Properties

Label 5408.2.a.bd.1.1
Level $5408$
Weight $2$
Character 5408.1
Self dual yes
Analytic conductor $43.183$
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] = [5408,2,Mod(1,5408)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5408, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("5408.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5408 = 2^{5} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5408.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: \(43.1830974131\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 416)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 5408.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.56155 q^{3} +3.56155 q^{5} -3.56155 q^{7} -0.561553 q^{9} +O(q^{10})\) \(q-1.56155 q^{3} +3.56155 q^{5} -3.56155 q^{7} -0.561553 q^{9} -2.00000 q^{11} -5.56155 q^{15} +3.56155 q^{17} -6.00000 q^{19} +5.56155 q^{21} +7.68466 q^{25} +5.56155 q^{27} +8.24621 q^{29} +1.12311 q^{31} +3.12311 q^{33} -12.6847 q^{35} -2.68466 q^{37} +1.12311 q^{41} +11.8078 q^{43} -2.00000 q^{45} +10.6847 q^{47} +5.68466 q^{49} -5.56155 q^{51} -13.1231 q^{53} -7.12311 q^{55} +9.36932 q^{57} +6.00000 q^{59} -11.3693 q^{61} +2.00000 q^{63} -6.00000 q^{67} -10.6847 q^{71} -10.0000 q^{73} -12.0000 q^{75} +7.12311 q^{77} +12.0000 q^{79} -7.00000 q^{81} -7.36932 q^{83} +12.6847 q^{85} -12.8769 q^{87} -8.24621 q^{89} -1.75379 q^{93} -21.3693 q^{95} +6.00000 q^{97} +1.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{3} + 3 q^{5} - 3 q^{7} + 3 q^{9} - 4 q^{11} - 7 q^{15} + 3 q^{17} - 12 q^{19} + 7 q^{21} + 3 q^{25} + 7 q^{27} - 6 q^{31} - 2 q^{33} - 13 q^{35} + 7 q^{37} - 6 q^{41} + 3 q^{43} - 4 q^{45} + 9 q^{47} - q^{49} - 7 q^{51} - 18 q^{53} - 6 q^{55} - 6 q^{57} + 12 q^{59} + 2 q^{61} + 4 q^{63} - 12 q^{67} - 9 q^{71} - 20 q^{73} - 24 q^{75} + 6 q^{77} + 24 q^{79} - 14 q^{81} + 10 q^{83} + 13 q^{85} - 34 q^{87} - 20 q^{93} - 18 q^{95} + 12 q^{97} - 6 q^{99}+O(q^{100}) \) 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
\(3\) −1.56155 −0.901563 −0.450781 0.892634i \(-0.648855\pi\)
−0.450781 + 0.892634i \(0.648855\pi\)
\(4\) 0 0
\(5\) 3.56155 1.59277 0.796387 0.604787i \(-0.206742\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) 0 0
\(7\) −3.56155 −1.34614 −0.673070 0.739579i \(-0.735025\pi\)
−0.673070 + 0.739579i \(0.735025\pi\)
\(8\) 0 0
\(9\) −0.561553 −0.187184
\(10\) 0 0
\(11\) −2.00000 −0.603023 −0.301511 0.953463i \(-0.597491\pi\)
−0.301511 + 0.953463i \(0.597491\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −5.56155 −1.43599
\(16\) 0 0
\(17\) 3.56155 0.863803 0.431902 0.901921i \(-0.357843\pi\)
0.431902 + 0.901921i \(0.357843\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 5.56155 1.21363
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 7.68466 1.53693
\(26\) 0 0
\(27\) 5.56155 1.07032
\(28\) 0 0
\(29\) 8.24621 1.53128 0.765641 0.643268i \(-0.222422\pi\)
0.765641 + 0.643268i \(0.222422\pi\)
\(30\) 0 0
\(31\) 1.12311 0.201716 0.100858 0.994901i \(-0.467841\pi\)
0.100858 + 0.994901i \(0.467841\pi\)
\(32\) 0 0
\(33\) 3.12311 0.543663
\(34\) 0 0
\(35\) −12.6847 −2.14410
\(36\) 0 0
\(37\) −2.68466 −0.441355 −0.220678 0.975347i \(-0.570827\pi\)
−0.220678 + 0.975347i \(0.570827\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.12311 0.175400 0.0876998 0.996147i \(-0.472048\pi\)
0.0876998 + 0.996147i \(0.472048\pi\)
\(42\) 0 0
\(43\) 11.8078 1.80067 0.900334 0.435200i \(-0.143323\pi\)
0.900334 + 0.435200i \(0.143323\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 10.6847 1.55852 0.779259 0.626702i \(-0.215596\pi\)
0.779259 + 0.626702i \(0.215596\pi\)
\(48\) 0 0
\(49\) 5.68466 0.812094
\(50\) 0 0
\(51\) −5.56155 −0.778773
\(52\) 0 0
\(53\) −13.1231 −1.80260 −0.901299 0.433198i \(-0.857385\pi\)
−0.901299 + 0.433198i \(0.857385\pi\)
\(54\) 0 0
\(55\) −7.12311 −0.960479
\(56\) 0 0
\(57\) 9.36932 1.24100
\(58\) 0 0
\(59\) 6.00000 0.781133 0.390567 0.920575i \(-0.372279\pi\)
0.390567 + 0.920575i \(0.372279\pi\)
\(60\) 0 0
\(61\) −11.3693 −1.45569 −0.727846 0.685741i \(-0.759478\pi\)
