Properties

Label 3040.2.a.x.1.5
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $5$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.387268.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.88930\) of defining polynomial
Character \(\chi\) \(=\) 3040.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.45876 q^{3} -1.00000 q^{5} -4.18444 q^{7} +8.96304 q^{9} +O(q^{10})\) \(q+3.45876 q^{3} -1.00000 q^{5} -4.18444 q^{7} +8.96304 q^{9} -1.91967 q^{11} -5.31770 q^{13} -3.45876 q^{15} -6.96519 q^{17} -1.00000 q^{19} -14.4730 q^{21} +2.55761 q^{23} +1.00000 q^{25} +20.6248 q^{27} +2.90445 q^{29} -4.26651 q^{31} -6.63968 q^{33} +4.18444 q^{35} +6.07710 q^{37} -18.3927 q^{39} -6.91753 q^{41} -3.64887 q^{43} -8.96304 q^{45} -6.29283 q^{47} +10.5095 q^{49} -24.0909 q^{51} -12.7150 q^{53} +1.91967 q^{55} -3.45876 q^{57} -5.24303 q^{59} -3.37102 q^{61} -37.5053 q^{63} +5.31770 q^{65} -8.20081 q^{67} +8.84616 q^{69} +16.3533 q^{71} -3.30133 q^{73} +3.45876 q^{75} +8.03274 q^{77} +1.02418 q^{79} +44.4470 q^{81} -4.34256 q^{83} +6.96519 q^{85} +10.0458 q^{87} -1.45564 q^{89} +22.2516 q^{91} -14.7569 q^{93} +1.00000 q^{95} -8.51561 q^{97} -17.2061 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 4 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9} + 2 q^{11} - 4 q^{13} - 4 q^{15} - 12 q^{17} - 5 q^{19} - 10 q^{21} - 8 q^{23} + 5 q^{25} + 16 q^{27} - 6 q^{29} - 10 q^{31} - 18 q^{33} + 4 q^{35} - 6 q^{37} - 18 q^{39} - 8 q^{41} + 12 q^{43} - 7 q^{45} - 16 q^{47} + 7 q^{49} - 14 q^{51} - 18 q^{53} - 2 q^{55} - 4 q^{57} + 8 q^{59} + 2 q^{61} - 36 q^{63} + 4 q^{65} + 10 q^{67} - 22 q^{69} + 18 q^{71} - 28 q^{73} + 4 q^{75} - 28 q^{77} - 14 q^{79} + 25 q^{81} + 8 q^{83} + 12 q^{85} - 24 q^{87} - 30 q^{89} + 28 q^{91} - 24 q^{93} + 5 q^{95} - 18 q^{97} - 14 q^{99}+O(q^{100}) \) 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
\(3\) 3.45876 1.99692 0.998459 0.0554941i \(-0.0176734\pi\)
0.998459 + 0.0554941i \(0.0176734\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.18444 −1.58157 −0.790785 0.612094i \(-0.790327\pi\)
−0.790785 + 0.612094i \(0.790327\pi\)
\(8\) 0 0
\(9\) 8.96304 2.98768
\(10\) 0 0
\(11\) −1.91967 −0.578802 −0.289401 0.957208i \(-0.593456\pi\)
−0.289401 + 0.957208i \(0.593456\pi\)
\(12\) 0 0
\(13\) −5.31770 −1.47486 −0.737432 0.675421i \(-0.763962\pi\)
−0.737432 + 0.675421i \(0.763962\pi\)
\(14\) 0 0
\(15\) −3.45876 −0.893049
\(16\) 0 0
\(17\) −6.96519 −1.68931 −0.844653 0.535314i \(-0.820193\pi\)
−0.844653 + 0.535314i \(0.820193\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −14.4730 −3.15827
\(22\) 0 0
\(23\) 2.55761 0.533298 0.266649 0.963794i \(-0.414084\pi\)
0.266649 + 0.963794i \(0.414084\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 20.6248 3.96924
\(28\) 0 0
\(29\) 2.90445 0.539343 0.269672 0.962952i \(-0.413085\pi\)
0.269672 + 0.962952i \(0.413085\pi\)
\(30\) 0 0
\(31\) −4.26651 −0.766289 −0.383144 0.923688i \(-0.625159\pi\)
−0.383144 + 0.923688i \(0.625159\pi\)
\(32\) 0 0
\(33\) −6.63968 −1.15582
\(34\) 0 0
\(35\) 4.18444 0.707300
\(36\) 0 0
\(37\) 6.07710 0.999070 0.499535 0.866294i \(-0.333504\pi\)
0.499535 + 0.866294i \(0.333504\pi\)
\(38\) 0 0
\(39\) −18.3927 −2.94518
\(40\) 0 0
\(41\) −6.91753 −1.08034 −0.540168 0.841557i \(-0.681639\pi\)
−0.540168 + 0.841557i \(0.681639\pi\)
\(42\) 0 0
\(43\) −3.64887 −0.556448 −0.278224 0.960516i \(-0.589746\pi\)
−0.278224 + 0.960516i \(0.589746\pi\)
\(44\) 0 0
\(45\) −8.96304 −1.33613
\(46\) 0 0
\(47\) −6.29283 −0.917904 −0.458952 0.888461i \(-0.651775\pi\)
−0.458952 + 0.888461i \(0.651775\pi\)
\(48\) 0 0
\(49\) 10.5095 1.50136
\(50\) 0 0
\(51\) −24.0909 −3.37341
\(52\) 0 0
\(53\) −12.7150 −1.74655 −0.873273 0.487232i \(-0.838007\pi\)
−0.873273 + 0.487232i \(0.838007\pi\)
\(54\) 0 0
\(55\) 1.91967 0.258848
\(56\) 0 0
\(57\) −3.45876 −0.458124
\(58\) 0 0
\(59\) −5.24303 −0.682585 −0.341292 0.939957i \(-0.610865\pi\)
−0.341292 + 0.939957i \(0.610865\pi\)
\(60\) 0 0
\(61\) −3.37102 −0.431615 −0.215808 0.976436i \(-0.569238\pi\)
−0.215808 + 0.976436i \(0.569238\pi\)
\(62\) 0 0
\(63\) −37.5053 −4.72523
\(64\) 0 0
\(65\) 5.31770 0.659579
\(66\) 0 0
