Properties

Label 6024.2.a.o.1.7
Level $6024$
Weight $2$
Character 6024.1
Self dual yes
Analytic conductor $48.102$
Analytic rank $1$
Dimension $14$
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] = [6024,2,Mod(1,6024)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6024, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6024.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6024.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: \(48.1018821776\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 3 x^{13} - 43 x^{12} + 119 x^{11} + 679 x^{10} - 1667 x^{9} - 4890 x^{8} + 9662 x^{7} + 16575 x^{6} - 20277 x^{5} - 25196 x^{4} + 8040 x^{3} + 10776 x^{2} + \cdots - 416 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(0.155434\) of defining polynomial
Character \(\chi\) \(=\) 6024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.155434 q^{5} -3.91825 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.155434 q^{5} -3.91825 q^{7} +1.00000 q^{9} -2.98599 q^{11} +3.24976 q^{13} +0.155434 q^{15} +4.57836 q^{17} +5.89924 q^{19} +3.91825 q^{21} -1.13728 q^{23} -4.97584 q^{25} -1.00000 q^{27} -5.58606 q^{29} -3.95207 q^{31} +2.98599 q^{33} +0.609029 q^{35} +4.88043 q^{37} -3.24976 q^{39} -3.92467 q^{41} +4.68821 q^{43} -0.155434 q^{45} -11.0714 q^{47} +8.35268 q^{49} -4.57836 q^{51} +11.2510 q^{53} +0.464124 q^{55} -5.89924 q^{57} +4.51311 q^{59} +6.90103 q^{61} -3.91825 q^{63} -0.505123 q^{65} -4.82967 q^{67} +1.13728 q^{69} +15.5370 q^{71} +4.21490 q^{73} +4.97584 q^{75} +11.6998 q^{77} +0.279589 q^{79} +1.00000 q^{81} -8.21288 q^{83} -0.711633 q^{85} +5.58606 q^{87} +0.919299 q^{89} -12.7334 q^{91} +3.95207 q^{93} -0.916943 q^{95} -6.23550 q^{97} -2.98599 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{3} - 3 q^{5} + 7 q^{7} + 14 q^{9} - 22 q^{11} + 9 q^{13} + 3 q^{15} - 7 q^{21} - 17 q^{23} + 25 q^{25} - 14 q^{27} - 18 q^{29} - 7 q^{31} + 22 q^{33} - 27 q^{35} + 9 q^{37} - 9 q^{39} - 6 q^{41} - 14 q^{43} - 3 q^{45} - 15 q^{47} + 15 q^{49} - 11 q^{53} + 2 q^{55} - 36 q^{59} + 2 q^{61} + 7 q^{63} + 8 q^{65} + 3 q^{67} + 17 q^{69} - 29 q^{71} + 2 q^{73} - 25 q^{75} + 8 q^{77} + 23 q^{79} + 14 q^{81} - 55 q^{83} + 7 q^{85} + 18 q^{87} + 9 q^{89} - 22 q^{91} + 7 q^{93} - 27 q^{95} + 17 q^{97} - 22 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −0.155434 −0.0695122 −0.0347561 0.999396i \(-0.511065\pi\)
−0.0347561 + 0.999396i \(0.511065\pi\)
\(6\) 0 0
\(7\) −3.91825 −1.48096 −0.740480 0.672079i \(-0.765402\pi\)
−0.740480 + 0.672079i \(0.765402\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.98599 −0.900309 −0.450154 0.892951i \(-0.648631\pi\)
−0.450154 + 0.892951i \(0.648631\pi\)
\(12\) 0 0
\(13\) 3.24976 0.901321 0.450661 0.892695i \(-0.351188\pi\)
0.450661 + 0.892695i \(0.351188\pi\)
\(14\) 0 0
\(15\) 0.155434 0.0401329
\(16\) 0 0
\(17\) 4.57836 1.11041 0.555207 0.831712i \(-0.312639\pi\)
0.555207 + 0.831712i \(0.312639\pi\)
\(18\) 0 0
\(19\) 5.89924 1.35338 0.676689 0.736269i \(-0.263414\pi\)
0.676689 + 0.736269i \(0.263414\pi\)
\(20\) 0 0
\(21\) 3.91825 0.855032
\(22\) 0 0
\(23\) −1.13728 −0.237139 −0.118569 0.992946i \(-0.537831\pi\)
−0.118569 + 0.992946i \(0.537831\pi\)
\(24\) 0 0
\(25\) −4.97584 −0.995168
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −5.58606 −1.03731 −0.518653 0.854985i \(-0.673566\pi\)
−0.518653 + 0.854985i \(0.673566\pi\)
\(30\) 0 0
\(31\) −3.95207 −0.709812 −0.354906 0.934902i \(-0.615487\pi\)
−0.354906 + 0.934902i \(0.615487\pi\)
\(32\) 0 0
\(33\) 2.98599 0.519793
\(34\) 0 0
\(35\) 0.609029 0.102945
\(36\) 0 0
\(37\) 4.88043 0.802337 0.401169 0.916004i \(-0.368604\pi\)
0.401169 + 0.916004i \(0.368604\pi\)
\(38\) 0 0
\(39\) −3.24976 −0.520378
\(40\) 0 0
\(41\) −3.92467 −0.612931 −0.306466 0.951882i \(-0.599146\pi\)
−0.306466 + 0.951882i \(0.599146\pi\)
\(42\) 0 0
\(43\) 4.68821 0.714945 0.357473 0.933924i \(-0.383639\pi\)
0.357473 + 0.933924i \(0.383639\pi\)
\(44\) 0 0
\(45\) −0.155434 −0.0231707
\(46\) 0 0
\(47\) −11.0714 −1.61493 −0.807467 0.589913i \(-0.799162\pi\)
−0.807467 + 0.589913i \(0.799162\pi\)
\(48\) 0 0
\(49\) 8.35268 1.19324
\(50\) 0 0
\(51\) −4.57836 −0.641098
\(52\) 0 0
\(53\) 11.2510 1.54544 0.772721 0.634746i \(-0.218895\pi\)