−0.727846 + 0.685741i \(0.759478\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −6.00000 −0.733017 −0.366508 0.930415i \(-0.619447\pi\)
−0.366508 + 0.930415i \(0.619447\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −10.6847 −1.26804 −0.634018 0.773318i \(-0.718595\pi\)
−0.634018 + 0.773318i \(0.718595\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) −12.0000 −1.38564
\(76\) 0 0
\(77\) 7.12311 0.811753
\(78\) 0 0
\(79\) 12.0000 1.35011 0.675053 0.737769i \(-0.264121\pi\)
0.675053 + 0.737769i \(0.264121\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −7.36932 −0.808888 −0.404444 0.914563i \(-0.632535\pi\)
−0.404444 + 0.914563i \(0.632535\pi\)
\(84\) 0 0
\(85\) 12.6847 1.37584
\(86\) 0 0
\(87\) −12.8769 −1.38055
\(88\) 0 0
\(89\) −8.24621 −0.874097 −0.437048 0.899438i \(-0.643976\pi\)
−0.437048 + 0.899438i \(0.643976\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.75379 −0.181859
\(94\) 0 0
\(95\) −21.3693 −2.19245
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 0 0
\(99\) 1.12311 0.112876
\(100\) 0 0
\(101\) 1.12311 0.111753 0.0558766 0.998438i \(-0.482205\pi\)
0.0558766 + 0.998438i \(0.482205\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 19.8078 1.93304
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −10.6847 −1.02340 −0.511702 0.859163i \(-0.670985\pi\)
−0.511702 + 0.859163i \(0.670985\pi\)
\(110\) 0 0
\(111\) 4.19224 0.397909
\(112\) 0 0
\(113\) −8.24621 −0.775738 −0.387869 0.921714i \(-0.626789\pi\)
−0.387869 + 0.921714i \(0.626789\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −12.6847 −1.16280
\(120\) 0 0
\(121\) −7.00000 −0.636364
\(122\) 0 0
\(123\) −1.75379 −0.158134
\(124\) 0 0
\(125\) 9.56155 0.855211
\(126\) 0 0
\(127\) −14.2462 −1.26415 −0.632073 0.774909i \(-0.717796\pi\)
−0.632073 + 0.774909i \(0.717796\pi\)
\(128\) 0 0
\(129\) −18.4384 −1.62341
\(130\) 0 0
\(131\) 3.31534 0.289663 0.144831 0.989456i \(-0.453736\pi\)
0.144831 + 0.989456i \(0.453736\pi\)
\(132\) 0 0
\(133\) 21.3693 1.85295
\(134\) 0 0
\(135\) 19.8078 1.70478
\(136\) 0 0
\(137\) −13.1231 −1.12118 −0.560591 0.828093i \(-0.689426\pi\)
−0.560591 + 0.828093i \(0.689426\pi\)
\(138\) 0 0
\(139\) −4.68466 −0.397348 −0.198674 0.980066i \(-0.563663\pi\)
−0.198674 + 0.980066i \(0.563663\pi\)
\(140\) 0 0
\(141\) −16.6847 −1.40510
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 29.3693 2.43899
\(146\) 0 0
\(147\) −8.87689 −0.732154
\(148\) 0 0
\(149\) −8.24621 −0.675556 −0.337778 0.941226i \(-0.609675\pi\)
−0.337778 + 0.941226i \(0.609675\pi\)
\(150\) 0 0
\(151\) −22.6847 −1.84605 −0.923026 0.384738i \(-0.874292\pi\)
−0.923026 + 0.384738i \(0.874292\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 4.00000 0.321288
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 0 0
\(159\) 20.4924 1.62515
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 15.3693 1.20382 0.601909 0.798565i \(-0.294407\pi\)
0.601909 + 0.798565i \(0.294407\pi\)
\(164\) 0 0
\(165\) 11.1231 0.865933
\(166\) 0 0
\(167\) −15.3693 −1.18931 −0.594657 0.803980i \(-0.702712\pi\)
−0.594657 + 0.803980i \(0.702712\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 3.36932 0.257658
\(172\) 0 0
\(173\) −3.36932 −0.256164 −0.128082 0.991764i \(-0.540882\pi\)
−0.128082 + 0.991764i \(0.540882\pi\)
\(174\) 0 0
\(175\) −27.3693 −2.06893
\(176\) 0 0
\(177\) −9.36932 −0.704241
\(178\) 0 0
\(179\) 18.0540 1.34942 0.674709 0.738084i \(-0.264269\pi\)
0.674709 + 0.738084i \(0.264269\pi\)
\(180\) 0 0
\(181\) −7.36932 −0.547757 −0.273879 0.961764i \(-0.588307\pi\)
−0.273879 + 0.961764i \(0.588307\pi\)
\(182\) 0 0
\(183\) 17.7538 1.31240
\(184\) 0 0
\(185\) −9.56155 −0.702979
\(186\) 0 0
\(187\) −7.12311 −0.520893
\(188\) 0 0
\(189\) −19.8078 −1.44080
\(190\) 0 0
\(191\) 5.36932 0.388510 0.194255 0.980951i \(-0.437771\pi\)
0.194255 + 0.980951i \(0.437771\pi\)
\(192\) 0 0
\(193\) 23.3693 1.68216 0.841080 0.540911i \(-0.181920\pi\)
0.841080 + 0.540911i \(0.181920\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.31534 −0.0937142 −0.0468571 0.998902i \(-0.514921\pi\)
−0.0468571 + 0.998902i \(0.514921\pi\)
\(198\) 0 0
\(199\) −9.36932 −0.664173 −0.332087 0.943249i \(-0.607753\pi\)
