Properties

Label 10000.2.a.bj.1.4
Level $10000$
Weight $2$
Character 10000.1
Self dual yes
Analytic conductor $79.850$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [10000,2,Mod(1,10000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10000.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10000 = 2^{4} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 10000.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: \(79.8504020213\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.14884000000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 25)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(2.30927\) of defining polynomial
Character \(\chi\) \(=\) 10000.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.474903 q^{3} -3.03582 q^{7} -2.77447 q^{9} +O(q^{10})\) \(q-0.474903 q^{3} -3.03582 q^{7} -2.77447 q^{9} -2.00000 q^{11} -1.42721 q^{13} +1.86025 q^{17} -0.903319 q^{19} +1.44172 q^{21} +3.32932 q^{23} +2.74231 q^{27} +3.96307 q^{29} +6.43997 q^{31} +0.949806 q^{33} +3.82022 q^{37} +0.677786 q^{39} -1.83422 q^{41} +3.59445 q^{43} +4.79995 q^{47} +2.21619 q^{49} -0.883436 q^{51} -9.50473 q^{53} +0.428989 q^{57} -10.6456 q^{59} +14.2742 q^{61} +8.42278 q^{63} +10.6902 q^{67} -1.58111 q^{69} -12.4598 q^{71} +0.267631 q^{73} +6.07163 q^{77} +8.57176 q^{79} +7.02107 q^{81} +12.6182 q^{83} -1.88207 q^{87} -4.76796 q^{89} +4.33275 q^{91} -3.05836 q^{93} -9.95805 q^{97} +5.54893 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} - 16 q^{11} - 10 q^{19} + 6 q^{21} + 20 q^{29} - 16 q^{31} - 18 q^{39} + 26 q^{41} - 14 q^{49} + 4 q^{51} - 30 q^{59} + 6 q^{61} + 8 q^{69} - 46 q^{71} - 10 q^{79} - 32 q^{81} + 30 q^{89} + 14 q^{91} - 8 q^{99}+O(q^{100}) \) 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
\(3\) −0.474903 −0.274185 −0.137093 0.990558i \(-0.543776\pi\)
−0.137093 + 0.990558i \(0.543776\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3.03582 −1.14743 −0.573716 0.819055i \(-0.694498\pi\)
−0.573716 + 0.819055i \(0.694498\pi\)
\(8\) 0 0
\(9\) −2.77447 −0.924822
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −1.42721 −0.395837 −0.197918 0.980218i \(-0.563418\pi\)
−0.197918 + 0.980218i \(0.563418\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.86025 0.451176 0.225588 0.974223i \(-0.427570\pi\)
0.225588 + 0.974223i \(0.427570\pi\)
\(18\) 0 0
\(19\) −0.903319 −0.207236 −0.103618 0.994617i \(-0.533042\pi\)
−0.103618 + 0.994617i \(0.533042\pi\)
\(20\) 0 0
\(21\) 1.44172 0.314609
\(22\) 0 0
\(23\) 3.32932 0.694212 0.347106 0.937826i \(-0.387164\pi\)
0.347106 + 0.937826i \(0.387164\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 2.74231 0.527758
\(28\) 0 0
\(29\) 3.96307 0.735924 0.367962 0.929841i \(-0.380056\pi\)
0.367962 + 0.929841i \(0.380056\pi\)
\(30\) 0 0
\(31\) 6.43997 1.15665 0.578326 0.815806i \(-0.303706\pi\)
0.578326 + 0.815806i \(0.303706\pi\)
\(32\) 0 0
\(33\) 0.949806 0.165340
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.82022 0.628040 0.314020 0.949416i \(-0.398324\pi\)
0.314020 + 0.949416i \(0.398324\pi\)
\(38\) 0 0
\(39\) 0.677786 0.108533
\(40\) 0 0
\(41\) −1.83422 −0.286457 −0.143228 0.989690i \(-0.545748\pi\)
−0.143228 + 0.989690i \(0.545748\pi\)
\(42\) 0 0
\(43\) 3.59445 0.548149 0.274074 0.961708i \(-0.411629\pi\)
0.274074 + 0.961708i \(0.411629\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.79995 0.700144 0.350072 0.936723i \(-0.386157\pi\)
0.350072 + 0.936723i \(0.386157\pi\)
\(48\) 0 0
\(49\) 2.21619 0.316598
\(50\) 0 0
\(51\) −0.883436 −0.123706
\(52\) 0 0
\(53\) −9.50473 −1.30558 −0.652788 0.757541i \(-0.726401\pi\)
−0.652788 + 0.757541i \(0.726401\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.428989 0.0568209
\(58\) 0 0
\(59\) −10.6456 −1.38594 −0.692971 0.720966i \(-0.743698\pi\)
−0.692971 + 0.720966i \(0.743698\pi\)
\(60\) 0 0
\(61\) 14.2742 1.82762 0.913811 0.406140i \(-0.133125\pi\)
0.913811 + 0.406140i \(0.133125\pi\)
\(62\) 0 0
\(63\) 8.42278 1.06117
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 10.6902 1.30601 0.653007 0.757352i \(-0.273507\pi\)
0.653007 + 0.757352i \(0.273507\pi\)
\(68\) 0 0
\(69\) −1.58111 −0.190343
\(70\) 0 0
\(71\) −12.4598 −1.47871 −0.739356 0.673315i \(-0.764870\pi\)
−0.739356 + 0.673315i \(0.764870\pi\)
\(72\) 0 0
