.543359
\(668\) 0 0
\(669\) 23324.0 1.34792
\(670\) 0 0
\(671\) 6426.00 0.369706
\(672\) 0 0
\(673\) −12602.0 −0.721800 −0.360900 − 0.932605i \(-0.617530\pi\)
−0.360900 + 0.932605i \(0.617530\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −25536.0 −1.44967 −0.724836 − 0.688921i \(-0.758085\pi\)
−0.724836 + 0.688921i \(0.758085\pi\)
\(678\) 0 0
\(679\) 37604.0 2.12534
\(680\) 0 0
\(681\) 11172.0 0.628652
\(682\) 0 0
\(683\) 2127.00 0.119162 0.0595808 − 0.998223i \(-0.481024\pi\)
0.0595808 + 0.998223i \(0.481024\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 30380.0 1.68715
\(688\) 0 0
\(689\) −19656.0 −1.08684
\(690\) 0 0
\(691\) 8953.00 0.492892 0.246446 − 0.969157i \(-0.420737\pi\)
0.246446 + 0.969157i \(0.420737\pi\)
\(692\) 0 0
\(693\) 20196.0 1.10705
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −7497.00 −0.407416
\(698\) 0 0
\(699\) 21126.0 1.14315
\(700\) 0 0
\(701\) 23172.0 1.24849 0.624247 − 0.781227i \(-0.285406\pi\)
0.624247 + 0.781227i \(0.285406\pi\)
\(702\) 0 0
\(703\) 5110.00 0.274150
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −15708.0 −0.835587
\(708\) 0 0
\(709\) 9620.00 0.509572 0.254786 − 0.966997i \(-0.417995\pi\)
0.254786 + 0.966997i \(0.417995\pi\)
\(710\) 0 0
\(711\) −14300.0 −0.754278
\(712\) 0 0
\(713\) 14196.0 0.745644
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −31080.0 −1.61883
\(718\) 0 0
\(719\) −4830.00 −0.250527 −0.125263 − 0.992124i \(-0.539978\pi\)
−0.125263 + 0.992124i \(0.539978\pi\)
\(720\) 0 0
\(721\) −27608.0 −1.42604
\(722\) 0 0
\(723\) −25921.0 −1.33335
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 11816.0 0.602794 0.301397 − 0.953499i \(-0.402547\pi\)
0.301397 + 0.953499i \(0.402547\pi\)
\(728\) 0 0
\(729\) −11843.0 −0.601687
\(730\) 0 0
\(731\) 3108.00 0.157255
\(732\) 0 0
\(733\) −23492.0 −1.18376 −0.591881 − 0.806026i \(-0.701614\pi\)
−0.591881 + 0.806026i \(0.701614\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −12447.0 −0.622105
\(738\) 0 0
\(739\) −35300.0 −1.75715 −0.878573 − 0.477607i \(-0.841504\pi\)
−0.878573 + 0.477607i \(0.841504\pi\)
\(740\) 0 0
\(741\) −6860.00 −0.340092
\(742\) 0 0
\(743\) 16242.0 0.801967 0.400983 − 0.916085i \(-0.368669\pi\)
0.400983 + 0.916085i \(0.368669\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −19866.0 −0.973037
\(748\) 0 0
\(749\) 26826.0 1.30868
\(750\) 0 0
\(751\) −10712.0 −0.520488 −0.260244 − 0.965543i \(-0.583803\pi\)
−0.260244 + 0.965543i \(0.583803\pi\)
\(752\) 0 0
\(753\) −49539.0 −2.39748
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 13504.0 0.648364 0.324182 − 0.945995i \(-0.394911\pi\)
0.324182 + 0.945995i \(0.394911\pi\)
\(758\) 0 0
\(759\) 14742.0 0.705008
\(760\) 0 0
