Properties

Label 1596.2.a.j.1.2
Level $1596$
Weight $2$
Character 1596.1
Self dual yes
Analytic conductor $12.744$
Analytic rank $0$
Dimension $3$
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: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1016.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.85577\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.29966 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.00000 q^{11} +5.71155 q^{13} +1.29966 q^{15} +1.29966 q^{17} -1.00000 q^{19} +1.00000 q^{21} +2.00000 q^{23} -3.31087 q^{25} +1.00000 q^{27} -10.7228 q^{29} -3.71155 q^{31} +2.00000 q^{33} +1.29966 q^{35} -0.599328 q^{37} +5.71155 q^{39} -4.00000 q^{41} +10.3109 q^{43} +1.29966 q^{45} +10.1234 q^{47} +1.00000 q^{49} +1.29966 q^{51} +0.700336 q^{53} +2.59933 q^{55} -1.00000 q^{57} +3.40067 q^{61} +1.00000 q^{63} +7.42309 q^{65} -2.59933 q^{67} +2.00000 q^{69} -0.700336 q^{71} +13.4231 q^{73} -3.31087 q^{75} +2.00000 q^{77} -6.02242 q^{79} +1.00000 q^{81} -12.7228 q^{83} +1.68913 q^{85} -10.7228 q^{87} -4.00000 q^{89} +5.71155 q^{91} -3.71155 q^{93} -1.29966 q^{95} -2.00000 q^{97} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 3 q^{7} + 3 q^{9} + 6 q^{11} + 2 q^{13} - 3 q^{19} + 3 q^{21} + 6 q^{23} + 13 q^{25} + 3 q^{27} + 2 q^{29} + 4 q^{31} + 6 q^{33} + 6 q^{37} + 2 q^{39} - 12 q^{41} + 8 q^{43} + 4 q^{47} + 3 q^{49}+ \cdots + 6 q^{99}+O(q^{100}) \) 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
\(5\) 1.29966 0.581227 0.290614 0.956840i \(-0.406141\pi\)
0.290614 + 0.956840i \(0.406141\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.00000 0.603023 0.301511 0.953463i \(-0.402509\pi\)
0.301511 + 0.953463i \(0.402509\pi\)
\(12\) 0 0
\(13\) 5.71155 1.58410 0.792049 0.610458i \(-0.209015\pi\)
0.792049 + 0.610458i \(0.209015\pi\)
\(14\) 0 0
\(15\) 1.29966 0.335572
\(16\) 0 0
\(17\) 1.29966 0.315215 0.157607 0.987502i \(-0.449622\pi\)
0.157607 + 0.987502i \(0.449622\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 2.00000 0.417029 0.208514 0.978019i \(-0.433137\pi\)
0.208514 + 0.978019i \(0.433137\pi\)
\(24\) 0 0
\(25\) −3.31087 −0.662175
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −10.7228 −1.99117 −0.995583 0.0938884i \(-0.970070\pi\)
−0.995583 + 0.0938884i \(0.970070\pi\)
\(30\) 0 0
\(31\) −3.71155 −0.666613 −0.333307 0.942818i \(-0.608164\pi\)
−0.333307 + 0.942818i \(0.608164\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 1.29966 0.219683
\(36\) 0 0
\(37\) −0.599328 −0.0985290 −0.0492645 0.998786i \(-0.515688\pi\)
−0.0492645 + 0.998786i \(0.515688\pi\)
\(38\) 0 0
\(39\) 5.71155 0.914579
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 10.3109 1.57239 0.786197 0.617976i \(-0.212047\pi\)
0.786197 + 0.617976i \(0.212047\pi\)
\(44\) 0 0
\(45\) 1.29966 0.193742
\(46\) 0 0
\(47\) 10.1234 1.47665 0.738327 0.674443i \(-0.235616\pi\)
0.738327 + 0.674443i \(0.235616\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 1.29966 0.181989
\(52\) 0 0
\(53\) 0.700336 0.0961985 0.0480993 0.998843i \(-0.484684\pi\)
0.0480993 + 0.998843i \(0.484684\pi\)
\(54\) 0 0
\(55\) 2.59933 0.350493
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 3.40067 0.435411 0.217706 0.976014i \(-0.430143\pi\)
0.217706 + 0.976014i \(0.430143\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 7.42309 0.920721
\(66\) 0 0
\(67\) −2.59933 −0.317558 −0.158779 0.987314i \(-0.550756\pi\)
−0.158779 + 0.987314i \(0.550756\pi\)
\(68\) 0 0
\(69\) 2.00000 0.240772
\(70\) 0 0
\(71\) −0.700336 −0.0831146 −0.0415573 0.999136i \(-0.513232\pi\)
−0.0415573 + 0.999136i \(0.513232\pi\)
\(72\) 0 0
\(73\) 13.4231 1.57105 0.785527 0.618827i \(-0.212392\pi\)
0.785527 + 0.618827i \(0.212392\pi\)
\(74\) 0 0
\(75\) −3.31087 −0.382307
\(76\) 0 0
\(77\) 2.00000 0.227921
\(78\) 0 0
\(79\) −6.02242 −0.677575 −0.338787 0.940863i \(-0.610017\pi\)
−0.338787 + 0.940863i \(0.610017\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −12.7228 −1.39650 −0.698252 0.715852i \(-0.746038\pi\)
−0.698252 + 0.715852i \(0.746038\pi\)
