Properties

Label 1596.2.a.f.1.2
Level $1596$
Weight $2$
Character 1596.1
Self dual yes
Analytic conductor $12.744$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1596,2,Mod(1,1596)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1596, 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("1596.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1596 = 2^{2} \cdot 3 \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1596.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: \(12.7441241626\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{7}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64575\) of defining polynomial
Character \(\chi\) \(=\) 1596.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +1.64575 q^{5} -1.00000 q^{7} +1.00000 q^{9} -5.29150 q^{11} -1.64575 q^{15} +5.64575 q^{17} +1.00000 q^{19} +1.00000 q^{21} -5.29150 q^{23} -2.29150 q^{25} -1.00000 q^{27} -4.35425 q^{29} -6.00000 q^{31} +5.29150 q^{33} -1.64575 q^{35} -5.29150 q^{37} +7.29150 q^{41} +9.29150 q^{43} +1.64575 q^{45} -9.64575 q^{47} +1.00000 q^{49} -5.64575 q^{51} -8.35425 q^{53} -8.70850 q^{55} -1.00000 q^{57} -8.00000 q^{59} -1.29150 q^{61} -1.00000 q^{63} -3.29150 q^{67} +5.29150 q^{69} +6.93725 q^{71} -2.00000 q^{73} +2.29150 q^{75} +5.29150 q^{77} +7.29150 q^{79} +1.00000 q^{81} +0.937254 q^{83} +9.29150 q^{85} +4.35425 q^{87} +7.29150 q^{89} +6.00000 q^{93} +1.64575 q^{95} -16.5830 q^{97} -5.29150 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{15} + 6 q^{17} + 2 q^{19} + 2 q^{21} + 6 q^{25} - 2 q^{27} - 14 q^{29} - 12 q^{31} + 2 q^{35} + 4 q^{41} + 8 q^{43} - 2 q^{45} - 14 q^{47} + 2 q^{49} - 6 q^{51}+ \cdots - 12 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 1.64575 0.736002 0.368001 0.929825i \(-0.380042\pi\)
0.368001 + 0.929825i \(0.380042\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.29150 −1.59545 −0.797724 0.603023i \(-0.793963\pi\)
−0.797724 + 0.603023i \(0.793963\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.64575 −0.424931
\(16\) 0 0
\(17\) 5.64575 1.36930 0.684648 0.728874i \(-0.259956\pi\)
0.684648 + 0.728874i \(0.259956\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −5.29150 −1.10335 −0.551677 0.834058i \(-0.686012\pi\)
−0.551677 + 0.834058i \(0.686012\pi\)
\(24\) 0 0
\(25\) −2.29150 −0.458301
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.35425 −0.808564 −0.404282 0.914634i \(-0.632479\pi\)
−0.404282 + 0.914634i \(0.632479\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) 5.29150 0.921132
\(34\) 0 0
\(35\) −1.64575 −0.278183
\(36\) 0 0
\(37\) −5.29150 −0.869918 −0.434959 0.900450i \(-0.643237\pi\)
−0.434959 + 0.900450i \(0.643237\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.29150 1.13874 0.569371 0.822081i \(-0.307187\pi\)
0.569371 + 0.822081i \(0.307187\pi\)
\(42\) 0 0
\(43\) 9.29150 1.41694 0.708470 0.705740i \(-0.249386\pi\)
0.708470 + 0.705740i \(0.249386\pi\)
\(44\) 0 0
\(45\) 1.64575 0.245334
\(46\) 0 0
\(47\) −9.64575 −1.40698 −0.703489 0.710706i \(-0.748375\pi\)
−0.703489 + 0.710706i \(0.748375\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.64575 −0.790563
\(52\) 0 0
\(53\) −8.35425 −1.14754 −0.573772 0.819015i \(-0.694521\pi\)
−0.573772 + 0.819015i \(0.694521\pi\)
\(54\) 0 0
\(55\) −8.70850 −1.17425
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) −8.00000 −1.04151 −0.520756 0.853706i \(-0.674350\pi\)
−0.520756 + 0.853706i \(0.674350\pi\)
\(60\) 0 0
\(61\) −1.29150 −0.165360 −0.0826800 0.996576i \(-0.526348\pi\)
−0.0826800 + 0.996576i \(0.526348\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −3.29150 −0.402121 −0.201061 0.979579i \(-0.564439\pi\)
−0.201061 + 0.979579i \(0.564439\pi\)
\(68\) 0 0
\(69\) 5.29150 0.637022
\(70\) 0 0
\(71\) 6.93725 0.823301 0.411650 0.911342i \(-0.364953\pi\)
0.411650 + 0.911342i \(0.364953\pi\)
\(72\) 0 0
\(73\) −2.00000 −0.234082 −0.117041 0.993127i \(-0.537341\pi\)
−0.117041 + 0.993127i \(0.537341\pi\)
\(74\) 0 0
\(75\) 2.29150 0.264600
\(76\) 0 0
\(77\) 5.29150 0.603023
\(78\) 0 0