\(73\) 0.267631 0.0313239 0.0156619 0.999877i \(-0.495014\pi\)
0.0156619 + 0.999877i \(0.495014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 6.07163 0.691927
\(78\) 0 0
\(79\) 8.57176 0.964398 0.482199 0.876062i \(-0.339838\pi\)
0.482199 + 0.876062i \(0.339838\pi\)
\(80\) 0 0
\(81\) 7.02107 0.780119
\(82\) 0 0
\(83\) 12.6182 1.38502 0.692512 0.721406i \(-0.256504\pi\)
0.692512 + 0.721406i \(0.256504\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −1.88207 −0.201779
\(88\) 0 0
\(89\) −4.76796 −0.505402 −0.252701 0.967544i \(-0.581319\pi\)
−0.252701 + 0.967544i \(0.581319\pi\)
\(90\) 0 0
\(91\) 4.33275 0.454196
\(92\) 0 0
\(93\) −3.05836 −0.317137
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −9.95805 −1.01109 −0.505543 0.862801i \(-0.668708\pi\)
−0.505543 + 0.862801i \(0.668708\pi\)
\(98\) 0 0
\(99\) 5.54893 0.557689
\(100\) 0 0
\(101\) 9.34612 0.929974 0.464987 0.885318i \(-0.346059\pi\)
0.464987 + 0.885318i \(0.346059\pi\)
\(102\) 0 0
\(103\) −9.08408 −0.895081 −0.447540 0.894264i \(-0.647700\pi\)
−0.447540 + 0.894264i \(0.647700\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −5.62871 −0.544148 −0.272074 0.962276i \(-0.587710\pi\)
−0.272074 + 0.962276i \(0.587710\pi\)
\(108\) 0 0
\(109\) −10.1130 −0.968649 −0.484325 0.874888i \(-0.660935\pi\)
−0.484325 + 0.874888i \(0.660935\pi\)
\(110\) 0 0
\(111\) −1.81423 −0.172199
\(112\) 0 0
\(113\) 10.7120 1.00770 0.503851 0.863791i \(-0.331916\pi\)
0.503851 + 0.863791i \(0.331916\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.95975 0.366079
\(118\) 0 0
\(119\) −5.64737 −0.517693
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 0.871076 0.0785423
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.3609 1.00812 0.504060 0.863669i \(-0.331839\pi\)
0.504060 + 0.863669i \(0.331839\pi\)
\(128\) 0 0
\(129\) −1.70702 −0.150294
\(130\) 0 0
\(131\) −7.98771 −0.697890 −0.348945 0.937143i \(-0.613460\pi\)
−0.348945 + 0.937143i \(0.613460\pi\)
\(132\) 0 0
\(133\) 2.74231 0.237789
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.33726 0.797736 0.398868 0.917008i \(-0.369403\pi\)
0.398868 + 0.917008i \(0.369403\pi\)
\(138\) 0 0
\(139\) −17.9150 −1.51953 −0.759767 0.650195i \(-0.774687\pi\)
−0.759767 + 0.650195i \(0.774687\pi\)
\(140\) 0 0
\(141\) −2.27951 −0.191969
\(142\) 0 0
\(143\) 2.85442 0.238699
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1.05247 −0.0868065
\(148\) 0 0
\(149\) −6.31395 −0.517259 −0.258629 0.965977i \(-0.583271\pi\)
−0.258629 + 0.965977i \(0.583271\pi\)
\(150\) 0 0
\(151\) −4.71947 −0.384065 −0.192033 0.981389i \(-0.561508\pi\)
−0.192033 + 0.981389i \(0.561508\pi\)
\(152\) 0 0
\(153\) −5.16119 −0.417258
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.46908 0.117245 0.0586225 0.998280i \(-0.481329\pi\)
0.0586225 + 0.998280i \(0.481329\pi\)
\(158\) 0 0
\(159\) 4.51382 0.357969
\(160\) 0 0
\(161\) −10.1072 −0.796560
\(162\) 0 0
\(163\) −4.45969 −0.349310 −0.174655 0.984630i \(-0.555881\pi\)
−0.174655 + 0.984630i \(0.555881\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −10.4337 −0.807381 −0.403691 0.914896i \(-0.632273\pi\)
−0.403691 + 0.914896i \(0.632273\pi\)
\(168\) 0 0
\(169\) −10.9631 −0.843313
\(170\) 0 0
\(171\) 2.50623 0.191656
\(172\) 0 0
\(173\) −7.67619 −0.583610 −0.291805 0.956478i \(-0.594256\pi\)
−0.291805 + 0.956478i \(0.594256\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 5.05563 0.380005
\(178\) 0 0
\(179\) −15.5168 −1.15978 −0.579889 0.814696i \(-0.696904\pi\)
−0.579889 + 0.814696i \(0.696904\pi\)
\(180\) 0 0
\(181\) −1.59056 −0.118225 −0.0591126 0.998251i \(-0.518827\pi\)
−0.0591126 + 0.998251i \(0.518827\pi\)
\(182\) 0 0
\(183\) −6.77885 −0.501107
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.72049 −0.272069
\(188\) 0 0
\(189\) −8.32515 −0.605566
\(190\) 0 0
\(191\) −19.6684 −1.42316 −0.711579 0.702606i \(-0.752020\pi\)
−0.711579 + 0.702606i \(0.752020\pi\)
\(192\) 0 0
\(193\) 13.1100 0.943680 0.471840 0.881684i \(-0.343590\pi\)
0.471840 + 0.881684i \(0.343590\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −3.42949 −0.244341 −0.122170 0.992509i \(-0.538985\pi\)
−0.122170 + 0.992509i \(0.538985\pi\)
\(198\) 0 0
\(199\) −17.6959 −1.25443 −0.627215 0.778846i \(-0.715805\pi\)
−0.627215 + 0.778846i \(0.715805\pi\)
\(200\) 0 0
\(201\) −5.07680 −0.358090
\(202\) 0 0
