Properties

Label 5520.2.a.bn.1.1
Level $5520$
Weight $2$
Character 5520.1
Self dual yes
Analytic conductor $44.077$
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] = [5520,2,Mod(1,5520)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5520, 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("5520.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5520.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: \(44.0774219157\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2760)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 5520.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -0.561553 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -0.561553 q^{7} +1.00000 q^{9} -2.00000 q^{11} -2.00000 q^{13} -1.00000 q^{15} -2.56155 q^{17} +1.12311 q^{19} -0.561553 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +4.56155 q^{29} +3.68466 q^{31} -2.00000 q^{33} +0.561553 q^{35} +3.68466 q^{37} -2.00000 q^{39} +5.68466 q^{41} -1.00000 q^{45} +10.2462 q^{47} -6.68466 q^{49} -2.56155 q^{51} -3.43845 q^{53} +2.00000 q^{55} +1.12311 q^{57} -2.56155 q^{59} -14.2462 q^{61} -0.561553 q^{63} +2.00000 q^{65} -11.6847 q^{67} +1.00000 q^{69} -13.9309 q^{71} -10.0000 q^{73} +1.00000 q^{75} +1.12311 q^{77} -13.3693 q^{79} +1.00000 q^{81} +5.68466 q^{83} +2.56155 q^{85} +4.56155 q^{87} -6.24621 q^{89} +1.12311 q^{91} +3.68466 q^{93} -1.12311 q^{95} -14.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 3 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} - 2 q^{5} + 3 q^{7} + 2 q^{9} - 4 q^{11} - 4 q^{13} - 2 q^{15} - q^{17} - 6 q^{19} + 3 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 5 q^{29} - 5 q^{31} - 4 q^{33} - 3 q^{35} - 5 q^{37} - 4 q^{39} - q^{41} - 2 q^{45} + 4 q^{47} - q^{49} - q^{51} - 11 q^{53} + 4 q^{55} - 6 q^{57} - q^{59} - 12 q^{61} + 3 q^{63} + 4 q^{65} - 11 q^{67} + 2 q^{69} + q^{71} - 20 q^{73} + 2 q^{75} - 6 q^{77} - 2 q^{79} + 2 q^{81} - q^{83} + q^{85} + 5 q^{87} + 4 q^{89} - 6 q^{91} - 5 q^{93} + 6 q^{95} - 28 q^{97} - 4 q^{99}+O(q^{100}) \) 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
\(7\) −0.561553 −0.212247 −0.106124 0.994353i \(-0.533844\pi\)
−0.106124 + 0.994353i \(0.533844\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −2.56155 −0.621268 −0.310634 0.950530i \(-0.600541\pi\)
−0.310634 + 0.950530i \(0.600541\pi\)
\(18\) 0 0
\(19\) 1.12311 0.257658 0.128829 0.991667i \(-0.458878\pi\)
0.128829 + 0.991667i \(0.458878\pi\)
\(20\) 0 0
\(21\) −0.561553 −0.122541
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.56155 0.847059 0.423530 0.905882i \(-0.360791\pi\)
0.423530 + 0.905882i \(0.360791\pi\)
\(30\) 0 0
\(31\) 3.68466 0.661784 0.330892 0.943669i \(-0.392650\pi\)
0.330892 + 0.943669i \(0.392650\pi\)
\(32\) 0 0
\(33\) −2.00000 −0.348155
\(34\) 0 0
\(35\) 0.561553 0.0949197
\(36\) 0 0
\(37\) 3.68466 0.605754 0.302877 0.953030i \(-0.402053\pi\)
0.302877 + 0.953030i \(0.402053\pi\)
\(38\) 0 0
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) 5.68466 0.887794 0.443897 0.896078i \(-0.353596\pi\)
0.443897 + 0.896078i \(0.353596\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 10.2462 1.49456 0.747282 0.664507i \(-0.231359\pi\)
0.747282 + 0.664507i \(0.231359\pi\)
\(48\) 0 0
\(49\) −6.68466 −0.954951
\(50\) 0 0
\(51\) −2.56155 −0.358689
\(52\) 0 0
\(53\) −3.43845 −0.472307 −0.236154 0.971716i \(-0.575887\pi\)
−0.236154 + 0.971716i \(0.575887\pi\)
\(54\) 0 0
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) 1.12311 0.148759
\(58\) 0 0
\(59\) −2.56155 −0.333486 −0.166743 0.986000i \(-0.553325\pi\)
−0.166743 + 0.986000i \(0.553325\pi\)
\(60\) 0 0
\(61\) −14.2462 −1.82404 −0.912020 0.410145i \(-0.865478\pi\)
−0.912020 + 0.410145i \(0.865478\pi\)
\(62\) 0 0
\(63\) −0.561553 −0.0707490
\(64\) 0 0
\(65\) 2.00000 0.248069
\(66\) 0 0
\(67\) −11.6847 −1.42751 −0.713754 0.700396i \(-0.753007\pi\)
−0.713754 + 0.700396i \(0.753007\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −13.9309 −1.65329 −0.826645 0.562724i \(-0.809753\pi\)
−0.826645 + 0.562724i \(0.809753\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 1.12311 0.127990