−0.332087 + 0.943249i \(0.607753\pi\)
\(200\) 0 0
\(201\) 9.36932 0.660861
\(202\) 0 0
\(203\) −29.3693 −2.06132
\(204\) 0 0
\(205\) 4.00000 0.279372
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) 23.8078 1.63899 0.819497 0.573083i \(-0.194253\pi\)
0.819497 + 0.573083i \(0.194253\pi\)
\(212\) 0 0
\(213\) 16.6847 1.14321
\(214\) 0 0
\(215\) 42.0540 2.86806
\(216\) 0 0
\(217\) −4.00000 −0.271538
\(218\) 0 0
\(219\) 15.6155 1.05520
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −6.19224 −0.414663 −0.207331 0.978271i \(-0.566478\pi\)
−0.207331 + 0.978271i \(0.566478\pi\)
\(224\) 0 0
\(225\) −4.31534 −0.287689
\(226\) 0 0
\(227\) 3.36932 0.223629 0.111815 0.993729i \(-0.464334\pi\)
0.111815 + 0.993729i \(0.464334\pi\)
\(228\) 0 0
\(229\) −1.31534 −0.0869202 −0.0434601 0.999055i \(-0.513838\pi\)
−0.0434601 + 0.999055i \(0.513838\pi\)
\(230\) 0 0
\(231\) −11.1231 −0.731847
\(232\) 0 0
\(233\) 1.31534 0.0861709 0.0430854 0.999071i \(-0.486281\pi\)
0.0430854 + 0.999071i \(0.486281\pi\)
\(234\) 0 0
\(235\) 38.0540 2.48237
\(236\) 0 0
\(237\) −18.7386 −1.21721
\(238\) 0 0
\(239\) −9.31534 −0.602559 −0.301280 0.953536i \(-0.597414\pi\)
−0.301280 + 0.953536i \(0.597414\pi\)
\(240\) 0 0
\(241\) −2.00000 −0.128831 −0.0644157 0.997923i \(-0.520518\pi\)
−0.0644157 + 0.997923i \(0.520518\pi\)
\(242\) 0 0
\(243\) −5.75379 −0.369106
\(244\) 0 0
\(245\) 20.2462 1.29348
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 11.5076 0.729263
\(250\) 0 0
\(251\) −16.0000 −1.00991 −0.504956 0.863145i \(-0.668491\pi\)
−0.504956 + 0.863145i \(0.668491\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −19.8078 −1.24041
\(256\) 0 0
\(257\) −12.9309 −0.806605 −0.403303 0.915067i \(-0.632138\pi\)
−0.403303 + 0.915067i \(0.632138\pi\)
\(258\) 0 0
\(259\) 9.56155 0.594126
\(260\) 0 0
\(261\) −4.63068 −0.286632
\(262\) 0 0
\(263\) −22.7386 −1.40212 −0.701062 0.713100i \(-0.747290\pi\)
−0.701062 + 0.713100i \(0.747290\pi\)
\(264\) 0 0
\(265\) −46.7386 −2.87113
\(266\) 0 0
\(267\) 12.8769 0.788053
\(268\) 0 0
\(269\) 3.36932 0.205431 0.102715 0.994711i \(-0.467247\pi\)
0.102715 + 0.994711i \(0.467247\pi\)
\(270\) 0 0
\(271\) 10.6847 0.649047 0.324523 0.945878i \(-0.394796\pi\)
0.324523 + 0.945878i \(0.394796\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −15.3693 −0.926805
\(276\) 0 0
\(277\) 23.3693 1.40413 0.702063 0.712115i \(-0.252262\pi\)
0.702063 + 0.712115i \(0.252262\pi\)
\(278\) 0 0
\(279\) −0.630683 −0.0377580
\(280\) 0 0
\(281\) −3.75379 −0.223932 −0.111966 0.993712i \(-0.535715\pi\)
−0.111966 + 0.993712i \(0.535715\pi\)
\(282\) 0 0
\(283\) −14.2462 −0.846849 −0.423425 0.905931i \(-0.639172\pi\)
−0.423425 + 0.905931i \(0.639172\pi\)
\(284\) 0 0
\(285\) 33.3693 1.97663
\(286\) 0 0
\(287\) −4.00000 −0.236113
\(288\) 0 0
\(289\) −4.31534 −0.253844
\(290\) 0 0
\(291\) −9.36932 −0.549239
\(292\) 0 0
\(293\) 27.5616 1.61016 0.805082 0.593164i \(-0.202121\pi\)
0.805082 + 0.593164i \(0.202121\pi\)
\(294\) 0 0
\(295\) 21.3693 1.24417
\(296\) 0 0
\(297\) −11.1231 −0.645428
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −42.0540 −2.42395
\(302\) 0 0
\(303\) −1.75379 −0.100753
\(304\) 0 0
\(305\) −40.4924 −2.31859
\(306\) 0 0
\(307\) 6.00000 0.342438 0.171219 0.985233i \(-0.445229\pi\)
0.171219 + 0.985233i \(0.445229\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.36932 0.531285 0.265643 0.964072i \(-0.414416\pi\)
0.265643 + 0.964072i \(0.414416\pi\)
\(312\) 0 0
\(313\) −20.0540 −1.13352 −0.566759 0.823884i \(-0.691803\pi\)
−0.566759 + 0.823884i \(0.691803\pi\)
\(314\) 0 0
\(315\) 7.12311 0.401342
\(316\) 0 0
\(317\) 15.7538 0.884821 0.442410 0.896813i \(-0.354123\pi\)
0.442410 + 0.896813i \(0.354123\pi\)
\(318\) 0 0
\(319\) −16.4924 −0.923398
\(320\) 0 0
\(321\) 18.7386 1.04589
\(322\) 0 0
\(323\) −21.3693 −1.18902
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 16.6847 0.922664
\(328\) 0 0
\(329\) −38.0540 −2.09798
\(330\) 0 0
\(331\) 3.36932 0.185194 0.0925972 0.995704i \(-0.470483\pi\)
0.0925972 + 0.995704i \(0.470483\pi\)
\(332\) 0 0
\(333\) 1.50758 0.0826147
\(334\) 0 0
\(335\) −21.3693 −1.16753
