Properties

Label 3720.2.a.o.1.3
Level $3720$
Weight $2$
Character 3720.1
Self dual yes
Analytic conductor $29.704$
Analytic rank $1$
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] = [3720,2,Mod(1,3720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("3720.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3720.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: \(29.7043495519\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 3720.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +1.17009 q^{7} +1.00000 q^{9} -5.41855 q^{11} -1.17009 q^{13} -1.00000 q^{15} +7.12783 q^{17} -4.00000 q^{19} +1.17009 q^{21} -6.97107 q^{23} +1.00000 q^{25} +1.00000 q^{27} +9.21953 q^{29} -1.00000 q^{31} -5.41855 q^{33} -1.17009 q^{35} -2.09171 q^{37} -1.17009 q^{39} -5.26180 q^{41} -6.00000 q^{43} -1.00000 q^{45} -1.52586 q^{47} -5.63090 q^{49} +7.12783 q^{51} +0.474142 q^{53} +5.41855 q^{55} -4.00000 q^{57} +14.6381 q^{59} -11.9421 q^{61} +1.17009 q^{63} +1.17009 q^{65} -13.3268 q^{67} -6.97107 q^{69} -8.14116 q^{71} -8.74539 q^{73} +1.00000 q^{75} -6.34017 q^{77} +9.46800 q^{79} +1.00000 q^{81} -3.86603 q^{83} -7.12783 q^{85} +9.21953 q^{87} -2.04226 q^{89} -1.36910 q^{91} -1.00000 q^{93} +4.00000 q^{95} -2.73820 q^{97} -5.41855 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} - 2 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) 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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.17009 0.442251 0.221126 0.975245i \(-0.429027\pi\)
0.221126 + 0.975245i \(0.429027\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.41855 −1.63375 −0.816877 0.576812i \(-0.804297\pi\)
−0.816877 + 0.576812i \(0.804297\pi\)
\(12\) 0 0
\(13\) −1.17009 −0.324524 −0.162262 0.986748i \(-0.551879\pi\)
−0.162262 + 0.986748i \(0.551879\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) 7.12783 1.72875 0.864376 0.502846i \(-0.167714\pi\)
0.864376 + 0.502846i \(0.167714\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) 1.17009 0.255334
\(22\) 0 0
\(23\) −6.97107 −1.45357 −0.726784 0.686866i \(-0.758986\pi\)
−0.726784 + 0.686866i \(0.758986\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 9.21953 1.71202 0.856012 0.516955i \(-0.172935\pi\)
0.856012 + 0.516955i \(0.172935\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −5.41855 −0.943249
\(34\) 0 0
\(35\) −1.17009 −0.197781
\(36\) 0 0
\(37\) −2.09171 −0.343875 −0.171937 0.985108i \(-0.555003\pi\)
−0.171937 + 0.985108i \(0.555003\pi\)
\(38\) 0 0
\(39\) −1.17009 −0.187364
\(40\) 0 0
\(41\) −5.26180 −0.821754 −0.410877 0.911691i \(-0.634777\pi\)
−0.410877 + 0.911691i \(0.634777\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −1.52586 −0.222569 −0.111285 0.993789i \(-0.535497\pi\)
−0.111285 + 0.993789i \(0.535497\pi\)
\(48\) 0 0
\(49\) −5.63090 −0.804414
\(50\) 0 0
\(51\) 7.12783 0.998095
\(52\) 0 0
\(53\) 0.474142 0.0651284 0.0325642 0.999470i \(-0.489633\pi\)
0.0325642 + 0.999470i \(0.489633\pi\)
\(54\) 0 0
\(55\) 5.41855 0.730637
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 14.6381 1.90572 0.952858 0.303416i \(-0.0981272\pi\)
0.952858 + 0.303416i \(0.0981272\pi\)
\(60\) 0 0
\(61\) −11.9421 −1.52903 −0.764517 0.644603i \(-0.777023\pi\)
−0.764517 + 0.644603i \(0.777023\pi\)
\(62\) 0 0
\(63\) 1.17009 0.147417
\(64\) 0 0
\(65\) 1.17009 0.145131
\(66\) 0 0
\(67\) −13.3268 −1.62813 −0.814066 0.580772i \(-0.802751\pi\)
−0.814066 + 0.580772i \(0.802751\pi\)
\(68\) 0 0
\(69\) −6.97107 −0.839218
\(70\) 0 0
\(71\) −8.14116 −0.966178 −0.483089 0.875571i \(-0.660485\pi\)
−0.483089 + 0.875571i \(0.660485\pi\)
\(72\) 0 0
\(73\) −8.74539 −1.02357 −0.511785 0.859113i \(-0.671016\pi\)
−0.511785 + 0.859113i \(0.671016\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) −6.34017 −0.722530
\(78\) 0 0
\(79\) 9.46800 1.06523 0.532617 0.846357i \(-0.321209\pi\)
0.532617 + 0.846357i \(0.321209\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.86603 −0.424352 −0.212176 0.977231i \(-0.568055\pi\)
−0.212176 + 0.977231i \(0.568055\pi\)