0.772721 + 0.634746i \(0.218895\pi\)
\(54\) 0 0
\(55\) 0.464124 0.0625824
\(56\) 0 0
\(57\) −5.89924 −0.781373
\(58\) 0 0
\(59\) 4.51311 0.587557 0.293779 0.955873i \(-0.405087\pi\)
0.293779 + 0.955873i \(0.405087\pi\)
\(60\) 0 0
\(61\) 6.90103 0.883586 0.441793 0.897117i \(-0.354343\pi\)
0.441793 + 0.897117i \(0.354343\pi\)
\(62\) 0 0
\(63\) −3.91825 −0.493653
\(64\) 0 0
\(65\) −0.505123 −0.0626528
\(66\) 0 0
\(67\) −4.82967 −0.590038 −0.295019 0.955491i \(-0.595326\pi\)
−0.295019 + 0.955491i \(0.595326\pi\)
\(68\) 0 0
\(69\) 1.13728 0.136912
\(70\) 0 0
\(71\) 15.5370 1.84390 0.921949 0.387312i \(-0.126597\pi\)
0.921949 + 0.387312i \(0.126597\pi\)
\(72\) 0 0
\(73\) 4.21490 0.493317 0.246658 0.969102i \(-0.420667\pi\)
0.246658 + 0.969102i \(0.420667\pi\)
\(74\) 0 0
\(75\) 4.97584 0.574561
\(76\) 0 0
\(77\) 11.6998 1.33332
\(78\) 0 0
\(79\) 0.279589 0.0314563 0.0157281 0.999876i \(-0.494993\pi\)
0.0157281 + 0.999876i \(0.494993\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −8.21288 −0.901481 −0.450741 0.892655i \(-0.648840\pi\)
−0.450741 + 0.892655i \(0.648840\pi\)
\(84\) 0 0
\(85\) −0.711633 −0.0771874
\(86\) 0 0
\(87\) 5.58606 0.598889
\(88\) 0 0
\(89\) 0.919299 0.0974455 0.0487228 0.998812i \(-0.484485\pi\)
0.0487228 + 0.998812i \(0.484485\pi\)
\(90\) 0 0
\(91\) −12.7334 −1.33482
\(92\) 0 0
\(93\) 3.95207 0.409810
\(94\) 0 0
\(95\) −0.916943 −0.0940763
\(96\) 0 0
\(97\) −6.23550 −0.633119 −0.316559 0.948573i \(-0.602528\pi\)
−0.316559 + 0.948573i \(0.602528\pi\)
\(98\) 0 0
\(99\) −2.98599 −0.300103
\(100\) 0 0
\(101\) 13.2888 1.32228 0.661141 0.750262i \(-0.270073\pi\)
0.661141 + 0.750262i \(0.270073\pi\)
\(102\) 0 0
\(103\) 4.20637 0.414466 0.207233 0.978292i \(-0.433554\pi\)
0.207233 + 0.978292i \(0.433554\pi\)
\(104\) 0 0
\(105\) −0.609029 −0.0594352
\(106\) 0 0
\(107\) −7.56486 −0.731322 −0.365661 0.930748i \(-0.619157\pi\)
−0.365661 + 0.930748i \(0.619157\pi\)
\(108\) 0 0
\(109\) 6.16079 0.590097 0.295048 0.955482i \(-0.404664\pi\)
0.295048 + 0.955482i \(0.404664\pi\)
\(110\) 0 0
\(111\) −4.88043 −0.463230
\(112\) 0 0
\(113\) 5.52532 0.519778 0.259889 0.965639i \(-0.416314\pi\)
0.259889 + 0.965639i \(0.416314\pi\)
\(114\) 0 0
\(115\) 0.176772 0.0164840
\(116\) 0 0
\(117\) 3.24976 0.300440
\(118\) 0 0
\(119\) −17.9392 −1.64448
\(120\) 0 0
\(121\) −2.08389 −0.189444
\(122\) 0 0
\(123\) 3.92467 0.353876
\(124\) 0 0
\(125\) 1.55058 0.138689
\(126\) 0 0
\(127\) 4.74541 0.421087 0.210544 0.977584i \(-0.432477\pi\)
0.210544 + 0.977584i \(0.432477\pi\)
\(128\) 0 0
\(129\) −4.68821 −0.412774
\(130\) 0 0
\(131\) 7.77369 0.679191 0.339595 0.940572i \(-0.389710\pi\)
0.339595 + 0.940572i \(0.389710\pi\)
\(132\) 0 0
\(133\) −23.1147 −2.00430
\(134\) 0 0
\(135\) 0.155434 0.0133776
\(136\) 0 0
\(137\) −14.5804 −1.24569 −0.622843 0.782347i \(-0.714023\pi\)
−0.622843 + 0.782347i \(0.714023\pi\)
\(138\) 0 0
\(139\) −18.6727 −1.58380 −0.791900 0.610650i \(-0.790908\pi\)
−0.791900 + 0.610650i \(0.790908\pi\)
\(140\) 0 0
\(141\) 11.0714 0.932383
\(142\) 0 0
\(143\) −9.70374 −0.811467
\(144\) 0 0
\(145\) 0.868264 0.0721054
\(146\) 0 0
\(147\) −8.35268 −0.688917
\(148\) 0 0
\(149\) −14.0745 −1.15303 −0.576515 0.817087i \(-0.695588\pi\)
−0.576515 + 0.817087i \(0.695588\pi\)
\(150\) 0 0
\(151\) −8.47923 −0.690030 −0.345015 0.938597i \(-0.612126\pi\)
−0.345015 + 0.938597i \(0.612126\pi\)
\(152\) 0 0
\(153\) 4.57836 0.370138
\(154\) 0 0
\(155\) 0.614286 0.0493406
\(156\) 0 0
\(157\) 18.4129 1.46951 0.734755 0.678332i \(-0.237297\pi\)
0.734755 + 0.678332i \(0.237297\pi\)
\(158\) 0 0
\(159\) −11.2510 −0.892261
\(160\) 0 0
\(161\) 4.45614 0.351193
\(162\) 0 0
\(163\) −24.0573 −1.88431 −0.942157 0.335173i \(-0.891205\pi\)
−0.942157 + 0.335173i \(0.891205\pi\)
\(164\) 0 0
\(165\) −0.464124 −0.0361320
\(166\) 0 0
\(167\) 13.2283 1.02364 0.511820 0.859093i \(-0.328971\pi\)
0.511820 + 0.859093i \(0.328971\pi\)
\(168\) 0 0
\(169\) −2.43906 −0.187620
\(170\) 0 0
\(171\) 5.89924 0.451126
\(172\) 0 0
\(173\) −15.3763 −1.16904 −0.584520 0.811379i \(-0.698717\pi\)
−0.584520 + 0.811379i \(0.698717\pi\)
\(174\) 0 0
\(175\) 19.4966 1.47380
\(176\) 0 0
\(177\) −4.51311 −0.339226
\(178\) 0 0
\(179\) −24.6831 −1.84490 −0.922450 0.386116i \(-0.873817\pi\)