\(336\) 0 0
\(337\) −36.0540 −1.96399 −0.981993 0.188919i \(-0.939501\pi\)
−0.981993 + 0.188919i \(0.939501\pi\)
\(338\) 0 0
\(339\) 12.8769 0.699377
\(340\) 0 0
\(341\) −2.24621 −0.121639
\(342\) 0 0
\(343\) 4.68466 0.252948
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −28.6847 −1.53987 −0.769937 0.638120i \(-0.779712\pi\)
−0.769937 + 0.638120i \(0.779712\pi\)
\(348\) 0 0
\(349\) 8.05398 0.431119 0.215560 0.976491i \(-0.430842\pi\)
0.215560 + 0.976491i \(0.430842\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −31.8617 −1.69583 −0.847915 0.530133i \(-0.822142\pi\)
−0.847915 + 0.530133i \(0.822142\pi\)
\(354\) 0 0
\(355\) −38.0540 −2.01970
\(356\) 0 0
\(357\) 19.8078 1.04834
\(358\) 0 0
\(359\) −7.36932 −0.388938 −0.194469 0.980909i \(-0.562298\pi\)
−0.194469 + 0.980909i \(0.562298\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 0 0
\(363\) 10.9309 0.573722
\(364\) 0 0
\(365\) −35.6155 −1.86420
\(366\) 0 0
\(367\) −2.63068 −0.137321 −0.0686603 0.997640i \(-0.521872\pi\)
−0.0686603 + 0.997640i \(0.521872\pi\)
\(368\) 0 0
\(369\) −0.630683 −0.0328321
\(370\) 0 0
\(371\) 46.7386 2.42655
\(372\) 0 0
\(373\) 11.3693 0.588681 0.294340 0.955701i \(-0.404900\pi\)
0.294340 + 0.955701i \(0.404900\pi\)
\(374\) 0 0
\(375\) −14.9309 −0.771027
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −25.1231 −1.29049 −0.645244 0.763977i \(-0.723244\pi\)
−0.645244 + 0.763977i \(0.723244\pi\)
\(380\) 0 0
\(381\) 22.2462 1.13971
\(382\) 0 0
\(383\) −10.6847 −0.545961 −0.272980 0.962020i \(-0.588009\pi\)
−0.272980 + 0.962020i \(0.588009\pi\)
\(384\) 0 0
\(385\) 25.3693 1.29294
\(386\) 0 0
\(387\) −6.63068 −0.337057
\(388\) 0 0
\(389\) 3.75379 0.190325 0.0951623 0.995462i \(-0.469663\pi\)
0.0951623 + 0.995462i \(0.469663\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −5.17708 −0.261149
\(394\) 0 0
\(395\) 42.7386 2.15041
\(396\) 0 0
\(397\) 6.00000 0.301131 0.150566 0.988600i \(-0.451890\pi\)
0.150566 + 0.988600i \(0.451890\pi\)
\(398\) 0 0
\(399\) −33.3693 −1.67056
\(400\) 0 0
\(401\) −22.8769 −1.14242 −0.571209 0.820805i \(-0.693525\pi\)
−0.571209 + 0.820805i \(0.693525\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −24.9309 −1.23882
\(406\) 0 0
\(407\) 5.36932 0.266147
\(408\) 0 0
\(409\) 31.3693 1.55111 0.775556 0.631278i \(-0.217469\pi\)
0.775556 + 0.631278i \(0.217469\pi\)
\(410\) 0 0
\(411\) 20.4924 1.01082
\(412\) 0 0
\(413\) −21.3693 −1.05152
\(414\) 0 0
\(415\) −26.2462 −1.28838
\(416\) 0 0
\(417\) 7.31534 0.358234
\(418\) 0 0
\(419\) −22.0540 −1.07741 −0.538704 0.842495i \(-0.681086\pi\)
−0.538704 + 0.842495i \(0.681086\pi\)
\(420\) 0 0
\(421\) 21.3153 1.03885 0.519423 0.854517i \(-0.326147\pi\)
0.519423 + 0.854517i \(0.326147\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 27.3693 1.32761
\(426\) 0 0
\(427\) 40.4924 1.95957
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.68466 0.321989 0.160994 0.986955i \(-0.448530\pi\)
0.160994 + 0.986955i \(0.448530\pi\)
\(432\) 0 0
\(433\) 26.6847 1.28238 0.641191 0.767381i \(-0.278440\pi\)
0.641191 + 0.767381i \(0.278440\pi\)
\(434\) 0 0
\(435\) −45.8617 −2.19890
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −0.384472 −0.0183498 −0.00917492 0.999958i \(-0.502921\pi\)
−0.00917492 + 0.999958i \(0.502921\pi\)
\(440\) 0 0
\(441\) −3.19224 −0.152011
\(442\) 0 0
\(443\) 4.68466 0.222575 0.111287 0.993788i \(-0.464503\pi\)
0.111287 + 0.993788i \(0.464503\pi\)
\(444\) 0 0
\(445\) −29.3693 −1.39224
\(446\) 0 0
\(447\) 12.8769 0.609056
\(448\) 0 0
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −2.24621 −0.105770
\(452\) 0 0
\(453\) 35.4233 1.66433
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 14.0000 0.654892 0.327446 0.944870i \(-0.393812\pi\)
0.327446 + 0.944870i \(0.393812\pi\)
\(458\) 0 0
\(459\) 19.8078 0.924547
\(460\) 0 0
\(461\) 41.8078 1.94718 0.973591 0.228300i \(-0.0733167\pi\)
0.973591 + 0.228300i \(0.0733167\pi\)
\(462\) 0 0
\(463\) 17.6155 0.818663 0.409332 0.912386i \(-0.365762\pi\)
0.409332 + 0.912386i \(0.365762\pi\)
\(464\) 0 0
\(465\) −6.24621 −0.289661
\(466\) 0 0
\(467\) −34.7386 −1.60751 −0.803756 0.594959i \(-0.797168\pi\)