\(203\) −12.0312 −0.844422
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −9.23710 −0.642023
\(208\) 0 0
\(209\) 1.80664 0.124968
\(210\) 0 0
\(211\) 3.24366 0.223303 0.111651 0.993747i \(-0.464386\pi\)
0.111651 + 0.993747i \(0.464386\pi\)
\(212\) 0 0
\(213\) 5.91722 0.405441
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −19.5506 −1.32718
\(218\) 0 0
\(219\) −0.127099 −0.00858854
\(220\) 0 0
\(221\) −2.65496 −0.178592
\(222\) 0 0
\(223\) −28.7148 −1.92288 −0.961441 0.275010i \(-0.911319\pi\)
−0.961441 + 0.275010i \(0.911319\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.7206 −0.777926 −0.388963 0.921253i \(-0.627167\pi\)
−0.388963 + 0.921253i \(0.627167\pi\)
\(228\) 0 0
\(229\) −16.4013 −1.08383 −0.541914 0.840434i \(-0.682300\pi\)
−0.541914 + 0.840434i \(0.682300\pi\)
\(230\) 0 0
\(231\) −2.88344 −0.189716
\(232\) 0 0
\(233\) 22.5146 1.47498 0.737490 0.675358i \(-0.236011\pi\)
0.737490 + 0.675358i \(0.236011\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4.07075 −0.264424
\(238\) 0 0
\(239\) 6.63333 0.429074 0.214537 0.976716i \(-0.431176\pi\)
0.214537 + 0.976716i \(0.431176\pi\)
\(240\) 0 0
\(241\) 26.2261 1.68937 0.844684 0.535265i \(-0.179788\pi\)
0.844684 + 0.535265i \(0.179788\pi\)
\(242\) 0 0
\(243\) −11.5613 −0.741655
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.28923 0.0820315
\(248\) 0 0
\(249\) −5.99241 −0.379753
\(250\) 0 0
\(251\) 10.9121 0.688766 0.344383 0.938829i \(-0.388088\pi\)
0.344383 + 0.938829i \(0.388088\pi\)
\(252\) 0 0
\(253\) −6.65865 −0.418626
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.58051 0.410481 0.205240 0.978712i \(-0.434202\pi\)
0.205240 + 0.978712i \(0.434202\pi\)
\(258\) 0 0
\(259\) −11.5975 −0.720632
\(260\) 0 0
\(261\) −10.9954 −0.680599
\(262\) 0 0
\(263\) 27.1073 1.67151 0.835753 0.549106i \(-0.185032\pi\)
0.835753 + 0.549106i \(0.185032\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 2.26432 0.138574
\(268\) 0 0
\(269\) 1.00945 0.0615474 0.0307737 0.999526i \(-0.490203\pi\)
0.0307737 + 0.999526i \(0.490203\pi\)
\(270\) 0 0
\(271\) −6.25203 −0.379784 −0.189892 0.981805i \(-0.560814\pi\)
−0.189892 + 0.981805i \(0.560814\pi\)
\(272\) 0 0
\(273\) −2.05763 −0.124534
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −24.6703 −1.48230 −0.741148 0.671342i \(-0.765718\pi\)
−0.741148 + 0.671342i \(0.765718\pi\)
\(278\) 0 0
\(279\) −17.8675 −1.06970
\(280\) 0 0
\(281\) 1.83891 0.109700 0.0548502 0.998495i \(-0.482532\pi\)
0.0548502 + 0.998495i \(0.482532\pi\)
\(282\) 0 0
\(283\) 8.64116 0.513664 0.256832 0.966456i \(-0.417321\pi\)
0.256832 + 0.966456i \(0.417321\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.56835 0.328690
\(288\) 0 0
\(289\) −13.5395 −0.796440
\(290\) 0 0
\(291\) 4.72910 0.277225
\(292\) 0 0
\(293\) 6.29156 0.367557 0.183779 0.982968i \(-0.441167\pi\)
0.183779 + 0.982968i \(0.441167\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −5.48462 −0.318250
\(298\) 0 0
\(299\) −4.75164 −0.274795
\(300\) 0 0
\(301\) −10.9121 −0.628963
\(302\) 0 0
\(303\) −4.43850 −0.254985
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 28.6661 1.63606 0.818030 0.575175i \(-0.195066\pi\)
0.818030 + 0.575175i \(0.195066\pi\)
\(308\) 0 0
\(309\) 4.31405 0.245418
\(310\) 0 0
\(311\) 7.83649 0.444367 0.222183 0.975005i \(-0.428682\pi\)
0.222183 + 0.975005i \(0.428682\pi\)
\(312\) 0 0
\(313\) 21.4093 1.21012 0.605061 0.796179i \(-0.293149\pi\)
0.605061 + 0.796179i \(0.293149\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.01983 0.225776 0.112888 0.993608i \(-0.463990\pi\)
0.112888 + 0.993608i \(0.463990\pi\)
\(318\) 0 0
\(319\) −7.92614 −0.443779
\(320\) 0 0
\(321\) 2.67309 0.149197
\(322\) 0 0
\(323\) −1.68040 −0.0934997
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 4.80269 0.265589
\(328\) 0 0
\(329\) −14.5718 −0.803367
\(330\) 0 0
\(331\) −11.6439 −0.640005 −0.320002 0.947417i \(-0.603684\pi\)
−0.320002 + 0.947417i \(0.603684\pi\)
\(332\) 0 0
\(333\) −10.5991 −0.580825
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −21.5348 −1.17307 −0.586537 0.809922i \(-0.699509\pi\)
−0.586537 + 0.809922i \(0.699509\pi\)
\(338\) 0 0
\(339\) −5.08716 −0.276297
\(340\) 0 0
\(341\) −12.8799 −0.697487
\(342\) 0 0
\(343\) 14.5228 0.784157