\(67\) −8.20081 −1.00189 −0.500944 0.865480i \(-0.667014\pi\)
−0.500944 + 0.865480i \(0.667014\pi\)
\(68\) 0 0
\(69\) 8.84616 1.06495
\(70\) 0 0
\(71\) 16.3533 1.94078 0.970388 0.241552i \(-0.0776565\pi\)
0.970388 + 0.241552i \(0.0776565\pi\)
\(72\) 0 0
\(73\) −3.30133 −0.386391 −0.193196 0.981160i \(-0.561885\pi\)
−0.193196 + 0.981160i \(0.561885\pi\)
\(74\) 0 0
\(75\) 3.45876 0.399384
\(76\) 0 0
\(77\) 8.03274 0.915416
\(78\) 0 0
\(79\) 1.02418 0.115229 0.0576145 0.998339i \(-0.481651\pi\)
0.0576145 + 0.998339i \(0.481651\pi\)
\(80\) 0 0
\(81\) 44.4470 4.93856
\(82\) 0 0
\(83\) −4.34256 −0.476658 −0.238329 0.971184i \(-0.576600\pi\)
−0.238329 + 0.971184i \(0.576600\pi\)
\(84\) 0 0
\(85\) 6.96519 0.755481
\(86\) 0 0
\(87\) 10.0458 1.07702
\(88\) 0 0
\(89\) −1.45564 −0.154297 −0.0771487 0.997020i \(-0.524582\pi\)
−0.0771487 + 0.997020i \(0.524582\pi\)
\(90\) 0 0
\(91\) 22.2516 2.33260
\(92\) 0 0
\(93\) −14.7569 −1.53022
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −8.51561 −0.864630 −0.432315 0.901723i \(-0.642303\pi\)
−0.432315 + 0.901723i \(0.642303\pi\)
\(98\) 0 0
\(99\) −17.2061 −1.72928
\(100\) 0 0
\(101\) −2.08033 −0.207001 −0.103500 0.994629i \(-0.533004\pi\)
−0.103500 + 0.994629i \(0.533004\pi\)
\(102\) 0 0
\(103\) −14.2598 −1.40506 −0.702530 0.711654i \(-0.747946\pi\)
−0.702530 + 0.711654i \(0.747946\pi\)
\(104\) 0 0
\(105\) 14.4730 1.41242
\(106\) 0 0
\(107\) 8.01874 0.775201 0.387600 0.921827i \(-0.373304\pi\)
0.387600 + 0.921827i \(0.373304\pi\)
\(108\) 0 0
\(109\) −8.22558 −0.787867 −0.393934 0.919139i \(-0.628886\pi\)
−0.393934 + 0.919139i \(0.628886\pi\)
\(110\) 0 0
\(111\) 21.0193 1.99506
\(112\) 0 0
\(113\) 16.3280 1.53601 0.768005 0.640444i \(-0.221250\pi\)
0.768005 + 0.640444i \(0.221250\pi\)
\(114\) 0 0
\(115\) −2.55761 −0.238498
\(116\) 0 0
\(117\) −47.6628 −4.40642
\(118\) 0 0
\(119\) 29.1454 2.67176
\(120\) 0 0
\(121\) −7.31487 −0.664988
\(122\) 0 0
\(123\) −23.9261 −2.15734
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 9.89012 0.877606 0.438803 0.898583i \(-0.355403\pi\)
0.438803 + 0.898583i \(0.355403\pi\)
\(128\) 0 0
\(129\) −12.6206 −1.11118
\(130\) 0 0
\(131\) −5.98438 −0.522858 −0.261429 0.965223i \(-0.584194\pi\)
−0.261429 + 0.965223i \(0.584194\pi\)
\(132\) 0 0
\(133\) 4.18444 0.362837
\(134\) 0 0
\(135\) −20.6248 −1.77510
\(136\) 0 0
\(137\) −11.1258 −0.950545 −0.475273 0.879839i \(-0.657651\pi\)
−0.475273 + 0.879839i \(0.657651\pi\)
\(138\) 0 0
\(139\) 16.5011 1.39960 0.699801 0.714338i \(-0.253272\pi\)
0.699801 + 0.714338i \(0.253272\pi\)
\(140\) 0 0
\(141\) −21.7654 −1.83298
\(142\) 0 0
\(143\) 10.2082 0.853654
\(144\) 0 0
\(145\) −2.90445 −0.241202
\(146\) 0 0
\(147\) 36.3500 2.99810
\(148\) 0 0
\(149\) 17.7917 1.45756 0.728778 0.684750i \(-0.240088\pi\)
0.728778 + 0.684750i \(0.240088\pi\)
\(150\) 0 0
\(151\) 0.913243 0.0743187 0.0371593 0.999309i \(-0.488169\pi\)
0.0371593 + 0.999309i \(0.488169\pi\)
\(152\) 0 0
\(153\) −62.4293 −5.04711
\(154\) 0 0
\(155\) 4.26651 0.342695
\(156\) 0 0
\(157\) −2.28213 −0.182134 −0.0910669 0.995845i \(-0.529028\pi\)
−0.0910669 + 0.995845i \(0.529028\pi\)
\(158\) 0 0
\(159\) −43.9783 −3.48771
\(160\) 0 0
\(161\) −10.7022 −0.843448
\(162\) 0 0
\(163\) 2.34685 0.183819 0.0919096 0.995767i \(-0.470703\pi\)
0.0919096 + 0.995767i \(0.470703\pi\)
\(164\) 0 0
\(165\) 6.63968 0.516898
\(166\) 0 0
\(167\) 14.3130 1.10757 0.553787 0.832658i \(-0.313182\pi\)
0.553787 + 0.832658i \(0.313182\pi\)
\(168\) 0 0
\(169\) 15.2779 1.17522
\(170\) 0 0
\(171\) −8.96304 −0.685421
\(172\) 0 0
\(173\) 11.4702 0.872061 0.436030 0.899932i \(-0.356384\pi\)
0.436030 + 0.899932i \(0.356384\pi\)
\(174\) 0 0
\(175\) −4.18444 −0.316314
\(176\) 0 0
\(177\) −18.1344 −1.36307
\(178\) 0 0
\(179\) −0.494634 −0.0369707 −0.0184853 0.999829i \(-0.505884\pi\)
−0.0184853 + 0.999829i \(0.505884\pi\)
\(180\) 0 0
\(181\) 5.02418 0.373444 0.186722 0.982413i \(-0.440214\pi\)
0.186722 + 0.982413i \(0.440214\pi\)
\(182\) 0 0
\(183\) −11.6596 −0.861901
\(184\) 0 0
\(185\) −6.07710 −0.446798
\(186\) 0 0
\(187\) 13.3709 0.977774
\(188\) 0 0
\(189\) −86.3031 −6.27763
\(190\) 0 0
\(191\) 4.01736 0.290686 0.145343 0.989381i \(-0.453571\pi\)