−0.803756 + 0.594959i \(0.797168\pi\)
\(468\) 0 0
\(469\) 21.3693 0.986743
\(470\) 0 0
\(471\) −3.12311 −0.143905
\(472\) 0 0
\(473\) −23.6155 −1.08584
\(474\) 0 0
\(475\) −46.1080 −2.11558
\(476\) 0 0
\(477\) 7.36932 0.337418
\(478\) 0 0
\(479\) −21.3153 −0.973923 −0.486961 0.873423i \(-0.661895\pi\)
−0.486961 + 0.873423i \(0.661895\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 21.3693 0.970331
\(486\) 0 0
\(487\) 10.8769 0.492879 0.246440 0.969158i \(-0.420739\pi\)
0.246440 + 0.969158i \(0.420739\pi\)
\(488\) 0 0
\(489\) −24.0000 −1.08532
\(490\) 0 0
\(491\) 23.3153 1.05221 0.526103 0.850421i \(-0.323653\pi\)
0.526103 + 0.850421i \(0.323653\pi\)
\(492\) 0 0
\(493\) 29.3693 1.32273
\(494\) 0 0
\(495\) 4.00000 0.179787
\(496\) 0 0
\(497\) 38.0540 1.70695
\(498\) 0 0
\(499\) −25.1231 −1.12466 −0.562332 0.826911i \(-0.690096\pi\)
−0.562332 + 0.826911i \(0.690096\pi\)
\(500\) 0 0
\(501\) 24.0000 1.07224
\(502\) 0 0
\(503\) −29.3693 −1.30951 −0.654757 0.755840i \(-0.727229\pi\)
−0.654757 + 0.755840i \(0.727229\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −22.4924 −0.996959 −0.498480 0.866901i \(-0.666108\pi\)
−0.498480 + 0.866901i \(0.666108\pi\)
\(510\) 0 0
\(511\) 35.6155 1.57554
\(512\) 0 0
\(513\) −33.3693 −1.47329
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −21.3693 −0.939821
\(518\) 0 0
\(519\) 5.26137 0.230948
\(520\) 0 0
\(521\) 8.43845 0.369695 0.184848 0.982767i \(-0.440821\pi\)
0.184848 + 0.982767i \(0.440821\pi\)
\(522\) 0 0
\(523\) −7.50758 −0.328283 −0.164142 0.986437i \(-0.552485\pi\)
−0.164142 + 0.986437i \(0.552485\pi\)
\(524\) 0 0
\(525\) 42.7386 1.86527
\(526\) 0 0
\(527\) 4.00000 0.174243
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) −3.36932 −0.146216
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −42.7386 −1.84775
\(536\) 0 0
\(537\) −28.1922 −1.21658
\(538\) 0 0
\(539\) −11.3693 −0.489711
\(540\) 0 0
\(541\) 30.6847 1.31924 0.659618 0.751601i \(-0.270718\pi\)
0.659618 + 0.751601i \(0.270718\pi\)
\(542\) 0 0
\(543\) 11.5076 0.493837
\(544\) 0 0
\(545\) −38.0540 −1.63005
\(546\) 0 0
\(547\) −14.0540 −0.600905 −0.300452 0.953797i \(-0.597138\pi\)
−0.300452 + 0.953797i \(0.597138\pi\)
\(548\) 0 0
\(549\) 6.38447 0.272483
\(550\) 0 0
\(551\) −49.4773 −2.10780
\(552\) 0 0
\(553\) −42.7386 −1.81743
\(554\) 0 0
\(555\) 14.9309 0.633780
\(556\) 0 0
\(557\) −10.3002 −0.436433 −0.218216 0.975900i \(-0.570024\pi\)
−0.218216 + 0.975900i \(0.570024\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 11.1231 0.469618
\(562\) 0 0
\(563\) 31.3153 1.31978 0.659892 0.751360i \(-0.270602\pi\)
0.659892 + 0.751360i \(0.270602\pi\)
\(564\) 0 0
\(565\) −29.3693 −1.23558
\(566\) 0 0
\(567\) 24.9309 1.04700
\(568\) 0 0
\(569\) −13.3153 −0.558208 −0.279104 0.960261i \(-0.590037\pi\)
−0.279104 + 0.960261i \(0.590037\pi\)
\(570\) 0 0
\(571\) −21.5616 −0.902323 −0.451161 0.892442i \(-0.648990\pi\)
−0.451161 + 0.892442i \(0.648990\pi\)
\(572\) 0 0
\(573\) −8.38447 −0.350266
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −44.7386 −1.86249 −0.931247 0.364389i \(-0.881278\pi\)
−0.931247 + 0.364389i \(0.881278\pi\)
\(578\) 0 0
\(579\) −36.4924 −1.51657
\(580\) 0 0
\(581\) 26.2462 1.08888
\(582\) 0 0
\(583\) 26.2462 1.08701
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −15.3693 −0.634360 −0.317180 0.948365i \(-0.602736\pi\)
−0.317180 + 0.948365i \(0.602736\pi\)
\(588\) 0 0
\(589\) −6.73863 −0.277661
\(590\) 0 0
\(591\) 2.05398 0.0844893
\(592\) 0 0
\(593\) −3.75379 −0.154150 −0.0770748 0.997025i \(-0.524558\pi\)
−0.0770748 + 0.997025i \(0.524558\pi\)
\(594\) 0 0
\(595\) −45.1771 −1.85208
\(596\) 0 0
\(597\) 14.6307 0.598794
\(598\) 0 0
\(599\) 10.6307 0.434358 0.217179 0.976132i \(-0.430314\pi\)
0.217179 + 0.976132i \(0.430314\pi\)
\(600\) 0 0
\(601\) 16.0540 0.654855 0.327428 0.944876i \(-0.393818\pi\)
0.327428 + 0.944876i \(0.393818\pi\)
\(602\) 0 0
\(603\) 3.36932 0.137209
\(604\) 0 0
\(605\) −24.9309 −1.01358
\(606\) 0 0
\(607\) −25.8617 −1.04970 −0.524848 0.851196i \(-0.675878\pi\)
−0.524848 + 0.851196i \(0.675878\pi\)
\(608\) 0 0