\(79\) 7.29150 0.820358 0.410179 0.912005i \(-0.365466\pi\)
0.410179 + 0.912005i \(0.365466\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.937254 0.102877 0.0514385 0.998676i \(-0.483619\pi\)
0.0514385 + 0.998676i \(0.483619\pi\)
\(84\) 0 0
\(85\) 9.29150 1.00780
\(86\) 0 0
\(87\) 4.35425 0.466824
\(88\) 0 0
\(89\) 7.29150 0.772898 0.386449 0.922311i \(-0.373702\pi\)
0.386449 + 0.922311i \(0.373702\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 6.00000 0.622171
\(94\) 0 0
\(95\) 1.64575 0.168851
\(96\) 0 0
\(97\) −16.5830 −1.68375 −0.841875 0.539673i \(-0.818548\pi\)
−0.841875 + 0.539673i \(0.818548\pi\)
\(98\) 0 0
\(99\) −5.29150 −0.531816
\(100\) 0 0
\(101\) −17.6458 −1.75582 −0.877909 0.478828i \(-0.841062\pi\)
−0.877909 + 0.478828i \(0.841062\pi\)
\(102\) 0 0
\(103\) −6.58301 −0.648643 −0.324321 0.945947i \(-0.605136\pi\)
−0.324321 + 0.945947i \(0.605136\pi\)
\(104\) 0 0
\(105\) 1.64575 0.160609
\(106\) 0 0
\(107\) −8.35425 −0.807636 −0.403818 0.914839i \(-0.632317\pi\)
−0.403818 + 0.914839i \(0.632317\pi\)
\(108\) 0 0
\(109\) −6.70850 −0.642558 −0.321279 0.946985i \(-0.604113\pi\)
−0.321279 + 0.946985i \(0.604113\pi\)
\(110\) 0 0
\(111\) 5.29150 0.502247
\(112\) 0 0
\(113\) −6.93725 −0.652602 −0.326301 0.945266i \(-0.605802\pi\)
−0.326301 + 0.945266i \(0.605802\pi\)
\(114\) 0 0
\(115\) −8.70850 −0.812072
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.64575 −0.517545
\(120\) 0 0
\(121\) 17.0000 1.54545
\(122\) 0 0
\(123\) −7.29150 −0.657453
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 7.29150 0.647016 0.323508 0.946225i \(-0.395138\pi\)
0.323508 + 0.946225i \(0.395138\pi\)
\(128\) 0 0
\(129\) −9.29150 −0.818071
\(130\) 0 0
\(131\) −10.3542 −0.904655 −0.452327 0.891852i \(-0.649406\pi\)
−0.452327 + 0.891852i \(0.649406\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) −1.64575 −0.141644
\(136\) 0 0
\(137\) −19.1660 −1.63746 −0.818731 0.574177i \(-0.805322\pi\)
−0.818731 + 0.574177i \(0.805322\pi\)
\(138\) 0 0
\(139\) −10.5830 −0.897639 −0.448819 0.893622i \(-0.648155\pi\)
−0.448819 + 0.893622i \(0.648155\pi\)
\(140\) 0 0
\(141\) 9.64575 0.812319
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −7.16601 −0.595105
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 10.7085 0.877274 0.438637 0.898664i \(-0.355461\pi\)
0.438637 + 0.898664i \(0.355461\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) 5.64575 0.456432
\(154\) 0 0
\(155\) −9.87451 −0.793140
\(156\) 0 0
\(157\) 9.29150 0.741543 0.370771 0.928724i \(-0.379093\pi\)
0.370771 + 0.928724i \(0.379093\pi\)
\(158\) 0 0
\(159\) 8.35425 0.662535
\(160\) 0 0
\(161\) 5.29150 0.417029
\(162\) 0 0
\(163\) 23.8745 1.87000 0.934998 0.354653i \(-0.115401\pi\)
0.934998 + 0.354653i \(0.115401\pi\)
\(164\) 0 0
\(165\) 8.70850 0.677956
\(166\) 0 0
\(167\) −3.29150 −0.254704 −0.127352 0.991858i \(-0.540648\pi\)
−0.127352 + 0.991858i \(0.540648\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 1.00000 0.0764719
\(172\) 0 0
\(173\) −15.2915 −1.16259 −0.581296 0.813692i \(-0.697454\pi\)
−0.581296 + 0.813692i \(0.697454\pi\)
\(174\) 0 0
\(175\) 2.29150 0.173221
\(176\) 0 0
\(177\) 8.00000 0.601317
\(178\) 0 0
\(179\) −15.6458 −1.16942 −0.584709 0.811243i \(-0.698791\pi\)
−0.584709 + 0.811243i \(0.698791\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 0 0
\(183\) 1.29150 0.0954706
\(184\) 0 0
\(185\) −8.70850 −0.640261
\(186\) 0 0
\(187\) −29.8745 −2.18464
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 1.29150 0.0934499 0.0467249 0.998908i \(-0.485122\pi\)
0.0467249 + 0.998908i \(0.485122\pi\)
\(192\) 0 0
\(193\) 23.8745 1.71852 0.859262 0.511535i \(-0.170923\pi\)
0.859262 + 0.511535i \(0.170923\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 16.5830 1.18149 0.590745 0.806858i \(-0.298834\pi\)
0.590745 + 0.806858i \(0.298834\pi\)
\(198\) 0 0
\(199\) 17.8745 1.26709 0.633545 0.773706i \(-0.281599\pi\)
0.633545 + 0.773706i \(0.281599\pi\)
\(200\) 0 0
\(201\) 3.29150 0.232165
\(202\) 0 0
\(203\) 4.35425 0.305608
\(204\) 0 0