0.145343 + 0.989381i \(0.453571\pi\)
\(192\) 0 0
\(193\) −15.3265 −1.10322 −0.551612 0.834101i \(-0.685987\pi\)
−0.551612 + 0.834101i \(0.685987\pi\)
\(194\) 0 0
\(195\) 18.3927 1.31713
\(196\) 0 0
\(197\) −2.20197 −0.156884 −0.0784420 0.996919i \(-0.524995\pi\)
−0.0784420 + 0.996919i \(0.524995\pi\)
\(198\) 0 0
\(199\) 11.4972 0.815013 0.407506 0.913202i \(-0.366398\pi\)
0.407506 + 0.913202i \(0.366398\pi\)
\(200\) 0 0
\(201\) −28.3647 −2.00069
\(202\) 0 0
\(203\) −12.1535 −0.853009
\(204\) 0 0
\(205\) 6.91753 0.483141
\(206\) 0 0
\(207\) 22.9239 1.59332
\(208\) 0 0
\(209\) 1.91967 0.132786
\(210\) 0 0
\(211\) 2.98080 0.205207 0.102603 0.994722i \(-0.467283\pi\)
0.102603 + 0.994722i \(0.467283\pi\)
\(212\) 0 0
\(213\) 56.5621 3.87557
\(214\) 0 0
\(215\) 3.64887 0.248851
\(216\) 0 0
\(217\) 17.8530 1.21194
\(218\) 0 0
\(219\) −11.4185 −0.771592
\(220\) 0 0
\(221\) 37.0388 2.49150
\(222\) 0 0
\(223\) 16.1105 1.07884 0.539419 0.842038i \(-0.318644\pi\)
0.539419 + 0.842038i \(0.318644\pi\)
\(224\) 0 0
\(225\) 8.96304 0.597536
\(226\) 0 0
\(227\) 16.7992 1.11500 0.557500 0.830177i \(-0.311760\pi\)
0.557500 + 0.830177i \(0.311760\pi\)
\(228\) 0 0
\(229\) −17.1521 −1.13344 −0.566720 0.823910i \(-0.691788\pi\)
−0.566720 + 0.823910i \(0.691788\pi\)
\(230\) 0 0
\(231\) 27.7834 1.82801
\(232\) 0 0
\(233\) −12.4713 −0.817019 −0.408509 0.912754i \(-0.633951\pi\)
−0.408509 + 0.912754i \(0.633951\pi\)
\(234\) 0 0
\(235\) 6.29283 0.410499
\(236\) 0 0
\(237\) 3.54239 0.230103
\(238\) 0 0
\(239\) 0.600913 0.0388698 0.0194349 0.999811i \(-0.493813\pi\)
0.0194349 + 0.999811i \(0.493813\pi\)
\(240\) 0 0
\(241\) 21.5643 1.38908 0.694538 0.719456i \(-0.255609\pi\)
0.694538 + 0.719456i \(0.255609\pi\)
\(242\) 0 0
\(243\) 91.8575 5.89266
\(244\) 0 0
\(245\) −10.5095 −0.671430
\(246\) 0 0
\(247\) 5.31770 0.338357
\(248\) 0 0
\(249\) −15.0199 −0.951847
\(250\) 0 0
\(251\) 6.34667 0.400598 0.200299 0.979735i \(-0.435809\pi\)
0.200299 + 0.979735i \(0.435809\pi\)
\(252\) 0 0
\(253\) −4.90976 −0.308674
\(254\) 0 0
\(255\) 24.0909 1.50863
\(256\) 0 0
\(257\) −24.1830 −1.50849 −0.754246 0.656592i \(-0.771997\pi\)
−0.754246 + 0.656592i \(0.771997\pi\)
\(258\) 0 0
\(259\) −25.4293 −1.58010
\(260\) 0 0
\(261\) 26.0327 1.61139
\(262\) 0 0
\(263\) −4.80017 −0.295991 −0.147996 0.988988i \(-0.547282\pi\)
−0.147996 + 0.988988i \(0.547282\pi\)
\(264\) 0 0
\(265\) 12.7150 0.781079
\(266\) 0 0
\(267\) −5.03471 −0.308119
\(268\) 0 0
\(269\) −4.88050 −0.297569 −0.148785 0.988870i \(-0.547536\pi\)
−0.148785 + 0.988870i \(0.547536\pi\)
\(270\) 0 0
\(271\) 1.60226 0.0973301 0.0486651 0.998815i \(-0.484503\pi\)
0.0486651 + 0.998815i \(0.484503\pi\)
\(272\) 0 0
\(273\) 76.9630 4.65801
\(274\) 0 0
\(275\) −1.91967 −0.115760
\(276\) 0 0
\(277\) 17.3012 1.03953 0.519765 0.854309i \(-0.326020\pi\)
0.519765 + 0.854309i \(0.326020\pi\)
\(278\) 0 0
\(279\) −38.2410 −2.28943
\(280\) 0 0
\(281\) −24.7440 −1.47610 −0.738052 0.674744i \(-0.764254\pi\)
−0.738052 + 0.674744i \(0.764254\pi\)
\(282\) 0 0
\(283\) 3.78687 0.225106 0.112553 0.993646i \(-0.464097\pi\)
0.112553 + 0.993646i \(0.464097\pi\)
\(284\) 0 0
\(285\) 3.45876 0.204879
\(286\) 0 0
\(287\) 28.9460 1.70863
\(288\) 0 0
\(289\) 31.5138 1.85375
\(290\) 0 0
\(291\) −29.4535 −1.72659
\(292\) 0 0
\(293\) −14.1812 −0.828475 −0.414238 0.910169i \(-0.635952\pi\)
−0.414238 + 0.910169i \(0.635952\pi\)
\(294\) 0 0
\(295\) 5.24303 0.305261
\(296\) 0 0
\(297\) −39.5927 −2.29740
\(298\) 0 0
\(299\) −13.6006 −0.786542
\(300\) 0 0
\(301\) 15.2685 0.880061
\(302\) 0 0
\(303\) −7.19537 −0.413363
\(304\) 0 0
\(305\) 3.37102 0.193024
\(306\) 0 0
\(307\) −12.9379 −0.738404 −0.369202 0.929349i \(-0.620369\pi\)
−0.369202 + 0.929349i \(0.620369\pi\)
\(308\) 0 0
\(309\) −49.3213 −2.80579
\(310\) 0 0
\(311\) −9.62875 −0.545996 −0.272998 0.962015i \(-0.588015\pi\)
−0.272998 + 0.962015i \(0.588015\pi\)
\(312\) 0 0
\(313\) −17.5614 −0.992628 −0.496314 0.868143i \(-0.665314\pi\)
−0.496314 + 0.868143i \(0.665314\pi\)
\(314\) 0 0
\(315\) 37.5053 2.11319
\(316\) 0 0
\(317\) −7.85153 −0.440986 −0.220493 0.975389i \(-0.570767\pi\)
−0.220493 + 0.975389i \(0.570767\pi\)
\(318\) 0 0
\(319\) −5.57559 −0.312173