\(78\) 0 0
\(79\) −13.3693 −1.50417 −0.752083 0.659069i \(-0.770951\pi\)
−0.752083 + 0.659069i \(0.770951\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.68466 0.623972 0.311986 0.950087i \(-0.399006\pi\)
0.311986 + 0.950087i \(0.399006\pi\)
\(84\) 0 0
\(85\) 2.56155 0.277839
\(86\) 0 0
\(87\) 4.56155 0.489050
\(88\) 0 0
\(89\) −6.24621 −0.662097 −0.331049 0.943614i \(-0.607402\pi\)
−0.331049 + 0.943614i \(0.607402\pi\)
\(90\) 0 0
\(91\) 1.12311 0.117733
\(92\) 0 0
\(93\) 3.68466 0.382081
\(94\) 0 0
\(95\) −1.12311 −0.115228
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 0 0
\(99\) −2.00000 −0.201008
\(100\) 0 0
\(101\) −7.43845 −0.740153 −0.370077 0.929001i \(-0.620669\pi\)
−0.370077 + 0.929001i \(0.620669\pi\)
\(102\) 0 0
\(103\) 11.1231 1.09599 0.547996 0.836481i \(-0.315391\pi\)
0.547996 + 0.836481i \(0.315391\pi\)
\(104\) 0 0
\(105\) 0.561553 0.0548019
\(106\) 0 0
\(107\) −14.8078 −1.43152 −0.715760 0.698346i \(-0.753920\pi\)
−0.715760 + 0.698346i \(0.753920\pi\)
\(108\) 0 0
\(109\) −13.1231 −1.25697 −0.628483 0.777824i \(-0.716324\pi\)
−0.628483 + 0.777824i \(0.716324\pi\)
\(110\) 0 0
\(111\) 3.68466 0.349732
\(112\) 0 0
\(113\) −12.8078 −1.20485 −0.602427 0.798174i \(-0.705799\pi\)
−0.602427 + 0.798174i \(0.705799\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 1.43845 0.131862
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) 5.68466 0.512568
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 16.4924 1.46347 0.731733 0.681591i \(-0.238712\pi\)
0.731733 + 0.681591i \(0.238712\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 10.2462 0.895216 0.447608 0.894230i \(-0.352276\pi\)
0.447608 + 0.894230i \(0.352276\pi\)
\(132\) 0 0
\(133\) −0.630683 −0.0546872
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 2.87689 0.245790 0.122895 0.992420i \(-0.460782\pi\)
0.122895 + 0.992420i \(0.460782\pi\)
\(138\) 0 0
\(139\) −7.68466 −0.651804 −0.325902 0.945404i \(-0.605668\pi\)
−0.325902 + 0.945404i \(0.605668\pi\)
\(140\) 0 0
\(141\) 10.2462 0.862887
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 0 0
\(145\) −4.56155 −0.378816
\(146\) 0 0
\(147\) −6.68466 −0.551341
\(148\) 0 0
\(149\) −3.12311 −0.255855 −0.127927 0.991784i \(-0.540832\pi\)
−0.127927 + 0.991784i \(0.540832\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) −2.56155 −0.207089
\(154\) 0 0
\(155\) −3.68466 −0.295959
\(156\) 0 0
\(157\) −9.43845 −0.753270 −0.376635 0.926362i \(-0.622919\pi\)
−0.376635 + 0.926362i \(0.622919\pi\)
\(158\) 0 0
\(159\) −3.43845 −0.272687
\(160\) 0 0
\(161\) −0.561553 −0.0442566
\(162\) 0 0
\(163\) 9.12311 0.714577 0.357288 0.933994i \(-0.383701\pi\)
0.357288 + 0.933994i \(0.383701\pi\)
\(164\) 0 0
\(165\) 2.00000 0.155700
\(166\) 0 0
\(167\) −2.24621 −0.173817 −0.0869085 0.996216i \(-0.527699\pi\)
−0.0869085 + 0.996216i \(0.527699\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 1.12311 0.0858860
\(172\) 0 0
\(173\) −7.12311 −0.541560 −0.270780 0.962641i \(-0.587282\pi\)
−0.270780 + 0.962641i \(0.587282\pi\)
\(174\) 0 0
\(175\) −0.561553 −0.0424494
\(176\) 0 0
\(177\) −2.56155 −0.192538
\(178\) 0 0
\(179\) −8.00000 −0.597948 −0.298974 0.954261i \(-0.596644\pi\)
−0.298974 + 0.954261i \(0.596644\pi\)
\(180\) 0 0
\(181\) 2.24621 0.166960 0.0834798 0.996509i \(-0.473397\pi\)
0.0834798 + 0.996509i \(0.473397\pi\)
\(182\) 0 0
\(183\) −14.2462 −1.05311
\(184\) 0 0
\(185\) −3.68466 −0.270901
\(186\) 0 0
\(187\) 5.12311 0.374639
\(188\) 0 0
\(189\) −0.561553 −0.0408470
\(190\) 0 0
\(191\) 18.2462 1.32025 0.660125 0.751156i \(-0.270503\pi\)
0.660125 + 0.751156i \(0.270503\pi\)
\(192\) 0 0
\(193\) −15.1231 −1.08858 −0.544292 0.838896i \(-0.683202\pi\)
−0.544292 + 0.838896i \(0.683202\pi\)
\(194\) 0 0
\(195\) 2.00000 0.143223
\(196\) 0 0
\(197\) −0.246211 −0.0175418 −0.00877091 0.999962i \(-0.502792\pi\)
−0.00877091 + 0.999962i \(0.502792\pi\)
\(198\) 0 0
\(199\) 12.2462 0.868111 0.434055 0.900886i \(-0.357082\pi\)
0.434055 + 0.900886i \(0.357082\pi\)
\(200\) 0 0
\(201\) −11.6847 −0.824172
\(202\) 0 0
\(203\) −2.56155 −0.179786