\(609\) 45.8617 1.85841
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −10.0000 −0.403896 −0.201948 0.979396i \(-0.564727\pi\)
−0.201948 + 0.979396i \(0.564727\pi\)
\(614\) 0 0
\(615\) −6.24621 −0.251872
\(616\) 0 0
\(617\) −17.6155 −0.709174 −0.354587 0.935023i \(-0.615379\pi\)
−0.354587 + 0.935023i \(0.615379\pi\)
\(618\) 0 0
\(619\) 15.3693 0.617745 0.308873 0.951103i \(-0.400048\pi\)
0.308873 + 0.951103i \(0.400048\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 29.3693 1.17666
\(624\) 0 0
\(625\) −4.36932 −0.174773
\(626\) 0 0
\(627\) −18.7386 −0.748349
\(628\) 0 0
\(629\) −9.56155 −0.381244
\(630\) 0 0
\(631\) −3.94602 −0.157089 −0.0785444 0.996911i \(-0.525027\pi\)
−0.0785444 + 0.996911i \(0.525027\pi\)
\(632\) 0 0
\(633\) −37.1771 −1.47766
\(634\) 0 0
\(635\) −50.7386 −2.01350
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 6.00000 0.237356
\(640\) 0 0
\(641\) −22.4924 −0.888397 −0.444199 0.895928i \(-0.646512\pi\)
−0.444199 + 0.895928i \(0.646512\pi\)
\(642\) 0 0
\(643\) 18.0000 0.709851 0.354925 0.934895i \(-0.384506\pi\)
0.354925 + 0.934895i \(0.384506\pi\)
\(644\) 0 0
\(645\) −65.6695 −2.58573
\(646\) 0 0
\(647\) 44.1080 1.73406 0.867031 0.498254i \(-0.166025\pi\)
0.867031 + 0.498254i \(0.166025\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 6.24621 0.244808
\(652\) 0 0
\(653\) −6.38447 −0.249844 −0.124922 0.992167i \(-0.539868\pi\)
−0.124922 + 0.992167i \(0.539868\pi\)
\(654\) 0 0
\(655\) 11.8078 0.461368
\(656\) 0 0
\(657\) 5.61553 0.219083
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −28.7386 −1.11780 −0.558902 0.829234i \(-0.688777\pi\)
−0.558902 + 0.829234i \(0.688777\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 76.1080 2.95134
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 9.66950 0.373845
\(670\) 0 0
\(671\) 22.7386 0.877815
\(672\) 0 0
\(673\) 16.0540 0.618835 0.309418 0.950926i \(-0.399866\pi\)
0.309418 + 0.950926i \(0.399866\pi\)
\(674\) 0 0
\(675\) 42.7386 1.64501
\(676\) 0 0
\(677\) −3.36932 −0.129493 −0.0647467 0.997902i \(-0.520624\pi\)
−0.0647467 + 0.997902i \(0.520624\pi\)
\(678\) 0 0
\(679\) −21.3693 −0.820079
\(680\) 0 0
\(681\) −5.26137 −0.201616
\(682\) 0 0
\(683\) −19.3693 −0.741146 −0.370573 0.928803i \(-0.620839\pi\)
−0.370573 + 0.928803i \(0.620839\pi\)
\(684\) 0 0
\(685\) −46.7386 −1.78579
\(686\) 0 0
\(687\) 2.05398 0.0783640
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 13.1231 0.499226 0.249613 0.968346i \(-0.419697\pi\)
0.249613 + 0.968346i \(0.419697\pi\)
\(692\) 0 0
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) −16.6847 −0.632885
\(696\) 0 0
\(697\) 4.00000 0.151511
\(698\) 0 0
\(699\) −2.05398 −0.0776885
\(700\) 0 0
\(701\) −15.3693 −0.580491 −0.290246 0.956952i \(-0.593737\pi\)
−0.290246 + 0.956952i \(0.593737\pi\)
\(702\) 0 0
\(703\) 16.1080 0.607523
\(704\) 0 0
\(705\) −59.4233 −2.23801
\(706\) 0 0
\(707\) −4.00000 −0.150435
\(708\) 0 0
\(709\) 32.7386 1.22953 0.614763 0.788712i \(-0.289252\pi\)
0.614763 + 0.788712i \(0.289252\pi\)
\(710\) 0 0
\(711\) −6.73863 −0.252719
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 14.5464 0.543245
\(718\) 0 0
\(719\) 9.36932 0.349417 0.174708 0.984620i \(-0.444102\pi\)
0.174708 + 0.984620i \(0.444102\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 3.12311 0.116150
\(724\) 0 0
\(725\) 63.3693 2.35348
\(726\) 0 0
\(727\) −0.384472 −0.0142593 −0.00712964 0.999975i \(-0.502269\pi\)
−0.00712964 + 0.999975i \(0.502269\pi\)
\(728\) 0 0
\(729\) 29.9848 1.11055
\(730\) 0 0
\(731\) 42.0540 1.55542
\(732\) 0 0
\(733\) −12.0540 −0.445224 −0.222612 0.974907i \(-0.571458\pi\)
−0.222612 + 0.974907i \(0.571458\pi\)
\(734\) 0 0
\(735\) −31.6155 −1.16616
\(736\) 0 0
\(737\) 12.0000 0.442026
\(738\) 0 0
\(739\) −22.1080 −0.813254 −0.406627 0.913594i \(-0.633295\pi\)
−0.406627 + 0.913594i \(0.633295\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.31534 −0.0482552 −0.0241276 0.999709i \(-0.507681\pi\)
−0.0241276 + 0.999709i \(0.507681\pi\)
\(744\) 0 0
\(745\) −29.3693 −1.07601
\(746\) 0 0
\(747\) 4.13826 0.151411
\(748\) 0 0
\(749\) 42.7386 1.56164
\(750\) 0 0