\(320\) 0 0
\(321\) 27.7349 1.54801
\(322\) 0 0
\(323\) 6.96519 0.387553
\(324\) 0 0
\(325\) −5.31770 −0.294973
\(326\) 0 0
\(327\) −28.4503 −1.57331
\(328\) 0 0
\(329\) 26.3320 1.45173
\(330\) 0 0
\(331\) 22.3426 1.22806 0.614031 0.789282i \(-0.289547\pi\)
0.614031 + 0.789282i \(0.289547\pi\)
\(332\) 0 0
\(333\) 54.4694 2.98490
\(334\) 0 0
\(335\) 8.20081 0.448058
\(336\) 0 0
\(337\) 32.9016 1.79227 0.896133 0.443786i \(-0.146365\pi\)
0.896133 + 0.443786i \(0.146365\pi\)
\(338\) 0 0
\(339\) 56.4747 3.06728
\(340\) 0 0
\(341\) 8.19030 0.443529
\(342\) 0 0
\(343\) −14.6855 −0.792942
\(344\) 0 0
\(345\) −8.84616 −0.476261
\(346\) 0 0
\(347\) −8.38387 −0.450070 −0.225035 0.974351i \(-0.572250\pi\)
−0.225035 + 0.974351i \(0.572250\pi\)
\(348\) 0 0
\(349\) 4.05969 0.217310 0.108655 0.994080i \(-0.465346\pi\)
0.108655 + 0.994080i \(0.465346\pi\)
\(350\) 0 0
\(351\) −109.676 −5.85409
\(352\) 0 0
\(353\) −9.56784 −0.509245 −0.254622 0.967041i \(-0.581951\pi\)
−0.254622 + 0.967041i \(0.581951\pi\)
\(354\) 0 0
\(355\) −16.3533 −0.867941
\(356\) 0 0
\(357\) 100.807 5.33528
\(358\) 0 0
\(359\) −25.1347 −1.32656 −0.663280 0.748372i \(-0.730836\pi\)
−0.663280 + 0.748372i \(0.730836\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −25.3004 −1.32793
\(364\) 0 0
\(365\) 3.30133 0.172799
\(366\) 0 0
\(367\) −13.1236 −0.685047 −0.342523 0.939509i \(-0.611282\pi\)
−0.342523 + 0.939509i \(0.611282\pi\)
\(368\) 0 0
\(369\) −62.0021 −3.22770
\(370\) 0 0
\(371\) 53.2053 2.76228
\(372\) 0 0
\(373\) −19.6739 −1.01867 −0.509337 0.860567i \(-0.670109\pi\)
−0.509337 + 0.860567i \(0.670109\pi\)
\(374\) 0 0
\(375\) −3.45876 −0.178610
\(376\) 0 0
\(377\) −15.4450 −0.795458
\(378\) 0 0
\(379\) 33.9979 1.74636 0.873178 0.487401i \(-0.162055\pi\)
0.873178 + 0.487401i \(0.162055\pi\)
\(380\) 0 0
\(381\) 34.2076 1.75251
\(382\) 0 0
\(383\) −0.534885 −0.0273314 −0.0136657 0.999907i \(-0.504350\pi\)
−0.0136657 + 0.999907i \(0.504350\pi\)
\(384\) 0 0
\(385\) −8.03274 −0.409386
\(386\) 0 0
\(387\) −32.7050 −1.66249
\(388\) 0 0
\(389\) −13.2296 −0.670769 −0.335385 0.942081i \(-0.608866\pi\)
−0.335385 + 0.942081i \(0.608866\pi\)
\(390\) 0 0
\(391\) −17.8142 −0.900903
\(392\) 0 0
\(393\) −20.6986 −1.04410
\(394\) 0 0
\(395\) −1.02418 −0.0515320
\(396\) 0 0
\(397\) 11.5381 0.579081 0.289541 0.957166i \(-0.406498\pi\)
0.289541 + 0.957166i \(0.406498\pi\)
\(398\) 0 0
\(399\) 14.4730 0.724556
\(400\) 0 0
\(401\) −23.6658 −1.18181 −0.590907 0.806739i \(-0.701230\pi\)
−0.590907 + 0.806739i \(0.701230\pi\)
\(402\) 0 0
\(403\) 22.6880 1.13017
\(404\) 0 0
\(405\) −44.4470 −2.20859
\(406\) 0 0
\(407\) −11.6660 −0.578264
\(408\) 0 0
\(409\) −18.0206 −0.891062 −0.445531 0.895267i \(-0.646985\pi\)
−0.445531 + 0.895267i \(0.646985\pi\)
\(410\) 0 0
\(411\) −38.4817 −1.89816
\(412\) 0 0
\(413\) 21.9392 1.07956
\(414\) 0 0
\(415\) 4.34256 0.213168
\(416\) 0 0
\(417\) 57.0732 2.79489
\(418\) 0 0
\(419\) 18.6880 0.912970 0.456485 0.889731i \(-0.349108\pi\)
0.456485 + 0.889731i \(0.349108\pi\)
\(420\) 0 0
\(421\) −2.27530 −0.110892 −0.0554458 0.998462i \(-0.517658\pi\)
−0.0554458 + 0.998462i \(0.517658\pi\)
\(422\) 0 0
\(423\) −56.4030 −2.74241
\(424\) 0 0
\(425\) −6.96519 −0.337861
\(426\) 0 0
\(427\) 14.1059 0.682630
\(428\) 0 0
\(429\) 35.3078 1.70468
\(430\) 0 0
\(431\) 3.25578 0.156826 0.0784128 0.996921i \(-0.475015\pi\)
0.0784128 + 0.996921i \(0.475015\pi\)
\(432\) 0 0
\(433\) −0.690637 −0.0331899 −0.0165949 0.999862i \(-0.505283\pi\)
−0.0165949 + 0.999862i \(0.505283\pi\)
\(434\) 0 0
\(435\) −10.0458 −0.481660
\(436\) 0 0
\(437\) −2.55761 −0.122347
\(438\) 0 0
\(439\) −11.5529 −0.551391 −0.275695 0.961245i \(-0.588908\pi\)
−0.275695 + 0.961245i \(0.588908\pi\)
\(440\) 0 0
\(441\) 94.1975 4.48560
\(442\) 0 0
\(443\) 35.0042 1.66310 0.831551 0.555448i \(-0.187453\pi\)
0.831551 + 0.555448i \(0.187453\pi\)
\(444\) 0 0
\(445\) 1.45564 0.0690039
\(446\) 0 0
\(447\) 61.5374 2.91062
\(448\) 0 0
\(449\) −7.59989 −0.358661 −0.179330 0.983789i \(-0.557393\pi\)
−0.179330 + 0.983789i \(0.557393\pi\)
\(450\) 0 0
\(451\) 13.2794 0.625301
\(452\) 0 0
\(453\) 3.15869 0.148408
\(454\) 0 0
\(455\) −22.2516 −1.04317
\(456\) 0 0