\(84\) 0 0
\(85\) 1.68913 0.183212
\(86\) 0 0
\(87\) −10.7228 −1.14960
\(88\) 0 0
\(89\) −4.00000 −0.423999 −0.212000 0.977270i \(-0.567998\pi\)
−0.212000 + 0.977270i \(0.567998\pi\)
\(90\) 0 0
\(91\) 5.71155 0.598733
\(92\) 0 0
\(93\) −3.71155 −0.384869
\(94\) 0 0
\(95\) −1.29966 −0.133343
\(96\) 0 0
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 2.00000 0.201008
\(100\) 0 0
\(101\) 2.70034 0.268693 0.134347 0.990934i \(-0.457106\pi\)
0.134347 + 0.990934i \(0.457106\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 1.29966 0.126834
\(106\) 0 0
\(107\) 13.3221 1.28789 0.643947 0.765070i \(-0.277296\pi\)
0.643947 + 0.765070i \(0.277296\pi\)
\(108\) 0 0
\(109\) −4.59933 −0.440536 −0.220268 0.975439i \(-0.570693\pi\)
−0.220268 + 0.975439i \(0.570693\pi\)
\(110\) 0 0
\(111\) −0.599328 −0.0568857
\(112\) 0 0
\(113\) 8.12343 0.764188 0.382094 0.924124i \(-0.375203\pi\)
0.382094 + 0.924124i \(0.375203\pi\)
\(114\) 0 0
\(115\) 2.59933 0.242389
\(116\) 0 0
\(117\) 5.71155 0.528033
\(118\) 0 0
\(119\) 1.29966 0.119140
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −4.00000 −0.360668
\(124\) 0 0
\(125\) −10.8013 −0.966102
\(126\) 0 0
\(127\) 6.59933 0.585596 0.292798 0.956174i \(-0.405414\pi\)
0.292798 + 0.956174i \(0.405414\pi\)
\(128\) 0 0
\(129\) 10.3109 0.907822
\(130\) 0 0
\(131\) −7.32208 −0.639733 −0.319867 0.947463i \(-0.603638\pi\)
−0.319867 + 0.947463i \(0.603638\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) 1.29966 0.111857
\(136\) 0 0
\(137\) −17.4231 −1.48856 −0.744278 0.667870i \(-0.767206\pi\)
−0.744278 + 0.667870i \(0.767206\pi\)
\(138\) 0 0
\(139\) −6.22443 −0.527950 −0.263975 0.964530i \(-0.585034\pi\)
−0.263975 + 0.964530i \(0.585034\pi\)
\(140\) 0 0
\(141\) 10.1234 0.852546
\(142\) 0 0
\(143\) 11.4231 0.955247
\(144\) 0 0
\(145\) −13.9360 −1.15732
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) −16.0224 −1.31261 −0.656304 0.754497i \(-0.727881\pi\)
−0.656304 + 0.754497i \(0.727881\pi\)
\(150\) 0 0
\(151\) 7.42309 0.604082 0.302041 0.953295i \(-0.402332\pi\)
0.302041 + 0.953295i \(0.402332\pi\)
\(152\) 0 0
\(153\) 1.29966 0.105072
\(154\) 0 0
\(155\) −4.82376 −0.387454
\(156\) 0 0
\(157\) −6.82376 −0.544595 −0.272298 0.962213i \(-0.587784\pi\)
−0.272298 + 0.962213i \(0.587784\pi\)
\(158\) 0 0
\(159\) 0.700336 0.0555402
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 0 0
\(163\) 21.7340 1.70234 0.851168 0.524894i \(-0.175895\pi\)
0.851168 + 0.524894i \(0.175895\pi\)
\(164\) 0 0
\(165\) 2.59933 0.202357
\(166\) 0 0
\(167\) −1.97758 −0.153030 −0.0765149 0.997068i \(-0.524379\pi\)
−0.0765149 + 0.997068i \(0.524379\pi\)
\(168\) 0 0
\(169\) 19.6217 1.50937
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 0 0
\(173\) −1.19866 −0.0911322 −0.0455661 0.998961i \(-0.514509\pi\)
−0.0455661 + 0.998961i \(0.514509\pi\)
\(174\) 0 0
\(175\) −3.31087 −0.250278
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −13.3221 −0.995739 −0.497870 0.867252i \(-0.665884\pi\)
−0.497870 + 0.867252i \(0.665884\pi\)
\(180\) 0 0
\(181\) 1.42309 0.105777 0.0528887 0.998600i \(-0.483157\pi\)
0.0528887 + 0.998600i \(0.483157\pi\)
\(182\) 0 0
\(183\) 3.40067 0.251385
\(184\) 0 0
\(185\) −0.778925 −0.0572677
\(186\) 0 0
\(187\) 2.59933 0.190082
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −9.42309 −0.681831 −0.340915 0.940094i \(-0.610737\pi\)
−0.340915 + 0.940094i \(0.610737\pi\)
\(192\) 0 0
\(193\) 10.8238 0.779111 0.389556 0.921003i \(-0.372629\pi\)
0.389556 + 0.921003i \(0.372629\pi\)
\(194\) 0 0
\(195\) 7.42309 0.531579
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 13.4007 0.949948 0.474974 0.880000i \(-0.342457\pi\)
0.474974 + 0.880000i \(0.342457\pi\)
\(200\) 0 0
\(201\) −2.59933 −0.183342
\(202\) 0 0
\(203\) −10.7228 −0.752590
\(204\) 0 0
\(205\) −5.19866 −0.363090
\(206\) 0 0
\(207\) 2.00000 0.139010
\(208\) 0 0
\(209\) −2.00000 −0.138343
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 0 0
\(213\) −0.700336 −0.0479862
\(214\) 0 0
\(215\) 13.4007 0.913918