\(204\) 0 0
\(205\) −5.68466 −0.397034
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −2.24621 −0.155374
\(210\) 0 0
\(211\) −23.0540 −1.58710 −0.793551 0.608504i \(-0.791770\pi\)
−0.793551 + 0.608504i \(0.791770\pi\)
\(212\) 0 0
\(213\) −13.9309 −0.954527
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.06913 −0.140462
\(218\) 0 0
\(219\) −10.0000 −0.675737
\(220\) 0 0
\(221\) 5.12311 0.344617
\(222\) 0 0
\(223\) 14.8769 0.996231 0.498115 0.867111i \(-0.334026\pi\)
0.498115 + 0.867111i \(0.334026\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −0.876894 −0.0582015 −0.0291008 0.999576i \(-0.509264\pi\)
−0.0291008 + 0.999576i \(0.509264\pi\)
\(228\) 0 0
\(229\) 18.2462 1.20574 0.602872 0.797838i \(-0.294023\pi\)
0.602872 + 0.797838i \(0.294023\pi\)
\(230\) 0 0
\(231\) 1.12311 0.0738949
\(232\) 0 0
\(233\) 15.1231 0.990748 0.495374 0.868680i \(-0.335031\pi\)
0.495374 + 0.868680i \(0.335031\pi\)
\(234\) 0 0
\(235\) −10.2462 −0.668389
\(236\) 0 0
\(237\) −13.3693 −0.868430
\(238\) 0 0
\(239\) 9.43845 0.610522 0.305261 0.952269i \(-0.401256\pi\)
0.305261 + 0.952269i \(0.401256\pi\)
\(240\) 0 0
\(241\) −3.75379 −0.241803 −0.120901 0.992665i \(-0.538578\pi\)
−0.120901 + 0.992665i \(0.538578\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 6.68466 0.427067
\(246\) 0 0
\(247\) −2.24621 −0.142923
\(248\) 0 0
\(249\) 5.68466 0.360251
\(250\) 0 0
\(251\) 0.246211 0.0155407 0.00777036 0.999970i \(-0.497527\pi\)
0.00777036 + 0.999970i \(0.497527\pi\)
\(252\) 0 0
\(253\) −2.00000 −0.125739
\(254\) 0 0
\(255\) 2.56155 0.160411
\(256\) 0 0
\(257\) 27.6155 1.72261 0.861305 0.508089i \(-0.169648\pi\)
0.861305 + 0.508089i \(0.169648\pi\)
\(258\) 0 0
\(259\) −2.06913 −0.128570
\(260\) 0 0
\(261\) 4.56155 0.282353
\(262\) 0 0
\(263\) −2.56155 −0.157952 −0.0789761 0.996877i \(-0.525165\pi\)
−0.0789761 + 0.996877i \(0.525165\pi\)
\(264\) 0 0
\(265\) 3.43845 0.211222
\(266\) 0 0
\(267\) −6.24621 −0.382262
\(268\) 0 0
\(269\) 1.19224 0.0726919 0.0363460 0.999339i \(-0.488428\pi\)
0.0363460 + 0.999339i \(0.488428\pi\)
\(270\) 0 0
\(271\) −5.43845 −0.330362 −0.165181 0.986263i \(-0.552821\pi\)
−0.165181 + 0.986263i \(0.552821\pi\)
\(272\) 0 0
\(273\) 1.12311 0.0679734
\(274\) 0 0
\(275\) −2.00000 −0.120605
\(276\) 0 0
\(277\) −28.7386 −1.72674 −0.863369 0.504574i \(-0.831650\pi\)
−0.863369 + 0.504574i \(0.831650\pi\)
\(278\) 0 0
\(279\) 3.68466 0.220595
\(280\) 0 0
\(281\) 1.12311 0.0669989 0.0334994 0.999439i \(-0.489335\pi\)
0.0334994 + 0.999439i \(0.489335\pi\)
\(282\) 0 0
\(283\) 13.4384 0.798833 0.399416 0.916770i \(-0.369213\pi\)
0.399416 + 0.916770i \(0.369213\pi\)
\(284\) 0 0
\(285\) −1.12311 −0.0665270
\(286\) 0 0
\(287\) −3.19224 −0.188432
\(288\) 0 0
\(289\) −10.4384 −0.614026
\(290\) 0 0
\(291\) −14.0000 −0.820695
\(292\) 0 0
\(293\) −4.06913 −0.237721 −0.118861 0.992911i \(-0.537924\pi\)
−0.118861 + 0.992911i \(0.537924\pi\)
\(294\) 0 0
\(295\) 2.56155 0.149139
\(296\) 0 0
\(297\) −2.00000 −0.116052
\(298\) 0 0
\(299\) −2.00000 −0.115663
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −7.43845 −0.427328
\(304\) 0 0
\(305\) 14.2462 0.815736
\(306\) 0 0
\(307\) −25.1231 −1.43385 −0.716926 0.697150i \(-0.754451\pi\)
−0.716926 + 0.697150i \(0.754451\pi\)
\(308\) 0 0
\(309\) 11.1231 0.632771
\(310\) 0 0
\(311\) −32.9848 −1.87040 −0.935199 0.354121i \(-0.884780\pi\)
−0.935199 + 0.354121i \(0.884780\pi\)
\(312\) 0 0
\(313\) −10.8078 −0.610891 −0.305445 0.952210i \(-0.598805\pi\)
−0.305445 + 0.952210i \(0.598805\pi\)
\(314\) 0 0
\(315\) 0.561553 0.0316399
\(316\) 0 0
\(317\) −19.6155 −1.10172 −0.550859 0.834598i \(-0.685700\pi\)
−0.550859 + 0.834598i \(0.685700\pi\)
\(318\) 0 0
\(319\) −9.12311 −0.510796
\(320\) 0 0
\(321\) −14.8078 −0.826489
\(322\) 0 0
\(323\) −2.87689 −0.160075
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) −13.1231 −0.725709
\(328\) 0 0
\(329\) −5.75379 −0.317217
\(330\) 0 0
\(331\) 23.6847 1.30183 0.650913 0.759152i \(-0.274386\pi\)
0.650913 + 0.759152i \(0.274386\pi\)
\(332\) 0 0
\(333\) 3.68466 0.201918
\(334\) 0 0
\(335\) 11.6847 0.638401