\(84\) 0 0
\(85\) −7.12783 −0.773121
\(86\) 0 0
\(87\) 9.21953 0.988438
\(88\) 0 0
\(89\) −2.04226 −0.216479 −0.108240 0.994125i \(-0.534521\pi\)
−0.108240 + 0.994125i \(0.534521\pi\)
\(90\) 0 0
\(91\) −1.36910 −0.143521
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −2.73820 −0.278023 −0.139011 0.990291i \(-0.544392\pi\)
−0.139011 + 0.990291i \(0.544392\pi\)
\(98\) 0 0
\(99\) −5.41855 −0.544585
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) −8.92881 −0.879782 −0.439891 0.898051i \(-0.644983\pi\)
−0.439891 + 0.898051i \(0.644983\pi\)
\(104\) 0 0
\(105\) −1.17009 −0.114189
\(106\) 0 0
\(107\) −15.6514 −1.51308 −0.756540 0.653948i \(-0.773112\pi\)
−0.756540 + 0.653948i \(0.773112\pi\)
\(108\) 0 0
\(109\) −14.0228 −1.34314 −0.671570 0.740941i \(-0.734380\pi\)
−0.671570 + 0.740941i \(0.734380\pi\)
\(110\) 0 0
\(111\) −2.09171 −0.198536
\(112\) 0 0
\(113\) 5.23513 0.492480 0.246240 0.969209i \(-0.420805\pi\)
0.246240 + 0.969209i \(0.420805\pi\)
\(114\) 0 0
\(115\) 6.97107 0.650056
\(116\) 0 0
\(117\) −1.17009 −0.108175
\(118\) 0 0
\(119\) 8.34017 0.764542
\(120\) 0 0
\(121\) 18.3607 1.66915
\(122\) 0 0
\(123\) −5.26180 −0.474440
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −8.18342 −0.726161 −0.363080 0.931758i \(-0.618275\pi\)
−0.363080 + 0.931758i \(0.618275\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) 7.21953 0.630774 0.315387 0.948963i \(-0.397866\pi\)
0.315387 + 0.948963i \(0.397866\pi\)
\(132\) 0 0
\(133\) −4.68035 −0.405837
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 8.38962 0.716774 0.358387 0.933573i \(-0.383327\pi\)
0.358387 + 0.933573i \(0.383327\pi\)
\(138\) 0 0
\(139\) −18.3402 −1.55559 −0.777797 0.628516i \(-0.783663\pi\)
−0.777797 + 0.628516i \(0.783663\pi\)
\(140\) 0 0
\(141\) −1.52586 −0.128500
\(142\) 0 0
\(143\) 6.34017 0.530192
\(144\) 0 0
\(145\) −9.21953 −0.765641
\(146\) 0 0
\(147\) −5.63090 −0.464429
\(148\) 0 0
\(149\) 12.0989 0.991180 0.495590 0.868557i \(-0.334952\pi\)
0.495590 + 0.868557i \(0.334952\pi\)
\(150\) 0 0
\(151\) 22.3318 1.81733 0.908667 0.417523i \(-0.137102\pi\)
0.908667 + 0.417523i \(0.137102\pi\)
\(152\) 0 0
\(153\) 7.12783 0.576251
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 14.8371 1.18413 0.592065 0.805890i \(-0.298313\pi\)
0.592065 + 0.805890i \(0.298313\pi\)
\(158\) 0 0
\(159\) 0.474142 0.0376019
\(160\) 0 0
\(161\) −8.15676 −0.642842
\(162\) 0 0
\(163\) −1.59478 −0.124913 −0.0624564 0.998048i \(-0.519893\pi\)
−0.0624564 + 0.998048i \(0.519893\pi\)
\(164\) 0 0
\(165\) 5.41855 0.421834
\(166\) 0 0
\(167\) 3.91548 0.302989 0.151494 0.988458i \(-0.451591\pi\)
0.151494 + 0.988458i \(0.451591\pi\)
\(168\) 0 0
\(169\) −11.6309 −0.894684
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) −1.23513 −0.0939054 −0.0469527 0.998897i \(-0.514951\pi\)
−0.0469527 + 0.998897i \(0.514951\pi\)
\(174\) 0 0
\(175\) 1.17009 0.0884502
\(176\) 0 0
\(177\) 14.6381 1.10027
\(178\) 0 0
\(179\) −15.4186 −1.15244 −0.576218 0.817296i \(-0.695472\pi\)
−0.576218 + 0.817296i \(0.695472\pi\)
\(180\) 0 0
\(181\) −7.26180 −0.539765 −0.269882 0.962893i \(-0.586985\pi\)
−0.269882 + 0.962893i \(0.586985\pi\)
\(182\) 0 0
\(183\) −11.9421 −0.882788
\(184\) 0 0
\(185\) 2.09171 0.153785
\(186\) 0 0
\(187\) −38.6225 −2.82436
\(188\) 0 0
\(189\) 1.17009 0.0851113
\(190\) 0 0
\(191\) 17.8732 1.29326 0.646630 0.762803i \(-0.276178\pi\)
0.646630 + 0.762803i \(0.276178\pi\)
\(192\) 0 0
\(193\) 21.6742 1.56014 0.780072 0.625690i \(-0.215183\pi\)
0.780072 + 0.625690i \(0.215183\pi\)
\(194\) 0 0
\(195\) 1.17009 0.0837916
\(196\) 0 0
\(197\) −22.0638 −1.57198 −0.785991 0.618238i \(-0.787847\pi\)
−0.785991 + 0.618238i \(0.787847\pi\)
\(198\) 0 0
\(199\) −16.3896 −1.16183 −0.580915 0.813964i \(-0.697305\pi\)
−0.580915 + 0.813964i \(0.697305\pi\)
\(200\) 0 0
\(201\) −13.3268 −0.940003
\(202\) 0 0
\(203\) 10.7877 0.757145
\(204\) 0 0
\(205\) 5.26180 0.367500
\(206\) 0 0
\(207\) −6.97107 −0.484523
\(208\) 0 0
\(209\) 21.6742 1.49924
\(210\) 0 0
\(211\) −11.6020 −0.798712 −0.399356 0.916796i \(-0.630766\pi\)