\(761\) −18123.0 −0.863283 −0.431641 − 0.902045i \(-0.642065\pi\)
−0.431641 + 0.902045i \(0.642065\pi\)
\(762\) 0 0
\(763\) −7820.00 −0.371039
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 23520.0 1.10725
\(768\) 0 0
\(769\) 9485.00 0.444783 0.222391 − 0.974957i \(-0.428614\pi\)
0.222391 + 0.974957i \(0.428614\pi\)
\(770\) 0 0
\(771\) −47922.0 −2.23848
\(772\) 0 0
\(773\) 31248.0 1.45396 0.726981 − 0.686658i \(-0.240923\pi\)
0.726981 + 0.686658i \(0.240923\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 34748.0 1.60435
\(778\) 0 0
\(779\) −12495.0 −0.574685
\(780\) 0 0
\(781\) −19116.0 −0.875831
\(782\) 0 0
\(783\) 4200.00 0.191693
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −15484.0 −0.701328 −0.350664 − 0.936501i \(-0.614044\pi\)
−0.350664 + 0.936501i \(0.614044\pi\)
\(788\) 0 0
\(789\) −24066.0 −1.08590
\(790\) 0 0
\(791\) −70482.0 −3.16821
\(792\) 0 0
\(793\) −6664.00 −0.298418
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −34146.0 −1.51758 −0.758791 − 0.651334i \(-0.774210\pi\)
−0.758791 + 0.651334i \(0.774210\pi\)
\(798\) 0 0
\(799\) 1764.00 0.0781049
\(800\) 0 0
\(801\) 16170.0 0.713282
\(802\) 0 0
\(803\) −3591.00 −0.157813
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −11760.0 −0.512976
\(808\) 0 0
\(809\) −36030.0 −1.56582 −0.782909 − 0.622136i \(-0.786265\pi\)
−0.782909 + 0.622136i \(0.786265\pi\)
\(810\) 0 0
\(811\) 6748.00 0.292175 0.146088 − 0.989272i \(-0.453332\pi\)
0.146088 + 0.989272i \(0.453332\pi\)
\(812\) 0 0
\(813\) −36554.0 −1.57688
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 5180.00 0.221818
\(818\) 0 0
\(819\) −20944.0 −0.893581
\(820\) 0 0
\(821\) −5598.00 −0.237968 −0.118984 − 0.992896i \(-0.537964\pi\)
−0.118984 + 0.992896i \(0.537964\pi\)
\(822\) 0 0
\(823\) 5732.00 0.242776 0.121388 − 0.992605i \(-0.461265\pi\)
0.121388 + 0.992605i \(0.461265\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −39999.0 −1.68186 −0.840932 − 0.541141i \(-0.817993\pi\)
−0.840932 + 0.541141i \(0.817993\pi\)
\(828\) 0 0
\(829\) 16940.0 0.709711 0.354856 − 0.934921i \(-0.384530\pi\)
0.354856 + 0.934921i \(0.384530\pi\)
\(830\) 0 0
\(831\) 9688.00 0.404420
\(832\) 0 0
\(833\) −17073.0 −0.710137
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 6370.00 0.263058
\(838\) 0 0
\(839\) 45360.0 1.86651 0.933255 − 0.359216i \(-0.116956\pi\)
0.933255 + 0.359216i \(0.116956\pi\)
\(840\) 0 0
\(841\) −9989.00 −0.409570
\(842\) 0 0
\(843\) −27006.0 −1.10336
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 20468.0 0.830329
\(848\) 0 0
\(849\) 29939.0 1.21025
\(850\) 0 0
\(851\) 11388.0 0.458726
\(852\) 0 0
\(853\) 43918.0 1.76286 0.881432 − 0.472310i \(-0.156580\pi\)