\(336\) 0 0
\(337\) 8.24621 0.449200 0.224600 0.974451i \(-0.427892\pi\)
0.224600 + 0.974451i \(0.427892\pi\)
\(338\) 0 0
\(339\) −12.8078 −0.695622
\(340\) 0 0
\(341\) −7.36932 −0.399071
\(342\) 0 0
\(343\) 7.68466 0.414933
\(344\) 0 0
\(345\) −1.00000 −0.0538382
\(346\) 0 0
\(347\) 12.4924 0.670628 0.335314 0.942106i \(-0.391158\pi\)
0.335314 + 0.942106i \(0.391158\pi\)
\(348\) 0 0
\(349\) 2.80776 0.150296 0.0751481 0.997172i \(-0.476057\pi\)
0.0751481 + 0.997172i \(0.476057\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 0 0
\(353\) −10.6307 −0.565814 −0.282907 0.959147i \(-0.591299\pi\)
−0.282907 + 0.959147i \(0.591299\pi\)
\(354\) 0 0
\(355\) 13.9309 0.739374
\(356\) 0 0
\(357\) 1.43845 0.0761307
\(358\) 0 0
\(359\) 14.7386 0.777875 0.388938 0.921264i \(-0.372842\pi\)
0.388938 + 0.921264i \(0.372842\pi\)
\(360\) 0 0
\(361\) −17.7386 −0.933612
\(362\) 0 0
\(363\) −7.00000 −0.367405
\(364\) 0 0
\(365\) 10.0000 0.523424
\(366\) 0 0
\(367\) −25.6847 −1.34073 −0.670364 0.742032i \(-0.733862\pi\)
−0.670364 + 0.742032i \(0.733862\pi\)
\(368\) 0 0
\(369\) 5.68466 0.295931
\(370\) 0 0
\(371\) 1.93087 0.100246
\(372\) 0 0
\(373\) 23.3693 1.21002 0.605009 0.796219i \(-0.293170\pi\)
0.605009 + 0.796219i \(0.293170\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −9.12311 −0.469864
\(378\) 0 0
\(379\) −22.2462 −1.14271 −0.571356 0.820703i \(-0.693582\pi\)
−0.571356 + 0.820703i \(0.693582\pi\)
\(380\) 0 0
\(381\) 16.4924 0.844932
\(382\) 0 0
\(383\) 13.4384 0.686673 0.343336 0.939213i \(-0.388443\pi\)
0.343336 + 0.939213i \(0.388443\pi\)
\(384\) 0 0
\(385\) −1.12311 −0.0572388
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 30.4924 1.54603 0.773014 0.634389i \(-0.218748\pi\)
0.773014 + 0.634389i \(0.218748\pi\)
\(390\) 0 0
\(391\) −2.56155 −0.129543
\(392\) 0 0
\(393\) 10.2462 0.516853
\(394\) 0 0
\(395\) 13.3693 0.672683
\(396\) 0 0
\(397\) 15.6155 0.783721 0.391860 0.920025i \(-0.371832\pi\)
0.391860 + 0.920025i \(0.371832\pi\)
\(398\) 0 0
\(399\) −0.630683 −0.0315736
\(400\) 0 0
\(401\) −17.1231 −0.855087 −0.427544 0.903995i \(-0.640621\pi\)
−0.427544 + 0.903995i \(0.640621\pi\)
\(402\) 0 0
\(403\) −7.36932 −0.367092
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −7.36932 −0.365283
\(408\) 0 0
\(409\) −0.561553 −0.0277670 −0.0138835 0.999904i \(-0.504419\pi\)
−0.0138835 + 0.999904i \(0.504419\pi\)
\(410\) 0 0
\(411\) 2.87689 0.141907
\(412\) 0 0
\(413\) 1.43845 0.0707814
\(414\) 0 0
\(415\) −5.68466 −0.279049
\(416\) 0 0
\(417\) −7.68466 −0.376319
\(418\) 0 0
\(419\) −18.4924 −0.903414 −0.451707 0.892166i \(-0.649185\pi\)
−0.451707 + 0.892166i \(0.649185\pi\)
\(420\) 0 0
\(421\) −9.12311 −0.444633 −0.222316 0.974975i \(-0.571362\pi\)
−0.222316 + 0.974975i \(0.571362\pi\)
\(422\) 0 0
\(423\) 10.2462 0.498188
\(424\) 0 0
\(425\) −2.56155 −0.124254
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) 0 0
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 1.75379 0.0844770 0.0422385 0.999108i \(-0.486551\pi\)
0.0422385 + 0.999108i \(0.486551\pi\)
\(432\) 0 0
\(433\) 14.8078 0.711616 0.355808 0.934559i \(-0.384206\pi\)
0.355808 + 0.934559i \(0.384206\pi\)
\(434\) 0 0
\(435\) −4.56155 −0.218710
\(436\) 0 0
\(437\) 1.12311 0.0537254
\(438\) 0 0
\(439\) −32.9848 −1.57428 −0.787140 0.616774i \(-0.788439\pi\)
−0.787140 + 0.616774i \(0.788439\pi\)
\(440\) 0 0
\(441\) −6.68466 −0.318317
\(442\) 0 0
\(443\) 1.75379 0.0833250 0.0416625 0.999132i \(-0.486735\pi\)
0.0416625 + 0.999132i \(0.486735\pi\)
\(444\) 0 0
\(445\) 6.24621 0.296099
\(446\) 0 0
\(447\) −3.12311 −0.147718
\(448\) 0 0
\(449\) 31.3002 1.47715 0.738574 0.674173i \(-0.235500\pi\)
0.738574 + 0.674173i \(0.235500\pi\)
\(450\) 0 0
\(451\) −11.3693 −0.535360
\(452\) 0 0
\(453\) 12.0000 0.563809
\(454\) 0 0
\(455\) −1.12311 −0.0526520
\(456\) 0 0
\(457\) −30.8078 −1.44113 −0.720563 0.693389i \(-0.756117\pi\)
−0.720563 + 0.693389i \(0.756117\pi\)
\(458\) 0 0
\(459\) −2.56155 −0.119563
\(460\) 0 0
\(461\) 12.2462 0.570363 0.285181 0.958474i \(-0.407946\pi\)
0.285181 + 0.958474i \(0.407946\pi\)
\(462\) 0 0