−0.399356 + 0.916796i \(0.630766\pi\)
\(212\) 0 0
\(213\) −8.14116 −0.557823
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) −1.17009 −0.0794306
\(218\) 0 0
\(219\) −8.74539 −0.590959
\(220\) 0 0
\(221\) −8.34017 −0.561021
\(222\) 0 0
\(223\) 11.5441 0.773051 0.386525 0.922279i \(-0.373675\pi\)
0.386525 + 0.922279i \(0.373675\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −14.7031 −0.975881 −0.487941 0.872877i \(-0.662252\pi\)
−0.487941 + 0.872877i \(0.662252\pi\)
\(228\) 0 0
\(229\) −9.60197 −0.634516 −0.317258 0.948339i \(-0.602762\pi\)
−0.317258 + 0.948339i \(0.602762\pi\)
\(230\) 0 0
\(231\) −6.34017 −0.417153
\(232\) 0 0
\(233\) −10.4163 −0.682393 −0.341197 0.939992i \(-0.610832\pi\)
−0.341197 + 0.939992i \(0.610832\pi\)
\(234\) 0 0
\(235\) 1.52586 0.0995360
\(236\) 0 0
\(237\) 9.46800 0.615013
\(238\) 0 0
\(239\) −9.02052 −0.583489 −0.291744 0.956496i \(-0.594236\pi\)
−0.291744 + 0.956496i \(0.594236\pi\)
\(240\) 0 0
\(241\) 3.97334 0.255945 0.127973 0.991778i \(-0.459153\pi\)
0.127973 + 0.991778i \(0.459153\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 5.63090 0.359745
\(246\) 0 0
\(247\) 4.68035 0.297803
\(248\) 0 0
\(249\) −3.86603 −0.245000
\(250\) 0 0
\(251\) 5.94214 0.375065 0.187532 0.982258i \(-0.439951\pi\)
0.187532 + 0.982258i \(0.439951\pi\)
\(252\) 0 0
\(253\) 37.7731 2.37477
\(254\) 0 0
\(255\) −7.12783 −0.446362
\(256\) 0 0
\(257\) 13.7359 0.856824 0.428412 0.903583i \(-0.359073\pi\)
0.428412 + 0.903583i \(0.359073\pi\)
\(258\) 0 0
\(259\) −2.44748 −0.152079
\(260\) 0 0
\(261\) 9.21953 0.570675
\(262\) 0 0
\(263\) 16.8638 1.03986 0.519932 0.854208i \(-0.325957\pi\)
0.519932 + 0.854208i \(0.325957\pi\)
\(264\) 0 0
\(265\) −0.474142 −0.0291263
\(266\) 0 0
\(267\) −2.04226 −0.124984
\(268\) 0 0
\(269\) −17.5018 −1.06711 −0.533553 0.845766i \(-0.679144\pi\)
−0.533553 + 0.845766i \(0.679144\pi\)
\(270\) 0 0
\(271\) 6.02666 0.366094 0.183047 0.983104i \(-0.441404\pi\)
0.183047 + 0.983104i \(0.441404\pi\)
\(272\) 0 0
\(273\) −1.36910 −0.0828618
\(274\) 0 0
\(275\) −5.41855 −0.326751
\(276\) 0 0
\(277\) 16.2485 0.976276 0.488138 0.872767i \(-0.337676\pi\)
0.488138 + 0.872767i \(0.337676\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) −20.1711 −1.20331 −0.601654 0.798757i \(-0.705492\pi\)
−0.601654 + 0.798757i \(0.705492\pi\)
\(282\) 0 0
\(283\) 16.9021 1.00473 0.502364 0.864656i \(-0.332464\pi\)
0.502364 + 0.864656i \(0.332464\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) −6.15676 −0.363422
\(288\) 0 0
\(289\) 33.8059 1.98858
\(290\) 0 0
\(291\) −2.73820 −0.160516
\(292\) 0 0
\(293\) 13.3691 0.781031 0.390516 0.920596i \(-0.372297\pi\)
0.390516 + 0.920596i \(0.372297\pi\)
\(294\) 0 0
\(295\) −14.6381 −0.852262
\(296\) 0 0
\(297\) −5.41855 −0.314416
\(298\) 0 0
\(299\) 8.15676 0.471717
\(300\) 0 0
\(301\) −7.02052 −0.404656
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 11.9421 0.683805
\(306\) 0 0
\(307\) −9.48360 −0.541257 −0.270629 0.962684i \(-0.587232\pi\)
−0.270629 + 0.962684i \(0.587232\pi\)
\(308\) 0 0
\(309\) −8.92881 −0.507942
\(310\) 0 0
\(311\) −12.6381 −0.716640 −0.358320 0.933599i \(-0.616650\pi\)
−0.358320 + 0.933599i \(0.616650\pi\)
\(312\) 0 0
\(313\) −5.30018 −0.299584 −0.149792 0.988718i \(-0.547860\pi\)
−0.149792 + 0.988718i \(0.547860\pi\)
\(314\) 0 0
\(315\) −1.17009 −0.0659269
\(316\) 0 0
\(317\) −8.25953 −0.463901 −0.231951 0.972728i \(-0.574511\pi\)
−0.231951 + 0.972728i \(0.574511\pi\)
\(318\) 0 0
\(319\) −49.9565 −2.79703
\(320\) 0 0
\(321\) −15.6514 −0.873577
\(322\) 0 0
\(323\) −28.5113 −1.58641
\(324\) 0 0
\(325\) −1.17009 −0.0649047
\(326\) 0 0
\(327\) −14.0228 −0.775462
\(328\) 0 0
\(329\) −1.78539 −0.0984315
\(330\) 0 0
\(331\) −27.3112 −1.50116 −0.750581 0.660779i \(-0.770226\pi\)
−0.750581 + 0.660779i \(0.770226\pi\)
\(332\) 0 0
\(333\) −2.09171 −0.114625
\(334\) 0 0
\(335\) 13.3268 0.728123
\(336\) 0 0
\(337\) −6.77205 −0.368897 −0.184449 0.982842i \(-0.559050\pi\)
−0.184449 + 0.982842i \(0.559050\pi\)
\(338\) 0 0