\(216\) 0 0
\(217\) −3.71155 −0.251956
\(218\) 0 0
\(219\) 13.4231 0.907048
\(220\) 0 0
\(221\) 7.42309 0.499331
\(222\) 0 0
\(223\) −21.9360 −1.46894 −0.734471 0.678640i \(-0.762570\pi\)
−0.734471 + 0.678640i \(0.762570\pi\)
\(224\) 0 0
\(225\) −3.31087 −0.220725
\(226\) 0 0
\(227\) −3.17624 −0.210814 −0.105407 0.994429i \(-0.533615\pi\)
−0.105407 + 0.994429i \(0.533615\pi\)
\(228\) 0 0
\(229\) 17.2211 1.13800 0.569000 0.822337i \(-0.307330\pi\)
0.569000 + 0.822337i \(0.307330\pi\)
\(230\) 0 0
\(231\) 2.00000 0.131590
\(232\) 0 0
\(233\) −22.2469 −1.45744 −0.728720 0.684812i \(-0.759884\pi\)
−0.728720 + 0.684812i \(0.759884\pi\)
\(234\) 0 0
\(235\) 13.1571 0.858272
\(236\) 0 0
\(237\) −6.02242 −0.391198
\(238\) 0 0
\(239\) −1.17624 −0.0760845 −0.0380423 0.999276i \(-0.512112\pi\)
−0.0380423 + 0.999276i \(0.512112\pi\)
\(240\) 0 0
\(241\) −14.6217 −0.941869 −0.470935 0.882168i \(-0.656083\pi\)
−0.470935 + 0.882168i \(0.656083\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 1.29966 0.0830325
\(246\) 0 0
\(247\) −5.71155 −0.363417
\(248\) 0 0
\(249\) −12.7228 −0.806272
\(250\) 0 0
\(251\) −1.87657 −0.118448 −0.0592242 0.998245i \(-0.518863\pi\)
−0.0592242 + 0.998245i \(0.518863\pi\)
\(252\) 0 0
\(253\) 4.00000 0.251478
\(254\) 0 0
\(255\) 1.68913 0.105777
\(256\) 0 0
\(257\) −22.0224 −1.37372 −0.686860 0.726789i \(-0.741012\pi\)
−0.686860 + 0.726789i \(0.741012\pi\)
\(258\) 0 0
\(259\) −0.599328 −0.0372404
\(260\) 0 0
\(261\) −10.7228 −0.663722
\(262\) 0 0
\(263\) −0.599328 −0.0369562 −0.0184781 0.999829i \(-0.505882\pi\)
−0.0184781 + 0.999829i \(0.505882\pi\)
\(264\) 0 0
\(265\) 0.910201 0.0559132
\(266\) 0 0
\(267\) −4.00000 −0.244796
\(268\) 0 0
\(269\) −5.40067 −0.329285 −0.164642 0.986353i \(-0.552647\pi\)
−0.164642 + 0.986353i \(0.552647\pi\)
\(270\) 0 0
\(271\) 31.2211 1.89655 0.948273 0.317457i \(-0.102829\pi\)
0.948273 + 0.317457i \(0.102829\pi\)
\(272\) 0 0
\(273\) 5.71155 0.345678
\(274\) 0 0
\(275\) −6.62175 −0.399306
\(276\) 0 0
\(277\) −29.9584 −1.80003 −0.900013 0.435863i \(-0.856443\pi\)
−0.900013 + 0.435863i \(0.856443\pi\)
\(278\) 0 0
\(279\) −3.71155 −0.222204
\(280\) 0 0
\(281\) −8.70034 −0.519019 −0.259509 0.965741i \(-0.583561\pi\)
−0.259509 + 0.965741i \(0.583561\pi\)
\(282\) 0 0
\(283\) −6.02242 −0.357996 −0.178998 0.983849i \(-0.557286\pi\)
−0.178998 + 0.983849i \(0.557286\pi\)
\(284\) 0 0
\(285\) −1.29966 −0.0769855
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −15.3109 −0.900640
\(290\) 0 0
\(291\) −2.00000 −0.117242
\(292\) 0 0
\(293\) −29.8204 −1.74213 −0.871063 0.491171i \(-0.836569\pi\)
−0.871063 + 0.491171i \(0.836569\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 2.00000 0.116052
\(298\) 0 0
\(299\) 11.4231 0.660614
\(300\) 0 0
\(301\) 10.3109 0.594309
\(302\) 0 0
\(303\) 2.70034 0.155130
\(304\) 0 0
\(305\) 4.41973 0.253073
\(306\) 0 0
\(307\) −10.6858 −0.609869 −0.304934 0.952373i \(-0.598635\pi\)
−0.304934 + 0.952373i \(0.598635\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) −20.5207 −1.16362 −0.581812 0.813323i \(-0.697656\pi\)
−0.581812 + 0.813323i \(0.697656\pi\)
\(312\) 0 0
\(313\) 9.22107 0.521206 0.260603 0.965446i \(-0.416079\pi\)
0.260603 + 0.965446i \(0.416079\pi\)
\(314\) 0 0
\(315\) 1.29966 0.0732278
\(316\) 0 0
\(317\) 30.9696 1.73943 0.869713 0.493557i \(-0.164304\pi\)
0.869713 + 0.493557i \(0.164304\pi\)
\(318\) 0 0
\(319\) −21.4455 −1.20072
\(320\) 0 0
\(321\) 13.3221 0.743566
\(322\) 0 0
\(323\) −1.29966 −0.0723152
\(324\) 0 0
\(325\) −18.9102 −1.04895
\(326\) 0 0
\(327\) −4.59933 −0.254343
\(328\) 0 0
\(329\) 10.1234 0.558123
\(330\) 0 0
\(331\) 6.80134 0.373836 0.186918 0.982376i \(-0.440150\pi\)
0.186918 + 0.982376i \(0.440150\pi\)
\(332\) 0 0
\(333\) −0.599328 −0.0328430
\(334\) 0 0
\(335\) −3.37825 −0.184574
\(336\) 0 0
\(337\) 12.5993 0.686329 0.343165 0.939275i \(-0.388501\pi\)
0.343165 + 0.939275i \(0.388501\pi\)
\(338\) 0 0
\(339\) 8.12343 0.441204
\(340\) 0 0
\(341\) −7.42309 −0.401983