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −15.5972 −0.837303 −0.418652 0.908147i \(-0.637497\pi\)
−0.418652 + 0.908147i \(0.637497\pi\)
\(348\) 0 0
\(349\) 5.56598 0.297940 0.148970 0.988842i \(-0.452404\pi\)
0.148970 + 0.988842i \(0.452404\pi\)
\(350\) 0 0
\(351\) −3.91385 −0.208906
\(352\) 0 0
\(353\) 8.02216 0.426977 0.213488 0.976946i \(-0.431517\pi\)
0.213488 + 0.976946i \(0.431517\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 2.68195 0.141944
\(358\) 0 0
\(359\) 12.3427 0.651424 0.325712 0.945469i \(-0.394396\pi\)
0.325712 + 0.945469i \(0.394396\pi\)
\(360\) 0 0
\(361\) −18.1840 −0.957053
\(362\) 0 0
\(363\) 3.32432 0.174482
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −26.8749 −1.40286 −0.701430 0.712738i \(-0.747455\pi\)
−0.701430 + 0.712738i \(0.747455\pi\)
\(368\) 0 0
\(369\) 5.08898 0.264922
\(370\) 0 0
\(371\) 28.8546 1.49806
\(372\) 0 0
\(373\) −27.6389 −1.43109 −0.715544 0.698567i \(-0.753821\pi\)
−0.715544 + 0.698567i \(0.753821\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5.65614 −0.291306
\(378\) 0 0
\(379\) 3.47462 0.178479 0.0892397 0.996010i \(-0.471556\pi\)
0.0892397 + 0.996010i \(0.471556\pi\)
\(380\) 0 0
\(381\) −5.39534 −0.276412
\(382\) 0 0
\(383\) −27.3719 −1.39864 −0.699319 0.714810i \(-0.746513\pi\)
−0.699319 + 0.714810i \(0.746513\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −9.97269 −0.506940
\(388\) 0 0
\(389\) 10.8845 0.551867 0.275934 0.961177i \(-0.411013\pi\)
0.275934 + 0.961177i \(0.411013\pi\)
\(390\) 0 0
\(391\) 6.19336 0.313212
\(392\) 0 0
\(393\) 3.79339 0.191351
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 16.2212 0.814120 0.407060 0.913401i \(-0.366554\pi\)
0.407060 + 0.913401i \(0.366554\pi\)
\(398\) 0 0
\(399\) −1.30233 −0.0651981
\(400\) 0 0
\(401\) 3.78686 0.189107 0.0945534 0.995520i \(-0.469858\pi\)
0.0945534 + 0.995520i \(0.469858\pi\)
\(402\) 0 0
\(403\) −9.19118 −0.457845
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −7.64044 −0.378722
\(408\) 0 0
\(409\) 1.85585 0.0917661 0.0458831 0.998947i \(-0.485390\pi\)
0.0458831 + 0.998947i \(0.485390\pi\)
\(410\) 0 0
\(411\) −4.43429 −0.218728
\(412\) 0 0
\(413\) 32.3181 1.59027
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 8.50790 0.416634
\(418\) 0 0
\(419\) −14.3472 −0.700907 −0.350453 0.936580i \(-0.613972\pi\)
−0.350453 + 0.936580i \(0.613972\pi\)
\(420\) 0 0
\(421\) 15.4545 0.753207 0.376604 0.926374i \(-0.377092\pi\)
0.376604 + 0.926374i \(0.377092\pi\)
\(422\) 0 0
\(423\) −13.3173 −0.647509
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −43.3338 −2.09707
\(428\) 0 0
\(429\) −1.35557 −0.0654477
\(430\) 0 0
\(431\) −12.5043 −0.602311 −0.301156 0.953575i \(-0.597372\pi\)
−0.301156 + 0.953575i \(0.597372\pi\)
\(432\) 0 0
\(433\) −22.4951 −1.08105 −0.540524 0.841329i \(-0.681774\pi\)
−0.540524 + 0.841329i \(0.681774\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.00744 −0.143865
\(438\) 0 0
\(439\) −12.1776 −0.581204 −0.290602 0.956844i \(-0.593855\pi\)
−0.290602 + 0.956844i \(0.593855\pi\)
\(440\) 0 0
\(441\) −6.14873 −0.292797
\(442\) 0 0
\(443\) 20.7101 0.983968 0.491984 0.870604i \(-0.336272\pi\)
0.491984 + 0.870604i \(0.336272\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 2.99851 0.141825
\(448\) 0 0
\(449\) −25.9539 −1.22484 −0.612420 0.790533i \(-0.709804\pi\)
−0.612420 + 0.790533i \(0.709804\pi\)
\(450\) 0 0
\(451\) 3.66844 0.172740
\(452\) 0 0
\(453\) 2.24129 0.105305
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −8.50150 −0.397684 −0.198842 0.980032i \(-0.563718\pi\)
−0.198842 + 0.980032i \(0.563718\pi\)
\(458\) 0 0
\(459\) 5.10137 0.238112
\(460\) 0 0
\(461\) 14.7851 0.688609 0.344305 0.938858i \(-0.388115\pi\)
0.344305 + 0.938858i \(0.388115\pi\)
\(462\) 0 0
\(463\) 22.1921 1.03135 0.515677 0.856783i \(-0.327540\pi\)
0.515677 + 0.856783i \(0.327540\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 28.5014 1.31889 0.659443 0.751754i \(-0.270792\pi\)
0.659443 + 0.751754i \(0.270792\pi\)
\(468\) 0 0
\(469\) −32.4534 −1.49856
\(470\) 0 0
\(471\) −0.697669 −0.0321469
\(472\) 0 0
\(473\) −7.18891 −0.330546
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 26.3706 1.20743
\(478\) 0 0
\(479\) −25.0569 −1.14488 −0.572440 0.819947i \(-0.694003\pi\)
−0.572440 + 0.819947i \(0.694003\pi\)
\(480\) 0 0
\(481\) −5.45225 −0.248601
\(482\) 0 0