\(339\) 5.23513 0.284333
\(340\) 0 0
\(341\) 5.41855 0.293431
\(342\) 0 0
\(343\) −14.7792 −0.798004
\(344\) 0 0
\(345\) 6.97107 0.375310
\(346\) 0 0
\(347\) −3.71769 −0.199576 −0.0997879 0.995009i \(-0.531816\pi\)
−0.0997879 + 0.995009i \(0.531816\pi\)
\(348\) 0 0
\(349\) −6.60424 −0.353517 −0.176758 0.984254i \(-0.556561\pi\)
−0.176758 + 0.984254i \(0.556561\pi\)
\(350\) 0 0
\(351\) −1.17009 −0.0624546
\(352\) 0 0
\(353\) −32.2206 −1.71493 −0.857464 0.514544i \(-0.827961\pi\)
−0.857464 + 0.514544i \(0.827961\pi\)
\(354\) 0 0
\(355\) 8.14116 0.432088
\(356\) 0 0
\(357\) 8.34017 0.441409
\(358\) 0 0
\(359\) 2.93722 0.155021 0.0775103 0.996992i \(-0.475303\pi\)
0.0775103 + 0.996992i \(0.475303\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) 18.3607 0.963686
\(364\) 0 0
\(365\) 8.74539 0.457755
\(366\) 0 0
\(367\) 17.3074 0.903437 0.451719 0.892160i \(-0.350811\pi\)
0.451719 + 0.892160i \(0.350811\pi\)
\(368\) 0 0
\(369\) −5.26180 −0.273918
\(370\) 0 0
\(371\) 0.554787 0.0288031
\(372\) 0 0
\(373\) 30.5958 1.58419 0.792096 0.610397i \(-0.208990\pi\)
0.792096 + 0.610397i \(0.208990\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −10.7877 −0.555592
\(378\) 0 0
\(379\) 33.5585 1.72378 0.861892 0.507092i \(-0.169280\pi\)
0.861892 + 0.507092i \(0.169280\pi\)
\(380\) 0 0
\(381\) −8.18342 −0.419249
\(382\) 0 0
\(383\) 36.2784 1.85374 0.926871 0.375380i \(-0.122488\pi\)
0.926871 + 0.375380i \(0.122488\pi\)
\(384\) 0 0
\(385\) 6.34017 0.323125
\(386\) 0 0
\(387\) −6.00000 −0.304997
\(388\) 0 0
\(389\) 9.54533 0.483968 0.241984 0.970280i \(-0.422202\pi\)
0.241984 + 0.970280i \(0.422202\pi\)
\(390\) 0 0
\(391\) −49.6886 −2.51286
\(392\) 0 0
\(393\) 7.21953 0.364177
\(394\) 0 0
\(395\) −9.46800 −0.476387
\(396\) 0 0
\(397\) 4.05786 0.203658 0.101829 0.994802i \(-0.467531\pi\)
0.101829 + 0.994802i \(0.467531\pi\)
\(398\) 0 0
\(399\) −4.68035 −0.234310
\(400\) 0 0
\(401\) 32.0833 1.60216 0.801082 0.598555i \(-0.204258\pi\)
0.801082 + 0.598555i \(0.204258\pi\)
\(402\) 0 0
\(403\) 1.17009 0.0582862
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 11.3340 0.561807
\(408\) 0 0
\(409\) 2.89496 0.143147 0.0715733 0.997435i \(-0.477198\pi\)
0.0715733 + 0.997435i \(0.477198\pi\)
\(410\) 0 0
\(411\) 8.38962 0.413830
\(412\) 0 0
\(413\) 17.1278 0.842805
\(414\) 0 0
\(415\) 3.86603 0.189776
\(416\) 0 0
\(417\) −18.3402 −0.898122
\(418\) 0 0
\(419\) −36.6381 −1.78989 −0.894944 0.446179i \(-0.852785\pi\)
−0.894944 + 0.446179i \(0.852785\pi\)
\(420\) 0 0
\(421\) −8.26406 −0.402766 −0.201383 0.979513i \(-0.564544\pi\)
−0.201383 + 0.979513i \(0.564544\pi\)
\(422\) 0 0
\(423\) −1.52586 −0.0741897
\(424\) 0 0
\(425\) 7.12783 0.345750
\(426\) 0 0
\(427\) −13.9733 −0.676217
\(428\) 0 0
\(429\) 6.34017 0.306106
\(430\) 0 0
\(431\) 6.19902 0.298596 0.149298 0.988792i \(-0.452299\pi\)
0.149298 + 0.988792i \(0.452299\pi\)
\(432\) 0 0
\(433\) −10.1061 −0.485667 −0.242834 0.970068i \(-0.578077\pi\)
−0.242834 + 0.970068i \(0.578077\pi\)
\(434\) 0 0
\(435\) −9.21953 −0.442043
\(436\) 0 0
\(437\) 27.8843 1.33389
\(438\) 0 0
\(439\) 7.20394 0.343825 0.171913 0.985112i \(-0.445005\pi\)
0.171913 + 0.985112i \(0.445005\pi\)
\(440\) 0 0
\(441\) −5.63090 −0.268138
\(442\) 0 0
\(443\) 22.7031 1.07866 0.539329 0.842095i \(-0.318678\pi\)
0.539329 + 0.842095i \(0.318678\pi\)
\(444\) 0 0
\(445\) 2.04226 0.0968124
\(446\) 0 0
\(447\) 12.0989 0.572258
\(448\) 0 0
\(449\) 4.56585 0.215476 0.107738 0.994179i \(-0.465639\pi\)
0.107738 + 0.994179i \(0.465639\pi\)
\(450\) 0 0
\(451\) 28.5113 1.34254
\(452\) 0 0
\(453\) 22.3318 1.04924
\(454\) 0 0
\(455\) 1.36910 0.0641845
\(456\) 0 0
\(457\) 22.8710 1.06986 0.534929 0.844897i \(-0.320338\pi\)
0.534929 + 0.844897i \(0.320338\pi\)
\(458\) 0 0
\(459\) 7.12783 0.332698
\(460\) 0 0
\(461\) −32.0566 −1.49303 −0.746513 0.665371i \(-0.768274\pi\)
−0.746513 + 0.665371i \(0.768274\pi\)
\(462\) 0 0
\(463\) 10.0989 0.469336 0.234668 0.972076i \(-0.424600\pi\)
0.234668 + 0.972076i \(0.424600\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) −0.630898 −0.0291945 −0.0145972 0.999893i \(-0.504647\pi\)