\(457\) −4.49322 −0.210184 −0.105092 0.994463i \(-0.533514\pi\)
−0.105092 + 0.994463i \(0.533514\pi\)
\(458\) 0 0
\(459\) −143.655 −6.70526
\(460\) 0 0
\(461\) −28.3207 −1.31902 −0.659512 0.751694i \(-0.729237\pi\)
−0.659512 + 0.751694i \(0.729237\pi\)
\(462\) 0 0
\(463\) 11.3519 0.527569 0.263784 0.964582i \(-0.415029\pi\)
0.263784 + 0.964582i \(0.415029\pi\)
\(464\) 0 0
\(465\) 14.7569 0.684333
\(466\) 0 0
\(467\) −4.27998 −0.198054 −0.0990270 0.995085i \(-0.531573\pi\)
−0.0990270 + 0.995085i \(0.531573\pi\)
\(468\) 0 0
\(469\) 34.3158 1.58456
\(470\) 0 0
\(471\) −7.89335 −0.363706
\(472\) 0 0
\(473\) 7.00462 0.322073
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −113.966 −5.21812
\(478\) 0 0
\(479\) −18.9310 −0.864979 −0.432490 0.901639i \(-0.642365\pi\)
−0.432490 + 0.901639i \(0.642365\pi\)
\(480\) 0 0
\(481\) −32.3162 −1.47349
\(482\) 0 0
\(483\) −37.0162 −1.68430
\(484\) 0 0
\(485\) 8.51561 0.386674
\(486\) 0 0
\(487\) −29.3181 −1.32853 −0.664265 0.747497i \(-0.731255\pi\)
−0.664265 + 0.747497i \(0.731255\pi\)
\(488\) 0 0
\(489\) 8.11718 0.367072
\(490\) 0 0
\(491\) −3.21099 −0.144910 −0.0724550 0.997372i \(-0.523083\pi\)
−0.0724550 + 0.997372i \(0.523083\pi\)
\(492\) 0 0
\(493\) −20.2301 −0.911116
\(494\) 0 0
\(495\) 17.2061 0.773356
\(496\) 0 0
\(497\) −68.4293 −3.06947
\(498\) 0 0
\(499\) −3.85861 −0.172735 −0.0863675 0.996263i \(-0.527526\pi\)
−0.0863675 + 0.996263i \(0.527526\pi\)
\(500\) 0 0
\(501\) 49.5053 2.21174
\(502\) 0 0
\(503\) 21.4587 0.956797 0.478399 0.878143i \(-0.341217\pi\)
0.478399 + 0.878143i \(0.341217\pi\)
\(504\) 0 0
\(505\) 2.08033 0.0925735
\(506\) 0 0
\(507\) 52.8427 2.34683
\(508\) 0 0
\(509\) −13.6077 −0.603152 −0.301576 0.953442i \(-0.597513\pi\)
−0.301576 + 0.953442i \(0.597513\pi\)
\(510\) 0 0
\(511\) 13.8142 0.611105
\(512\) 0 0
\(513\) −20.6248 −0.910605
\(514\) 0 0
\(515\) 14.2598 0.628362
\(516\) 0 0
\(517\) 12.0802 0.531285
\(518\) 0 0
\(519\) 39.6726 1.74143
\(520\) 0 0
\(521\) −36.3168 −1.59107 −0.795535 0.605908i \(-0.792810\pi\)
−0.795535 + 0.605908i \(0.792810\pi\)
\(522\) 0 0
\(523\) −38.9982 −1.70527 −0.852636 0.522506i \(-0.824997\pi\)
−0.852636 + 0.522506i \(0.824997\pi\)
\(524\) 0 0
\(525\) −14.4730 −0.631653
\(526\) 0 0
\(527\) 29.7171 1.29450
\(528\) 0 0
\(529\) −16.4586 −0.715593
\(530\) 0 0
\(531\) −46.9935 −2.03935
\(532\) 0 0
\(533\) 36.7853 1.59335
\(534\) 0 0
\(535\) −8.01874 −0.346680
\(536\) 0 0
\(537\) −1.71082 −0.0738274
\(538\) 0 0
\(539\) −20.1749 −0.868992
\(540\) 0 0
\(541\) 18.8267 0.809421 0.404711 0.914445i \(-0.367372\pi\)
0.404711 + 0.914445i \(0.367372\pi\)
\(542\) 0 0
\(543\) 17.3774 0.745738
\(544\) 0 0
\(545\) 8.22558 0.352345
\(546\) 0 0
\(547\) 19.3493 0.827317 0.413659 0.910432i \(-0.364251\pi\)
0.413659 + 0.910432i \(0.364251\pi\)
\(548\) 0 0
\(549\) −30.2146 −1.28953
\(550\) 0 0
\(551\) −2.90445 −0.123734
\(552\) 0 0
\(553\) −4.28561 −0.182243
\(554\) 0 0
\(555\) −21.0193 −0.892218
\(556\) 0 0
\(557\) −29.3012 −1.24153 −0.620766 0.783996i \(-0.713178\pi\)
−0.620766 + 0.783996i \(0.713178\pi\)
\(558\) 0 0
\(559\) 19.4036 0.820685
\(560\) 0 0
\(561\) 46.2466 1.95253
\(562\) 0 0
\(563\) −20.2988 −0.855492 −0.427746 0.903899i \(-0.640692\pi\)
−0.427746 + 0.903899i \(0.640692\pi\)
\(564\) 0 0
\(565\) −16.3280 −0.686924
\(566\) 0 0
\(567\) −185.986 −7.81068
\(568\) 0 0
\(569\) −1.49266 −0.0625757 −0.0312879 0.999510i \(-0.509961\pi\)
−0.0312879 + 0.999510i \(0.509961\pi\)
\(570\) 0 0
\(571\) 23.4085 0.979615 0.489808 0.871830i \(-0.337067\pi\)
0.489808 + 0.871830i \(0.337067\pi\)
\(572\) 0 0
\(573\) 13.8951 0.580476
\(574\) 0 0
\(575\) 2.55761 0.106660
\(576\) 0 0
\(577\) −29.4279 −1.22510 −0.612549 0.790433i \(-0.709856\pi\)
−0.612549 + 0.790433i \(0.709856\pi\)
\(578\) 0 0
\(579\) −53.0107 −2.20305
\(580\) 0 0
\(581\) 18.1712 0.753868
\(582\) 0 0
\(583\) 24.4087 1.01090
\(584\) 0 0
\(585\) 47.6628 1.97061
\(586\) 0 0
\(587\) 38.3731 1.58383 0.791914 0.610632i \(-0.209085\pi\)
0.791914 + 0.610632i \(0.209085\pi\)
\(588\) 0 0
\(589\) 4.26651 0.175799
\(590\) 0 0
\(591\) −7.61610 −0.313284
\(592\) 0 0
\(593\) −12.1564 −0.499203 −0.249601 0.968349i \(-0.580300\pi\)