\(483\) 4.79995 0.218405
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.46479 0.0663758 0.0331879 0.999449i \(-0.489434\pi\)
0.0331879 + 0.999449i \(0.489434\pi\)
\(488\) 0 0
\(489\) 2.11792 0.0957757
\(490\) 0 0
\(491\) 20.0686 0.905685 0.452843 0.891591i \(-0.350410\pi\)
0.452843 + 0.891591i \(0.350410\pi\)
\(492\) 0 0
\(493\) 7.37229 0.332031
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 37.8258 1.69672
\(498\) 0 0
\(499\) −0.624999 −0.0279788 −0.0139894 0.999902i \(-0.504453\pi\)
−0.0139894 + 0.999902i \(0.504453\pi\)
\(500\) 0 0
\(501\) 4.95498 0.221372
\(502\) 0 0
\(503\) 19.3052 0.860776 0.430388 0.902644i \(-0.358377\pi\)
0.430388 + 0.902644i \(0.358377\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 5.20639 0.231224
\(508\) 0 0
\(509\) −10.5202 −0.466298 −0.233149 0.972441i \(-0.574903\pi\)
−0.233149 + 0.972441i \(0.574903\pi\)
\(510\) 0 0
\(511\) −0.812479 −0.0359420
\(512\) 0 0
\(513\) −2.47718 −0.109370
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −9.59989 −0.422203
\(518\) 0 0
\(519\) 3.64545 0.160017
\(520\) 0 0
\(521\) −10.0070 −0.438413 −0.219207 0.975678i \(-0.570347\pi\)
−0.219207 + 0.975678i \(0.570347\pi\)
\(522\) 0 0
\(523\) −22.7830 −0.996233 −0.498117 0.867110i \(-0.665975\pi\)
−0.498117 + 0.867110i \(0.665975\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 11.9799 0.521854
\(528\) 0 0
\(529\) −11.9156 −0.518070
\(530\) 0 0
\(531\) 29.5359 1.28175
\(532\) 0 0
\(533\) 2.61782 0.113390
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 7.36896 0.317994
\(538\) 0 0
\(539\) −4.43237 −0.190916
\(540\) 0 0
\(541\) 3.25900 0.140115 0.0700576 0.997543i \(-0.477682\pi\)
0.0700576 + 0.997543i \(0.477682\pi\)
\(542\) 0 0
\(543\) 0.755360 0.0324156
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 13.5883 0.580993 0.290497 0.956876i \(-0.406179\pi\)
0.290497 + 0.956876i \(0.406179\pi\)
\(548\) 0 0
\(549\) −39.6033 −1.69023
\(550\) 0 0
\(551\) −3.57992 −0.152510
\(552\) 0 0
\(553\) −26.0223 −1.10658
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −27.6399 −1.17114 −0.585571 0.810621i \(-0.699130\pi\)
−0.585571 + 0.810621i \(0.699130\pi\)
\(558\) 0 0
\(559\) −5.13004 −0.216978
\(560\) 0 0
\(561\) 1.76687 0.0745974
\(562\) 0 0
\(563\) 1.65925 0.0699291 0.0349646 0.999389i \(-0.488868\pi\)
0.0349646 + 0.999389i \(0.488868\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −21.3147 −0.895133
\(568\) 0 0
\(569\) 17.8828 0.749686 0.374843 0.927088i \(-0.377697\pi\)
0.374843 + 0.927088i \(0.377697\pi\)
\(570\) 0 0
\(571\) 36.8723 1.54306 0.771530 0.636193i \(-0.219492\pi\)
0.771530 + 0.636193i \(0.219492\pi\)
\(572\) 0 0
\(573\) 9.34060 0.390209
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −22.8137 −0.949746 −0.474873 0.880054i \(-0.657506\pi\)
−0.474873 + 0.880054i \(0.657506\pi\)
\(578\) 0 0
\(579\) −6.22599 −0.258743
\(580\) 0 0
\(581\) −38.3065 −1.58922
\(582\) 0 0
\(583\) 19.0095 0.787291
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 11.0855 0.457546 0.228773 0.973480i \(-0.426529\pi\)
0.228773 + 0.973480i \(0.426529\pi\)
\(588\) 0 0
\(589\) −5.81734 −0.239699
\(590\) 0 0
\(591\) 1.62867 0.0669947
\(592\) 0 0
\(593\) 11.1321 0.457139 0.228570 0.973528i \(-0.426595\pi\)
0.228570 + 0.973528i \(0.426595\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 8.40384 0.343946
\(598\) 0 0
\(599\) −36.2736 −1.48210 −0.741049 0.671451i \(-0.765671\pi\)
−0.741049 + 0.671451i \(0.765671\pi\)
\(600\) 0 0
\(601\) −15.1051 −0.616150 −0.308075 0.951362i \(-0.599685\pi\)
−0.308075 + 0.951362i \(0.599685\pi\)
\(602\) 0 0
\(603\) −29.6596 −1.20783
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 33.5066 1.35999 0.679996 0.733216i \(-0.261982\pi\)
0.679996 + 0.733216i \(0.261982\pi\)
\(608\) 0 0
\(609\) 5.71363 0.231528
\(610\) 0 0
\(611\) −6.85053 −0.277143
\(612\) 0 0
\(613\) −28.0289 −1.13208 −0.566038 0.824379i \(-0.691524\pi\)
−0.566038 + 0.824379i \(0.691524\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.5223 1.22878 0.614391 0.789002i \(-0.289402\pi\)
0.614391 + 0.789002i \(0.289402\pi\)
\(618\) 0 0
\(619\) 21.6971 0.872080 0.436040 0.899927i \(-0.356381\pi\)
0.436040 + 0.899927i \(0.356381\pi\)
\(620\) 0 0
\(621\) 9.13004 0.366376
\(622\) 0 0
\(623\) 14.4746 0.579914
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −0.857977 −0.0342643