\(751\) −9.75379 −0.355921 −0.177960 0.984038i \(-0.556950\pi\)
−0.177960 + 0.984038i \(0.556950\pi\)
\(752\) 0 0
\(753\) 24.9848 0.910498
\(754\) 0 0
\(755\) −80.7926 −2.94034
\(756\) 0 0
\(757\) 15.3693 0.558607 0.279304 0.960203i \(-0.409896\pi\)
0.279304 + 0.960203i \(0.409896\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 38.0540 1.37765
\(764\) 0 0
\(765\) −7.12311 −0.257536
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 15.3693 0.554232 0.277116 0.960837i \(-0.410621\pi\)
0.277116 + 0.960837i \(0.410621\pi\)
\(770\) 0 0
\(771\) 20.1922 0.727206
\(772\) 0 0
\(773\) 13.3153 0.478920 0.239460 0.970906i \(-0.423030\pi\)
0.239460 + 0.970906i \(0.423030\pi\)
\(774\) 0 0
\(775\) 8.63068 0.310023
\(776\) 0 0
\(777\) −14.9309 −0.535642
\(778\) 0 0
\(779\) −6.73863 −0.241437
\(780\) 0 0
\(781\) 21.3693 0.764654
\(782\) 0 0
\(783\) 45.8617 1.63896
\(784\) 0 0
\(785\) 7.12311 0.254235
\(786\) 0 0
\(787\) 5.61553 0.200172 0.100086 0.994979i \(-0.468088\pi\)
0.100086 + 0.994979i \(0.468088\pi\)
\(788\) 0 0
\(789\) 35.5076 1.26410
\(790\) 0 0
\(791\) 29.3693 1.04425
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 72.9848 2.58851
\(796\) 0 0
\(797\) 36.7386 1.30135 0.650675 0.759357i \(-0.274486\pi\)
0.650675 + 0.759357i \(0.274486\pi\)
\(798\) 0 0
\(799\) 38.0540 1.34625
\(800\) 0 0
\(801\) 4.63068 0.163617
\(802\) 0 0
\(803\) 20.0000 0.705785
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −5.26137 −0.185209
\(808\) 0 0
\(809\) −11.0691 −0.389170 −0.194585 0.980886i \(-0.562336\pi\)
−0.194585 + 0.980886i \(0.562336\pi\)
\(810\) 0 0
\(811\) 31.8617 1.11882 0.559408 0.828892i \(-0.311028\pi\)
0.559408 + 0.828892i \(0.311028\pi\)
\(812\) 0 0
\(813\) −16.6847 −0.585157
\(814\) 0 0
\(815\) 54.7386 1.91741
\(816\) 0 0
\(817\) −70.8466 −2.47861
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 32.0540 1.11869 0.559346 0.828934i \(-0.311052\pi\)
0.559346 + 0.828934i \(0.311052\pi\)
\(822\) 0 0
\(823\) −14.6307 −0.509994 −0.254997 0.966942i \(-0.582074\pi\)
−0.254997 + 0.966942i \(0.582074\pi\)
\(824\) 0 0
\(825\) 24.0000 0.835573
\(826\) 0 0
\(827\) −12.7386 −0.442966 −0.221483 0.975164i \(-0.571090\pi\)
−0.221483 + 0.975164i \(0.571090\pi\)
\(828\) 0 0
\(829\) 16.7386 0.581357 0.290678 0.956821i \(-0.406119\pi\)
0.290678 + 0.956821i \(0.406119\pi\)
\(830\) 0 0
\(831\) −36.4924 −1.26591
\(832\) 0 0
\(833\) 20.2462 0.701490
\(834\) 0 0
\(835\) −54.7386 −1.89431
\(836\) 0 0
\(837\) 6.24621 0.215901
\(838\) 0 0
\(839\) 38.1080 1.31563 0.657816 0.753178i \(-0.271480\pi\)
0.657816 + 0.753178i \(0.271480\pi\)
\(840\) 0 0
\(841\) 39.0000 1.34483
\(842\) 0 0
\(843\) 5.86174 0.201889
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 24.9309 0.856635
\(848\) 0 0
\(849\) 22.2462 0.763488
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −9.31534 −0.318951 −0.159476 0.987202i \(-0.550980\pi\)
−0.159476 + 0.987202i \(0.550980\pi\)
\(854\) 0 0
\(855\) 12.0000 0.410391
\(856\) 0 0
\(857\) −15.7538 −0.538139 −0.269070 0.963121i \(-0.586716\pi\)
−0.269070 + 0.963121i \(0.586716\pi\)
\(858\) 0 0
\(859\) −36.0000 −1.22830 −0.614152 0.789188i \(-0.710502\pi\)
−0.614152 + 0.789188i \(0.710502\pi\)
\(860\) 0 0
\(861\) 6.24621 0.212870
\(862\) 0 0
\(863\) −12.0540 −0.410322 −0.205161 0.978728i \(-0.565772\pi\)
−0.205161 + 0.978728i \(0.565772\pi\)
\(864\) 0 0
\(865\) −12.0000 −0.408012
\(866\) 0 0
\(867\) 6.73863 0.228856
\(868\) 0 0
\(869\) −24.0000 −0.814144
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −3.36932 −0.114034
\(874\) 0 0
\(875\) −34.0540 −1.15123
\(876\) 0 0
\(877\) −54.7926 −1.85021 −0.925107 0.379705i \(-0.876025\pi\)
−0.925107 + 0.379705i \(0.876025\pi\)
\(878\) 0 0
\(879\) −43.0388 −1.45166
\(880\) 0 0
\(881\) 34.3002 1.15560 0.577801 0.816177i \(-0.303911\pi\)
0.577801 + 0.816177i \(0.303911\pi\)
\(882\) 0 0
\(883\) −21.5616 −0.725604 −0.362802 0.931866i \(-0.618180\pi\)
−0.362802 + 0.931866i \(0.618180\pi\)
\(884\) 0 0
\(885\) −33.3693 −1.12170
\(886\) 0 0
\(887\) 12.0000 0.402921 0.201460 0.979497i \(-0.435431\pi\)
0.201460 + 0.979497i \(0.435431\pi\)
\(888\) 0 0
\(889\) 50.7386 1.70172