−0.249601 + 0.968349i \(0.580300\pi\)
\(594\) 0 0
\(595\) −29.1454 −1.19485
\(596\) 0 0
\(597\) 39.7660 1.62751
\(598\) 0 0
\(599\) −4.87208 −0.199068 −0.0995340 0.995034i \(-0.531735\pi\)
−0.0995340 + 0.995034i \(0.531735\pi\)
\(600\) 0 0
\(601\) −4.82141 −0.196669 −0.0983347 0.995153i \(-0.531352\pi\)
−0.0983347 + 0.995153i \(0.531352\pi\)
\(602\) 0 0
\(603\) −73.5042 −2.99332
\(604\) 0 0
\(605\) 7.31487 0.297392
\(606\) 0 0
\(607\) 14.9897 0.608414 0.304207 0.952606i \(-0.401609\pi\)
0.304207 + 0.952606i \(0.401609\pi\)
\(608\) 0 0
\(609\) −42.0361 −1.70339
\(610\) 0 0
\(611\) 33.4634 1.35378
\(612\) 0 0
\(613\) 11.0133 0.444823 0.222412 0.974953i \(-0.428607\pi\)
0.222412 + 0.974953i \(0.428607\pi\)
\(614\) 0 0
\(615\) 23.9261 0.964793
\(616\) 0 0
\(617\) 11.3041 0.455088 0.227544 0.973768i \(-0.426931\pi\)
0.227544 + 0.973768i \(0.426931\pi\)
\(618\) 0 0
\(619\) 15.8466 0.636927 0.318463 0.947935i \(-0.396833\pi\)
0.318463 + 0.947935i \(0.396833\pi\)
\(620\) 0 0
\(621\) 52.7500 2.11679
\(622\) 0 0
\(623\) 6.09104 0.244032
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 6.63968 0.265163
\(628\) 0 0
\(629\) −42.3282 −1.68773
\(630\) 0 0
\(631\) −22.3835 −0.891074 −0.445537 0.895263i \(-0.646987\pi\)
−0.445537 + 0.895263i \(0.646987\pi\)
\(632\) 0 0
\(633\) 10.3099 0.409781
\(634\) 0 0
\(635\) −9.89012 −0.392478
\(636\) 0 0
\(637\) −55.8866 −2.21431
\(638\) 0 0
\(639\) 146.575 5.79842
\(640\) 0 0
\(641\) 31.2552 1.23451 0.617253 0.786765i \(-0.288246\pi\)
0.617253 + 0.786765i \(0.288246\pi\)
\(642\) 0 0
\(643\) −28.6100 −1.12827 −0.564134 0.825683i \(-0.690790\pi\)
−0.564134 + 0.825683i \(0.690790\pi\)
\(644\) 0 0
\(645\) 12.6206 0.496935
\(646\) 0 0
\(647\) −17.0323 −0.669610 −0.334805 0.942287i \(-0.608671\pi\)
−0.334805 + 0.942287i \(0.608671\pi\)
\(648\) 0 0
\(649\) 10.0649 0.395081
\(650\) 0 0
\(651\) 61.7492 2.42014
\(652\) 0 0
\(653\) −27.2694 −1.06714 −0.533568 0.845757i \(-0.679149\pi\)
−0.533568 + 0.845757i \(0.679149\pi\)
\(654\) 0 0
\(655\) 5.98438 0.233829
\(656\) 0 0
\(657\) −29.5899 −1.15441
\(658\) 0 0
\(659\) −37.0311 −1.44253 −0.721264 0.692660i \(-0.756438\pi\)
−0.721264 + 0.692660i \(0.756438\pi\)
\(660\) 0 0
\(661\) 6.03237 0.234632 0.117316 0.993095i \(-0.462571\pi\)
0.117316 + 0.993095i \(0.462571\pi\)
\(662\) 0 0
\(663\) 128.108 4.97531
\(664\) 0 0
\(665\) −4.18444 −0.162266
\(666\) 0 0
\(667\) 7.42845 0.287631
\(668\) 0 0
\(669\) 55.7223 2.15435
\(670\) 0 0
\(671\) 6.47125 0.249820
\(672\) 0 0
\(673\) −37.8118 −1.45754 −0.728768 0.684761i \(-0.759907\pi\)
−0.728768 + 0.684761i \(0.759907\pi\)
\(674\) 0 0
\(675\) 20.6248 0.793847
\(676\) 0 0
\(677\) −31.0356 −1.19279 −0.596397 0.802690i \(-0.703402\pi\)
−0.596397 + 0.802690i \(0.703402\pi\)
\(678\) 0 0
\(679\) 35.6331 1.36747
\(680\) 0 0
\(681\) 58.1044 2.22657
\(682\) 0 0
\(683\) −6.68835 −0.255923 −0.127961 0.991779i \(-0.540843\pi\)
−0.127961 + 0.991779i \(0.540843\pi\)
\(684\) 0 0
\(685\) 11.1258 0.425097
\(686\) 0 0
\(687\) −59.3249 −2.26339
\(688\) 0 0
\(689\) 67.6148 2.57592
\(690\) 0 0
\(691\) 27.6759 1.05284 0.526421 0.850224i \(-0.323534\pi\)
0.526421 + 0.850224i \(0.323534\pi\)
\(692\) 0 0
\(693\) 71.9978 2.73497
\(694\) 0 0
\(695\) −16.5011 −0.625921
\(696\) 0 0
\(697\) 48.1819 1.82502
\(698\) 0 0
\(699\) −43.1351 −1.63152
\(700\) 0 0
\(701\) −10.6234 −0.401242 −0.200621 0.979669i \(-0.564296\pi\)
−0.200621 + 0.979669i \(0.564296\pi\)
\(702\) 0 0
\(703\) −6.07710 −0.229202
\(704\) 0 0
\(705\) 21.7654 0.819733
\(706\) 0 0
\(707\) 8.70502 0.327386
\(708\) 0 0
\(709\) 34.0768 1.27978 0.639891 0.768466i \(-0.278980\pi\)
0.639891 + 0.768466i \(0.278980\pi\)
\(710\) 0 0
\(711\) 9.17976 0.344268
\(712\) 0 0
\(713\) −10.9121 −0.408660
\(714\) 0 0
\(715\) −10.2082 −0.381766
\(716\) 0 0
\(717\) 2.07842 0.0776199
\(718\) 0 0
\(719\) −25.8319 −0.963367 −0.481683 0.876345i \(-0.659974\pi\)
−0.481683 + 0.876345i \(0.659974\pi\)
\(720\) 0 0
\(721\) 59.6693 2.22220
\(722\) 0 0
\(723\) 74.5857 2.77387
\(724\) 0 0
\(725\) 2.90445 0.107869
\(726\) 0 0
\(727\) −41.7046 −1.54674 −0.773369 0.633956i \(-0.781430\pi\)
−0.773369 + 0.633956i \(0.781430\pi\)
\(728\) 0 0
\(729\) 184.372 6.82860
\(730\) 0 0