\(463\) 31.3693 1.45786 0.728928 0.684590i \(-0.240019\pi\)
0.728928 + 0.684590i \(0.240019\pi\)
\(464\) 0 0
\(465\) −3.68466 −0.170872
\(466\) 0 0
\(467\) −16.5616 −0.766377 −0.383189 0.923670i \(-0.625174\pi\)
−0.383189 + 0.923670i \(0.625174\pi\)
\(468\) 0 0
\(469\) 6.56155 0.302984
\(470\) 0 0
\(471\) −9.43845 −0.434901
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 1.12311 0.0515316
\(476\) 0 0
\(477\) −3.43845 −0.157436
\(478\) 0 0
\(479\) −28.4924 −1.30185 −0.650926 0.759141i \(-0.725619\pi\)
−0.650926 + 0.759141i \(0.725619\pi\)
\(480\) 0 0
\(481\) −7.36932 −0.336012
\(482\) 0 0
\(483\) −0.561553 −0.0255515
\(484\) 0 0
\(485\) 14.0000 0.635707
\(486\) 0 0
\(487\) 22.7386 1.03039 0.515193 0.857074i \(-0.327720\pi\)
0.515193 + 0.857074i \(0.327720\pi\)
\(488\) 0 0
\(489\) 9.12311 0.412561
\(490\) 0 0
\(491\) 19.0540 0.859894 0.429947 0.902854i \(-0.358532\pi\)
0.429947 + 0.902854i \(0.358532\pi\)
\(492\) 0 0
\(493\) −11.6847 −0.526251
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) 7.82292 0.350906
\(498\) 0 0
\(499\) −21.4384 −0.959717 −0.479858 0.877346i \(-0.659312\pi\)
−0.479858 + 0.877346i \(0.659312\pi\)
\(500\) 0 0
\(501\) −2.24621 −0.100353
\(502\) 0 0
\(503\) 36.1771 1.61306 0.806528 0.591196i \(-0.201344\pi\)
0.806528 + 0.591196i \(0.201344\pi\)
\(504\) 0 0
\(505\) 7.43845 0.331007
\(506\) 0 0
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) 19.7538 0.875571 0.437786 0.899079i \(-0.355763\pi\)
0.437786 + 0.899079i \(0.355763\pi\)
\(510\) 0 0
\(511\) 5.61553 0.248416
\(512\) 0 0
\(513\) 1.12311 0.0495863
\(514\) 0 0
\(515\) −11.1231 −0.490143
\(516\) 0 0
\(517\) −20.4924 −0.901256
\(518\) 0 0
\(519\) −7.12311 −0.312670
\(520\) 0 0
\(521\) 15.3693 0.673342 0.336671 0.941622i \(-0.390699\pi\)
0.336671 + 0.941622i \(0.390699\pi\)
\(522\) 0 0
\(523\) 5.75379 0.251596 0.125798 0.992056i \(-0.459851\pi\)
0.125798 + 0.992056i \(0.459851\pi\)
\(524\) 0 0
\(525\) −0.561553 −0.0245082
\(526\) 0 0
\(527\) −9.43845 −0.411145
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −2.56155 −0.111162
\(532\) 0 0
\(533\) −11.3693 −0.492460
\(534\) 0 0
\(535\) 14.8078 0.640195
\(536\) 0 0
\(537\) −8.00000 −0.345225
\(538\) 0 0
\(539\) 13.3693 0.575857
\(540\) 0 0
\(541\) −8.24621 −0.354532 −0.177266 0.984163i \(-0.556725\pi\)
−0.177266 + 0.984163i \(0.556725\pi\)
\(542\) 0 0
\(543\) 2.24621 0.0963942
\(544\) 0 0
\(545\) 13.1231 0.562132
\(546\) 0 0
\(547\) 10.7386 0.459151 0.229575 0.973291i \(-0.426266\pi\)
0.229575 + 0.973291i \(0.426266\pi\)
\(548\) 0 0
\(549\) −14.2462 −0.608013
\(550\) 0 0
\(551\) 5.12311 0.218252
\(552\) 0 0
\(553\) 7.50758 0.319255
\(554\) 0 0
\(555\) −3.68466 −0.156405
\(556\) 0 0
\(557\) 22.3153 0.945531 0.472766 0.881188i \(-0.343256\pi\)
0.472766 + 0.881188i \(0.343256\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 5.12311 0.216298
\(562\) 0 0
\(563\) 16.5616 0.697986 0.348993 0.937125i \(-0.386524\pi\)
0.348993 + 0.937125i \(0.386524\pi\)
\(564\) 0 0
\(565\) 12.8078 0.538827
\(566\) 0 0
\(567\) −0.561553 −0.0235830
\(568\) 0 0
\(569\) −1.12311 −0.0470830 −0.0235415 0.999723i \(-0.507494\pi\)
−0.0235415 + 0.999723i \(0.507494\pi\)
\(570\) 0 0
\(571\) −12.4924 −0.522792 −0.261396 0.965232i \(-0.584183\pi\)
−0.261396 + 0.965232i \(0.584183\pi\)
\(572\) 0 0
\(573\) 18.2462 0.762246
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 13.8617 0.577072 0.288536 0.957469i \(-0.406832\pi\)
0.288536 + 0.957469i \(0.406832\pi\)
\(578\) 0 0
\(579\) −15.1231 −0.628495
\(580\) 0 0
\(581\) −3.19224 −0.132436
\(582\) 0 0
\(583\) 6.87689 0.284812
\(584\) 0 0
\(585\) 2.00000 0.0826898
\(586\) 0 0
\(587\) −34.8769 −1.43952 −0.719762 0.694221i \(-0.755749\pi\)
−0.719762 + 0.694221i \(0.755749\pi\)
\(588\) 0 0
\(589\) 4.13826 0.170514
\(590\) 0 0
\(591\) −0.246211 −0.0101278
\(592\) 0 0
\(593\) 2.63068 0.108029 0.0540146 0.998540i \(-0.482798\pi\)
0.0540146 + 0.998540i \(0.482798\pi\)
\(594\) 0 0
\(595\) −1.43845 −0.0589706
\(596\) 0 0
\(597\) 12.2462 0.501204
\(598\) 0 0
\(599\) 4.49242 0.183555 0.0917777 0.995780i \(-0.470745\pi\)