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) −5.29150 −0.367785
\(208\) 0 0
\(209\) −5.29150 −0.366021
\(210\) 0 0
\(211\) 10.5830 0.728564 0.364282 0.931289i \(-0.381314\pi\)
0.364282 + 0.931289i \(0.381314\pi\)
\(212\) 0 0
\(213\) −6.93725 −0.475333
\(214\) 0 0
\(215\) 15.2915 1.04287
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) 2.00000 0.135147
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 19.1660 1.28345 0.641725 0.766934i \(-0.278219\pi\)
0.641725 + 0.766934i \(0.278219\pi\)
\(224\) 0 0
\(225\) −2.29150 −0.152767
\(226\) 0 0
\(227\) 16.7085 1.10898 0.554491 0.832190i \(-0.312913\pi\)
0.554491 + 0.832190i \(0.312913\pi\)
\(228\) 0 0
\(229\) 7.87451 0.520362 0.260181 0.965560i \(-0.416218\pi\)
0.260181 + 0.965560i \(0.416218\pi\)
\(230\) 0 0
\(231\) −5.29150 −0.348155
\(232\) 0 0
\(233\) 9.29150 0.608707 0.304353 0.952559i \(-0.401560\pi\)
0.304353 + 0.952559i \(0.401560\pi\)
\(234\) 0 0
\(235\) −15.8745 −1.03554
\(236\) 0 0
\(237\) −7.29150 −0.473634
\(238\) 0 0
\(239\) −6.00000 −0.388108 −0.194054 0.980991i \(-0.562164\pi\)
−0.194054 + 0.980991i \(0.562164\pi\)
\(240\) 0 0
\(241\) 16.5830 1.06821 0.534103 0.845420i \(-0.320650\pi\)
0.534103 + 0.845420i \(0.320650\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 1.64575 0.105143
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −0.937254 −0.0593961
\(250\) 0 0
\(251\) −7.06275 −0.445797 −0.222898 0.974842i \(-0.571552\pi\)
−0.222898 + 0.974842i \(0.571552\pi\)
\(252\) 0 0
\(253\) 28.0000 1.76034
\(254\) 0 0
\(255\) −9.29150 −0.581856
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 5.29150 0.328798
\(260\) 0 0
\(261\) −4.35425 −0.269521
\(262\) 0 0
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) −13.7490 −0.844595
\(266\) 0 0
\(267\) −7.29150 −0.446233
\(268\) 0 0
\(269\) −22.5830 −1.37691 −0.688455 0.725279i \(-0.741711\pi\)
−0.688455 + 0.725279i \(0.741711\pi\)
\(270\) 0 0
\(271\) 4.70850 0.286021 0.143010 0.989721i \(-0.454322\pi\)
0.143010 + 0.989721i \(0.454322\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 12.1255 0.731195
\(276\) 0 0
\(277\) 32.4575 1.95018 0.975091 0.221803i \(-0.0711942\pi\)
0.975091 + 0.221803i \(0.0711942\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) 1.77124 0.105664 0.0528318 0.998603i \(-0.483175\pi\)
0.0528318 + 0.998603i \(0.483175\pi\)
\(282\) 0 0
\(283\) 25.8745 1.53808 0.769040 0.639201i \(-0.220735\pi\)
0.769040 + 0.639201i \(0.220735\pi\)
\(284\) 0 0
\(285\) −1.64575 −0.0974859
\(286\) 0 0
\(287\) −7.29150 −0.430404
\(288\) 0 0
\(289\) 14.8745 0.874971
\(290\) 0 0
\(291\) 16.5830 0.972113
\(292\) 0 0
\(293\) −19.2915 −1.12702 −0.563511 0.826109i \(-0.690550\pi\)
−0.563511 + 0.826109i \(0.690550\pi\)
\(294\) 0 0
\(295\) −13.1660 −0.766555
\(296\) 0 0
\(297\) 5.29150 0.307044
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −9.29150 −0.535553
\(302\) 0 0
\(303\) 17.6458 1.01372
\(304\) 0 0
\(305\) −2.12549 −0.121705
\(306\) 0 0
\(307\) 0.583005 0.0332739 0.0166369 0.999862i \(-0.494704\pi\)
0.0166369 + 0.999862i \(0.494704\pi\)
\(308\) 0 0
\(309\) 6.58301 0.374494
\(310\) 0 0
\(311\) −0.937254 −0.0531468 −0.0265734 0.999647i \(-0.508460\pi\)
−0.0265734 + 0.999647i \(0.508460\pi\)
\(312\) 0 0
\(313\) 14.7085 0.831373 0.415687 0.909508i \(-0.363541\pi\)
0.415687 + 0.909508i \(0.363541\pi\)
\(314\) 0 0
\(315\) −1.64575 −0.0927276
\(316\) 0 0
\(317\) 1.06275 0.0596898 0.0298449 0.999555i \(-0.490499\pi\)
0.0298449 + 0.999555i \(0.490499\pi\)
\(318\) 0 0
\(319\) 23.0405 1.29002
\(320\) 0 0
\(321\) 8.35425 0.466289
\(322\) 0 0
\(323\) 5.64575 0.314138
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6.70850 0.370981
\(328\) 0 0
\(329\) 9.64575 0.531787
\(330\) 0 0
\(331\) −30.5830 −1.68099 −0.840497 0.541816i \(-0.817737\pi\)
−0.840497 + 0.541816i \(0.817737\pi\)
\(332\) 0 0
\(333\) −5.29150 −0.289973
\(334\) 0 0
\(335\) −5.41699 −0.295962
\(336\) 0 0
\(337\) 22.7085 1.23701 0.618505 0.785781i \(-0.287739\pi\)
0.618505 + 0.785781i \(0.287739\pi\)
\(338\) 0 0