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 2.59933 0.139943
\(346\) 0 0
\(347\) −4.02242 −0.215935 −0.107967 0.994154i \(-0.534434\pi\)
−0.107967 + 0.994154i \(0.534434\pi\)
\(348\) 0 0
\(349\) −27.8204 −1.48919 −0.744596 0.667515i \(-0.767358\pi\)
−0.744596 + 0.667515i \(0.767358\pi\)
\(350\) 0 0
\(351\) 5.71155 0.304860
\(352\) 0 0
\(353\) −17.2997 −0.920768 −0.460384 0.887720i \(-0.652288\pi\)
−0.460384 + 0.887720i \(0.652288\pi\)
\(354\) 0 0
\(355\) −0.910201 −0.0483085
\(356\) 0 0
\(357\) 1.29966 0.0687855
\(358\) 0 0
\(359\) 31.8204 1.67942 0.839708 0.543038i \(-0.182726\pi\)
0.839708 + 0.543038i \(0.182726\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 17.4455 0.913140
\(366\) 0 0
\(367\) 24.8686 1.29813 0.649065 0.760733i \(-0.275160\pi\)
0.649065 + 0.760733i \(0.275160\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 0.700336 0.0363596
\(372\) 0 0
\(373\) 26.0448 1.34855 0.674275 0.738480i \(-0.264456\pi\)
0.674275 + 0.738480i \(0.264456\pi\)
\(374\) 0 0
\(375\) −10.8013 −0.557779
\(376\) 0 0
\(377\) −61.2435 −3.15420
\(378\) 0 0
\(379\) −31.4679 −1.61640 −0.808199 0.588909i \(-0.799558\pi\)
−0.808199 + 0.588909i \(0.799558\pi\)
\(380\) 0 0
\(381\) 6.59933 0.338094
\(382\) 0 0
\(383\) 21.4007 1.09352 0.546762 0.837288i \(-0.315860\pi\)
0.546762 + 0.837288i \(0.315860\pi\)
\(384\) 0 0
\(385\) 2.59933 0.132474
\(386\) 0 0
\(387\) 10.3109 0.524131
\(388\) 0 0
\(389\) 20.6442 1.04670 0.523350 0.852118i \(-0.324682\pi\)
0.523350 + 0.852118i \(0.324682\pi\)
\(390\) 0 0
\(391\) 2.59933 0.131454
\(392\) 0 0
\(393\) −7.32208 −0.369350
\(394\) 0 0
\(395\) −7.82712 −0.393825
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 0 0
\(401\) −19.2997 −0.963779 −0.481890 0.876232i \(-0.660049\pi\)
−0.481890 + 0.876232i \(0.660049\pi\)
\(402\) 0 0
\(403\) −21.1987 −1.05598
\(404\) 0 0
\(405\) 1.29966 0.0645808
\(406\) 0 0
\(407\) −1.19866 −0.0594152
\(408\) 0 0
\(409\) −3.48711 −0.172427 −0.0862133 0.996277i \(-0.527477\pi\)
−0.0862133 + 0.996277i \(0.527477\pi\)
\(410\) 0 0
\(411\) −17.4231 −0.859418
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −16.5353 −0.811686
\(416\) 0 0
\(417\) −6.22443 −0.304812
\(418\) 0 0
\(419\) −32.5207 −1.58874 −0.794371 0.607433i \(-0.792199\pi\)
−0.794371 + 0.607433i \(0.792199\pi\)
\(420\) 0 0
\(421\) 17.4231 0.849149 0.424575 0.905393i \(-0.360424\pi\)
0.424575 + 0.905393i \(0.360424\pi\)
\(422\) 0 0
\(423\) 10.1234 0.492218
\(424\) 0 0
\(425\) −4.30302 −0.208727
\(426\) 0 0
\(427\) 3.40067 0.164570
\(428\) 0 0
\(429\) 11.4231 0.551512
\(430\) 0 0
\(431\) −0.325441 −0.0156760 −0.00783798 0.999969i \(-0.502495\pi\)
−0.00783798 + 0.999969i \(0.502495\pi\)
\(432\) 0 0
\(433\) −31.8204 −1.52919 −0.764595 0.644510i \(-0.777061\pi\)
−0.764595 + 0.644510i \(0.777061\pi\)
\(434\) 0 0
\(435\) −13.9360 −0.668179
\(436\) 0 0
\(437\) −2.00000 −0.0956730
\(438\) 0 0
\(439\) 23.7564 1.13383 0.566915 0.823776i \(-0.308137\pi\)
0.566915 + 0.823776i \(0.308137\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 24.5993 1.16875 0.584375 0.811484i \(-0.301340\pi\)
0.584375 + 0.811484i \(0.301340\pi\)
\(444\) 0 0
\(445\) −5.19866 −0.246440
\(446\) 0 0
\(447\) −16.0224 −0.757834
\(448\) 0 0
\(449\) −27.3445 −1.29047 −0.645233 0.763986i \(-0.723240\pi\)
−0.645233 + 0.763986i \(0.723240\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) 0 0
\(453\) 7.42309 0.348767
\(454\) 0 0
\(455\) 7.42309 0.348000
\(456\) 0 0
\(457\) 8.31087 0.388766 0.194383 0.980926i \(-0.437730\pi\)
0.194383 + 0.980926i \(0.437730\pi\)
\(458\) 0 0
\(459\) 1.29966 0.0606631
\(460\) 0 0
\(461\) 9.54652 0.444626 0.222313 0.974975i \(-0.428639\pi\)
0.222313 + 0.974975i \(0.428639\pi\)
\(462\) 0 0
\(463\) −10.8462 −0.504065 −0.252032 0.967719i \(-0.581099\pi\)
−0.252032 + 0.967719i \(0.581099\pi\)
\(464\) 0 0
\(465\) −4.82376 −0.223697
\(466\) 0 0
\(467\) 11.5241 0.533272 0.266636 0.963797i \(-0.414088\pi\)
0.266636 + 0.963797i \(0.414088\pi\)
\(468\) 0 0
\(469\) −2.59933 −0.120026
\(470\) 0 0
\(471\) −6.82376 −0.314422
\(472\) 0 0