\(731\) 25.4151 0.940010
\(732\) 0 0
\(733\) 13.3033 0.491370 0.245685 0.969350i \(-0.420987\pi\)
0.245685 + 0.969350i \(0.420987\pi\)
\(734\) 0 0
\(735\) −36.3500 −1.34079
\(736\) 0 0
\(737\) 15.7428 0.579895
\(738\) 0 0
\(739\) 37.7826 1.38985 0.694927 0.719080i \(-0.255437\pi\)
0.694927 + 0.719080i \(0.255437\pi\)
\(740\) 0 0
\(741\) 18.3927 0.675671
\(742\) 0 0
\(743\) 9.99052 0.366517 0.183258 0.983065i \(-0.441335\pi\)
0.183258 + 0.983065i \(0.441335\pi\)
\(744\) 0 0
\(745\) −17.7917 −0.651839
\(746\) 0 0
\(747\) −38.9226 −1.42410
\(748\) 0 0
\(749\) −33.5539 −1.22603
\(750\) 0 0
\(751\) −44.6739 −1.63017 −0.815087 0.579338i \(-0.803311\pi\)
−0.815087 + 0.579338i \(0.803311\pi\)
\(752\) 0 0
\(753\) 21.9516 0.799962
\(754\) 0 0
\(755\) −0.913243 −0.0332363
\(756\) 0 0
\(757\) 42.9608 1.56144 0.780718 0.624883i \(-0.214853\pi\)
0.780718 + 0.624883i \(0.214853\pi\)
\(758\) 0 0
\(759\) −16.9817 −0.616396
\(760\) 0 0
\(761\) −45.0314 −1.63239 −0.816194 0.577778i \(-0.803920\pi\)
−0.816194 + 0.577778i \(0.803920\pi\)
\(762\) 0 0
\(763\) 34.4194 1.24607
\(764\) 0 0
\(765\) 62.4293 2.25714
\(766\) 0 0
\(767\) 27.8809 1.00672
\(768\) 0 0
\(769\) 41.6483 1.50188 0.750938 0.660372i \(-0.229602\pi\)
0.750938 + 0.660372i \(0.229602\pi\)
\(770\) 0 0
\(771\) −83.6431 −3.01233
\(772\) 0 0
\(773\) 42.6831 1.53520 0.767602 0.640927i \(-0.221450\pi\)
0.767602 + 0.640927i \(0.221450\pi\)
\(774\) 0 0
\(775\) −4.26651 −0.153258
\(776\) 0 0
\(777\) −87.9539 −3.15533
\(778\) 0 0
\(779\) 6.91753 0.247846
\(780\) 0 0
\(781\) −31.3929 −1.12332
\(782\) 0 0
\(783\) 59.9036 2.14078
\(784\) 0 0
\(785\) 2.28213 0.0814527
\(786\) 0 0
\(787\) −16.4161 −0.585170 −0.292585 0.956240i \(-0.594515\pi\)
−0.292585 + 0.956240i \(0.594515\pi\)
\(788\) 0 0
\(789\) −16.6027 −0.591070
\(790\) 0 0
\(791\) −68.3236 −2.42931
\(792\) 0 0
\(793\) 17.9261 0.636574
\(794\) 0 0
\(795\) 43.9783 1.55975
\(796\) 0 0
\(797\) 6.04841 0.214246 0.107123 0.994246i \(-0.465836\pi\)
0.107123 + 0.994246i \(0.465836\pi\)
\(798\) 0 0
\(799\) 43.8308 1.55062
\(800\) 0 0
\(801\) −13.0470 −0.460992
\(802\) 0 0
\(803\) 6.33746 0.223644
\(804\) 0 0
\(805\) 10.7022 0.377201
\(806\) 0 0
\(807\) −16.8805 −0.594222
\(808\) 0 0
\(809\) −40.1166 −1.41043 −0.705213 0.708996i \(-0.749149\pi\)
−0.705213 + 0.708996i \(0.749149\pi\)
\(810\) 0 0
\(811\) −50.6027 −1.77690 −0.888450 0.458973i \(-0.848218\pi\)
−0.888450 + 0.458973i \(0.848218\pi\)
\(812\) 0 0
\(813\) 5.54182 0.194360
\(814\) 0 0
\(815\) −2.34685 −0.0822064
\(816\) 0 0
\(817\) 3.64887 0.127658
\(818\) 0 0
\(819\) 199.442 6.96907
\(820\) 0 0
\(821\) −7.91047 −0.276077 −0.138039 0.990427i \(-0.544080\pi\)
−0.138039 + 0.990427i \(0.544080\pi\)
\(822\) 0 0
\(823\) 18.5164 0.645442 0.322721 0.946494i \(-0.395402\pi\)
0.322721 + 0.946494i \(0.395402\pi\)
\(824\) 0 0
\(825\) −6.63968 −0.231164
\(826\) 0 0
\(827\) −16.9938 −0.590931 −0.295466 0.955353i \(-0.595475\pi\)
−0.295466 + 0.955353i \(0.595475\pi\)
\(828\) 0 0
\(829\) 17.7915 0.617926 0.308963 0.951074i \(-0.400018\pi\)
0.308963 + 0.951074i \(0.400018\pi\)
\(830\) 0 0
\(831\) 59.8408 2.07586
\(832\) 0 0
\(833\) −73.2010 −2.53626
\(834\) 0 0
\(835\) −14.3130 −0.495322
\(836\) 0 0
\(837\) −87.9958 −3.04158
\(838\) 0 0
\(839\) −42.6250 −1.47158 −0.735789 0.677211i \(-0.763189\pi\)
−0.735789 + 0.677211i \(0.763189\pi\)
\(840\) 0 0
\(841\) −20.5642 −0.709109
\(842\) 0 0
\(843\) −85.5837 −2.94766
\(844\) 0 0
\(845\) −15.2779 −0.525576
\(846\) 0 0
\(847\) 30.6086 1.05173
\(848\) 0 0
\(849\) 13.0979 0.449518
\(850\) 0 0
\(851\) 15.5428 0.532802
\(852\) 0 0
\(853\) −9.19246 −0.314744 −0.157372 0.987539i \(-0.550302\pi\)
−0.157372 + 0.987539i \(0.550302\pi\)
\(854\) 0 0
\(855\) 8.96304 0.306530
\(856\) 0 0
\(857\) 4.17610 0.142653 0.0713265 0.997453i \(-0.477277\pi\)
0.0713265 + 0.997453i \(0.477277\pi\)
\(858\) 0 0
\(859\) 26.0354 0.888317 0.444159 0.895948i \(-0.353503\pi\)
0.444159 + 0.895948i \(0.353503\pi\)
\(860\) 0 0
\(861\) 100.117 3.41199
\(862\) 0 0
\(863\) −34.0880 −1.16037 −0.580185 0.814485i \(-0.697020\pi\)
−0.580185 + 0.814485i \(0.697020\pi\)
\(864\) 0 0
\(865\) −11.4702 −0.389997
\(866\) 0 0
\(867\) 108.999 3.70180