\(339\) 6.93725 0.376780
\(340\) 0 0
\(341\) 31.7490 1.71931
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 8.70850 0.468850
\(346\) 0 0
\(347\) 11.1660 0.599423 0.299711 0.954030i \(-0.403110\pi\)
0.299711 + 0.954030i \(0.403110\pi\)
\(348\) 0 0
\(349\) −14.0000 −0.749403 −0.374701 0.927146i \(-0.622255\pi\)
−0.374701 + 0.927146i \(0.622255\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 28.9373 1.54017 0.770087 0.637939i \(-0.220213\pi\)
0.770087 + 0.637939i \(0.220213\pi\)
\(354\) 0 0
\(355\) 11.4170 0.605951
\(356\) 0 0
\(357\) 5.64575 0.298805
\(358\) 0 0
\(359\) 22.4575 1.18526 0.592631 0.805474i \(-0.298089\pi\)
0.592631 + 0.805474i \(0.298089\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −17.0000 −0.892269
\(364\) 0 0
\(365\) −3.29150 −0.172285
\(366\) 0 0
\(367\) −13.8745 −0.724243 −0.362122 0.932131i \(-0.617948\pi\)
−0.362122 + 0.932131i \(0.617948\pi\)
\(368\) 0 0
\(369\) 7.29150 0.379580
\(370\) 0 0
\(371\) 8.35425 0.433731
\(372\) 0 0
\(373\) −16.5830 −0.858635 −0.429318 0.903154i \(-0.641246\pi\)
−0.429318 + 0.903154i \(0.641246\pi\)
\(374\) 0 0
\(375\) 12.0000 0.619677
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −10.5830 −0.543612 −0.271806 0.962352i \(-0.587621\pi\)
−0.271806 + 0.962352i \(0.587621\pi\)
\(380\) 0 0
\(381\) −7.29150 −0.373555
\(382\) 0 0
\(383\) −1.87451 −0.0957829 −0.0478914 0.998853i \(-0.515250\pi\)
−0.0478914 + 0.998853i \(0.515250\pi\)
\(384\) 0 0
\(385\) 8.70850 0.443826
\(386\) 0 0
\(387\) 9.29150 0.472314
\(388\) 0 0
\(389\) −26.4575 −1.34145 −0.670725 0.741707i \(-0.734017\pi\)
−0.670725 + 0.741707i \(0.734017\pi\)
\(390\) 0 0
\(391\) −29.8745 −1.51082
\(392\) 0 0
\(393\) 10.3542 0.522303
\(394\) 0 0
\(395\) 12.0000 0.603786
\(396\) 0 0
\(397\) 30.0000 1.50566 0.752828 0.658217i \(-0.228689\pi\)
0.752828 + 0.658217i \(0.228689\pi\)
\(398\) 0 0
\(399\) 1.00000 0.0500626
\(400\) 0 0
\(401\) −10.9373 −0.546180 −0.273090 0.961988i \(-0.588046\pi\)
−0.273090 + 0.961988i \(0.588046\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 1.64575 0.0817780
\(406\) 0 0
\(407\) 28.0000 1.38791
\(408\) 0 0
\(409\) −18.5830 −0.918870 −0.459435 0.888211i \(-0.651948\pi\)
−0.459435 + 0.888211i \(0.651948\pi\)
\(410\) 0 0
\(411\) 19.1660 0.945390
\(412\) 0 0
\(413\) 8.00000 0.393654
\(414\) 0 0
\(415\) 1.54249 0.0757177
\(416\) 0 0
\(417\) 10.5830 0.518252
\(418\) 0 0
\(419\) −16.9373 −0.827439 −0.413720 0.910404i \(-0.635771\pi\)
−0.413720 + 0.910404i \(0.635771\pi\)
\(420\) 0 0
\(421\) −20.5830 −1.00315 −0.501577 0.865113i \(-0.667247\pi\)
−0.501577 + 0.865113i \(0.667247\pi\)
\(422\) 0 0
\(423\) −9.64575 −0.468992
\(424\) 0 0
\(425\) −12.9373 −0.627549
\(426\) 0 0
\(427\) 1.29150 0.0625002
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 3.64575 0.175610 0.0878048 0.996138i \(-0.472015\pi\)
0.0878048 + 0.996138i \(0.472015\pi\)
\(432\) 0 0
\(433\) −24.5830 −1.18138 −0.590692 0.806897i \(-0.701145\pi\)
−0.590692 + 0.806897i \(0.701145\pi\)
\(434\) 0 0
\(435\) 7.16601 0.343584
\(436\) 0 0
\(437\) −5.29150 −0.253127
\(438\) 0 0
\(439\) 11.1660 0.532925 0.266462 0.963845i \(-0.414145\pi\)
0.266462 + 0.963845i \(0.414145\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 31.1660 1.48074 0.740371 0.672199i \(-0.234650\pi\)
0.740371 + 0.672199i \(0.234650\pi\)
\(444\) 0 0
\(445\) 12.0000 0.568855
\(446\) 0 0
\(447\) −10.7085 −0.506495
\(448\) 0 0
\(449\) 27.3948 1.29284 0.646419 0.762982i \(-0.276266\pi\)
0.646419 + 0.762982i \(0.276266\pi\)
\(450\) 0 0
\(451\) −38.5830 −1.81680
\(452\) 0 0
\(453\) 8.00000 0.375873
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 23.2915 1.08953 0.544765 0.838589i \(-0.316619\pi\)
0.544765 + 0.838589i \(0.316619\pi\)
\(458\) 0 0
\(459\) −5.64575 −0.263521
\(460\) 0 0
\(461\) 35.5203 1.65434 0.827172 0.561949i \(-0.189948\pi\)
0.827172 + 0.561949i \(0.189948\pi\)
\(462\) 0 0
\(463\) −25.1660 −1.16956 −0.584782 0.811191i \(-0.698820\pi\)
−0.584782 + 0.811191i \(0.698820\pi\)
\(464\) 0 0
\(465\) 9.87451 0.457919
\(466\) 0 0