\(473\) 20.6217 0.948189
\(474\) 0 0
\(475\) 3.31087 0.151913
\(476\) 0 0
\(477\) 0.700336 0.0320662
\(478\) 0 0
\(479\) −35.3669 −1.61596 −0.807978 0.589213i \(-0.799438\pi\)
−0.807978 + 0.589213i \(0.799438\pi\)
\(480\) 0 0
\(481\) −3.42309 −0.156079
\(482\) 0 0
\(483\) 2.00000 0.0910032
\(484\) 0 0
\(485\) −2.59933 −0.118029
\(486\) 0 0
\(487\) −11.3783 −0.515598 −0.257799 0.966199i \(-0.582997\pi\)
−0.257799 + 0.966199i \(0.582997\pi\)
\(488\) 0 0
\(489\) 21.7340 0.982844
\(490\) 0 0
\(491\) 21.2211 0.957694 0.478847 0.877898i \(-0.341055\pi\)
0.478847 + 0.877898i \(0.341055\pi\)
\(492\) 0 0
\(493\) −13.9360 −0.627645
\(494\) 0 0
\(495\) 2.59933 0.116831
\(496\) 0 0
\(497\) −0.700336 −0.0314144
\(498\) 0 0
\(499\) 37.2435 1.66725 0.833624 0.552333i \(-0.186262\pi\)
0.833624 + 0.552333i \(0.186262\pi\)
\(500\) 0 0
\(501\) −1.97758 −0.0883518
\(502\) 0 0
\(503\) −0.924770 −0.0412334 −0.0206167 0.999787i \(-0.506563\pi\)
−0.0206167 + 0.999787i \(0.506563\pi\)
\(504\) 0 0
\(505\) 3.50953 0.156172
\(506\) 0 0
\(507\) 19.6217 0.871432
\(508\) 0 0
\(509\) −20.2469 −0.897426 −0.448713 0.893676i \(-0.648117\pi\)
−0.448713 + 0.893676i \(0.648117\pi\)
\(510\) 0 0
\(511\) 13.4231 0.593803
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 10.3973 0.458160
\(516\) 0 0
\(517\) 20.2469 0.890456
\(518\) 0 0
\(519\) −1.19866 −0.0526152
\(520\) 0 0
\(521\) −44.2469 −1.93849 −0.969245 0.246098i \(-0.920851\pi\)
−0.969245 + 0.246098i \(0.920851\pi\)
\(522\) 0 0
\(523\) −37.9808 −1.66079 −0.830393 0.557179i \(-0.811884\pi\)
−0.830393 + 0.557179i \(0.811884\pi\)
\(524\) 0 0
\(525\) −3.31087 −0.144498
\(526\) 0 0
\(527\) −4.82376 −0.210126
\(528\) 0 0
\(529\) −19.0000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −22.8462 −0.989578
\(534\) 0 0
\(535\) 17.3142 0.748560
\(536\) 0 0
\(537\) −13.3221 −0.574890
\(538\) 0 0
\(539\) 2.00000 0.0861461
\(540\) 0 0
\(541\) −9.46793 −0.407058 −0.203529 0.979069i \(-0.565241\pi\)
−0.203529 + 0.979069i \(0.565241\pi\)
\(542\) 0 0
\(543\) 1.42309 0.0610706
\(544\) 0 0
\(545\) −5.97758 −0.256051
\(546\) 0 0
\(547\) −11.1762 −0.477861 −0.238931 0.971037i \(-0.576797\pi\)
−0.238931 + 0.971037i \(0.576797\pi\)
\(548\) 0 0
\(549\) 3.40067 0.145137
\(550\) 0 0
\(551\) 10.7228 0.456805
\(552\) 0 0
\(553\) −6.02242 −0.256099
\(554\) 0 0
\(555\) −0.778925 −0.0330635
\(556\) 0 0
\(557\) 36.0224 1.52632 0.763159 0.646210i \(-0.223647\pi\)
0.763159 + 0.646210i \(0.223647\pi\)
\(558\) 0 0
\(559\) 58.8910 2.49082
\(560\) 0 0
\(561\) 2.59933 0.109744
\(562\) 0 0
\(563\) −10.0224 −0.422395 −0.211197 0.977443i \(-0.567736\pi\)
−0.211197 + 0.977443i \(0.567736\pi\)
\(564\) 0 0
\(565\) 10.5577 0.444167
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −11.0976 −0.465238 −0.232619 0.972568i \(-0.574729\pi\)
−0.232619 + 0.972568i \(0.574729\pi\)
\(570\) 0 0
\(571\) −28.0448 −1.17364 −0.586820 0.809717i \(-0.699620\pi\)
−0.586820 + 0.809717i \(0.699620\pi\)
\(572\) 0 0
\(573\) −9.42309 −0.393655
\(574\) 0 0
\(575\) −6.62175 −0.276146
\(576\) 0 0
\(577\) −17.1762 −0.715056 −0.357528 0.933902i \(-0.616380\pi\)
−0.357528 + 0.933902i \(0.616380\pi\)
\(578\) 0 0
\(579\) 10.8238 0.449820
\(580\) 0 0
\(581\) −12.7228 −0.527829
\(582\) 0 0
\(583\) 1.40067 0.0580099
\(584\) 0 0
\(585\) 7.42309 0.306907
\(586\) 0 0
\(587\) 16.5207 0.681884 0.340942 0.940084i \(-0.389254\pi\)
0.340942 + 0.940084i \(0.389254\pi\)
\(588\) 0 0
\(589\) 3.71155 0.152932
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 36.1907 1.48617 0.743087 0.669195i \(-0.233361\pi\)
0.743087 + 0.669195i \(0.233361\pi\)
\(594\) 0 0
\(595\) 1.68913 0.0692474
\(596\) 0 0
\(597\) 13.4007 0.548453
\(598\) 0 0
\(599\) 4.32544 0.176733 0.0883664 0.996088i \(-0.471835\pi\)
0.0883664 + 0.996088i \(0.471835\pi\)
\(600\) 0 0
\(601\) −31.6475 −1.29093 −0.645465 0.763790i \(-0.723336\pi\)
−0.645465 + 0.763790i \(0.723336\pi\)
\(602\) 0 0
\(603\) −2.59933 −0.105853
\(604\) 0 0
\(605\) −9.09765 −0.369872
\(606\) 0 0
\(607\) −14.8462 −0.602588 −0.301294 0.953531i \(-0.597419\pi\)