−0.922450 + 0.386116i \(0.873817\pi\)
\(180\) 0 0
\(181\) −23.0257 −1.71149 −0.855745 0.517398i \(-0.826901\pi\)
−0.855745 + 0.517398i \(0.826901\pi\)
\(182\) 0 0
\(183\) −6.90103 −0.510139
\(184\) 0 0
\(185\) −0.758584 −0.0557722
\(186\) 0 0
\(187\) −13.6709 −0.999716
\(188\) 0 0
\(189\) 3.91825 0.285011
\(190\) 0 0
\(191\) −13.4253 −0.971418 −0.485709 0.874121i \(-0.661439\pi\)
−0.485709 + 0.874121i \(0.661439\pi\)
\(192\) 0 0
\(193\) −11.8267 −0.851304 −0.425652 0.904887i \(-0.639955\pi\)
−0.425652 + 0.904887i \(0.639955\pi\)
\(194\) 0 0
\(195\) 0.505123 0.0361726
\(196\) 0 0
\(197\) −15.8298 −1.12782 −0.563912 0.825835i \(-0.690704\pi\)
−0.563912 + 0.825835i \(0.690704\pi\)
\(198\) 0 0
\(199\) −4.91582 −0.348473 −0.174237 0.984704i \(-0.555746\pi\)
−0.174237 + 0.984704i \(0.555746\pi\)
\(200\) 0 0
\(201\) 4.82967 0.340659
\(202\) 0 0
\(203\) 21.8876 1.53621
\(204\) 0 0
\(205\) 0.610028 0.0426062
\(206\) 0 0
\(207\) −1.13728 −0.0790462
\(208\) 0 0
\(209\) −17.6151 −1.21846
\(210\) 0 0
\(211\) 11.8470 0.815584 0.407792 0.913075i \(-0.366299\pi\)
0.407792 + 0.913075i \(0.366299\pi\)
\(212\) 0 0
\(213\) −15.5370 −1.06457
\(214\) 0 0
\(215\) −0.728707 −0.0496974
\(216\) 0 0
\(217\) 15.4852 1.05120
\(218\) 0 0
\(219\) −4.21490 −0.284817
\(220\) 0 0
\(221\) 14.8786 1.00084
\(222\) 0 0
\(223\) 16.6435 1.11453 0.557264 0.830335i \(-0.311851\pi\)
0.557264 + 0.830335i \(0.311851\pi\)
\(224\) 0 0
\(225\) −4.97584 −0.331723
\(226\) 0 0
\(227\) −15.8781 −1.05387 −0.526934 0.849906i \(-0.676659\pi\)
−0.526934 + 0.849906i \(0.676659\pi\)
\(228\) 0 0
\(229\) −10.5448 −0.696819 −0.348410 0.937342i \(-0.613278\pi\)
−0.348410 + 0.937342i \(0.613278\pi\)
\(230\) 0 0
\(231\) −11.6998 −0.769793
\(232\) 0 0
\(233\) −24.2058 −1.58578 −0.792888 0.609367i \(-0.791424\pi\)
−0.792888 + 0.609367i \(0.791424\pi\)
\(234\) 0 0
\(235\) 1.72088 0.112258
\(236\) 0 0
\(237\) −0.279589 −0.0181613
\(238\) 0 0
\(239\) −27.3928 −1.77189 −0.885947 0.463786i \(-0.846491\pi\)
−0.885947 + 0.463786i \(0.846491\pi\)
\(240\) 0 0
\(241\) −7.05270 −0.454305 −0.227152 0.973859i \(-0.572942\pi\)
−0.227152 + 0.973859i \(0.572942\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.29829 −0.0829447
\(246\) 0 0
\(247\) 19.1711 1.21983
\(248\) 0 0
\(249\) 8.21288 0.520470
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) 3.39589 0.213498
\(254\) 0 0
\(255\) 0.711633 0.0445642
\(256\) 0 0
\(257\) −2.02421 −0.126267 −0.0631333 0.998005i \(-0.520109\pi\)
−0.0631333 + 0.998005i \(0.520109\pi\)
\(258\) 0 0
\(259\) −19.1227 −1.18823
\(260\) 0 0
\(261\) −5.58606 −0.345768
\(262\) 0 0
\(263\) −14.6471 −0.903181 −0.451590 0.892225i \(-0.649143\pi\)
−0.451590 + 0.892225i \(0.649143\pi\)
\(264\) 0 0
\(265\) −1.74879 −0.107427
\(266\) 0 0
\(267\) −0.919299 −0.0562602
\(268\) 0 0
\(269\) 5.66846 0.345612 0.172806 0.984956i \(-0.444717\pi\)
0.172806 + 0.984956i \(0.444717\pi\)
\(270\) 0 0
\(271\) 23.8654 1.44972 0.724861 0.688895i \(-0.241904\pi\)
0.724861 + 0.688895i \(0.241904\pi\)
\(272\) 0 0
\(273\) 12.7334 0.770659
\(274\) 0 0
\(275\) 14.8578 0.895958
\(276\) 0 0
\(277\) −31.6053 −1.89898 −0.949491 0.313796i \(-0.898399\pi\)
−0.949491 + 0.313796i \(0.898399\pi\)
\(278\) 0 0
\(279\) −3.95207 −0.236604
\(280\) 0 0
\(281\) 0.130628 0.00779262 0.00389631 0.999992i \(-0.498760\pi\)
0.00389631 + 0.999992i \(0.498760\pi\)
\(282\) 0 0
\(283\) −0.836580 −0.0497295 −0.0248647 0.999691i \(-0.507916\pi\)
−0.0248647 + 0.999691i \(0.507916\pi\)
\(284\) 0 0
\(285\) 0.916943 0.0543150
\(286\) 0 0
\(287\) 15.3779 0.907726
\(288\) 0 0
\(289\) 3.96136 0.233021
\(290\) 0 0
\(291\) 6.23550 0.365531
\(292\) 0 0
\(293\) −8.78602 −0.513285 −0.256642 0.966506i \(-0.582616\pi\)
−0.256642 + 0.966506i \(0.582616\pi\)
\(294\) 0 0
\(295\) −0.701491 −0.0408424
\(296\) 0 0
\(297\) 2.98599 0.173264
\(298\) 0 0
\(299\) −3.69588 −0.213738
\(300\) 0 0
\(301\) −18.3696 −1.05881
\(302\) 0 0
\(303\) −13.2888 −0.763420
\(304\) 0 0
\(305\) −1.07265 −0.0614200
\(306\) 0 0
\(307\) 22.3921 1.27799 0.638994 0.769212i \(-0.279351\pi\)
0.638994 + 0.769212i \(0.279351\pi\)
\(308\) 0 0
\(309\) −4.20637 −0.239292
\(310\) 0 0
\(311\) 4.44519 0.252064 0.126032 0.992026i \(-0.459776\pi\)
0.126032 + 0.992026i \(0.459776\pi\)