−0.0145972 + 0.999893i \(0.504647\pi\)
\(468\) 0 0
\(469\) −15.5936 −0.720044
\(470\) 0 0
\(471\) 14.8371 0.683658
\(472\) 0 0
\(473\) 32.5113 1.49487
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 0.474142 0.0217095
\(478\) 0 0
\(479\) 10.9639 0.500953 0.250476 0.968123i \(-0.419413\pi\)
0.250476 + 0.968123i \(0.419413\pi\)
\(480\) 0 0
\(481\) 2.44748 0.111595
\(482\) 0 0
\(483\) −8.15676 −0.371145
\(484\) 0 0
\(485\) 2.73820 0.124335
\(486\) 0 0
\(487\) −37.0784 −1.68018 −0.840091 0.542446i \(-0.817498\pi\)
−0.840091 + 0.542446i \(0.817498\pi\)
\(488\) 0 0
\(489\) −1.59478 −0.0721185
\(490\) 0 0
\(491\) 25.6430 1.15725 0.578626 0.815593i \(-0.303589\pi\)
0.578626 + 0.815593i \(0.303589\pi\)
\(492\) 0 0
\(493\) 65.7152 2.95967
\(494\) 0 0
\(495\) 5.41855 0.243546
\(496\) 0 0
\(497\) −9.52586 −0.427293
\(498\) 0 0
\(499\) −13.1629 −0.589252 −0.294626 0.955613i \(-0.595195\pi\)
−0.294626 + 0.955613i \(0.595195\pi\)
\(500\) 0 0
\(501\) 3.91548 0.174931
\(502\) 0 0
\(503\) −0.362959 −0.0161836 −0.00809178 0.999967i \(-0.502576\pi\)
−0.00809178 + 0.999967i \(0.502576\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −11.6309 −0.516546
\(508\) 0 0
\(509\) 35.8154 1.58749 0.793744 0.608252i \(-0.208129\pi\)
0.793744 + 0.608252i \(0.208129\pi\)
\(510\) 0 0
\(511\) −10.2329 −0.452675
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) 8.92881 0.393450
\(516\) 0 0
\(517\) 8.26794 0.363624
\(518\) 0 0
\(519\) −1.23513 −0.0542163
\(520\) 0 0
\(521\) 23.7854 1.04206 0.521028 0.853539i \(-0.325549\pi\)
0.521028 + 0.853539i \(0.325549\pi\)
\(522\) 0 0
\(523\) 6.24128 0.272912 0.136456 0.990646i \(-0.456429\pi\)
0.136456 + 0.990646i \(0.456429\pi\)
\(524\) 0 0
\(525\) 1.17009 0.0510668
\(526\) 0 0
\(527\) −7.12783 −0.310493
\(528\) 0 0
\(529\) 25.5958 1.11286
\(530\) 0 0
\(531\) 14.6381 0.635239
\(532\) 0 0
\(533\) 6.15676 0.266679
\(534\) 0 0
\(535\) 15.6514 0.676670
\(536\) 0 0
\(537\) −15.4186 −0.665360
\(538\) 0 0
\(539\) 30.5113 1.31421
\(540\) 0 0
\(541\) 1.04331 0.0448552 0.0224276 0.999748i \(-0.492860\pi\)
0.0224276 + 0.999748i \(0.492860\pi\)
\(542\) 0 0
\(543\) −7.26180 −0.311633
\(544\) 0 0
\(545\) 14.0228 0.600670
\(546\) 0 0
\(547\) −35.6358 −1.52368 −0.761839 0.647767i \(-0.775703\pi\)
−0.761839 + 0.647767i \(0.775703\pi\)
\(548\) 0 0
\(549\) −11.9421 −0.509678
\(550\) 0 0
\(551\) −36.8781 −1.57106
\(552\) 0 0
\(553\) 11.0784 0.471101
\(554\) 0 0
\(555\) 2.09171 0.0887881
\(556\) 0 0
\(557\) −6.64527 −0.281569 −0.140785 0.990040i \(-0.544963\pi\)
−0.140785 + 0.990040i \(0.544963\pi\)
\(558\) 0 0
\(559\) 7.02052 0.296936
\(560\) 0 0
\(561\) −38.6225 −1.63064
\(562\) 0 0
\(563\) −21.2800 −0.896847 −0.448424 0.893821i \(-0.648014\pi\)
−0.448424 + 0.893821i \(0.648014\pi\)
\(564\) 0 0
\(565\) −5.23513 −0.220244
\(566\) 0 0
\(567\) 1.17009 0.0491390
\(568\) 0 0
\(569\) 32.7226 1.37180 0.685902 0.727694i \(-0.259408\pi\)
0.685902 + 0.727694i \(0.259408\pi\)
\(570\) 0 0
\(571\) −13.1773 −0.551452 −0.275726 0.961236i \(-0.588918\pi\)
−0.275726 + 0.961236i \(0.588918\pi\)
\(572\) 0 0
\(573\) 17.8732 0.746664
\(574\) 0 0
\(575\) −6.97107 −0.290714
\(576\) 0 0
\(577\) −10.8059 −0.449856 −0.224928 0.974375i \(-0.572215\pi\)
−0.224928 + 0.974375i \(0.572215\pi\)
\(578\) 0 0
\(579\) 21.6742 0.900749
\(580\) 0 0
\(581\) −4.52359 −0.187670
\(582\) 0 0
\(583\) −2.56916 −0.106404
\(584\) 0 0
\(585\) 1.17009 0.0483771
\(586\) 0 0
\(587\) 37.6886 1.55557 0.777787 0.628528i \(-0.216342\pi\)
0.777787 + 0.628528i \(0.216342\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) −22.0638 −0.907584
\(592\) 0 0
\(593\) −19.8576 −0.815455 −0.407727 0.913104i \(-0.633679\pi\)
−0.407727 + 0.913104i \(0.633679\pi\)
\(594\) 0 0
\(595\) −8.34017 −0.341914
\(596\) 0 0
\(597\) −16.3896 −0.670783
\(598\) 0 0
\(599\) 41.1350 1.68073 0.840366 0.542020i \(-0.182340\pi\)
0.840366 + 0.542020i \(0.182340\pi\)
\(600\) 0 0
\(601\) 17.3874 0.709245 0.354622 0.935010i \(-0.384609\pi\)