\(467\) 16.2288 0.750977 0.375489 0.926827i \(-0.377475\pi\)
0.375489 + 0.926827i \(0.377475\pi\)
\(468\) 0 0
\(469\) 3.29150 0.151987
\(470\) 0 0
\(471\) −9.29150 −0.428130
\(472\) 0 0
\(473\) −49.1660 −2.26066
\(474\) 0 0
\(475\) −2.29150 −0.105141
\(476\) 0 0
\(477\) −8.35425 −0.382515
\(478\) 0 0
\(479\) −11.5203 −0.526374 −0.263187 0.964745i \(-0.584774\pi\)
−0.263187 + 0.964745i \(0.584774\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −5.29150 −0.240772
\(484\) 0 0
\(485\) −27.2915 −1.23924
\(486\) 0 0
\(487\) −4.00000 −0.181257 −0.0906287 0.995885i \(-0.528888\pi\)
−0.0906287 + 0.995885i \(0.528888\pi\)
\(488\) 0 0
\(489\) −23.8745 −1.07964
\(490\) 0 0
\(491\) 6.00000 0.270776 0.135388 0.990793i \(-0.456772\pi\)
0.135388 + 0.990793i \(0.456772\pi\)
\(492\) 0 0
\(493\) −24.5830 −1.10716
\(494\) 0 0
\(495\) −8.70850 −0.391418
\(496\) 0 0
\(497\) −6.93725 −0.311178
\(498\) 0 0
\(499\) 13.4170 0.600627 0.300314 0.953841i \(-0.402909\pi\)
0.300314 + 0.953841i \(0.402909\pi\)
\(500\) 0 0
\(501\) 3.29150 0.147053
\(502\) 0 0
\(503\) 24.2288 1.08031 0.540153 0.841567i \(-0.318366\pi\)
0.540153 + 0.841567i \(0.318366\pi\)
\(504\) 0 0
\(505\) −29.0405 −1.29229
\(506\) 0 0
\(507\) 13.0000 0.577350
\(508\) 0 0
\(509\) −9.41699 −0.417401 −0.208700 0.977980i \(-0.566923\pi\)
−0.208700 + 0.977980i \(0.566923\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) −10.8340 −0.477403
\(516\) 0 0
\(517\) 51.0405 2.24476
\(518\) 0 0
\(519\) 15.2915 0.671222
\(520\) 0 0
\(521\) −14.5830 −0.638893 −0.319447 0.947604i \(-0.603497\pi\)
−0.319447 + 0.947604i \(0.603497\pi\)
\(522\) 0 0
\(523\) −27.1660 −1.18789 −0.593943 0.804507i \(-0.702430\pi\)
−0.593943 + 0.804507i \(0.702430\pi\)
\(524\) 0 0
\(525\) −2.29150 −0.100009
\(526\) 0 0
\(527\) −33.8745 −1.47560
\(528\) 0 0
\(529\) 5.00000 0.217391
\(530\) 0 0
\(531\) −8.00000 −0.347170
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −13.7490 −0.594422
\(536\) 0 0
\(537\) 15.6458 0.675164
\(538\) 0 0
\(539\) −5.29150 −0.227921
\(540\) 0 0
\(541\) −23.1660 −0.995985 −0.497992 0.867181i \(-0.665929\pi\)
−0.497992 + 0.867181i \(0.665929\pi\)
\(542\) 0 0
\(543\) −6.00000 −0.257485
\(544\) 0 0
\(545\) −11.0405 −0.472924
\(546\) 0 0
\(547\) 24.4575 1.04573 0.522864 0.852416i \(-0.324864\pi\)
0.522864 + 0.852416i \(0.324864\pi\)
\(548\) 0 0
\(549\) −1.29150 −0.0551200
\(550\) 0 0
\(551\) −4.35425 −0.185497
\(552\) 0 0
\(553\) −7.29150 −0.310066
\(554\) 0 0
\(555\) 8.70850 0.369655
\(556\) 0 0
\(557\) 7.87451 0.333654 0.166827 0.985986i \(-0.446648\pi\)
0.166827 + 0.985986i \(0.446648\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 29.8745 1.26130
\(562\) 0 0
\(563\) −8.70850 −0.367019 −0.183510 0.983018i \(-0.558746\pi\)
−0.183510 + 0.983018i \(0.558746\pi\)
\(564\) 0 0
\(565\) −11.4170 −0.480317
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 22.9373 0.961580 0.480790 0.876836i \(-0.340350\pi\)
0.480790 + 0.876836i \(0.340350\pi\)
\(570\) 0 0
\(571\) −29.1660 −1.22056 −0.610280 0.792186i \(-0.708943\pi\)
−0.610280 + 0.792186i \(0.708943\pi\)
\(572\) 0 0
\(573\) −1.29150 −0.0539533
\(574\) 0 0
\(575\) 12.1255 0.505668
\(576\) 0 0
\(577\) 19.8745 0.827387 0.413693 0.910416i \(-0.364239\pi\)
0.413693 + 0.910416i \(0.364239\pi\)
\(578\) 0 0
\(579\) −23.8745 −0.992191
\(580\) 0 0
\(581\) −0.937254 −0.0388838
\(582\) 0 0
\(583\) 44.2065 1.83085
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −28.2288 −1.16513 −0.582563 0.812786i \(-0.697950\pi\)
−0.582563 + 0.812786i \(0.697950\pi\)
\(588\) 0 0
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) −16.5830 −0.682134
\(592\) 0 0
\(593\) 0.228757 0.00939391 0.00469695 0.999989i \(-0.498505\pi\)
0.00469695 + 0.999989i \(0.498505\pi\)
\(594\) 0 0
\(595\) −9.29150 −0.380914
\(596\) 0 0
\(597\) −17.8745 −0.731555
\(598\) 0 0
\(599\) −30.2288 −1.23511 −0.617557 0.786526i \(-0.711877\pi\)
−0.617557 + 0.786526i \(0.711877\pi\)
\(600\) 0 0
\(601\) −36.5830 −1.49225 −0.746126 0.665805i \(-0.768088\pi\)
−0.746126 + 0.665805i \(0.768088\pi\)