\(312\) 0 0
\(313\) −19.7404 −1.11579 −0.557895 0.829911i \(-0.688391\pi\)
−0.557895 + 0.829911i \(0.688391\pi\)
\(314\) 0 0
\(315\) 0.609029 0.0343149
\(316\) 0 0
\(317\) 20.6169 1.15796 0.578980 0.815342i \(-0.303451\pi\)
0.578980 + 0.815342i \(0.303451\pi\)
\(318\) 0 0
\(319\) 16.6799 0.933895
\(320\) 0 0
\(321\) 7.56486 0.422229
\(322\) 0 0
\(323\) 27.0088 1.50281
\(324\) 0 0
\(325\) −16.1703 −0.896966
\(326\) 0 0
\(327\) −6.16079 −0.340692
\(328\) 0 0
\(329\) 43.3806 2.39165
\(330\) 0 0
\(331\) 15.0072 0.824868 0.412434 0.910987i \(-0.364679\pi\)
0.412434 + 0.910987i \(0.364679\pi\)
\(332\) 0 0
\(333\) 4.88043 0.267446
\(334\) 0 0
\(335\) 0.750695 0.0410149
\(336\) 0 0
\(337\) 24.7044 1.34573 0.672867 0.739763i \(-0.265062\pi\)
0.672867 + 0.739763i \(0.265062\pi\)
\(338\) 0 0
\(339\) −5.52532 −0.300094
\(340\) 0 0
\(341\) 11.8008 0.639050
\(342\) 0 0
\(343\) −5.30014 −0.286181
\(344\) 0 0
\(345\) −0.176772 −0.00951706
\(346\) 0 0
\(347\) −36.5680 −1.96307 −0.981536 0.191280i \(-0.938736\pi\)
−0.981536 + 0.191280i \(0.938736\pi\)
\(348\) 0 0
\(349\) 20.8305 1.11503 0.557515 0.830167i \(-0.311755\pi\)
0.557515 + 0.830167i \(0.311755\pi\)
\(350\) 0 0
\(351\) −3.24976 −0.173459
\(352\) 0 0
\(353\) 32.5901 1.73460 0.867298 0.497789i \(-0.165855\pi\)
0.867298 + 0.497789i \(0.165855\pi\)
\(354\) 0 0
\(355\) −2.41497 −0.128173
\(356\) 0 0
\(357\) 17.9392 0.949441
\(358\) 0 0
\(359\) −14.7854 −0.780342 −0.390171 0.920742i \(-0.627584\pi\)
−0.390171 + 0.920742i \(0.627584\pi\)
\(360\) 0 0
\(361\) 15.8010 0.831634
\(362\) 0 0
\(363\) 2.08389 0.109376
\(364\) 0 0
\(365\) −0.655139 −0.0342915
\(366\) 0 0
\(367\) −16.9599 −0.885299 −0.442649 0.896695i \(-0.645961\pi\)
−0.442649 + 0.896695i \(0.645961\pi\)
\(368\) 0 0
\(369\) −3.92467 −0.204310
\(370\) 0 0
\(371\) −44.0842 −2.28874
\(372\) 0 0
\(373\) 0.655098 0.0339197 0.0169598 0.999856i \(-0.494601\pi\)
0.0169598 + 0.999856i \(0.494601\pi\)
\(374\) 0 0
\(375\) −1.55058 −0.0800719
\(376\) 0 0
\(377\) −18.1534 −0.934946
\(378\) 0 0
\(379\) −5.04647 −0.259220 −0.129610 0.991565i \(-0.541372\pi\)
−0.129610 + 0.991565i \(0.541372\pi\)
\(380\) 0 0
\(381\) −4.74541 −0.243115
\(382\) 0 0
\(383\) 2.65268 0.135545 0.0677727 0.997701i \(-0.478411\pi\)
0.0677727 + 0.997701i \(0.478411\pi\)
\(384\) 0 0
\(385\) −1.81855 −0.0926820
\(386\) 0 0
\(387\) 4.68821 0.238315
\(388\) 0 0
\(389\) 21.0810 1.06885 0.534425 0.845216i \(-0.320528\pi\)
0.534425 + 0.845216i \(0.320528\pi\)
\(390\) 0 0
\(391\) −5.20686 −0.263322
\(392\) 0 0
\(393\) −7.77369 −0.392131
\(394\) 0 0
\(395\) −0.0434577 −0.00218659
\(396\) 0 0
\(397\) −24.0049 −1.20477 −0.602385 0.798206i \(-0.705783\pi\)
−0.602385 + 0.798206i \(0.705783\pi\)
\(398\) 0 0
\(399\) 23.1147 1.15718
\(400\) 0 0
\(401\) −12.2594 −0.612208 −0.306104 0.951998i \(-0.599025\pi\)
−0.306104 + 0.951998i \(0.599025\pi\)
\(402\) 0 0
\(403\) −12.8433 −0.639769
\(404\) 0 0
\(405\) −0.155434 −0.00772358
\(406\) 0 0
\(407\) −14.5729 −0.722351
\(408\) 0 0
\(409\) −2.99591 −0.148138 −0.0740692 0.997253i \(-0.523599\pi\)
−0.0740692 + 0.997253i \(0.523599\pi\)
\(410\) 0 0
\(411\) 14.5804 0.719197
\(412\) 0 0
\(413\) −17.6835 −0.870148
\(414\) 0 0
\(415\) 1.27656 0.0626639
\(416\) 0 0
\(417\) 18.6727 0.914408
\(418\) 0 0
\(419\) −12.6776 −0.619343 −0.309672 0.950844i \(-0.600219\pi\)
−0.309672 + 0.950844i \(0.600219\pi\)
\(420\) 0 0
\(421\) −32.8534 −1.60118 −0.800588 0.599215i \(-0.795479\pi\)
−0.800588 + 0.599215i \(0.795479\pi\)
\(422\) 0 0
\(423\) −11.0714 −0.538311
\(424\) 0 0
\(425\) −22.7812 −1.10505
\(426\) 0 0
\(427\) −27.0400 −1.30856
\(428\) 0 0
\(429\) 9.70374 0.468501
\(430\) 0 0
\(431\) −5.23222 −0.252027 −0.126014 0.992029i \(-0.540218\pi\)
−0.126014 + 0.992029i \(0.540218\pi\)
\(432\) 0 0
\(433\) 3.18616 0.153117 0.0765586 0.997065i \(-0.475607\pi\)
0.0765586 + 0.997065i \(0.475607\pi\)
\(434\) 0 0
\(435\) −0.868264 −0.0416301
\(436\) 0 0
\(437\) −6.70907 −0.320938
\(438\) 0 0
\(439\) 19.4806 0.929756 0.464878 0.885375i \(-0.346098\pi\)
0.464878 + 0.885375i \(0.346098\pi\)
\(440\) 0 0
\(441\) 8.35268 0.397747
\(442\) 0 0
\(443\) 8.11370 0.385494 0.192747 0.981249i \(-0.438260\pi\)
0.192747 + 0.981249i \(0.438260\pi\)
\(444\) 0 0