0.354622 + 0.935010i \(0.384609\pi\)
\(602\) 0 0
\(603\) −13.3268 −0.542711
\(604\) 0 0
\(605\) −18.3607 −0.746468
\(606\) 0 0
\(607\) −21.5558 −0.874924 −0.437462 0.899237i \(-0.644123\pi\)
−0.437462 + 0.899237i \(0.644123\pi\)
\(608\) 0 0
\(609\) 10.7877 0.437138
\(610\) 0 0
\(611\) 1.78539 0.0722290
\(612\) 0 0
\(613\) 33.0544 1.33505 0.667527 0.744586i \(-0.267353\pi\)
0.667527 + 0.744586i \(0.267353\pi\)
\(614\) 0 0
\(615\) 5.26180 0.212176
\(616\) 0 0
\(617\) −38.9315 −1.56732 −0.783661 0.621189i \(-0.786650\pi\)
−0.783661 + 0.621189i \(0.786650\pi\)
\(618\) 0 0
\(619\) 32.6719 1.31320 0.656598 0.754241i \(-0.271995\pi\)
0.656598 + 0.754241i \(0.271995\pi\)
\(620\) 0 0
\(621\) −6.97107 −0.279739
\(622\) 0 0
\(623\) −2.38962 −0.0957382
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 21.6742 0.865584
\(628\) 0 0
\(629\) −14.9093 −0.594474
\(630\) 0 0
\(631\) −4.68035 −0.186322 −0.0931608 0.995651i \(-0.529697\pi\)
−0.0931608 + 0.995651i \(0.529697\pi\)
\(632\) 0 0
\(633\) −11.6020 −0.461137
\(634\) 0 0
\(635\) 8.18342 0.324749
\(636\) 0 0
\(637\) 6.58864 0.261051
\(638\) 0 0
\(639\) −8.14116 −0.322059
\(640\) 0 0
\(641\) −16.0111 −0.632399 −0.316199 0.948693i \(-0.602407\pi\)
−0.316199 + 0.948693i \(0.602407\pi\)
\(642\) 0 0
\(643\) 6.24128 0.246132 0.123066 0.992398i \(-0.460727\pi\)
0.123066 + 0.992398i \(0.460727\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) 7.73206 0.303979 0.151989 0.988382i \(-0.451432\pi\)
0.151989 + 0.988382i \(0.451432\pi\)
\(648\) 0 0
\(649\) −79.3172 −3.11347
\(650\) 0 0
\(651\) −1.17009 −0.0458593
\(652\) 0 0
\(653\) 8.16063 0.319350 0.159675 0.987170i \(-0.448955\pi\)
0.159675 + 0.987170i \(0.448955\pi\)
\(654\) 0 0
\(655\) −7.21953 −0.282091
\(656\) 0 0
\(657\) −8.74539 −0.341190
\(658\) 0 0
\(659\) −40.2544 −1.56809 −0.784045 0.620704i \(-0.786847\pi\)
−0.784045 + 0.620704i \(0.786847\pi\)
\(660\) 0 0
\(661\) −49.7464 −1.93491 −0.967456 0.253039i \(-0.918570\pi\)
−0.967456 + 0.253039i \(0.918570\pi\)
\(662\) 0 0
\(663\) −8.34017 −0.323905
\(664\) 0 0
\(665\) 4.68035 0.181496
\(666\) 0 0
\(667\) −64.2700 −2.48855
\(668\) 0 0
\(669\) 11.5441 0.446321
\(670\) 0 0
\(671\) 64.7091 2.49807
\(672\) 0 0
\(673\) −46.4606 −1.79093 −0.895463 0.445136i \(-0.853155\pi\)
−0.895463 + 0.445136i \(0.853155\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 19.6248 0.754241 0.377120 0.926164i \(-0.376914\pi\)
0.377120 + 0.926164i \(0.376914\pi\)
\(678\) 0 0
\(679\) −3.20394 −0.122956
\(680\) 0 0
\(681\) −14.7031 −0.563425
\(682\) 0 0
\(683\) −6.23740 −0.238668 −0.119334 0.992854i \(-0.538076\pi\)
−0.119334 + 0.992854i \(0.538076\pi\)
\(684\) 0 0
\(685\) −8.38962 −0.320551
\(686\) 0 0
\(687\) −9.60197 −0.366338
\(688\) 0 0
\(689\) −0.554787 −0.0211357
\(690\) 0 0
\(691\) −40.2823 −1.53241 −0.766206 0.642595i \(-0.777858\pi\)
−0.766206 + 0.642595i \(0.777858\pi\)
\(692\) 0 0
\(693\) −6.34017 −0.240843
\(694\) 0 0
\(695\) 18.3402 0.695682
\(696\) 0 0
\(697\) −37.5052 −1.42061
\(698\) 0 0
\(699\) −10.4163 −0.393980
\(700\) 0 0
\(701\) 31.7998 1.20106 0.600530 0.799602i \(-0.294956\pi\)
0.600530 + 0.799602i \(0.294956\pi\)
\(702\) 0 0
\(703\) 8.36683 0.315561
\(704\) 0 0
\(705\) 1.52586 0.0574671
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −5.68649 −0.213561 −0.106780 0.994283i \(-0.534054\pi\)
−0.106780 + 0.994283i \(0.534054\pi\)
\(710\) 0 0
\(711\) 9.46800 0.355078
\(712\) 0 0
\(713\) 6.97107 0.261069
\(714\) 0 0
\(715\) −6.34017 −0.237109
\(716\) 0 0
\(717\) −9.02052 −0.336877
\(718\) 0 0
\(719\) 19.6286 0.732024 0.366012 0.930610i \(-0.380723\pi\)
0.366012 + 0.930610i \(0.380723\pi\)
\(720\) 0 0
\(721\) −10.4475 −0.389084
\(722\) 0 0
\(723\) 3.97334 0.147770
\(724\) 0 0
\(725\) 9.21953 0.342405
\(726\) 0 0
\(727\) 28.1783 1.04508 0.522538 0.852616i \(-0.324985\pi\)
0.522538 + 0.852616i \(0.324985\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −42.7670 −1.58179
\(732\) 0 0
\(733\) 4.63931 0.171357 0.0856784 0.996323i \(-0.472694\pi\)
0.0856784 + 0.996323i \(0.472694\pi\)
\(734\) 0 0