0.0917777 + 0.995780i \(0.470745\pi\)
\(600\) 0 0
\(601\) 25.5464 1.04206 0.521030 0.853539i \(-0.325548\pi\)
0.521030 + 0.853539i \(0.325548\pi\)
\(602\) 0 0
\(603\) −11.6847 −0.475836
\(604\) 0 0
\(605\) 7.00000 0.284590
\(606\) 0 0
\(607\) 17.7538 0.720604 0.360302 0.932836i \(-0.382674\pi\)
0.360302 + 0.932836i \(0.382674\pi\)
\(608\) 0 0
\(609\) −2.56155 −0.103799
\(610\) 0 0
\(611\) −20.4924 −0.829035
\(612\) 0 0
\(613\) −15.8617 −0.640650 −0.320325 0.947308i \(-0.603792\pi\)
−0.320325 + 0.947308i \(0.603792\pi\)
\(614\) 0 0
\(615\) −5.68466 −0.229228
\(616\) 0 0
\(617\) 18.5616 0.747260 0.373630 0.927578i \(-0.378113\pi\)
0.373630 + 0.927578i \(0.378113\pi\)
\(618\) 0 0
\(619\) −9.75379 −0.392038 −0.196019 0.980600i \(-0.562801\pi\)
−0.196019 + 0.980600i \(0.562801\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 3.50758 0.140528
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −2.24621 −0.0897050
\(628\) 0 0
\(629\) −9.43845 −0.376336
\(630\) 0 0
\(631\) −8.38447 −0.333781 −0.166890 0.985975i \(-0.553373\pi\)
−0.166890 + 0.985975i \(0.553373\pi\)
\(632\) 0 0
\(633\) −23.0540 −0.916313
\(634\) 0 0
\(635\) −16.4924 −0.654482
\(636\) 0 0
\(637\) 13.3693 0.529712
\(638\) 0 0
\(639\) −13.9309 −0.551097
\(640\) 0 0
\(641\) 35.3693 1.39700 0.698502 0.715608i \(-0.253850\pi\)
0.698502 + 0.715608i \(0.253850\pi\)
\(642\) 0 0
\(643\) −1.43845 −0.0567268 −0.0283634 0.999598i \(-0.509030\pi\)
−0.0283634 + 0.999598i \(0.509030\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −29.1231 −1.14495 −0.572474 0.819923i \(-0.694016\pi\)
−0.572474 + 0.819923i \(0.694016\pi\)
\(648\) 0 0
\(649\) 5.12311 0.201099
\(650\) 0 0
\(651\) −2.06913 −0.0810956
\(652\) 0 0
\(653\) −3.75379 −0.146897 −0.0734486 0.997299i \(-0.523400\pi\)
−0.0734486 + 0.997299i \(0.523400\pi\)
\(654\) 0 0
\(655\) −10.2462 −0.400353
\(656\) 0 0
\(657\) −10.0000 −0.390137
\(658\) 0 0
\(659\) 4.73863 0.184591 0.0922955 0.995732i \(-0.470580\pi\)
0.0922955 + 0.995732i \(0.470580\pi\)
\(660\) 0 0
\(661\) −25.1231 −0.977176 −0.488588 0.872515i \(-0.662488\pi\)
−0.488588 + 0.872515i \(0.662488\pi\)
\(662\) 0 0
\(663\) 5.12311 0.198965
\(664\) 0 0
\(665\) 0.630683 0.0244568
\(666\) 0 0
\(667\) 4.56155 0.176624
\(668\) 0 0
\(669\) 14.8769 0.575174
\(670\) 0 0
\(671\) 28.4924 1.09994
\(672\) 0 0
\(673\) 20.8769 0.804745 0.402373 0.915476i \(-0.368186\pi\)
0.402373 + 0.915476i \(0.368186\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 20.4233 0.784931 0.392465 0.919767i \(-0.371622\pi\)
0.392465 + 0.919767i \(0.371622\pi\)
\(678\) 0 0
\(679\) 7.86174 0.301706
\(680\) 0 0
\(681\) −0.876894 −0.0336027
\(682\) 0 0
\(683\) −6.73863 −0.257847 −0.128923 0.991655i \(-0.541152\pi\)
−0.128923 + 0.991655i \(0.541152\pi\)
\(684\) 0 0
\(685\) −2.87689 −0.109920
\(686\) 0 0
\(687\) 18.2462 0.696136
\(688\) 0 0
\(689\) 6.87689 0.261989
\(690\) 0 0
\(691\) 7.50758 0.285602 0.142801 0.989751i \(-0.454389\pi\)
0.142801 + 0.989751i \(0.454389\pi\)
\(692\) 0 0
\(693\) 1.12311 0.0426633
\(694\) 0 0
\(695\) 7.68466 0.291496
\(696\) 0 0
\(697\) −14.5616 −0.551558
\(698\) 0 0
\(699\) 15.1231 0.572008
\(700\) 0 0
\(701\) 0.876894 0.0331198 0.0165599 0.999863i \(-0.494729\pi\)
0.0165599 + 0.999863i \(0.494729\pi\)
\(702\) 0 0
\(703\) 4.13826 0.156077
\(704\) 0 0
\(705\) −10.2462 −0.385895
\(706\) 0 0
\(707\) 4.17708 0.157095
\(708\) 0 0
\(709\) −20.9848 −0.788102 −0.394051 0.919088i \(-0.628927\pi\)
−0.394051 + 0.919088i \(0.628927\pi\)
\(710\) 0 0
\(711\) −13.3693 −0.501389
\(712\) 0 0
\(713\) 3.68466 0.137992
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 0 0
\(717\) 9.43845 0.352485
\(718\) 0 0
\(719\) 17.4384 0.650344 0.325172 0.945655i \(-0.394578\pi\)
0.325172 + 0.945655i \(0.394578\pi\)
\(720\) 0 0
\(721\) −6.24621 −0.232621
\(722\) 0 0
\(723\) −3.75379 −0.139605
\(724\) 0 0
\(725\) 4.56155 0.169412
\(726\) 0 0
\(727\) 24.0691 0.892675 0.446337 0.894865i \(-0.352728\pi\)
0.446337 + 0.894865i \(0.352728\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 5.43845 0.200874 0.100437 0.994943i \(-0.467976\pi\)