\(445\) −0.142890 −0.00677365
\(446\) 0 0
\(447\) 14.0745 0.665702
\(448\) 0 0
\(449\) 14.7683 0.696957 0.348479 0.937317i \(-0.386698\pi\)
0.348479 + 0.937317i \(0.386698\pi\)
\(450\) 0 0
\(451\) 11.7190 0.551827
\(452\) 0 0
\(453\) 8.47923 0.398389
\(454\) 0 0
\(455\) 1.97920 0.0927863
\(456\) 0 0
\(457\) −11.9120 −0.557220 −0.278610 0.960404i \(-0.589874\pi\)
−0.278610 + 0.960404i \(0.589874\pi\)
\(458\) 0 0
\(459\) −4.57836 −0.213699
\(460\) 0 0
\(461\) 26.2214 1.22125 0.610626 0.791919i \(-0.290918\pi\)
0.610626 + 0.791919i \(0.290918\pi\)
\(462\) 0 0
\(463\) 31.0737 1.44412 0.722059 0.691832i \(-0.243196\pi\)
0.722059 + 0.691832i \(0.243196\pi\)
\(464\) 0 0
\(465\) −0.614286 −0.0284868
\(466\) 0 0
\(467\) 10.8183 0.500610 0.250305 0.968167i \(-0.419469\pi\)
0.250305 + 0.968167i \(0.419469\pi\)
\(468\) 0 0
\(469\) 18.9239 0.873823
\(470\) 0 0
\(471\) −18.4129 −0.848423
\(472\) 0 0
\(473\) −13.9989 −0.643672
\(474\) 0 0
\(475\) −29.3537 −1.34684
\(476\) 0 0
\(477\) 11.2510 0.515147
\(478\) 0 0
\(479\) −28.1839 −1.28776 −0.643878 0.765128i \(-0.722676\pi\)
−0.643878 + 0.765128i \(0.722676\pi\)
\(480\) 0 0
\(481\) 15.8602 0.723164
\(482\) 0 0
\(483\) −4.45614 −0.202761
\(484\) 0 0
\(485\) 0.969208 0.0440095
\(486\) 0 0
\(487\) 5.28447 0.239462 0.119731 0.992806i \(-0.461797\pi\)
0.119731 + 0.992806i \(0.461797\pi\)
\(488\) 0 0
\(489\) 24.0573 1.08791
\(490\) 0 0
\(491\) −23.6779 −1.06857 −0.534285 0.845305i \(-0.679419\pi\)
−0.534285 + 0.845305i \(0.679419\pi\)
\(492\) 0 0
\(493\) −25.5750 −1.15184
\(494\) 0 0
\(495\) 0.464124 0.0208608
\(496\) 0 0
\(497\) −60.8777 −2.73074
\(498\) 0 0
\(499\) −41.1262 −1.84106 −0.920530 0.390671i \(-0.872243\pi\)
−0.920530 + 0.390671i \(0.872243\pi\)
\(500\) 0 0
\(501\) −13.2283 −0.590999
\(502\) 0 0
\(503\) −8.31325 −0.370670 −0.185335 0.982675i \(-0.559337\pi\)
−0.185335 + 0.982675i \(0.559337\pi\)
\(504\) 0 0
\(505\) −2.06553 −0.0919147
\(506\) 0 0
\(507\) 2.43906 0.108322
\(508\) 0 0
\(509\) −21.2694 −0.942751 −0.471375 0.881933i \(-0.656242\pi\)
−0.471375 + 0.881933i \(0.656242\pi\)
\(510\) 0 0
\(511\) −16.5150 −0.730582
\(512\) 0 0
\(513\) −5.89924 −0.260458
\(514\) 0 0
\(515\) −0.653813 −0.0288104
\(516\) 0 0
\(517\) 33.0591 1.45394
\(518\) 0 0
\(519\) 15.3763 0.674946
\(520\) 0 0
\(521\) −19.8865 −0.871242 −0.435621 0.900130i \(-0.643471\pi\)
−0.435621 + 0.900130i \(0.643471\pi\)
\(522\) 0 0
\(523\) 34.4890 1.50810 0.754050 0.656817i \(-0.228097\pi\)
0.754050 + 0.656817i \(0.228097\pi\)
\(524\) 0 0
\(525\) −19.4966 −0.850901
\(526\) 0 0
\(527\) −18.0940 −0.788186
\(528\) 0 0
\(529\) −21.7066 −0.943765
\(530\) 0 0
\(531\) 4.51311 0.195852
\(532\) 0 0
\(533\) −12.7543 −0.552448
\(534\) 0 0
\(535\) 1.17584 0.0508358
\(536\) 0 0
\(537\) 24.6831 1.06515
\(538\) 0 0
\(539\) −24.9410 −1.07428
\(540\) 0 0
\(541\) 22.8257 0.981352 0.490676 0.871342i \(-0.336750\pi\)
0.490676 + 0.871342i \(0.336750\pi\)
\(542\) 0 0
\(543\) 23.0257 0.988129
\(544\) 0 0
\(545\) −0.957596 −0.0410189
\(546\) 0 0
\(547\) 22.2778 0.952532 0.476266 0.879301i \(-0.341990\pi\)
0.476266 + 0.879301i \(0.341990\pi\)
\(548\) 0 0
\(549\) 6.90103 0.294529
\(550\) 0 0
\(551\) −32.9535 −1.40387
\(552\) 0 0
\(553\) −1.09550 −0.0465854
\(554\) 0 0
\(555\) 0.758584 0.0322001
\(556\) 0 0
\(557\) 18.8688 0.799496 0.399748 0.916625i \(-0.369098\pi\)
0.399748 + 0.916625i \(0.369098\pi\)
\(558\) 0 0
\(559\) 15.2356 0.644396
\(560\) 0 0
\(561\) 13.6709 0.577186
\(562\) 0 0
\(563\) −20.7575 −0.874822 −0.437411 0.899262i \(-0.644105\pi\)
−0.437411 + 0.899262i \(0.644105\pi\)
\(564\) 0 0
\(565\) −0.858822 −0.0361309
\(566\) 0 0
\(567\) −3.91825 −0.164551
\(568\) 0 0
\(569\) −0.492918 −0.0206642 −0.0103321 0.999947i \(-0.503289\pi\)
−0.0103321 + 0.999947i \(0.503289\pi\)
\(570\) 0 0
\(571\) −32.7383 −1.37006 −0.685028 0.728517i \(-0.740210\pi\)
−0.685028 + 0.728517i \(0.740210\pi\)
\(572\) 0 0
\(573\) 13.4253 0.560849
\(574\) 0 0
\(575\) 5.65891 0.235993
\(576\) 0 0
\(577\) 15.7733 0.656650 0.328325 0.944565i \(-0.393516\pi\)
0.328325 + 0.944565i \(0.393516\pi\)
\(578\) 0 0
\(579\) 11.8267 0.491501
\(580\) 0 0
\(581\) 32.1801 1.33506
\(582\) 0 0
\(583\) −33.5953 −1.39137
\(584\) 0 0
\(585\) −0.505123 −0.0208843
\(586\) 0 0