−0.301294 + 0.953531i \(0.597419\pi\)
\(608\) 0 0
\(609\) −10.7228 −0.434508
\(610\) 0 0
\(611\) 57.8204 2.33916
\(612\) 0 0
\(613\) 12.8878 0.520533 0.260266 0.965537i \(-0.416190\pi\)
0.260266 + 0.965537i \(0.416190\pi\)
\(614\) 0 0
\(615\) −5.19866 −0.209630
\(616\) 0 0
\(617\) 3.60269 0.145039 0.0725194 0.997367i \(-0.476896\pi\)
0.0725194 + 0.997367i \(0.476896\pi\)
\(618\) 0 0
\(619\) 6.22443 0.250181 0.125091 0.992145i \(-0.460078\pi\)
0.125091 + 0.992145i \(0.460078\pi\)
\(620\) 0 0
\(621\) 2.00000 0.0802572
\(622\) 0 0
\(623\) −4.00000 −0.160257
\(624\) 0 0
\(625\) 2.51625 0.100650
\(626\) 0 0
\(627\) −2.00000 −0.0798723
\(628\) 0 0
\(629\) −0.778925 −0.0310578
\(630\) 0 0
\(631\) −8.08644 −0.321916 −0.160958 0.986961i \(-0.551458\pi\)
−0.160958 + 0.986961i \(0.551458\pi\)
\(632\) 0 0
\(633\) 4.00000 0.158986
\(634\) 0 0
\(635\) 8.57691 0.340364
\(636\) 0 0
\(637\) 5.71155 0.226300
\(638\) 0 0
\(639\) −0.700336 −0.0277049
\(640\) 0 0
\(641\) −20.1683 −0.796598 −0.398299 0.917256i \(-0.630399\pi\)
−0.398299 + 0.917256i \(0.630399\pi\)
\(642\) 0 0
\(643\) 20.8686 0.822977 0.411489 0.911415i \(-0.365009\pi\)
0.411489 + 0.911415i \(0.365009\pi\)
\(644\) 0 0
\(645\) 13.4007 0.527651
\(646\) 0 0
\(647\) −5.87657 −0.231032 −0.115516 0.993306i \(-0.536852\pi\)
−0.115516 + 0.993306i \(0.536852\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −3.71155 −0.145467
\(652\) 0 0
\(653\) 19.1987 0.751301 0.375651 0.926761i \(-0.377419\pi\)
0.375651 + 0.926761i \(0.377419\pi\)
\(654\) 0 0
\(655\) −9.51625 −0.371831
\(656\) 0 0
\(657\) 13.4231 0.523685
\(658\) 0 0
\(659\) 41.9438 1.63390 0.816950 0.576709i \(-0.195663\pi\)
0.816950 + 0.576709i \(0.195663\pi\)
\(660\) 0 0
\(661\) 26.7822 1.04171 0.520853 0.853647i \(-0.325614\pi\)
0.520853 + 0.853647i \(0.325614\pi\)
\(662\) 0 0
\(663\) 7.42309 0.288289
\(664\) 0 0
\(665\) −1.29966 −0.0503988
\(666\) 0 0
\(667\) −21.4455 −0.830373
\(668\) 0 0
\(669\) −21.9360 −0.848094
\(670\) 0 0
\(671\) 6.80134 0.262563
\(672\) 0 0
\(673\) 25.1762 0.970473 0.485236 0.874383i \(-0.338734\pi\)
0.485236 + 0.874383i \(0.338734\pi\)
\(674\) 0 0
\(675\) −3.31087 −0.127436
\(676\) 0 0
\(677\) −31.8428 −1.22382 −0.611910 0.790928i \(-0.709598\pi\)
−0.611910 + 0.790928i \(0.709598\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) −3.17624 −0.121714
\(682\) 0 0
\(683\) 38.7228 1.48169 0.740843 0.671679i \(-0.234426\pi\)
0.740843 + 0.671679i \(0.234426\pi\)
\(684\) 0 0
\(685\) −22.6442 −0.865189
\(686\) 0 0
\(687\) 17.2211 0.657025
\(688\) 0 0
\(689\) 4.00000 0.152388
\(690\) 0 0
\(691\) −30.0224 −1.14211 −0.571053 0.820913i \(-0.693465\pi\)
−0.571053 + 0.820913i \(0.693465\pi\)
\(692\) 0 0
\(693\) 2.00000 0.0759737
\(694\) 0 0
\(695\) −8.08967 −0.306859
\(696\) 0 0
\(697\) −5.19866 −0.196913
\(698\) 0 0
\(699\) −22.2469 −0.841453
\(700\) 0 0
\(701\) 47.6475 1.79962 0.899811 0.436280i \(-0.143704\pi\)
0.899811 + 0.436280i \(0.143704\pi\)
\(702\) 0 0
\(703\) 0.599328 0.0226041
\(704\) 0 0
\(705\) 13.1571 0.495523
\(706\) 0 0
\(707\) 2.70034 0.101557
\(708\) 0 0
\(709\) 25.5095 0.958030 0.479015 0.877807i \(-0.340994\pi\)
0.479015 + 0.877807i \(0.340994\pi\)
\(710\) 0 0
\(711\) −6.02242 −0.225858
\(712\) 0 0
\(713\) −7.42309 −0.277997
\(714\) 0 0
\(715\) 14.8462 0.555216
\(716\) 0 0
\(717\) −1.17624 −0.0439274
\(718\) 0 0
\(719\) 9.67456 0.360800 0.180400 0.983593i \(-0.442261\pi\)
0.180400 + 0.983593i \(0.442261\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 0 0
\(723\) −14.6217 −0.543789
\(724\) 0 0
\(725\) 35.5017 1.31850
\(726\) 0 0
\(727\) −45.4903 −1.68714 −0.843572 0.537016i \(-0.819551\pi\)
−0.843572 + 0.537016i \(0.819551\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 13.4007 0.495642
\(732\) 0 0
\(733\) 15.7756 0.582684 0.291342 0.956619i \(-0.405898\pi\)
0.291342 + 0.956619i \(0.405898\pi\)
\(734\) 0 0
\(735\) 1.29966 0.0479388
\(736\) 0 0
\(737\) −5.19866 −0.191495
\(738\) 0 0
\(739\) −50.3557 −1.85236 −0.926182 0.377076i \(-0.876930\pi\)