0.100437 + 0.994943i \(0.467976\pi\)
\(734\) 0 0
\(735\) 6.68466 0.246567
\(736\) 0 0
\(737\) 23.3693 0.860820
\(738\) 0 0
\(739\) −23.0540 −0.848054 −0.424027 0.905650i \(-0.639384\pi\)
−0.424027 + 0.905650i \(0.639384\pi\)
\(740\) 0 0
\(741\) −2.24621 −0.0825166
\(742\) 0 0
\(743\) −10.7386 −0.393962 −0.196981 0.980407i \(-0.563114\pi\)
−0.196981 + 0.980407i \(0.563114\pi\)
\(744\) 0 0
\(745\) 3.12311 0.114422
\(746\) 0 0
\(747\) 5.68466 0.207991
\(748\) 0 0
\(749\) 8.31534 0.303836
\(750\) 0 0
\(751\) −18.4924 −0.674798 −0.337399 0.941362i \(-0.609547\pi\)
−0.337399 + 0.941362i \(0.609547\pi\)
\(752\) 0 0
\(753\) 0.246211 0.00897244
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) −48.1771 −1.75103 −0.875513 0.483195i \(-0.839476\pi\)
−0.875513 + 0.483195i \(0.839476\pi\)
\(758\) 0 0
\(759\) −2.00000 −0.0725954
\(760\) 0 0
\(761\) −31.3002 −1.13463 −0.567315 0.823501i \(-0.692018\pi\)
−0.567315 + 0.823501i \(0.692018\pi\)
\(762\) 0 0
\(763\) 7.36932 0.266787
\(764\) 0 0
\(765\) 2.56155 0.0926131
\(766\) 0 0
\(767\) 5.12311 0.184985
\(768\) 0 0
\(769\) −27.6155 −0.995841 −0.497921 0.867223i \(-0.665903\pi\)
−0.497921 + 0.867223i \(0.665903\pi\)
\(770\) 0 0
\(771\) 27.6155 0.994549
\(772\) 0 0
\(773\) 50.4924 1.81609 0.908043 0.418877i \(-0.137576\pi\)
0.908043 + 0.418877i \(0.137576\pi\)
\(774\) 0 0
\(775\) 3.68466 0.132357
\(776\) 0 0
\(777\) −2.06913 −0.0742296
\(778\) 0 0
\(779\) 6.38447 0.228747
\(780\) 0 0
\(781\) 27.8617 0.996971
\(782\) 0 0
\(783\) 4.56155 0.163017
\(784\) 0 0
\(785\) 9.43845 0.336873
\(786\) 0 0
\(787\) −4.94602 −0.176307 −0.0881534 0.996107i \(-0.528097\pi\)
−0.0881534 + 0.996107i \(0.528097\pi\)
\(788\) 0 0
\(789\) −2.56155 −0.0911937
\(790\) 0 0
\(791\) 7.19224 0.255726
\(792\) 0 0
\(793\) 28.4924 1.01180
\(794\) 0 0
\(795\) 3.43845 0.121949
\(796\) 0 0
\(797\) −45.5464 −1.61334 −0.806668 0.591005i \(-0.798731\pi\)
−0.806668 + 0.591005i \(0.798731\pi\)
\(798\) 0 0
\(799\) −26.2462 −0.928524
\(800\) 0 0
\(801\) −6.24621 −0.220699
\(802\) 0 0
\(803\) 20.0000 0.705785
\(804\) 0 0
\(805\) 0.561553 0.0197921
\(806\) 0 0
\(807\) 1.19224 0.0419687
\(808\) 0 0
\(809\) 43.3002 1.52235 0.761177 0.648544i \(-0.224622\pi\)
0.761177 + 0.648544i \(0.224622\pi\)
\(810\) 0 0
\(811\) 0.946025 0.0332194 0.0166097 0.999862i \(-0.494713\pi\)
0.0166097 + 0.999862i \(0.494713\pi\)
\(812\) 0 0
\(813\) −5.43845 −0.190735
\(814\) 0 0
\(815\) −9.12311 −0.319568
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 1.12311 0.0392445
\(820\) 0 0
\(821\) 6.49242 0.226587 0.113294 0.993562i \(-0.463860\pi\)
0.113294 + 0.993562i \(0.463860\pi\)
\(822\) 0 0
\(823\) −6.73863 −0.234894 −0.117447 0.993079i \(-0.537471\pi\)
−0.117447 + 0.993079i \(0.537471\pi\)
\(824\) 0 0
\(825\) −2.00000 −0.0696311
\(826\) 0 0
\(827\) −23.9309 −0.832158 −0.416079 0.909328i \(-0.636596\pi\)
−0.416079 + 0.909328i \(0.636596\pi\)
\(828\) 0 0
\(829\) 47.9309 1.66471 0.832354 0.554244i \(-0.186993\pi\)
0.832354 + 0.554244i \(0.186993\pi\)
\(830\) 0 0
\(831\) −28.7386 −0.996932
\(832\) 0 0
\(833\) 17.1231 0.593280
\(834\) 0 0
\(835\) 2.24621 0.0777333
\(836\) 0 0
\(837\) 3.68466 0.127360
\(838\) 0 0
\(839\) −25.1231 −0.867346 −0.433673 0.901070i \(-0.642783\pi\)
−0.433673 + 0.901070i \(0.642783\pi\)
\(840\) 0 0
\(841\) −8.19224 −0.282491
\(842\) 0 0
\(843\) 1.12311 0.0386818
\(844\) 0 0
\(845\) 9.00000 0.309609
\(846\) 0 0
\(847\) 3.93087 0.135066
\(848\) 0 0
\(849\) 13.4384 0.461206
\(850\) 0 0
\(851\) 3.68466 0.126308
\(852\) 0 0
\(853\) −26.9848 −0.923943 −0.461972 0.886895i \(-0.652858\pi\)
−0.461972 + 0.886895i \(0.652858\pi\)
\(854\) 0 0
\(855\) −1.12311 −0.0384094
\(856\) 0 0
\(857\) −52.7386 −1.80152 −0.900759 0.434320i \(-0.856989\pi\)
−0.900759 + 0.434320i \(0.856989\pi\)
\(858\) 0 0
\(859\) −28.8078 −0.982908 −0.491454 0.870903i \(-0.663534\pi\)
−0.491454 + 0.870903i \(0.663534\pi\)
\(860\) 0 0
\(861\) −3.19224 −0.108791
\(862\) 0 0
\(863\) 31.3693 1.06782 0.533912 0.845540i \(-0.320721\pi\)
0.533912 + 0.845540i \(0.320721\pi\)
\(864\) 0 0
\(865\) 7.12311 0.242193
\(866\) 0 0
\(867\) −10.4384 −0.354508