\(868\) 0 0
\(869\) −1.96608 −0.0666948
\(870\) 0 0
\(871\) 43.6094 1.47765
\(872\) 0 0
\(873\) −76.3258 −2.58324
\(874\) 0 0
\(875\) 4.18444 0.141460
\(876\) 0 0
\(877\) −17.2442 −0.582297 −0.291148 0.956678i \(-0.594037\pi\)
−0.291148 + 0.956678i \(0.594037\pi\)
\(878\) 0 0
\(879\) −49.0495 −1.65440
\(880\) 0 0
\(881\) 1.50151 0.0505873 0.0252937 0.999680i \(-0.491948\pi\)
0.0252937 + 0.999680i \(0.491948\pi\)
\(882\) 0 0
\(883\) 9.28230 0.312374 0.156187 0.987727i \(-0.450080\pi\)
0.156187 + 0.987727i \(0.450080\pi\)
\(884\) 0 0
\(885\) 18.1344 0.609582
\(886\) 0 0
\(887\) 26.3714 0.885467 0.442733 0.896653i \(-0.354009\pi\)
0.442733 + 0.896653i \(0.354009\pi\)
\(888\) 0 0
\(889\) −41.3846 −1.38800
\(890\) 0 0
\(891\) −85.3236 −2.85845
\(892\) 0 0
\(893\) 6.29283 0.210582
\(894\) 0 0
\(895\) 0.494634 0.0165338
\(896\) 0 0
\(897\) −47.0412 −1.57066
\(898\) 0 0
\(899\) −12.3919 −0.413293
\(900\) 0 0
\(901\) 88.5626 2.95045
\(902\) 0 0
\(903\) 52.8101 1.75741
\(904\) 0 0
\(905\) −5.02418 −0.167009
\(906\) 0 0
\(907\) 15.6537 0.519772 0.259886 0.965639i \(-0.416315\pi\)
0.259886 + 0.965639i \(0.416315\pi\)
\(908\) 0 0
\(909\) −18.6461 −0.618452
\(910\) 0 0
\(911\) −55.5024 −1.83888 −0.919438 0.393235i \(-0.871356\pi\)
−0.919438 + 0.393235i \(0.871356\pi\)
\(912\) 0 0
\(913\) 8.33628 0.275891
\(914\) 0 0
\(915\) 11.6596 0.385454
\(916\) 0 0
\(917\) 25.0413 0.826937
\(918\) 0 0
\(919\) 49.9719 1.64842 0.824211 0.566284i \(-0.191619\pi\)
0.824211 + 0.566284i \(0.191619\pi\)
\(920\) 0 0
\(921\) −44.7491 −1.47453
\(922\) 0 0
\(923\) −86.9617 −2.86238
\(924\) 0 0
\(925\) 6.07710 0.199814
\(926\) 0 0
\(927\) −127.811 −4.19787
\(928\) 0 0
\(929\) 4.21840 0.138401 0.0692006 0.997603i \(-0.477955\pi\)
0.0692006 + 0.997603i \(0.477955\pi\)
\(930\) 0 0
\(931\) −10.5095 −0.344437
\(932\) 0 0
\(933\) −33.3036 −1.09031
\(934\) 0 0
\(935\) −13.3709 −0.437274
\(936\) 0 0
\(937\) 7.31403 0.238939 0.119469 0.992838i \(-0.461881\pi\)
0.119469 + 0.992838i \(0.461881\pi\)
\(938\) 0 0
\(939\) −60.7407 −1.98220
\(940\) 0 0
\(941\) −47.3976 −1.54512 −0.772558 0.634944i \(-0.781023\pi\)
−0.772558 + 0.634944i \(0.781023\pi\)
\(942\) 0 0
\(943\) −17.6923 −0.576141
\(944\) 0 0
\(945\) 86.3031 2.80744
\(946\) 0 0
\(947\) −34.0009 −1.10488 −0.552440 0.833553i \(-0.686303\pi\)
−0.552440 + 0.833553i \(0.686303\pi\)
\(948\) 0 0
\(949\) 17.5555 0.569875
\(950\) 0 0
\(951\) −27.1566 −0.880612
\(952\) 0 0
\(953\) −56.6039 −1.83358 −0.916790 0.399370i \(-0.869229\pi\)
−0.916790 + 0.399370i \(0.869229\pi\)
\(954\) 0 0
\(955\) −4.01736 −0.129999
\(956\) 0 0
\(957\) −19.2846 −0.623384
\(958\) 0 0
\(959\) 46.5555 1.50335
\(960\) 0 0
\(961\) −12.7969 −0.412802
\(962\) 0 0
\(963\) 71.8723 2.31605
\(964\) 0 0
\(965\) 15.3265 0.493377
\(966\) 0 0
\(967\) −33.7007 −1.08374 −0.541871 0.840462i \(-0.682284\pi\)
−0.541871 + 0.840462i \(0.682284\pi\)
\(968\) 0 0
\(969\) 24.0909 0.773912
\(970\) 0 0
\(971\) 14.4952 0.465173 0.232587 0.972576i \(-0.425281\pi\)
0.232587 + 0.972576i \(0.425281\pi\)
\(972\) 0 0
\(973\) −69.0477 −2.21357
\(974\) 0 0
\(975\) −18.3927 −0.589037
\(976\) 0 0
\(977\) 8.67759 0.277621 0.138810 0.990319i \(-0.455672\pi\)
0.138810 + 0.990319i \(0.455672\pi\)
\(978\) 0 0
\(979\) 2.79434 0.0893077
\(980\) 0 0
\(981\) −73.7262 −2.35390
\(982\) 0 0
\(983\) −15.5754 −0.496778 −0.248389 0.968660i \(-0.579901\pi\)
−0.248389 + 0.968660i \(0.579901\pi\)
\(984\) 0 0
\(985\) 2.20197 0.0701606
\(986\) 0 0
\(987\) 91.0762 2.89899
\(988\) 0 0
\(989\) −9.33238 −0.296752
\(990\) 0 0
\(991\) 35.5941 1.13068 0.565342 0.824857i \(-0.308744\pi\)
0.565342 + 0.824857i \(0.308744\pi\)
\(992\) 0 0
\(993\) 77.2779 2.45234
\(994\) 0 0
\(995\) −11.4972 −0.364485
\(996\) 0 0
\(997\) 32.2203 1.02043 0.510213 0.860048i \(-0.329567\pi\)
0.510213 + 0.860048i \(0.329567\pi\)
\(998\) 0 0
\(999\) 125.339 3.96555
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 3040.2.a.x.1.5 yes 5
4.3 odd 2 3040.2.a.u.1.1 5
8.3 odd 2 6080.2.a.cl.1.5 5
8.5 even 2 6080.2.a.ci.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.u.1.1 5 4.3 odd 2
3040.2.a.x.1.5 yes 5 1.1 even 1 trivial
6080.2.a.ci.1.1 5 8.5 even 2
6080.2.a.cl.1.5 5 8.3 odd 2