−0.926182 + 0.377076i \(0.876930\pi\)
\(740\) 0 0
\(741\) −5.71155 −0.209819
\(742\) 0 0
\(743\) 29.7709 1.09219 0.546095 0.837723i \(-0.316114\pi\)
0.546095 + 0.837723i \(0.316114\pi\)
\(744\) 0 0
\(745\) −20.8238 −0.762924
\(746\) 0 0
\(747\) −12.7228 −0.465501
\(748\) 0 0
\(749\) 13.3221 0.486778
\(750\) 0 0
\(751\) 25.8204 0.942200 0.471100 0.882080i \(-0.343857\pi\)
0.471100 + 0.882080i \(0.343857\pi\)
\(752\) 0 0
\(753\) −1.87657 −0.0683862
\(754\) 0 0
\(755\) 9.64752 0.351109
\(756\) 0 0
\(757\) −21.8720 −0.794950 −0.397475 0.917613i \(-0.630113\pi\)
−0.397475 + 0.917613i \(0.630113\pi\)
\(758\) 0 0
\(759\) 4.00000 0.145191
\(760\) 0 0
\(761\) 28.1458 1.02029 0.510143 0.860090i \(-0.329592\pi\)
0.510143 + 0.860090i \(0.329592\pi\)
\(762\) 0 0
\(763\) −4.59933 −0.166507
\(764\) 0 0
\(765\) 1.68913 0.0610705
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 30.8238 1.11153 0.555767 0.831338i \(-0.312425\pi\)
0.555767 + 0.831338i \(0.312425\pi\)
\(770\) 0 0
\(771\) −22.0224 −0.793118
\(772\) 0 0
\(773\) −18.8462 −0.677850 −0.338925 0.940813i \(-0.610063\pi\)
−0.338925 + 0.940813i \(0.610063\pi\)
\(774\) 0 0
\(775\) 12.2885 0.441414
\(776\) 0 0
\(777\) −0.599328 −0.0215008
\(778\) 0 0
\(779\) 4.00000 0.143315
\(780\) 0 0
\(781\) −1.40067 −0.0501200
\(782\) 0 0
\(783\) −10.7228 −0.383200
\(784\) 0 0
\(785\) −8.86860 −0.316534
\(786\) 0 0
\(787\) 10.6858 0.380906 0.190453 0.981696i \(-0.439004\pi\)
0.190453 + 0.981696i \(0.439004\pi\)
\(788\) 0 0
\(789\) −0.599328 −0.0213366
\(790\) 0 0
\(791\) 8.12343 0.288836
\(792\) 0 0
\(793\) 19.4231 0.689734
\(794\) 0 0
\(795\) 0.910201 0.0322815
\(796\) 0 0
\(797\) −39.8720 −1.41234 −0.706169 0.708044i \(-0.749578\pi\)
−0.706169 + 0.708044i \(0.749578\pi\)
\(798\) 0 0
\(799\) 13.1571 0.465463
\(800\) 0 0
\(801\) −4.00000 −0.141333
\(802\) 0 0
\(803\) 26.8462 0.947381
\(804\) 0 0
\(805\) 2.59933 0.0916143
\(806\) 0 0
\(807\) −5.40067 −0.190113
\(808\) 0 0
\(809\) 30.4489 1.07053 0.535263 0.844686i \(-0.320213\pi\)
0.535263 + 0.844686i \(0.320213\pi\)
\(810\) 0 0
\(811\) −0.448867 −0.0157619 −0.00788093 0.999969i \(-0.502509\pi\)
−0.00788093 + 0.999969i \(0.502509\pi\)
\(812\) 0 0
\(813\) 31.2211 1.09497
\(814\) 0 0
\(815\) 28.2469 0.989444
\(816\) 0 0
\(817\) −10.3109 −0.360732
\(818\) 0 0
\(819\) 5.71155 0.199578
\(820\) 0 0
\(821\) −21.0482 −0.734587 −0.367294 0.930105i \(-0.619716\pi\)
−0.367294 + 0.930105i \(0.619716\pi\)
\(822\) 0 0
\(823\) 35.9584 1.25343 0.626715 0.779248i \(-0.284399\pi\)
0.626715 + 0.779248i \(0.284399\pi\)
\(824\) 0 0
\(825\) −6.62175 −0.230540
\(826\) 0 0
\(827\) 23.3445 0.811768 0.405884 0.913925i \(-0.366964\pi\)
0.405884 + 0.913925i \(0.366964\pi\)
\(828\) 0 0
\(829\) 32.4421 1.12676 0.563381 0.826197i \(-0.309500\pi\)
0.563381 + 0.826197i \(0.309500\pi\)
\(830\) 0 0
\(831\) −29.9584 −1.03925
\(832\) 0 0
\(833\) 1.29966 0.0450307
\(834\) 0 0
\(835\) −2.57019 −0.0889452
\(836\) 0 0
\(837\) −3.71155 −0.128290
\(838\) 0 0
\(839\) −8.04484 −0.277739 −0.138869 0.990311i \(-0.544347\pi\)
−0.138869 + 0.990311i \(0.544347\pi\)
\(840\) 0 0
\(841\) 85.9775 2.96474
\(842\) 0 0
\(843\) −8.70034 −0.299655
\(844\) 0 0
\(845\) 25.5017 0.877284
\(846\) 0 0
\(847\) −7.00000 −0.240523
\(848\) 0 0
\(849\) −6.02242 −0.206689
\(850\) 0 0
\(851\) −1.19866 −0.0410894
\(852\) 0 0
\(853\) −24.2693 −0.830964 −0.415482 0.909601i \(-0.636387\pi\)
−0.415482 + 0.909601i \(0.636387\pi\)
\(854\) 0 0
\(855\) −1.29966 −0.0444476
\(856\) 0 0
\(857\) −17.7756 −0.607202 −0.303601 0.952799i \(-0.598189\pi\)
−0.303601 + 0.952799i \(0.598189\pi\)
\(858\) 0 0
\(859\) 9.02578 0.307956 0.153978 0.988074i \(-0.450792\pi\)
0.153978 + 0.988074i \(0.450792\pi\)
\(860\) 0 0
\(861\) −4.00000 −0.136320
\(862\) 0 0
\(863\) 42.3187 1.44055 0.720273 0.693691i \(-0.244017\pi\)
0.720273 + 0.693691i \(0.244017\pi\)
\(864\) 0 0
\(865\) −1.55785 −0.0529685
\(866\) 0 0
\(867\) −15.3109 −0.519985
\(868\) 0 0
\(869\) −12.0448 −0.408593
\(870\) 0 0
\(871\) −14.8462 −0.503044
\(872\) 0 0