\(868\) 0 0
\(869\) 26.7386 0.907046
\(870\) 0 0
\(871\) 23.3693 0.791839
\(872\) 0 0
\(873\) −14.0000 −0.473828
\(874\) 0 0
\(875\) 0.561553 0.0189839
\(876\) 0 0
\(877\) 16.7386 0.565224 0.282612 0.959234i \(-0.408799\pi\)
0.282612 + 0.959234i \(0.408799\pi\)
\(878\) 0 0
\(879\) −4.06913 −0.137248
\(880\) 0 0
\(881\) −10.7386 −0.361794 −0.180897 0.983502i \(-0.557900\pi\)
−0.180897 + 0.983502i \(0.557900\pi\)
\(882\) 0 0
\(883\) 30.2462 1.01787 0.508933 0.860806i \(-0.330040\pi\)
0.508933 + 0.860806i \(0.330040\pi\)
\(884\) 0 0
\(885\) 2.56155 0.0861057
\(886\) 0 0
\(887\) 16.6307 0.558404 0.279202 0.960232i \(-0.409930\pi\)
0.279202 + 0.960232i \(0.409930\pi\)
\(888\) 0 0
\(889\) −9.26137 −0.310616
\(890\) 0 0
\(891\) −2.00000 −0.0670025
\(892\) 0 0
\(893\) 11.5076 0.385086
\(894\) 0 0
\(895\) 8.00000 0.267411
\(896\) 0 0
\(897\) −2.00000 −0.0667781
\(898\) 0 0
\(899\) 16.8078 0.560570
\(900\) 0 0
\(901\) 8.80776 0.293429
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.24621 −0.0746666
\(906\) 0 0
\(907\) −38.4233 −1.27582 −0.637912 0.770109i \(-0.720202\pi\)
−0.637912 + 0.770109i \(0.720202\pi\)
\(908\) 0 0
\(909\) −7.43845 −0.246718
\(910\) 0 0
\(911\) 28.9848 0.960311 0.480155 0.877183i \(-0.340580\pi\)
0.480155 + 0.877183i \(0.340580\pi\)
\(912\) 0 0
\(913\) −11.3693 −0.376269
\(914\) 0 0
\(915\) 14.2462 0.470965
\(916\) 0 0
\(917\) −5.75379 −0.190007
\(918\) 0 0
\(919\) 1.36932 0.0451696 0.0225848 0.999745i \(-0.492810\pi\)
0.0225848 + 0.999745i \(0.492810\pi\)
\(920\) 0 0
\(921\) −25.1231 −0.827834
\(922\) 0 0
\(923\) 27.8617 0.917080
\(924\) 0 0
\(925\) 3.68466 0.121151
\(926\) 0 0
\(927\) 11.1231 0.365331
\(928\) 0 0
\(929\) −54.1771 −1.77749 −0.888746 0.458400i \(-0.848423\pi\)
−0.888746 + 0.458400i \(0.848423\pi\)
\(930\) 0 0
\(931\) −7.50758 −0.246051
\(932\) 0 0
\(933\) −32.9848 −1.07988
\(934\) 0 0
\(935\) −5.12311 −0.167543
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) −10.8078 −0.352698
\(940\) 0 0
\(941\) −43.1231 −1.40577 −0.702887 0.711302i \(-0.748106\pi\)
−0.702887 + 0.711302i \(0.748106\pi\)
\(942\) 0 0
\(943\) 5.68466 0.185118
\(944\) 0 0
\(945\) 0.561553 0.0182673
\(946\) 0 0
\(947\) −43.2311 −1.40482 −0.702410 0.711772i \(-0.747893\pi\)
−0.702410 + 0.711772i \(0.747893\pi\)
\(948\) 0 0
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) −19.6155 −0.636077
\(952\) 0 0
\(953\) 14.8769 0.481910 0.240955 0.970536i \(-0.422539\pi\)
0.240955 + 0.970536i \(0.422539\pi\)
\(954\) 0 0
\(955\) −18.2462 −0.590434
\(956\) 0 0
\(957\) −9.12311 −0.294908
\(958\) 0 0
\(959\) −1.61553 −0.0521681
\(960\) 0 0
\(961\) −17.4233 −0.562042
\(962\) 0 0
\(963\) −14.8078 −0.477174
\(964\) 0 0
\(965\) 15.1231 0.486830
\(966\) 0 0
\(967\) −16.9848 −0.546196 −0.273098 0.961986i \(-0.588048\pi\)
−0.273098 + 0.961986i \(0.588048\pi\)
\(968\) 0 0
\(969\) −2.87689 −0.0924192
\(970\) 0 0
\(971\) −38.3542 −1.23084 −0.615422 0.788198i \(-0.711014\pi\)
−0.615422 + 0.788198i \(0.711014\pi\)
\(972\) 0 0
\(973\) 4.31534 0.138343
\(974\) 0 0
\(975\) −2.00000 −0.0640513
\(976\) 0 0
\(977\) 30.5616 0.977751 0.488875 0.872354i \(-0.337407\pi\)
0.488875 + 0.872354i \(0.337407\pi\)
\(978\) 0 0
\(979\) 12.4924 0.399260
\(980\) 0 0
\(981\) −13.1231 −0.418989
\(982\) 0 0
\(983\) 33.3002 1.06211 0.531056 0.847337i \(-0.321796\pi\)
0.531056 + 0.847337i \(0.321796\pi\)
\(984\) 0 0
\(985\) 0.246211 0.00784494
\(986\) 0 0
\(987\) −5.75379 −0.183145
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 42.4233 1.34762 0.673810 0.738905i \(-0.264657\pi\)
0.673810 + 0.738905i \(0.264657\pi\)
\(992\) 0 0
\(993\) 23.6847 0.751610
\(994\) 0 0
\(995\) −12.2462 −0.388231
\(996\) 0 0
\(997\) −19.1231 −0.605635 −0.302817 0.953049i \(-0.597927\pi\)
−0.302817 + 0.953049i \(0.597927\pi\)
\(998\) 0 0
\(999\) 3.68466 0.116577
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 5520.2.a.bn.1.1 2
4.3 odd 2 2760.2.a.m.1.2 2
12.11 even 2 8280.2.a.bd.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2760.2.a.m.1.2 2 4.3 odd 2
5520.2.a.bn.1.1 2 1.1 even 1 trivial
8280.2.a.bd.1.2 2 12.11 even 2