\(602\) 0 0
\(603\) −3.29150 −0.134040
\(604\) 0 0
\(605\) 27.9778 1.13746
\(606\) 0 0
\(607\) 38.5830 1.56604 0.783018 0.621999i \(-0.213679\pi\)
0.783018 + 0.621999i \(0.213679\pi\)
\(608\) 0 0
\(609\) −4.35425 −0.176443
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 36.4575 1.47251 0.736253 0.676707i \(-0.236594\pi\)
0.736253 + 0.676707i \(0.236594\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) 43.1660 1.73780 0.868899 0.494989i \(-0.164828\pi\)
0.868899 + 0.494989i \(0.164828\pi\)
\(618\) 0 0
\(619\) −15.7490 −0.633006 −0.316503 0.948591i \(-0.602509\pi\)
−0.316503 + 0.948591i \(0.602509\pi\)
\(620\) 0 0
\(621\) 5.29150 0.212341
\(622\) 0 0
\(623\) −7.29150 −0.292128
\(624\) 0 0
\(625\) −8.29150 −0.331660
\(626\) 0 0
\(627\) 5.29150 0.211322
\(628\) 0 0
\(629\) −29.8745 −1.19117
\(630\) 0 0
\(631\) 23.8745 0.950429 0.475215 0.879870i \(-0.342370\pi\)
0.475215 + 0.879870i \(0.342370\pi\)
\(632\) 0 0
\(633\) −10.5830 −0.420637
\(634\) 0 0
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.93725 0.274434
\(640\) 0 0
\(641\) 1.77124 0.0699599 0.0349800 0.999388i \(-0.488863\pi\)
0.0349800 + 0.999388i \(0.488863\pi\)
\(642\) 0 0
\(643\) 8.45751 0.333532 0.166766 0.985997i \(-0.446668\pi\)
0.166766 + 0.985997i \(0.446668\pi\)
\(644\) 0 0
\(645\) −15.2915 −0.602102
\(646\) 0 0
\(647\) −14.8118 −0.582310 −0.291155 0.956676i \(-0.594040\pi\)
−0.291155 + 0.956676i \(0.594040\pi\)
\(648\) 0 0
\(649\) 42.3320 1.66168
\(650\) 0 0
\(651\) −6.00000 −0.235159
\(652\) 0 0
\(653\) −37.7490 −1.47723 −0.738617 0.674126i \(-0.764521\pi\)
−0.738617 + 0.674126i \(0.764521\pi\)
\(654\) 0 0
\(655\) −17.0405 −0.665828
\(656\) 0 0
\(657\) −2.00000 −0.0780274
\(658\) 0 0
\(659\) 19.3948 0.755513 0.377756 0.925905i \(-0.376696\pi\)
0.377756 + 0.925905i \(0.376696\pi\)
\(660\) 0 0
\(661\) −32.0000 −1.24466 −0.622328 0.782757i \(-0.713813\pi\)
−0.622328 + 0.782757i \(0.713813\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.64575 −0.0638195
\(666\) 0 0
\(667\) 23.0405 0.892132
\(668\) 0 0
\(669\) −19.1660 −0.741001
\(670\) 0 0
\(671\) 6.83399 0.263823
\(672\) 0 0
\(673\) −43.8745 −1.69124 −0.845619 0.533787i \(-0.820768\pi\)
−0.845619 + 0.533787i \(0.820768\pi\)
\(674\) 0 0
\(675\) 2.29150 0.0882000
\(676\) 0 0
\(677\) −36.0000 −1.38359 −0.691796 0.722093i \(-0.743180\pi\)
−0.691796 + 0.722093i \(0.743180\pi\)
\(678\) 0 0
\(679\) 16.5830 0.636397
\(680\) 0 0
\(681\) −16.7085 −0.640271
\(682\) 0 0
\(683\) 22.9373 0.877670 0.438835 0.898568i \(-0.355391\pi\)
0.438835 + 0.898568i \(0.355391\pi\)
\(684\) 0 0
\(685\) −31.5425 −1.20518
\(686\) 0 0
\(687\) −7.87451 −0.300431
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0.708497 0.0269525 0.0134763 0.999909i \(-0.495710\pi\)
0.0134763 + 0.999909i \(0.495710\pi\)
\(692\) 0 0
\(693\) 5.29150 0.201008
\(694\) 0 0
\(695\) −17.4170 −0.660664
\(696\) 0 0
\(697\) 41.1660 1.55927
\(698\) 0 0
\(699\) −9.29150 −0.351437
\(700\) 0 0
\(701\) 22.0000 0.830929 0.415464 0.909610i \(-0.363619\pi\)
0.415464 + 0.909610i \(0.363619\pi\)
\(702\) 0 0
\(703\) −5.29150 −0.199573
\(704\) 0 0
\(705\) 15.8745 0.597869
\(706\) 0 0
\(707\) 17.6458 0.663637
\(708\) 0 0
\(709\) 17.8745 0.671291 0.335646 0.941988i \(-0.391046\pi\)
0.335646 + 0.941988i \(0.391046\pi\)
\(710\) 0 0
\(711\) 7.29150 0.273453
\(712\) 0 0
\(713\) 31.7490 1.18901
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6.00000 0.224074
\(718\) 0 0
\(719\) 14.3542 0.535323 0.267662 0.963513i \(-0.413749\pi\)
0.267662 + 0.963513i \(0.413749\pi\)
\(720\) 0 0
\(721\) 6.58301 0.245164
\(722\) 0 0
\(723\) −16.5830 −0.616729
\(724\) 0 0
\(725\) 9.97777 0.370565
\(726\) 0 0
\(727\) −24.7085 −0.916387 −0.458194 0.888852i \(-0.651503\pi\)
−0.458194 + 0.888852i \(0.651503\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 52.4575 1.94021
\(732\) 0 0
\(733\) 27.4170 1.01267 0.506335 0.862337i \(-0.331000\pi\)
0.506335 + 0.862337i \(0.331000\pi\)
\(734\) 0 0
\(735\) −1.64575 −0.0607045
\(736\) 0 0
\(737\) 17.4170 0.641563
\(738\) 0 0