\(873\) −2.00000 −0.0676897
\(874\) 0 0
\(875\) −10.8013 −0.365152
\(876\) 0 0
\(877\) 29.5960 0.999385 0.499692 0.866203i \(-0.333446\pi\)
0.499692 + 0.866203i \(0.333446\pi\)
\(878\) 0 0
\(879\) −29.8204 −1.00582
\(880\) 0 0
\(881\) 16.3479 0.550773 0.275387 0.961334i \(-0.411194\pi\)
0.275387 + 0.961334i \(0.411194\pi\)
\(882\) 0 0
\(883\) 21.6475 0.728497 0.364249 0.931302i \(-0.381326\pi\)
0.364249 + 0.931302i \(0.381326\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 19.2211 0.645381 0.322690 0.946505i \(-0.395413\pi\)
0.322690 + 0.946505i \(0.395413\pi\)
\(888\) 0 0
\(889\) 6.59933 0.221334
\(890\) 0 0
\(891\) 2.00000 0.0670025
\(892\) 0 0
\(893\) −10.1234 −0.338768
\(894\) 0 0
\(895\) −17.3142 −0.578751
\(896\) 0 0
\(897\) 11.4231 0.381406
\(898\) 0 0
\(899\) 39.7980 1.32734
\(900\) 0 0
\(901\) 0.910201 0.0303232
\(902\) 0 0
\(903\) 10.3109 0.343124
\(904\) 0 0
\(905\) 1.84954 0.0614808
\(906\) 0 0
\(907\) −25.7756 −0.855864 −0.427932 0.903811i \(-0.640758\pi\)
−0.427932 + 0.903811i \(0.640758\pi\)
\(908\) 0 0
\(909\) 2.70034 0.0895645
\(910\) 0 0
\(911\) −8.49832 −0.281562 −0.140781 0.990041i \(-0.544961\pi\)
−0.140781 + 0.990041i \(0.544961\pi\)
\(912\) 0 0
\(913\) −25.4455 −0.842123
\(914\) 0 0
\(915\) 4.41973 0.146112
\(916\) 0 0
\(917\) −7.32208 −0.241796
\(918\) 0 0
\(919\) 41.2435 1.36050 0.680249 0.732981i \(-0.261872\pi\)
0.680249 + 0.732981i \(0.261872\pi\)
\(920\) 0 0
\(921\) −10.6858 −0.352108
\(922\) 0 0
\(923\) −4.00000 −0.131662
\(924\) 0 0
\(925\) 1.98430 0.0652434
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) −21.5914 −0.708389 −0.354195 0.935172i \(-0.615245\pi\)
−0.354195 + 0.935172i \(0.615245\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) 0 0
\(933\) −20.5207 −0.671819
\(934\) 0 0
\(935\) 3.37825 0.110481
\(936\) 0 0
\(937\) 38.0448 1.24287 0.621435 0.783465i \(-0.286550\pi\)
0.621435 + 0.783465i \(0.286550\pi\)
\(938\) 0 0
\(939\) 9.22107 0.300918
\(940\) 0 0
\(941\) 12.4197 0.404872 0.202436 0.979296i \(-0.435114\pi\)
0.202436 + 0.979296i \(0.435114\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 1.29966 0.0422781
\(946\) 0 0
\(947\) 42.6666 1.38648 0.693239 0.720708i \(-0.256183\pi\)
0.693239 + 0.720708i \(0.256183\pi\)
\(948\) 0 0
\(949\) 76.6666 2.48870
\(950\) 0 0
\(951\) 30.9696 1.00426
\(952\) 0 0
\(953\) 32.7003 1.05927 0.529634 0.848226i \(-0.322329\pi\)
0.529634 + 0.848226i \(0.322329\pi\)
\(954\) 0 0
\(955\) −12.2469 −0.396299
\(956\) 0 0
\(957\) −21.4455 −0.693235
\(958\) 0 0
\(959\) −17.4231 −0.562621
\(960\) 0 0
\(961\) −17.2244 −0.555627
\(962\) 0 0
\(963\) 13.3221 0.429298
\(964\) 0 0
\(965\) 14.0673 0.452841
\(966\) 0 0
\(967\) −25.1571 −0.808996 −0.404498 0.914539i \(-0.632554\pi\)
−0.404498 + 0.914539i \(0.632554\pi\)
\(968\) 0 0
\(969\) −1.29966 −0.0417512
\(970\) 0 0
\(971\) −28.0897 −0.901441 −0.450720 0.892665i \(-0.648833\pi\)
−0.450720 + 0.892665i \(0.648833\pi\)
\(972\) 0 0
\(973\) −6.22443 −0.199546
\(974\) 0 0
\(975\) −18.9102 −0.605611
\(976\) 0 0
\(977\) −14.1010 −0.451131 −0.225566 0.974228i \(-0.572423\pi\)
−0.225566 + 0.974228i \(0.572423\pi\)
\(978\) 0 0
\(979\) −8.00000 −0.255681
\(980\) 0 0
\(981\) −4.59933 −0.146845
\(982\) 0 0
\(983\) −39.4231 −1.25740 −0.628701 0.777647i \(-0.716413\pi\)
−0.628701 + 0.777647i \(0.716413\pi\)
\(984\) 0 0
\(985\) −7.79798 −0.248464
\(986\) 0 0
\(987\) 10.1234 0.322232
\(988\) 0 0
\(989\) 20.6217 0.655733
\(990\) 0 0
\(991\) −43.4679 −1.38080 −0.690402 0.723426i \(-0.742566\pi\)
−0.690402 + 0.723426i \(0.742566\pi\)
\(992\) 0 0
\(993\) 6.80134 0.215834
\(994\) 0 0
\(995\) 17.4164 0.552136
\(996\) 0 0
\(997\) 31.6184 1.00136 0.500682 0.865631i \(-0.333083\pi\)
0.500682 + 0.865631i \(0.333083\pi\)
\(998\) 0 0
\(999\) −0.599328 −0.0189619
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.j.1.2 3
3.2 odd 2 4788.2.a.p.1.2 3
4.3 odd 2 6384.2.a.bt.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1596.2.a.j.1.2 3 1.1 even 1 trivial
4788.2.a.p.1.2 3 3.2 odd 2
6384.2.a.bt.1.2 3 4.3 odd 2