\(735\) 5.63090 0.207699
\(736\) 0 0
\(737\) 72.2122 2.65997
\(738\) 0 0
\(739\) −32.0761 −1.17994 −0.589969 0.807426i \(-0.700860\pi\)
−0.589969 + 0.807426i \(0.700860\pi\)
\(740\) 0 0
\(741\) 4.68035 0.171937
\(742\) 0 0
\(743\) 12.5503 0.460424 0.230212 0.973140i \(-0.426058\pi\)
0.230212 + 0.973140i \(0.426058\pi\)
\(744\) 0 0
\(745\) −12.0989 −0.443269
\(746\) 0 0
\(747\) −3.86603 −0.141451
\(748\) 0 0
\(749\) −18.3135 −0.669161
\(750\) 0 0
\(751\) 45.4908 1.65998 0.829991 0.557777i \(-0.188345\pi\)
0.829991 + 0.557777i \(0.188345\pi\)
\(752\) 0 0
\(753\) 5.94214 0.216544
\(754\) 0 0
\(755\) −22.3318 −0.812736
\(756\) 0 0
\(757\) 1.93495 0.0703271 0.0351635 0.999382i \(-0.488805\pi\)
0.0351635 + 0.999382i \(0.488805\pi\)
\(758\) 0 0
\(759\) 37.7731 1.37108
\(760\) 0 0
\(761\) −38.2134 −1.38523 −0.692617 0.721305i \(-0.743542\pi\)
−0.692617 + 0.721305i \(0.743542\pi\)
\(762\) 0 0
\(763\) −16.4079 −0.594005
\(764\) 0 0
\(765\) −7.12783 −0.257707
\(766\) 0 0
\(767\) −17.1278 −0.618450
\(768\) 0 0
\(769\) 30.6042 1.10362 0.551808 0.833971i \(-0.313938\pi\)
0.551808 + 0.833971i \(0.313938\pi\)
\(770\) 0 0
\(771\) 13.7359 0.494688
\(772\) 0 0
\(773\) 33.7959 1.21555 0.607777 0.794108i \(-0.292062\pi\)
0.607777 + 0.794108i \(0.292062\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) −2.44748 −0.0878029
\(778\) 0 0
\(779\) 21.0472 0.754094
\(780\) 0 0
\(781\) 44.1133 1.57850
\(782\) 0 0
\(783\) 9.21953 0.329479
\(784\) 0 0
\(785\) −14.8371 −0.529559
\(786\) 0 0
\(787\) −15.5897 −0.555712 −0.277856 0.960623i \(-0.589624\pi\)
−0.277856 + 0.960623i \(0.589624\pi\)
\(788\) 0 0
\(789\) 16.8638 0.600366
\(790\) 0 0
\(791\) 6.12556 0.217800
\(792\) 0 0
\(793\) 13.9733 0.496208
\(794\) 0 0
\(795\) −0.474142 −0.0168161
\(796\) 0 0
\(797\) 43.4245 1.53818 0.769088 0.639143i \(-0.220711\pi\)
0.769088 + 0.639143i \(0.220711\pi\)
\(798\) 0 0
\(799\) −10.8760 −0.384767
\(800\) 0 0
\(801\) −2.04226 −0.0721597
\(802\) 0 0
\(803\) 47.3874 1.67226
\(804\) 0 0
\(805\) 8.15676 0.287488
\(806\) 0 0
\(807\) −17.5018 −0.616094
\(808\) 0 0
\(809\) −48.9471 −1.72089 −0.860444 0.509546i \(-0.829813\pi\)
−0.860444 + 0.509546i \(0.829813\pi\)
\(810\) 0 0
\(811\) −33.1773 −1.16501 −0.582506 0.812827i \(-0.697928\pi\)
−0.582506 + 0.812827i \(0.697928\pi\)
\(812\) 0 0
\(813\) 6.02666 0.211364
\(814\) 0 0
\(815\) 1.59478 0.0558627
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) 0 0
\(819\) −1.36910 −0.0478403
\(820\) 0 0
\(821\) −1.62985 −0.0568823 −0.0284411 0.999595i \(-0.509054\pi\)
−0.0284411 + 0.999595i \(0.509054\pi\)
\(822\) 0 0
\(823\) 34.2434 1.19365 0.596824 0.802372i \(-0.296429\pi\)
0.596824 + 0.802372i \(0.296429\pi\)
\(824\) 0 0
\(825\) −5.41855 −0.188650
\(826\) 0 0
\(827\) 49.5234 1.72210 0.861049 0.508522i \(-0.169808\pi\)
0.861049 + 0.508522i \(0.169808\pi\)
\(828\) 0 0
\(829\) 35.4740 1.23206 0.616031 0.787722i \(-0.288740\pi\)
0.616031 + 0.787722i \(0.288740\pi\)
\(830\) 0 0
\(831\) 16.2485 0.563653
\(832\) 0 0
\(833\) −40.1361 −1.39063
\(834\) 0 0
\(835\) −3.91548 −0.135501
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) −26.2544 −0.906404 −0.453202 0.891408i \(-0.649718\pi\)
−0.453202 + 0.891408i \(0.649718\pi\)
\(840\) 0 0
\(841\) 55.9998 1.93103
\(842\) 0 0
\(843\) −20.1711 −0.694731
\(844\) 0 0
\(845\) 11.6309 0.400115
\(846\) 0 0
\(847\) 21.4836 0.738185
\(848\) 0 0
\(849\) 16.9021 0.580080
\(850\) 0 0
\(851\) 14.5814 0.499846
\(852\) 0 0
\(853\) −45.5729 −1.56039 −0.780193 0.625539i \(-0.784879\pi\)
−0.780193 + 0.625539i \(0.784879\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 3.24742 0.110930 0.0554649 0.998461i \(-0.482336\pi\)
0.0554649 + 0.998461i \(0.482336\pi\)
\(858\) 0 0
\(859\) −28.1978 −0.962096 −0.481048 0.876694i \(-0.659744\pi\)
−0.481048 + 0.876694i \(0.659744\pi\)
\(860\) 0 0
\(861\) −6.15676 −0.209822
\(862\) 0 0
\(863\) −34.4079 −1.17126 −0.585629 0.810579i \(-0.699152\pi\)
−0.585629 + 0.810579i \(0.699152\pi\)
\(864\) 0 0
\(865\) 1.23513 0.0419958
\(866\) 0 0
\(867\) 33.8059 1.14811
\(868\) 0 0