\(587\) −3.18009 −0.131257 −0.0656283 0.997844i \(-0.520905\pi\)
−0.0656283 + 0.997844i \(0.520905\pi\)
\(588\) 0 0
\(589\) −23.3142 −0.960645
\(590\) 0 0
\(591\) 15.8298 0.651149
\(592\) 0 0
\(593\) −31.7048 −1.30196 −0.650979 0.759096i \(-0.725642\pi\)
−0.650979 + 0.759096i \(0.725642\pi\)
\(594\) 0 0
\(595\) 2.78835 0.114311
\(596\) 0 0
\(597\) 4.91582 0.201191
\(598\) 0 0
\(599\) −26.4566 −1.08099 −0.540495 0.841348i \(-0.681763\pi\)
−0.540495 + 0.841348i \(0.681763\pi\)
\(600\) 0 0
\(601\) 20.4907 0.835832 0.417916 0.908486i \(-0.362761\pi\)
0.417916 + 0.908486i \(0.362761\pi\)
\(602\) 0 0
\(603\) −4.82967 −0.196679
\(604\) 0 0
\(605\) 0.323907 0.0131687
\(606\) 0 0
\(607\) 35.2895 1.43236 0.716179 0.697917i \(-0.245890\pi\)
0.716179 + 0.697917i \(0.245890\pi\)
\(608\) 0 0
\(609\) −21.8876 −0.886930
\(610\) 0 0
\(611\) −35.9795 −1.45557
\(612\) 0 0
\(613\) 30.5780 1.23503 0.617517 0.786557i \(-0.288139\pi\)
0.617517 + 0.786557i \(0.288139\pi\)
\(614\) 0 0
\(615\) −0.610028 −0.0245987
\(616\) 0 0
\(617\) −36.3421 −1.46308 −0.731539 0.681800i \(-0.761198\pi\)
−0.731539 + 0.681800i \(0.761198\pi\)
\(618\) 0 0
\(619\) −25.1128 −1.00937 −0.504684 0.863304i \(-0.668391\pi\)
−0.504684 + 0.863304i \(0.668391\pi\)
\(620\) 0 0
\(621\) 1.13728 0.0456374
\(622\) 0 0
\(623\) −3.60204 −0.144313
\(624\) 0 0
\(625\) 24.6382 0.985528
\(626\) 0 0
\(627\) 17.6151 0.703477
\(628\) 0 0
\(629\) 22.3443 0.890927
\(630\) 0 0
\(631\) 14.7218 0.586065 0.293032 0.956102i \(-0.405336\pi\)
0.293032 + 0.956102i \(0.405336\pi\)
\(632\) 0 0
\(633\) −11.8470 −0.470878
\(634\) 0 0
\(635\) −0.737598 −0.0292707
\(636\) 0 0
\(637\) 27.1442 1.07549
\(638\) 0 0
\(639\) 15.5370 0.614632
\(640\) 0 0
\(641\) 40.2029 1.58792 0.793959 0.607971i \(-0.208016\pi\)
0.793959 + 0.607971i \(0.208016\pi\)
\(642\) 0 0
\(643\) 33.8577 1.33522 0.667608 0.744513i \(-0.267318\pi\)
0.667608 + 0.744513i \(0.267318\pi\)
\(644\) 0 0
\(645\) 0.728707 0.0286928
\(646\) 0 0
\(647\) −29.9308 −1.17670 −0.588350 0.808606i \(-0.700222\pi\)
−0.588350 + 0.808606i \(0.700222\pi\)
\(648\) 0 0
\(649\) −13.4761 −0.528983
\(650\) 0 0
\(651\) −15.4852 −0.606912
\(652\) 0 0
\(653\) 14.4369 0.564961 0.282480 0.959273i \(-0.408843\pi\)
0.282480 + 0.959273i \(0.408843\pi\)
\(654\) 0 0
\(655\) −1.20830 −0.0472120
\(656\) 0 0
\(657\) 4.21490 0.164439
\(658\) 0 0
\(659\) 29.8984 1.16468 0.582339 0.812946i \(-0.302138\pi\)
0.582339 + 0.812946i \(0.302138\pi\)
\(660\) 0 0
\(661\) −40.5483 −1.57715 −0.788573 0.614942i \(-0.789180\pi\)
−0.788573 + 0.614942i \(0.789180\pi\)
\(662\) 0 0
\(663\) −14.8786 −0.577836
\(664\) 0 0
\(665\) 3.59281 0.139323
\(666\) 0 0
\(667\) 6.35290 0.245985
\(668\) 0 0
\(669\) −16.6435 −0.643473
\(670\) 0 0
\(671\) −20.6064 −0.795500
\(672\) 0 0
\(673\) 6.77832 0.261285 0.130643 0.991430i \(-0.458296\pi\)
0.130643 + 0.991430i \(0.458296\pi\)
\(674\) 0 0
\(675\) 4.97584 0.191520
\(676\) 0 0
\(677\) −13.3266 −0.512184 −0.256092 0.966652i \(-0.582435\pi\)
−0.256092 + 0.966652i \(0.582435\pi\)
\(678\) 0 0
\(679\) 24.4322 0.937623
\(680\) 0 0
\(681\) 15.8781 0.608451
\(682\) 0 0
\(683\) 14.1796 0.542568 0.271284 0.962499i \(-0.412552\pi\)
0.271284 + 0.962499i \(0.412552\pi\)
\(684\) 0 0
\(685\) 2.26629 0.0865904
\(686\) 0 0
\(687\) 10.5448 0.402309
\(688\) 0 0
\(689\) 36.5630 1.39294
\(690\) 0 0
\(691\) 33.6675 1.28077 0.640385 0.768054i \(-0.278775\pi\)
0.640385 + 0.768054i \(0.278775\pi\)
\(692\) 0 0
\(693\) 11.6998 0.444440
\(694\) 0 0
\(695\) 2.90238 0.110093
\(696\) 0 0
\(697\) −17.9686 −0.680608
\(698\) 0 0
\(699\) 24.2058 0.915548
\(700\) 0 0
\(701\) −26.6588 −1.00689 −0.503445 0.864027i \(-0.667934\pi\)
−0.503445 + 0.864027i \(0.667934\pi\)
\(702\) 0 0
\(703\) 28.7908 1.08587
\(704\) 0 0
\(705\) −1.72088 −0.0648120
\(706\) 0 0
\(707\) −52.0687 −1.95825
\(708\) 0 0
\(709\) 26.2754 0.986794 0.493397 0.869804i \(-0.335755\pi\)
0.493397 + 0.869804i \(0.335755\pi\)
\(710\) 0 0
\(711\) 0.279589 0.0104854
\(712\) 0 0
\(713\) 4.49460 0.168324
\(714\) 0 0
\(715\) 1.50829 0.0564069
\(716\) 0 0
\(717\) 27.3928 1.02300
\(718\) 0 0
\(719\) −6.97360 −0.260071 −0.130036 0.991509i \(-0.541509\pi\)
−0.130036 + 0.991509i \(0.541509\pi\)
\(720\) 0 0
\(721\) −16.4816 −0.613807
\(722\) 0 0