\(739\) −42.4575 −1.56182 −0.780912 0.624641i \(-0.785246\pi\)
−0.780912 + 0.624641i \(0.785246\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −34.9373 −1.28172 −0.640862 0.767656i \(-0.721423\pi\)
−0.640862 + 0.767656i \(0.721423\pi\)
\(744\) 0 0
\(745\) 17.6235 0.645676
\(746\) 0 0
\(747\) 0.937254 0.0342923
\(748\) 0 0
\(749\) 8.35425 0.305258
\(750\) 0 0
\(751\) 53.1660 1.94006 0.970028 0.242995i \(-0.0781297\pi\)
0.970028 + 0.242995i \(0.0781297\pi\)
\(752\) 0 0
\(753\) 7.06275 0.257381
\(754\) 0 0
\(755\) −13.1660 −0.479160
\(756\) 0 0
\(757\) 8.58301 0.311955 0.155977 0.987761i \(-0.450147\pi\)
0.155977 + 0.987761i \(0.450147\pi\)
\(758\) 0 0
\(759\) −28.0000 −1.01634
\(760\) 0 0
\(761\) 15.0627 0.546024 0.273012 0.962011i \(-0.411980\pi\)
0.273012 + 0.962011i \(0.411980\pi\)
\(762\) 0 0
\(763\) 6.70850 0.242864
\(764\) 0 0
\(765\) 9.29150 0.335935
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −9.29150 −0.335060 −0.167530 0.985867i \(-0.553579\pi\)
−0.167530 + 0.985867i \(0.553579\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −35.0405 −1.26032 −0.630160 0.776465i \(-0.717011\pi\)
−0.630160 + 0.776465i \(0.717011\pi\)
\(774\) 0 0
\(775\) 13.7490 0.493879
\(776\) 0 0
\(777\) −5.29150 −0.189832
\(778\) 0 0
\(779\) 7.29150 0.261245
\(780\) 0 0
\(781\) −36.7085 −1.31353
\(782\) 0 0
\(783\) 4.35425 0.155608
\(784\) 0 0
\(785\) 15.2915 0.545777
\(786\) 0 0
\(787\) 35.1660 1.25353 0.626766 0.779207i \(-0.284378\pi\)
0.626766 + 0.779207i \(0.284378\pi\)
\(788\) 0 0
\(789\) 18.0000 0.640817
\(790\) 0 0
\(791\) 6.93725 0.246660
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 13.7490 0.487627
\(796\) 0 0
\(797\) −39.0405 −1.38289 −0.691443 0.722431i \(-0.743025\pi\)
−0.691443 + 0.722431i \(0.743025\pi\)
\(798\) 0 0
\(799\) −54.4575 −1.92657
\(800\) 0 0
\(801\) 7.29150 0.257633
\(802\) 0 0
\(803\) 10.5830 0.373466
\(804\) 0 0
\(805\) 8.70850 0.306934
\(806\) 0 0
\(807\) 22.5830 0.794959
\(808\) 0 0
\(809\) −23.4170 −0.823298 −0.411649 0.911343i \(-0.635047\pi\)
−0.411649 + 0.911343i \(0.635047\pi\)
\(810\) 0 0
\(811\) −39.7490 −1.39578 −0.697888 0.716207i \(-0.745877\pi\)
−0.697888 + 0.716207i \(0.745877\pi\)
\(812\) 0 0
\(813\) −4.70850 −0.165134
\(814\) 0 0
\(815\) 39.2915 1.37632
\(816\) 0 0
\(817\) 9.29150 0.325069
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.70850 −0.0945272 −0.0472636 0.998882i \(-0.515050\pi\)
−0.0472636 + 0.998882i \(0.515050\pi\)
\(822\) 0 0
\(823\) 12.1255 0.422668 0.211334 0.977414i \(-0.432219\pi\)
0.211334 + 0.977414i \(0.432219\pi\)
\(824\) 0 0
\(825\) −12.1255 −0.422155
\(826\) 0 0
\(827\) −13.0627 −0.454236 −0.227118 0.973867i \(-0.572930\pi\)
−0.227118 + 0.973867i \(0.572930\pi\)
\(828\) 0 0
\(829\) −48.3320 −1.67864 −0.839320 0.543637i \(-0.817047\pi\)
−0.839320 + 0.543637i \(0.817047\pi\)
\(830\) 0 0
\(831\) −32.4575 −1.12594
\(832\) 0 0
\(833\) 5.64575 0.195614
\(834\) 0 0
\(835\) −5.41699 −0.187463
\(836\) 0 0
\(837\) 6.00000 0.207390
\(838\) 0 0
\(839\) −42.5830 −1.47013 −0.735064 0.677997i \(-0.762848\pi\)
−0.735064 + 0.677997i \(0.762848\pi\)
\(840\) 0 0
\(841\) −10.0405 −0.346225
\(842\) 0 0
\(843\) −1.77124 −0.0610049
\(844\) 0 0
\(845\) −21.3948 −0.736002
\(846\) 0 0
\(847\) −17.0000 −0.584127
\(848\) 0 0
\(849\) −25.8745 −0.888011
\(850\) 0 0
\(851\) 28.0000 0.959828
\(852\) 0 0
\(853\) 7.16601 0.245360 0.122680 0.992446i \(-0.460851\pi\)
0.122680 + 0.992446i \(0.460851\pi\)
\(854\) 0 0
\(855\) 1.64575 0.0562835
\(856\) 0 0
\(857\) −14.1255 −0.482518 −0.241259 0.970461i \(-0.577560\pi\)
−0.241259 + 0.970461i \(0.577560\pi\)
\(858\) 0 0
\(859\) −45.1660 −1.54104 −0.770522 0.637413i \(-0.780004\pi\)
−0.770522 + 0.637413i \(0.780004\pi\)
\(860\) 0 0
\(861\) 7.29150 0.248494
\(862\) 0 0
\(863\) −31.3948 −1.06869 −0.534345 0.845266i \(-0.679442\pi\)
−0.534345 + 0.845266i \(0.679442\pi\)
\(864\) 0 0
\(865\) −25.1660 −0.855670
\(866\) 0 0
\(867\) −14.8745 −0.505165
\(868\) 0 0
\(869\) −38.5830 −1.30884
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −16.5830 −0.561250