\(890\) 0 0
\(891\) 14.0000 0.469018
\(892\) 0 0
\(893\) −64.1080 −2.14529
\(894\) 0 0
\(895\) 64.3002 2.14932
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 9.26137 0.308884
\(900\) 0 0
\(901\) −46.7386 −1.55709
\(902\) 0 0
\(903\) 65.6695 2.18534
\(904\) 0 0
\(905\) −26.2462 −0.872454
\(906\) 0 0
\(907\) −35.4233 −1.17621 −0.588106 0.808784i \(-0.700126\pi\)
−0.588106 + 0.808784i \(0.700126\pi\)
\(908\) 0 0
\(909\) −0.630683 −0.0209184
\(910\) 0 0
\(911\) −2.63068 −0.0871584 −0.0435792 0.999050i \(-0.513876\pi\)
−0.0435792 + 0.999050i \(0.513876\pi\)
\(912\) 0 0
\(913\) 14.7386 0.487778
\(914\) 0 0
\(915\) 63.2311 2.09035
\(916\) 0 0
\(917\) −11.8078 −0.389927
\(918\) 0 0
\(919\) 50.2462 1.65747 0.828735 0.559642i \(-0.189061\pi\)
0.828735 + 0.559642i \(0.189061\pi\)
\(920\) 0 0
\(921\) −9.36932 −0.308729
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −20.6307 −0.678333
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 5.61553 0.184240 0.0921198 0.995748i \(-0.470636\pi\)
0.0921198 + 0.995748i \(0.470636\pi\)
\(930\) 0 0
\(931\) −34.1080 −1.11784
\(932\) 0 0
\(933\) −14.6307 −0.478987
\(934\) 0 0
\(935\) −25.3693 −0.829665
\(936\) 0 0
\(937\) −8.73863 −0.285479 −0.142739 0.989760i \(-0.545591\pi\)
−0.142739 + 0.989760i \(0.545591\pi\)
\(938\) 0 0
\(939\) 31.3153 1.02194
\(940\) 0 0
\(941\) −25.3153 −0.825257 −0.412628 0.910900i \(-0.635389\pi\)
−0.412628 + 0.910900i \(0.635389\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −70.5464 −2.29487
\(946\) 0 0
\(947\) −2.00000 −0.0649913 −0.0324956 0.999472i \(-0.510346\pi\)
−0.0324956 + 0.999472i \(0.510346\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −24.6004 −0.797722
\(952\) 0 0
\(953\) −41.8078 −1.35429 −0.677143 0.735851i \(-0.736782\pi\)
−0.677143 + 0.735851i \(0.736782\pi\)
\(954\) 0 0
\(955\) 19.1231 0.618809
\(956\) 0 0
\(957\) 25.7538 0.832502
\(958\) 0 0
\(959\) 46.7386 1.50927
\(960\) 0 0
\(961\) −29.7386 −0.959311
\(962\) 0 0
\(963\) 6.73863 0.217149
\(964\) 0 0
\(965\) 83.2311 2.67930
\(966\) 0 0
\(967\) −53.4233 −1.71798 −0.858989 0.511995i \(-0.828907\pi\)
−0.858989 + 0.511995i \(0.828907\pi\)
\(968\) 0 0
\(969\) 33.3693 1.07198
\(970\) 0 0
\(971\) −3.31534 −0.106394 −0.0531972 0.998584i \(-0.516941\pi\)
−0.0531972 + 0.998584i \(0.516941\pi\)
\(972\) 0 0
\(973\) 16.6847 0.534886
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 14.9848 0.479408 0.239704 0.970846i \(-0.422950\pi\)
0.239704 + 0.970846i \(0.422950\pi\)
\(978\) 0 0
\(979\) 16.4924 0.527100
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) 0 0
\(983\) 21.4233 0.683297 0.341648 0.939828i \(-0.389015\pi\)
0.341648 + 0.939828i \(0.389015\pi\)
\(984\) 0 0
\(985\) −4.68466 −0.149266
\(986\) 0 0
\(987\) 59.4233 1.89146
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −21.3693 −0.678819 −0.339409 0.940639i \(-0.610227\pi\)
−0.339409 + 0.940639i \(0.610227\pi\)
\(992\) 0 0
\(993\) −5.26137 −0.166964
\(994\) 0 0
\(995\) −33.3693 −1.05788
\(996\) 0 0
\(997\) −4.73863 −0.150074 −0.0750370 0.997181i \(-0.523907\pi\)
−0.0750370 + 0.997181i \(0.523907\pi\)
\(998\) 0 0
\(999\) −14.9309 −0.472392
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 5408.2.a.bd.1.1 2
4.3 odd 2 5408.2.a.p.1.2 2
13.12 even 2 416.2.a.e.1.1 yes 2
39.38 odd 2 3744.2.a.y.1.2 2
52.51 odd 2 416.2.a.c.1.2 2
104.51 odd 2 832.2.a.o.1.1 2
104.77 even 2 832.2.a.l.1.2 2
156.155 even 2 3744.2.a.x.1.2 2
208.51 odd 4 3328.2.b.ba.1665.3 4
208.77 even 4 3328.2.b.u.1665.2 4
208.155 odd 4 3328.2.b.ba.1665.2 4
208.181 even 4 3328.2.b.u.1665.3 4
312.77 odd 2 7488.2.a.cf.1.1 2
312.155 even 2 7488.2.a.ce.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
416.2.a.c.1.2 2 52.51 odd 2
416.2.a.e.1.1 yes 2 13.12 even 2
832.2.a.l.1.2 2 104.77 even 2
832.2.a.o.1.1 2 104.51 odd 2
3328.2.b.u.1665.2 4 208.77 even 4
3328.2.b.u.1665.3 4 208.181 even 4
3328.2.b.ba.1665.2 4 208.155 odd 4
3328.2.b.ba.1665.3 4 208.51 odd 4
3744.2.a.x.1.2 2 156.155 even 2
3744.2.a.y.1.2 2 39.38 odd 2
5408.2.a.p.1.2 2 4.3 odd 2
5408.2.a.bd.1.1 2 1.1 even 1 trivial
7488.2.a.ce.1.1 2 312.155 even 2
7488.2.a.cf.1.1 2 312.77 odd 2