\(628\) 0 0
\(629\) 7.10655 0.283357
\(630\) 0 0
\(631\) 16.2277 0.646015 0.323007 0.946396i \(-0.395306\pi\)
0.323007 + 0.946396i \(0.395306\pi\)
\(632\) 0 0
\(633\) −1.54042 −0.0612264
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −3.16296 −0.125321
\(638\) 0 0
\(639\) 34.5694 1.36755
\(640\) 0 0
\(641\) −22.1774 −0.875956 −0.437978 0.898986i \(-0.644305\pi\)
−0.437978 + 0.898986i \(0.644305\pi\)
\(642\) 0 0
\(643\) 13.2767 0.523583 0.261792 0.965124i \(-0.415687\pi\)
0.261792 + 0.965124i \(0.415687\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 11.2853 0.443671 0.221835 0.975084i \(-0.428795\pi\)
0.221835 + 0.975084i \(0.428795\pi\)
\(648\) 0 0
\(649\) 21.2912 0.835754
\(650\) 0 0
\(651\) 9.28462 0.363893
\(652\) 0 0
\(653\) −35.8134 −1.40149 −0.700743 0.713414i \(-0.747148\pi\)
−0.700743 + 0.713414i \(0.747148\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −0.742534 −0.0289690
\(658\) 0 0
\(659\) −39.7655 −1.54905 −0.774523 0.632546i \(-0.782010\pi\)
−0.774523 + 0.632546i \(0.782010\pi\)
\(660\) 0 0
\(661\) −6.24734 −0.242993 −0.121497 0.992592i \(-0.538769\pi\)
−0.121497 + 0.992592i \(0.538769\pi\)
\(662\) 0 0
\(663\) 1.26085 0.0489673
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 13.1943 0.510887
\(668\) 0 0
\(669\) 13.6367 0.527226
\(670\) 0 0
\(671\) −28.5484 −1.10210
\(672\) 0 0
\(673\) −41.4627 −1.59827 −0.799135 0.601151i \(-0.794709\pi\)
−0.799135 + 0.601151i \(0.794709\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.43915 −0.0553112 −0.0276556 0.999618i \(-0.508804\pi\)
−0.0276556 + 0.999618i \(0.508804\pi\)
\(678\) 0 0
\(679\) 30.2308 1.16015
\(680\) 0 0
\(681\) 5.56616 0.213296
\(682\) 0 0
\(683\) 8.48623 0.324716 0.162358 0.986732i \(-0.448090\pi\)
0.162358 + 0.986732i \(0.448090\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 7.78902 0.297170
\(688\) 0 0
\(689\) 13.5652 0.516795
\(690\) 0 0
\(691\) 43.7797 1.66546 0.832730 0.553679i \(-0.186777\pi\)
0.832730 + 0.553679i \(0.186777\pi\)
\(692\) 0 0
\(693\) −16.8456 −0.639910
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −3.41210 −0.129243
\(698\) 0 0
\(699\) −10.6922 −0.404418
\(700\) 0 0
\(701\) 0.840795 0.0317564 0.0158782 0.999874i \(-0.494946\pi\)
0.0158782 + 0.999874i \(0.494946\pi\)
\(702\) 0 0
\(703\) −3.45087 −0.130152
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −28.3731 −1.06708
\(708\) 0 0
\(709\) −13.3812 −0.502543 −0.251271 0.967917i \(-0.580849\pi\)
−0.251271 + 0.967917i \(0.580849\pi\)
\(710\) 0 0
\(711\) −23.7821 −0.891897
\(712\) 0 0
\(713\) 21.4407 0.802962
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −3.15019 −0.117646
\(718\) 0 0
\(719\) −43.4148 −1.61910 −0.809550 0.587051i \(-0.800289\pi\)
−0.809550 + 0.587051i \(0.800289\pi\)
\(720\) 0 0
\(721\) 27.5776 1.02704
\(722\) 0 0
\(723\) −12.4548 −0.463200
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 32.0480 1.18859 0.594297 0.804245i \(-0.297430\pi\)
0.594297 + 0.804245i \(0.297430\pi\)
\(728\) 0 0
\(729\) −15.5727 −0.576768
\(730\) 0 0
\(731\) 6.68657 0.247312
\(732\) 0 0
\(733\) 8.13928 0.300631 0.150316 0.988638i \(-0.451971\pi\)
0.150316 + 0.988638i \(0.451971\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −21.3804 −0.787556
\(738\) 0 0
\(739\) 7.12714 0.262176 0.131088 0.991371i \(-0.458153\pi\)
0.131088 + 0.991371i \(0.458153\pi\)
\(740\) 0 0
\(741\) −0.612257 −0.0224918
\(742\) 0 0
\(743\) −21.9040 −0.803578 −0.401789 0.915732i \(-0.631612\pi\)
−0.401789 + 0.915732i \(0.631612\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −35.0087 −1.28090
\(748\) 0 0
\(749\) 17.0877 0.624373
\(750\) 0 0
\(751\) −9.21909 −0.336409 −0.168205 0.985752i \(-0.553797\pi\)
−0.168205 + 0.985752i \(0.553797\pi\)
\(752\) 0 0
\(753\) −5.18219 −0.188849
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 45.6524 1.65926 0.829632 0.558311i \(-0.188550\pi\)
0.829632 + 0.558311i \(0.188550\pi\)
\(758\) 0 0
\(759\) 3.16221 0.114781
\(760\) 0 0
\(761\) 39.9058 1.44658 0.723291 0.690543i \(-0.242628\pi\)
0.723291 + 0.690543i \(0.242628\pi\)
\(762\) 0 0
\(763\) 30.7012 1.11146
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.1935 0.548607
\(768\) 0 0
\(769\) −44.3420 −1.59901 −0.799506 0.600658i \(-0.794906\pi\)
−0.799506 + 0.600658i \(0.794906\pi\)