\(874\) 0 0
\(875\) 12.0000 0.405674
\(876\) 0 0
\(877\) −16.5830 −0.559968 −0.279984 0.960005i \(-0.590329\pi\)
−0.279984 + 0.960005i \(0.590329\pi\)
\(878\) 0 0
\(879\) 19.2915 0.650686
\(880\) 0 0
\(881\) 24.6863 0.831702 0.415851 0.909433i \(-0.363484\pi\)
0.415851 + 0.909433i \(0.363484\pi\)
\(882\) 0 0
\(883\) 1.16601 0.0392394 0.0196197 0.999808i \(-0.493754\pi\)
0.0196197 + 0.999808i \(0.493754\pi\)
\(884\) 0 0
\(885\) 13.1660 0.442571
\(886\) 0 0
\(887\) 33.8745 1.13739 0.568697 0.822547i \(-0.307448\pi\)
0.568697 + 0.822547i \(0.307448\pi\)
\(888\) 0 0
\(889\) −7.29150 −0.244549
\(890\) 0 0
\(891\) −5.29150 −0.177272
\(892\) 0 0
\(893\) −9.64575 −0.322783
\(894\) 0 0
\(895\) −25.7490 −0.860695
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 26.1255 0.871334
\(900\) 0 0
\(901\) −47.1660 −1.57133
\(902\) 0 0
\(903\) 9.29150 0.309202
\(904\) 0 0
\(905\) 9.87451 0.328240
\(906\) 0 0
\(907\) −58.3320 −1.93688 −0.968441 0.249241i \(-0.919819\pi\)
−0.968441 + 0.249241i \(0.919819\pi\)
\(908\) 0 0
\(909\) −17.6458 −0.585273
\(910\) 0 0
\(911\) 37.0627 1.22794 0.613972 0.789328i \(-0.289571\pi\)
0.613972 + 0.789328i \(0.289571\pi\)
\(912\) 0 0
\(913\) −4.95948 −0.164135
\(914\) 0 0
\(915\) 2.12549 0.0702666
\(916\) 0 0
\(917\) 10.3542 0.341927
\(918\) 0 0
\(919\) −56.0000 −1.84727 −0.923635 0.383274i \(-0.874797\pi\)
−0.923635 + 0.383274i \(0.874797\pi\)
\(920\) 0 0
\(921\) −0.583005 −0.0192107
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 12.1255 0.398684
\(926\) 0 0
\(927\) −6.58301 −0.216214
\(928\) 0 0
\(929\) −24.6863 −0.809930 −0.404965 0.914332i \(-0.632716\pi\)
−0.404965 + 0.914332i \(0.632716\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) 0 0
\(933\) 0.937254 0.0306843
\(934\) 0 0
\(935\) −49.1660 −1.60790
\(936\) 0 0
\(937\) 45.7490 1.49456 0.747278 0.664512i \(-0.231361\pi\)
0.747278 + 0.664512i \(0.231361\pi\)
\(938\) 0 0
\(939\) −14.7085 −0.479994
\(940\) 0 0
\(941\) 47.7490 1.55657 0.778287 0.627909i \(-0.216089\pi\)
0.778287 + 0.627909i \(0.216089\pi\)
\(942\) 0 0
\(943\) −38.5830 −1.25644
\(944\) 0 0
\(945\) 1.64575 0.0535363
\(946\) 0 0
\(947\) 43.8745 1.42573 0.712865 0.701301i \(-0.247397\pi\)
0.712865 + 0.701301i \(0.247397\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −1.06275 −0.0344619
\(952\) 0 0
\(953\) −5.52026 −0.178819 −0.0894094 0.995995i \(-0.528498\pi\)
−0.0894094 + 0.995995i \(0.528498\pi\)
\(954\) 0 0
\(955\) 2.12549 0.0687793
\(956\) 0 0
\(957\) −23.0405 −0.744794
\(958\) 0 0
\(959\) 19.1660 0.618903
\(960\) 0 0
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) −8.35425 −0.269212
\(964\) 0 0
\(965\) 39.2915 1.26484
\(966\) 0 0
\(967\) 21.2915 0.684689 0.342344 0.939575i \(-0.388779\pi\)
0.342344 + 0.939575i \(0.388779\pi\)
\(968\) 0 0
\(969\) −5.64575 −0.181368
\(970\) 0 0
\(971\) −31.7490 −1.01887 −0.509437 0.860508i \(-0.670146\pi\)
−0.509437 + 0.860508i \(0.670146\pi\)
\(972\) 0 0
\(973\) 10.5830 0.339276
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 32.8118 1.04974 0.524871 0.851182i \(-0.324114\pi\)
0.524871 + 0.851182i \(0.324114\pi\)
\(978\) 0 0
\(979\) −38.5830 −1.23312
\(980\) 0 0
\(981\) −6.70850 −0.214186
\(982\) 0 0
\(983\) 55.7490 1.77812 0.889059 0.457793i \(-0.151360\pi\)
0.889059 + 0.457793i \(0.151360\pi\)
\(984\) 0 0
\(985\) 27.2915 0.869580
\(986\) 0 0
\(987\) −9.64575 −0.307028
\(988\) 0 0
\(989\) −49.1660 −1.56339
\(990\) 0 0
\(991\) 54.3320 1.72591 0.862957 0.505278i \(-0.168610\pi\)
0.862957 + 0.505278i \(0.168610\pi\)
\(992\) 0 0
\(993\) 30.5830 0.970522
\(994\) 0 0
\(995\) 29.4170 0.932581
\(996\) 0 0
\(997\) −39.6235 −1.25489 −0.627445 0.778661i \(-0.715899\pi\)
−0.627445 + 0.778661i \(0.715899\pi\)
\(998\) 0 0
\(999\) 5.29150 0.167416
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 1596.2.a.f.1.2 2
3.2 odd 2 4788.2.a.k.1.1 2
4.3 odd 2 6384.2.a.bo.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1596.2.a.f.1.2 2 1.1 even 1 trivial
4788.2.a.k.1.1 2 3.2 odd 2
6384.2.a.bo.1.2 2 4.3 odd 2