0.881432 + 0.472310i \(0.156580\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3339.00 0.133090 0.0665450 − 0.997783i \(-0.478802\pi\)
0.0665450 + 0.997783i \(0.478802\pi\)
\(858\) 0 0
\(859\) 36925.0 1.46666 0.733332 − 0.679870i \(-0.237964\pi\)
0.733332 + 0.679870i \(0.237964\pi\)
\(860\) 0 0
\(861\) −84966.0 −3.36311
\(862\) 0 0
\(863\) −40608.0 −1.60175 −0.800876 − 0.598830i \(-0.795633\pi\)
−0.800876 + 0.598830i \(0.795633\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −31304.0 −1.22623
\(868\) 0 0
\(869\) 17550.0 0.685090
\(870\) 0 0
\(871\) 12908.0 0.502148
\(872\) 0 0
\(873\) −24332.0 −0.943314
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −35156.0 −1.35363 −0.676815 − 0.736153i \(-0.736640\pi\)
−0.676815 + 0.736153i \(0.736640\pi\)
\(878\) 0 0
\(879\) −44394.0 −1.70350
\(880\) 0 0
\(881\) 2142.00 0.0819135 0.0409568 − 0.999161i \(-0.486959\pi\)
0.0409568 + 0.999161i \(0.486959\pi\)
\(882\) 0 0
\(883\) −19153.0 −0.729954 −0.364977 − 0.931016i \(-0.618923\pi\)
−0.364977 + 0.931016i \(0.618923\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −24444.0 −0.925309 −0.462655 − 0.886539i \(-0.653103\pi\)
−0.462655 + 0.886539i \(0.653103\pi\)
\(888\) 0 0
\(889\) 37876.0 1.42893
\(890\) 0 0
\(891\) 22653.0 0.851744
\(892\) 0 0
\(893\) 2940.00 0.110172
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −15288.0 −0.569065
\(898\) 0 0
\(899\) 21840.0 0.810239
\(900\) 0 0
\(901\) 14742.0 0.545091
\(902\) 0 0
\(903\) 35224.0 1.29810
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 20036.0 0.733500 0.366750 − 0.930320i \(-0.380470\pi\)
0.366750 + 0.930320i \(0.380470\pi\)
\(908\) 0 0
\(909\) 10164.0 0.370868
\(910\) 0 0
\(911\) 33468.0 1.21717 0.608586 − 0.793488i \(-0.291737\pi\)
0.608586 + 0.793488i \(0.291737\pi\)
\(912\) 0 0
\(913\) 24381.0 0.883782
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8568.00 0.308550
\(918\) 0 0
\(919\) −35090.0 −1.25953 −0.629767 − 0.776784i \(-0.716850\pi\)
−0.629767 + 0.776784i \(0.716850\pi\)
\(920\) 0 0
\(921\) 40817.0 1.46033
\(922\) 0 0
\(923\) 19824.0 0.706950
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 17864.0 0.632935
\(928\) 0 0
\(929\) −32130.0 −1.13472 −0.567358 − 0.823471i \(-0.692034\pi\)
−0.567358 + 0.823471i \(0.692034\pi\)
\(930\) 0 0
\(931\) −28455.0 −1.00169
\(932\) 0 0
\(933\) 17346.0 0.608663
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −20531.0 −0.715815 −0.357907 − 0.933757i \(-0.616510\pi\)
−0.357907 + 0.933757i \(0.616510\pi\)
\(938\) 0 0
\(939\) 19306.0 0.670956
\(940\) 0 0
\(941\) 7812.00 0.270631 0.135316 − 0.990803i \(-0.456795\pi\)
0.135316 + 0.990803i \(0.456795\pi\)
\(942\) 0 0
\(943\) −27846.0 −0.961602
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 23256.0 0.798013 0.399007 − 0.916948i \(-0.369355\pi\)