\(770\) 0 0
\(771\) −3.12510 −0.112548
\(772\) 0 0
\(773\) −38.4944 −1.38455 −0.692273 0.721635i \(-0.743391\pi\)
−0.692273 + 0.721635i \(0.743391\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 5.50768 0.197587
\(778\) 0 0
\(779\) 1.65689 0.0593641
\(780\) 0 0
\(781\) 24.9197 0.891697
\(782\) 0 0
\(783\) 10.8680 0.388390
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 53.0202 1.88997 0.944983 0.327120i \(-0.106078\pi\)
0.944983 + 0.327120i \(0.106078\pi\)
\(788\) 0 0
\(789\) −12.8733 −0.458302
\(790\) 0 0
\(791\) −32.5197 −1.15627
\(792\) 0 0
\(793\) −20.3723 −0.723440
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −12.6769 −0.449037 −0.224519 0.974470i \(-0.572081\pi\)
−0.224519 + 0.974470i \(0.572081\pi\)
\(798\) 0 0
\(799\) 8.92908 0.315888
\(800\) 0 0
\(801\) 13.2285 0.467407
\(802\) 0 0
\(803\) −0.535262 −0.0188890
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −0.479392 −0.0168754
\(808\) 0 0
\(809\) −41.8935 −1.47290 −0.736449 0.676493i \(-0.763499\pi\)
−0.736449 + 0.676493i \(0.763499\pi\)
\(810\) 0 0
\(811\) 34.5486 1.21317 0.606583 0.795020i \(-0.292540\pi\)
0.606583 + 0.795020i \(0.292540\pi\)
\(812\) 0 0
\(813\) 2.96911 0.104131
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3.24694 −0.113596
\(818\) 0 0
\(819\) −12.0211 −0.420050
\(820\) 0 0
\(821\) −21.1349 −0.737612 −0.368806 0.929506i \(-0.620233\pi\)
−0.368806 + 0.929506i \(0.620233\pi\)
\(822\) 0 0
\(823\) −3.17714 −0.110748 −0.0553740 0.998466i \(-0.517635\pi\)
−0.0553740 + 0.998466i \(0.517635\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −9.74482 −0.338860 −0.169430 0.985542i \(-0.554193\pi\)
−0.169430 + 0.985542i \(0.554193\pi\)
\(828\) 0 0
\(829\) 23.4352 0.813938 0.406969 0.913442i \(-0.366586\pi\)
0.406969 + 0.913442i \(0.366586\pi\)
\(830\) 0 0
\(831\) 11.7160 0.406424
\(832\) 0 0
\(833\) 4.12265 0.142841
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 17.6604 0.610432
\(838\) 0 0
\(839\) −42.6819 −1.47354 −0.736771 0.676142i \(-0.763650\pi\)
−0.736771 + 0.676142i \(0.763650\pi\)
\(840\) 0 0
\(841\) −13.2941 −0.458416
\(842\) 0 0
\(843\) −0.873305 −0.0300782
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 21.2507 0.730183
\(848\) 0 0
\(849\) −4.10371 −0.140839
\(850\) 0 0
\(851\) 12.7187 0.435993
\(852\) 0 0
\(853\) 16.9610 0.580733 0.290367 0.956915i \(-0.406223\pi\)
0.290367 + 0.956915i \(0.406223\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −39.3176 −1.34306 −0.671531 0.740976i \(-0.734363\pi\)
−0.671531 + 0.740976i \(0.734363\pi\)
\(858\) 0 0
\(859\) −0.707056 −0.0241244 −0.0120622 0.999927i \(-0.503840\pi\)
−0.0120622 + 0.999927i \(0.503840\pi\)
\(860\) 0 0
\(861\) −2.64443 −0.0901219
\(862\) 0 0
\(863\) −0.909409 −0.0309567 −0.0154783 0.999880i \(-0.504927\pi\)
−0.0154783 + 0.999880i \(0.504927\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 6.42994 0.218372
\(868\) 0 0
\(869\) −17.1435 −0.581554
\(870\) 0 0
\(871\) −15.2571 −0.516968
\(872\) 0 0
\(873\) 27.6283 0.935075
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 33.6613 1.13666 0.568331 0.822800i \(-0.307589\pi\)
0.568331 + 0.822800i \(0.307589\pi\)
\(878\) 0 0
\(879\) −2.98788 −0.100779
\(880\) 0 0
\(881\) −19.9283 −0.671402 −0.335701 0.941969i \(-0.608973\pi\)
−0.335701 + 0.941969i \(0.608973\pi\)
\(882\) 0 0
\(883\) −15.6081 −0.525255 −0.262627 0.964897i \(-0.584589\pi\)
−0.262627 + 0.964897i \(0.584589\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −46.8968 −1.57464 −0.787320 0.616544i \(-0.788532\pi\)
−0.787320 + 0.616544i \(0.788532\pi\)
\(888\) 0 0
\(889\) −34.4897 −1.15675
\(890\) 0 0
\(891\) −14.0421 −0.470429
\(892\) 0 0
\(893\) −4.33588 −0.145095
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 2.25657 0.0753447
\(898\) 0 0
\(899\) 25.5220 0.851208
\(900\) 0 0
\(901\) −17.6811 −0.589044
\(902\) 0 0
\(903\) 5.18219 0.172452
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.43447 −0.0476308 −0.0238154 0.999716i \(-0.507581\pi\)
−0.0238154 + 0.999716i \(0.507581\pi\)
\(908\) 0 0
\(909\) −25.9305 −0.860061
\(910\) 0 0
\(911\) 2.81129 0.0931422 0.0465711 0.998915i \(-0.485171\pi\)
0.0465711 + 0.998915i \(0.485171\pi\)
\(912\) 0 0
\(913\) −25.2363 −0.835201
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 24.2492 0.800780
\(918\) 0 0