\(723\) 7.05270 0.262293
\(724\) 0 0
\(725\) 27.7953 1.03229
\(726\) 0 0
\(727\) 32.2590 1.19642 0.598210 0.801339i \(-0.295879\pi\)
0.598210 + 0.801339i \(0.295879\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 21.4643 0.793886
\(732\) 0 0
\(733\) 47.4501 1.75261 0.876305 0.481756i \(-0.160001\pi\)
0.876305 + 0.481756i \(0.160001\pi\)
\(734\) 0 0
\(735\) 1.29829 0.0478882
\(736\) 0 0
\(737\) 14.4213 0.531217
\(738\) 0 0
\(739\) −45.4672 −1.67254 −0.836268 0.548320i \(-0.815267\pi\)
−0.836268 + 0.548320i \(0.815267\pi\)
\(740\) 0 0
\(741\) −19.1711 −0.704269
\(742\) 0 0
\(743\) −2.95302 −0.108336 −0.0541679 0.998532i \(-0.517251\pi\)
−0.0541679 + 0.998532i \(0.517251\pi\)
\(744\) 0 0
\(745\) 2.18766 0.0801496
\(746\) 0 0
\(747\) −8.21288 −0.300494
\(748\) 0 0
\(749\) 29.6410 1.08306
\(750\) 0 0
\(751\) 46.8627 1.71005 0.855023 0.518590i \(-0.173543\pi\)
0.855023 + 0.518590i \(0.173543\pi\)
\(752\) 0 0
\(753\) −1.00000 −0.0364420
\(754\) 0 0
\(755\) 1.31796 0.0479655
\(756\) 0 0
\(757\) 28.3044 1.02874 0.514370 0.857568i \(-0.328026\pi\)
0.514370 + 0.857568i \(0.328026\pi\)
\(758\) 0 0
\(759\) −3.39589 −0.123263
\(760\) 0 0
\(761\) −23.0132 −0.834227 −0.417113 0.908855i \(-0.636958\pi\)
−0.417113 + 0.908855i \(0.636958\pi\)
\(762\) 0 0
\(763\) −24.1395 −0.873909
\(764\) 0 0
\(765\) −0.711633 −0.0257291
\(766\) 0 0
\(767\) 14.6665 0.529578
\(768\) 0 0
\(769\) 2.42938 0.0876057 0.0438029 0.999040i \(-0.486053\pi\)
0.0438029 + 0.999040i \(0.486053\pi\)
\(770\) 0 0
\(771\) 2.02421 0.0729001
\(772\) 0 0
\(773\) −48.6153 −1.74857 −0.874286 0.485411i \(-0.838670\pi\)
−0.874286 + 0.485411i \(0.838670\pi\)
\(774\) 0 0
\(775\) 19.6649 0.706383
\(776\) 0 0
\(777\) 19.1227 0.686024
\(778\) 0 0
\(779\) −23.1526 −0.829528
\(780\) 0 0
\(781\) −46.3931 −1.66008
\(782\) 0 0
\(783\) 5.58606 0.199630
\(784\) 0 0
\(785\) −2.86199 −0.102149
\(786\) 0 0
\(787\) −36.7728 −1.31081 −0.655405 0.755277i \(-0.727502\pi\)
−0.655405 + 0.755277i \(0.727502\pi\)
\(788\) 0 0
\(789\) 14.6471 0.521452
\(790\) 0 0
\(791\) −21.6496 −0.769770
\(792\) 0 0
\(793\) 22.4267 0.796395
\(794\) 0 0
\(795\) 1.74879 0.0620230
\(796\) 0 0
\(797\) −1.36397 −0.0483143 −0.0241572 0.999708i \(-0.507690\pi\)
−0.0241572 + 0.999708i \(0.507690\pi\)
\(798\) 0 0
\(799\) −50.6890 −1.79325
\(800\) 0 0
\(801\) 0.919299 0.0324818
\(802\) 0 0
\(803\) −12.5856 −0.444137
\(804\) 0 0
\(805\) −0.692635 −0.0244122
\(806\) 0 0
\(807\) −5.66846 −0.199539
\(808\) 0 0
\(809\) −6.83746 −0.240392 −0.120196 0.992750i \(-0.538352\pi\)
−0.120196 + 0.992750i \(0.538352\pi\)
\(810\) 0 0
\(811\) −4.61278 −0.161976 −0.0809882 0.996715i \(-0.525808\pi\)
−0.0809882 + 0.996715i \(0.525808\pi\)
\(812\) 0 0
\(813\) −23.8654 −0.836997
\(814\) 0 0
\(815\) 3.73932 0.130983
\(816\) 0 0
\(817\) 27.6569 0.967592
\(818\) 0 0
\(819\) −12.7334 −0.444940
\(820\) 0 0
\(821\) −47.8754 −1.67086 −0.835431 0.549595i \(-0.814782\pi\)
−0.835431 + 0.549595i \(0.814782\pi\)
\(822\) 0 0
\(823\) 4.56063 0.158974 0.0794868 0.996836i \(-0.474672\pi\)
0.0794868 + 0.996836i \(0.474672\pi\)
\(824\) 0 0
\(825\) −14.8578 −0.517282
\(826\) 0 0
\(827\) −35.3776 −1.23020 −0.615099 0.788450i \(-0.710884\pi\)
−0.615099 + 0.788450i \(0.710884\pi\)
\(828\) 0 0
\(829\) 7.69107 0.267122 0.133561 0.991041i \(-0.457359\pi\)
0.133561 + 0.991041i \(0.457359\pi\)
\(830\) 0 0
\(831\) 31.6053 1.09638
\(832\) 0 0
\(833\) 38.2416 1.32499
\(834\) 0 0
\(835\) −2.05613 −0.0711555
\(836\) 0 0
\(837\) 3.95207 0.136603
\(838\) 0 0
\(839\) −47.8718 −1.65272 −0.826359 0.563144i \(-0.809592\pi\)
−0.826359 + 0.563144i \(0.809592\pi\)
\(840\) 0 0
\(841\) 2.20407 0.0760025
\(842\) 0 0
\(843\) −0.130628 −0.00449907
\(844\) 0 0
\(845\) 0.379112 0.0130419
\(846\) 0 0
\(847\) 8.16519 0.280559
\(848\) 0 0
\(849\) 0.836580 0.0287113
\(850\) 0 0
\(851\) −5.55040 −0.190265
\(852\) 0 0
\(853\) −46.4819 −1.59151 −0.795755 0.605618i \(-0.792926\pi\)
−0.795755 + 0.605618i \(0.792926\pi\)
\(854\) 0 0
\(855\) −0.916943 −0.0313588
\(856\) 0 0
\(857\) 21.4642 0.733204 0.366602 0.930378i \(-0.380521\pi\)
0.366602 + 0.930378i \(0.380521\pi\)
\(858\) 0 0
\(859\) 32.6830 1.11513 0.557565 0.830133i \(-0.311736\pi\)
0.557565 + 0.830133i \(0.311736\pi\)
\(860\) 0 0
\(861\) −15.3779 −0.524076