\(869\) −51.3028 −1.74033
\(870\) 0 0
\(871\) 15.5936 0.528368
\(872\) 0 0
\(873\) −2.73820 −0.0926742
\(874\) 0 0
\(875\) −1.17009 −0.0395561
\(876\) 0 0
\(877\) 45.5897 1.53945 0.769727 0.638373i \(-0.220392\pi\)
0.769727 + 0.638373i \(0.220392\pi\)
\(878\) 0 0
\(879\) 13.3691 0.450929
\(880\) 0 0
\(881\) 23.1662 0.780489 0.390245 0.920711i \(-0.372390\pi\)
0.390245 + 0.920711i \(0.372390\pi\)
\(882\) 0 0
\(883\) 17.4863 0.588459 0.294230 0.955735i \(-0.404937\pi\)
0.294230 + 0.955735i \(0.404937\pi\)
\(884\) 0 0
\(885\) −14.6381 −0.492054
\(886\) 0 0
\(887\) −11.1012 −0.372741 −0.186370 0.982480i \(-0.559672\pi\)
−0.186370 + 0.982480i \(0.559672\pi\)
\(888\) 0 0
\(889\) −9.57531 −0.321145
\(890\) 0 0
\(891\) −5.41855 −0.181528
\(892\) 0 0
\(893\) 6.10343 0.204244
\(894\) 0 0
\(895\) 15.4186 0.515385
\(896\) 0 0
\(897\) 8.15676 0.272346
\(898\) 0 0
\(899\) −9.21953 −0.307489
\(900\) 0 0
\(901\) 3.37960 0.112591
\(902\) 0 0
\(903\) −7.02052 −0.233628
\(904\) 0 0
\(905\) 7.26180 0.241390
\(906\) 0 0
\(907\) 50.4463 1.67504 0.837520 0.546406i \(-0.184005\pi\)
0.837520 + 0.546406i \(0.184005\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 43.4596 1.43988 0.719940 0.694036i \(-0.244169\pi\)
0.719940 + 0.694036i \(0.244169\pi\)
\(912\) 0 0
\(913\) 20.9483 0.693287
\(914\) 0 0
\(915\) 11.9421 0.394795
\(916\) 0 0
\(917\) 8.44748 0.278960
\(918\) 0 0
\(919\) −17.4596 −0.575939 −0.287969 0.957640i \(-0.592980\pi\)
−0.287969 + 0.957640i \(0.592980\pi\)
\(920\) 0 0
\(921\) −9.48360 −0.312495
\(922\) 0 0
\(923\) 9.52586 0.313547
\(924\) 0 0
\(925\) −2.09171 −0.0687750
\(926\) 0 0
\(927\) −8.92881 −0.293261
\(928\) 0 0
\(929\) −9.13501 −0.299710 −0.149855 0.988708i \(-0.547881\pi\)
−0.149855 + 0.988708i \(0.547881\pi\)
\(930\) 0 0
\(931\) 22.5236 0.738181
\(932\) 0 0
\(933\) −12.6381 −0.413752
\(934\) 0 0
\(935\) 38.6225 1.26309
\(936\) 0 0
\(937\) 18.9893 0.620354 0.310177 0.950679i \(-0.399612\pi\)
0.310177 + 0.950679i \(0.399612\pi\)
\(938\) 0 0
\(939\) −5.30018 −0.172965
\(940\) 0 0
\(941\) 3.12064 0.101730 0.0508649 0.998706i \(-0.483802\pi\)
0.0508649 + 0.998706i \(0.483802\pi\)
\(942\) 0 0
\(943\) 36.6803 1.19448
\(944\) 0 0
\(945\) −1.17009 −0.0380629
\(946\) 0 0
\(947\) −6.60424 −0.214609 −0.107304 0.994226i \(-0.534222\pi\)
−0.107304 + 0.994226i \(0.534222\pi\)
\(948\) 0 0
\(949\) 10.2329 0.332173
\(950\) 0 0
\(951\) −8.25953 −0.267834
\(952\) 0 0
\(953\) 12.8554 0.416426 0.208213 0.978084i \(-0.433235\pi\)
0.208213 + 0.978084i \(0.433235\pi\)
\(954\) 0 0
\(955\) −17.8732 −0.578364
\(956\) 0 0
\(957\) −49.9565 −1.61486
\(958\) 0 0
\(959\) 9.81658 0.316994
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) −15.6514 −0.504360
\(964\) 0 0
\(965\) −21.6742 −0.697717
\(966\) 0 0
\(967\) 36.7982 1.18335 0.591674 0.806177i \(-0.298467\pi\)
0.591674 + 0.806177i \(0.298467\pi\)
\(968\) 0 0
\(969\) −28.5113 −0.915915
\(970\) 0 0
\(971\) −37.4707 −1.20249 −0.601245 0.799065i \(-0.705329\pi\)
−0.601245 + 0.799065i \(0.705329\pi\)
\(972\) 0 0
\(973\) −21.4596 −0.687963
\(974\) 0 0
\(975\) −1.17009 −0.0374728
\(976\) 0 0
\(977\) 42.5113 1.36006 0.680029 0.733186i \(-0.261967\pi\)
0.680029 + 0.733186i \(0.261967\pi\)
\(978\) 0 0
\(979\) 11.0661 0.353674
\(980\) 0 0
\(981\) −14.0228 −0.447713
\(982\) 0 0
\(983\) 12.8950 0.411285 0.205643 0.978627i \(-0.434072\pi\)
0.205643 + 0.978627i \(0.434072\pi\)
\(984\) 0 0
\(985\) 22.0638 0.703012
\(986\) 0 0
\(987\) −1.78539 −0.0568295
\(988\) 0 0
\(989\) 41.8264 1.33000
\(990\) 0 0
\(991\) −8.55025 −0.271608 −0.135804 0.990736i \(-0.543362\pi\)
−0.135804 + 0.990736i \(0.543362\pi\)
\(992\) 0 0
\(993\) −27.3112 −0.866696
\(994\) 0 0
\(995\) 16.3896 0.519586
\(996\) 0 0
\(997\) 23.5897 0.747093 0.373546 0.927612i \(-0.378142\pi\)
0.373546 + 0.927612i \(0.378142\pi\)
\(998\) 0 0
\(999\) −2.09171 −0.0661787
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 3720.2.a.o.1.3 3
4.3 odd 2 7440.2.a.bo.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.o.1.3 3 1.1 even 1 trivial
7440.2.a.bo.1.1 3 4.3 odd 2