\(919\) −0.992236 −0.0327308 −0.0163654 0.999866i \(-0.505210\pi\)
−0.0163654 + 0.999866i \(0.505210\pi\)
\(920\) 0 0
\(921\) −13.6136 −0.448584
\(922\) 0 0
\(923\) 17.7828 0.585329
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 25.2035 0.827791
\(928\) 0 0
\(929\) 33.3227 1.09328 0.546642 0.837367i \(-0.315906\pi\)
0.546642 + 0.837367i \(0.315906\pi\)
\(930\) 0 0
\(931\) −2.00192 −0.0656104
\(932\) 0 0
\(933\) −3.72157 −0.121839
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 13.3675 0.436698 0.218349 0.975871i \(-0.429933\pi\)
0.218349 + 0.975871i \(0.429933\pi\)
\(938\) 0 0
\(939\) −10.1673 −0.331798
\(940\) 0 0
\(941\) −2.14982 −0.0700820 −0.0350410 0.999386i \(-0.511156\pi\)
−0.0350410 + 0.999386i \(0.511156\pi\)
\(942\) 0 0
\(943\) −6.10671 −0.198862
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −34.9183 −1.13469 −0.567346 0.823479i \(-0.692030\pi\)
−0.567346 + 0.823479i \(0.692030\pi\)
\(948\) 0 0
\(949\) −0.381966 −0.0123991
\(950\) 0 0
\(951\) −1.90903 −0.0619046
\(952\) 0 0
\(953\) −8.27883 −0.268178 −0.134089 0.990969i \(-0.542811\pi\)
−0.134089 + 0.990969i \(0.542811\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 3.76415 0.121678
\(958\) 0 0
\(959\) −28.3462 −0.915347
\(960\) 0 0
\(961\) 10.4732 0.337844
\(962\) 0 0
\(963\) 15.6167 0.503241
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −29.4871 −0.948241 −0.474121 0.880460i \(-0.657234\pi\)
−0.474121 + 0.880460i \(0.657234\pi\)
\(968\) 0 0
\(969\) 0.798025 0.0256363
\(970\) 0 0
\(971\) −21.8666 −0.701733 −0.350867 0.936425i \(-0.614113\pi\)
−0.350867 + 0.936425i \(0.614113\pi\)
\(972\) 0 0
\(973\) 54.3868 1.74356
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 13.8482 0.443043 0.221521 0.975155i \(-0.428898\pi\)
0.221521 + 0.975155i \(0.428898\pi\)
\(978\) 0 0
\(979\) 9.53591 0.304769
\(980\) 0 0
\(981\) 28.0582 0.895828
\(982\) 0 0
\(983\) −2.93538 −0.0936241 −0.0468120 0.998904i \(-0.514906\pi\)
−0.0468120 + 0.998904i \(0.514906\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 6.92017 0.220271
\(988\) 0 0
\(989\) 11.9671 0.380531
\(990\) 0 0
\(991\) −31.6137 −1.00424 −0.502122 0.864797i \(-0.667447\pi\)
−0.502122 + 0.864797i \(0.667447\pi\)
\(992\) 0 0
\(993\) 5.52970 0.175480
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −12.0885 −0.382845 −0.191423 0.981508i \(-0.561310\pi\)
−0.191423 + 0.981508i \(0.561310\pi\)
\(998\) 0 0
\(999\) 10.4762 0.331453
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 10000.2.a.bj.1.4 8
4.3 odd 2 625.2.a.f.1.8 8
5.4 even 2 inner 10000.2.a.bj.1.5 8
12.11 even 2 5625.2.a.x.1.1 8
20.3 even 4 625.2.b.c.624.1 8
20.7 even 4 625.2.b.c.624.8 8
20.19 odd 2 625.2.a.f.1.1 8
25.3 odd 20 400.2.y.c.209.1 8
25.17 odd 20 400.2.y.c.289.1 8
60.59 even 2 5625.2.a.x.1.8 8
100.3 even 20 25.2.e.a.9.1 8
100.11 odd 10 625.2.d.o.501.4 16
100.19 odd 10 125.2.d.b.51.4 16
100.23 even 20 625.2.e.i.124.1 8
100.27 even 20 625.2.e.a.124.2 8
100.31 odd 10 125.2.d.b.51.1 16
100.39 odd 10 625.2.d.o.501.1 16
100.47 even 20 125.2.e.b.49.2 8
100.59 odd 10 625.2.d.o.126.1 16
100.63 even 20 625.2.e.a.499.2 8
100.67 even 20 25.2.e.a.14.1 yes 8
100.71 odd 10 125.2.d.b.76.1 16
100.79 odd 10 125.2.d.b.76.4 16
100.83 even 20 125.2.e.b.74.2 8
100.87 even 20 625.2.e.i.499.1 8
100.91 odd 10 625.2.d.o.126.4 16
300.167 odd 20 225.2.m.a.64.2 8
300.203 odd 20 225.2.m.a.109.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.e.a.9.1 8 100.3 even 20
25.2.e.a.14.1 yes 8 100.67 even 20
125.2.d.b.51.1 16 100.31 odd 10
125.2.d.b.51.4 16 100.19 odd 10
125.2.d.b.76.1 16 100.71 odd 10
125.2.d.b.76.4 16 100.79 odd 10
125.2.e.b.49.2 8 100.47 even 20
125.2.e.b.74.2 8 100.83 even 20
225.2.m.a.64.2 8 300.167 odd 20
225.2.m.a.109.2 8 300.203 odd 20
400.2.y.c.209.1 8 25.3 odd 20
400.2.y.c.289.1 8 25.17 odd 20
625.2.a.f.1.1 8 20.19 odd 2
625.2.a.f.1.8 8 4.3 odd 2
625.2.b.c.624.1 8 20.3 even 4
625.2.b.c.624.8 8 20.7 even 4
625.2.d.o.126.1 16 100.59 odd 10
625.2.d.o.126.4 16 100.91 odd 10
625.2.d.o.501.1 16 100.39 odd 10
625.2.d.o.501.4 16 100.11 odd 10
625.2.e.a.124.2 8 100.27 even 20
625.2.e.a.499.2 8 100.63 even 20
625.2.e.i.124.1 8 100.23 even 20
625.2.e.i.499.1 8 100.87 even 20
5625.2.a.x.1.1 8 12.11 even 2
5625.2.a.x.1.8 8 60.59 even 2
10000.2.a.bj.1.4 8 1.1 even 1 trivial
10000.2.a.bj.1.5 8 5.4 even 2 inner