\(862\) 0 0
\(863\) 45.7235 1.55645 0.778223 0.627989i \(-0.216122\pi\)
0.778223 + 0.627989i \(0.216122\pi\)
\(864\) 0 0
\(865\) 2.39000 0.0812626
\(866\) 0 0
\(867\) −3.96136 −0.134535
\(868\) 0 0
\(869\) −0.834850 −0.0283203
\(870\) 0 0
\(871\) −15.6953 −0.531814
\(872\) 0 0
\(873\) −6.23550 −0.211040
\(874\) 0 0
\(875\) −6.07558 −0.205392
\(876\) 0 0
\(877\) −18.7830 −0.634258 −0.317129 0.948382i \(-0.602719\pi\)
−0.317129 + 0.948382i \(0.602719\pi\)
\(878\) 0 0
\(879\) 8.78602 0.296345
\(880\) 0 0
\(881\) 22.8918 0.771244 0.385622 0.922657i \(-0.373987\pi\)
0.385622 + 0.922657i \(0.373987\pi\)
\(882\) 0 0
\(883\) −12.8753 −0.433287 −0.216644 0.976251i \(-0.569511\pi\)
−0.216644 + 0.976251i \(0.569511\pi\)
\(884\) 0 0
\(885\) 0.701491 0.0235804
\(886\) 0 0
\(887\) 0.866963 0.0291098 0.0145549 0.999894i \(-0.495367\pi\)
0.0145549 + 0.999894i \(0.495367\pi\)
\(888\) 0 0
\(889\) −18.5937 −0.623613
\(890\) 0 0
\(891\) −2.98599 −0.100034
\(892\) 0 0
\(893\) −65.3130 −2.18562
\(894\) 0 0
\(895\) 3.83659 0.128243
\(896\) 0 0
\(897\) 3.69588 0.123402
\(898\) 0 0
\(899\) 22.0765 0.736292
\(900\) 0 0
\(901\) 51.5110 1.71608
\(902\) 0 0
\(903\) 18.3696 0.611301
\(904\) 0 0
\(905\) 3.57898 0.118969
\(906\) 0 0
\(907\) −48.9486 −1.62531 −0.812656 0.582743i \(-0.801979\pi\)
−0.812656 + 0.582743i \(0.801979\pi\)
\(908\) 0 0
\(909\) 13.2888 0.440761
\(910\) 0 0
\(911\) 15.3830 0.509661 0.254830 0.966986i \(-0.417980\pi\)
0.254830 + 0.966986i \(0.417980\pi\)
\(912\) 0 0
\(913\) 24.5236 0.811611
\(914\) 0 0
\(915\) 1.07265 0.0354609
\(916\) 0 0
\(917\) −30.4593 −1.00585
\(918\) 0 0
\(919\) −51.5782 −1.70141 −0.850704 0.525645i \(-0.823824\pi\)
−0.850704 + 0.525645i \(0.823824\pi\)
\(920\) 0 0
\(921\) −22.3921 −0.737846
\(922\) 0 0
\(923\) 50.4914 1.66194
\(924\) 0 0
\(925\) −24.2842 −0.798460
\(926\) 0 0
\(927\) 4.20637 0.138155
\(928\) 0 0
\(929\) 10.1433 0.332791 0.166396 0.986059i \(-0.446787\pi\)
0.166396 + 0.986059i \(0.446787\pi\)
\(930\) 0 0
\(931\) 49.2745 1.61491
\(932\) 0 0
\(933\) −4.44519 −0.145529
\(934\) 0 0
\(935\) 2.12492 0.0694925
\(936\) 0 0
\(937\) 40.6550 1.32814 0.664070 0.747671i \(-0.268828\pi\)
0.664070 + 0.747671i \(0.268828\pi\)
\(938\) 0 0
\(939\) 19.7404 0.644202
\(940\) 0 0
\(941\) −24.2211 −0.789586 −0.394793 0.918770i \(-0.629184\pi\)
−0.394793 + 0.918770i \(0.629184\pi\)
\(942\) 0 0
\(943\) 4.46344 0.145350
\(944\) 0 0
\(945\) −0.609029 −0.0198117
\(946\) 0 0
\(947\) 7.66373 0.249038 0.124519 0.992217i \(-0.460261\pi\)
0.124519 + 0.992217i \(0.460261\pi\)
\(948\) 0 0
\(949\) 13.6974 0.444637
\(950\) 0 0
\(951\) −20.6169 −0.668549
\(952\) 0 0
\(953\) 29.7925 0.965075 0.482537 0.875875i \(-0.339715\pi\)
0.482537 + 0.875875i \(0.339715\pi\)
\(954\) 0 0
\(955\) 2.08674 0.0675254
\(956\) 0 0
\(957\) −16.6799 −0.539185
\(958\) 0 0
\(959\) 57.1296 1.84481
\(960\) 0 0
\(961\) −15.3812 −0.496166
\(962\) 0 0
\(963\) −7.56486 −0.243774
\(964\) 0 0
\(965\) 1.83827 0.0591760
\(966\) 0 0
\(967\) 8.77563 0.282205 0.141103 0.989995i \(-0.454935\pi\)
0.141103 + 0.989995i \(0.454935\pi\)
\(968\) 0 0
\(969\) −27.0088 −0.867649
\(970\) 0 0
\(971\) −10.4608 −0.335702 −0.167851 0.985812i \(-0.553683\pi\)
−0.167851 + 0.985812i \(0.553683\pi\)
\(972\) 0 0
\(973\) 73.1644 2.34554
\(974\) 0 0
\(975\) 16.1703 0.517864
\(976\) 0 0
\(977\) −8.96714 −0.286884 −0.143442 0.989659i \(-0.545817\pi\)
−0.143442 + 0.989659i \(0.545817\pi\)
\(978\) 0 0
\(979\) −2.74502 −0.0877311
\(980\) 0 0
\(981\) 6.16079 0.196699
\(982\) 0 0
\(983\) 23.0221 0.734292 0.367146 0.930163i \(-0.380335\pi\)
0.367146 + 0.930163i \(0.380335\pi\)
\(984\) 0 0
\(985\) 2.46048 0.0783975
\(986\) 0 0
\(987\) −43.3806 −1.38082
\(988\) 0 0
\(989\) −5.33179 −0.169541
\(990\) 0 0
\(991\) −3.68998 −0.117216 −0.0586080 0.998281i \(-0.518666\pi\)
−0.0586080 + 0.998281i \(0.518666\pi\)
\(992\) 0 0
\(993\) −15.0072 −0.476238
\(994\) 0 0
\(995\) 0.764085 0.0242231
\(996\) 0 0
\(997\) 31.9206 1.01094 0.505468 0.862845i \(-0.331320\pi\)
0.505468 + 0.862845i \(0.331320\pi\)
\(998\) 0 0
\(999\) −4.88043 −0.154410
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 6024.2.a.o.1.7 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6024.2.a.o.1.7 14 1.1 even 1 trivial