0.399007 + 0.916948i \(0.369355\pi\)
\(948\) 0 0
\(949\) 3724.00 0.127383
\(950\) 0 0
\(951\) −4452.00 −0.151804
\(952\) 0 0
\(953\) −14097.0 −0.479167 −0.239584 − 0.970876i \(-0.577011\pi\)
−0.239584 + 0.970876i \(0.577011\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 22680.0 0.766082
\(958\) 0 0
\(959\) 65994.0 2.22217
\(960\) 0 0
\(961\) 3333.00 0.111879
\(962\) 0 0
\(963\) −17358.0 −0.580845
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −34144.0 −1.13547 −0.567734 − 0.823212i \(-0.692180\pi\)
−0.567734 + 0.823212i \(0.692180\pi\)
\(968\) 0 0
\(969\) 5145.00 0.170569
\(970\) 0 0
\(971\) 32613.0 1.07786 0.538929 − 0.842351i \(-0.318829\pi\)
0.538929 + 0.842351i \(0.318829\pi\)
\(972\) 0 0
\(973\) −55930.0 −1.84279
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 43359.0 1.41983 0.709917 − 0.704286i \(-0.248733\pi\)
0.709917 + 0.704286i \(0.248733\pi\)
\(978\) 0 0
\(979\) −19845.0 −0.647854
\(980\) 0 0
\(981\) 5060.00 0.164682
\(982\) 0 0
\(983\) −28518.0 −0.925313 −0.462657 − 0.886538i \(-0.653104\pi\)
−0.462657 + 0.886538i \(0.653104\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 19992.0 0.644734
\(988\) 0 0
\(989\) 11544.0 0.371161
\(990\) 0 0
\(991\) −18122.0 −0.580892 −0.290446 − 0.956891i \(-0.593804\pi\)
−0.290446 + 0.956891i \(0.593804\pi\)
\(992\) 0 0
\(993\) −48209.0 −1.54065
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 28924.0 0.918789 0.459394 − 0.888232i \(-0.348066\pi\)
0.459394 + 0.888232i \(0.348066\pi\)
\(998\) 0 0
\(999\) 5110.00 0.161835
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 400.4.a.r.1.1 1
4.3 odd 2 50.4.a.a.1.1 ✓ 1
5.2 odd 4 400.4.c.d.49.1 2
5.3 odd 4 400.4.c.d.49.2 2
5.4 even 2 400.4.a.d.1.1 1
8.3 odd 2 1600.4.a.bu.1.1 1
8.5 even 2 1600.4.a.g.1.1 1
12.11 even 2 450.4.a.t.1.1 1
20.3 even 4 50.4.b.b.49.2 2
20.7 even 4 50.4.b.b.49.1 2
20.19 odd 2 50.4.a.e.1.1 yes 1
28.27 even 2 2450.4.a.t.1.1 1
40.19 odd 2 1600.4.a.f.1.1 1
40.29 even 2 1600.4.a.bv.1.1 1
60.23 odd 4 450.4.c.c.199.1 2
60.47 odd 4 450.4.c.c.199.2 2
60.59 even 2 450.4.a.a.1.1 1
140.139 even 2 2450.4.a.y.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
50.4.a.a.1.1 ✓ 1 4.3 odd 2
50.4.a.e.1.1 yes 1 20.19 odd 2
50.4.b.b.49.1 2 20.7 even 4
50.4.b.b.49.2 2 20.3 even 4
400.4.a.d.1.1 1 5.4 even 2
400.4.a.r.1.1 1 1.1 even 1 trivial
400.4.c.d.49.1 2 5.2 odd 4
400.4.c.d.49.2 2 5.3 odd 4
450.4.a.a.1.1 1 60.59 even 2
450.4.a.t.1.1 1 12.11 even 2
450.4.c.c.199.1 2 60.23 odd 4
450.4.c.c.199.2 2 60.47 odd 4
1600.4.a.f.1.1 1 40.19 odd 2
1600.4.a.g.1.1 1 8.5 even 2
1600.4.a.bu.1.1 1 8.3 odd 2
1600.4.a.bv.1.1 1 40.29 even 2
2450.4.a.t.1.1 1 28.27 even 2
2450.4.a.y.1.1 1 140.139 even 2