Properties

Label 3040.2.a.q.1.1
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $0$
Dimension $4$
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: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.78292.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 10x^{2} + 8x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.68461\) 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)
 
\(f(q)\) \(=\) \(q-2.68461 q^{3} -1.00000 q^{5} +3.20715 q^{7} +4.20715 q^{9} +O(q^{10})\) \(q-2.68461 q^{3} -1.00000 q^{5} +3.20715 q^{7} +4.20715 q^{9} +6.44787 q^{11} +3.76325 q^{13} +2.68461 q^{15} +5.65501 q^{17} -1.00000 q^{19} -8.60995 q^{21} +7.20715 q^{23} +1.00000 q^{25} -3.24072 q^{27} -2.12851 q^{29} +10.8957 q^{31} -17.3100 q^{33} -3.20715 q^{35} -1.89176 q^{37} -10.1029 q^{39} -6.48144 q^{41} -5.49293 q^{43} -4.20715 q^{45} -2.86216 q^{47} +3.28579 q^{49} -15.1815 q^{51} -3.60597 q^{53} -6.44787 q^{55} +2.68461 q^{57} +10.7672 q^{59} -6.44787 q^{61} +13.4929 q^{63} -3.76325 q^{65} +6.62541 q^{67} -19.3484 q^{69} +4.41429 q^{71} +13.1815 q^{73} -2.68461 q^{75} +20.6793 q^{77} -9.78352 q^{79} -3.92136 q^{81} -1.49293 q^{83} -5.65501 q^{85} +5.71421 q^{87} -3.94080 q^{89} +12.0693 q^{91} -29.2508 q^{93} +1.00000 q^{95} -5.09411 q^{97} +27.1271 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{3} - 4 q^{5} + 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{3} - 4 q^{5} + 5 q^{7} + 9 q^{9} + 6 q^{11} + 5 q^{13} + q^{15} - 5 q^{17} - 4 q^{19} - 3 q^{21} + 21 q^{23} + 4 q^{25} - q^{27} - q^{29} + 4 q^{31} - 14 q^{33} - 5 q^{35} + 10 q^{37} + 7 q^{39} - 2 q^{41} - 6 q^{43} - 9 q^{45} + 24 q^{47} + 5 q^{49} - 13 q^{51} - 5 q^{53} - 6 q^{55} + q^{57} + 11 q^{59} - 6 q^{61} + 38 q^{63} - 5 q^{65} - 19 q^{67} - 7 q^{69} + 2 q^{71} + 5 q^{73} - q^{75} + 8 q^{77} - 4 q^{79} - 16 q^{81} + 10 q^{83} + 5 q^{85} + 31 q^{87} + 20 q^{89} + 5 q^{91} - 26 q^{93} + 4 q^{95} - 6 q^{97} + 14 q^{99}+O(q^{100}) \) 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
\(3\) −2.68461 −1.54996 −0.774981 0.631985i \(-0.782241\pi\)
−0.774981 + 0.631985i \(0.782241\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.20715 1.21219 0.606094 0.795393i \(-0.292736\pi\)
0.606094 + 0.795393i \(0.292736\pi\)
\(8\) 0 0
\(9\) 4.20715 1.40238
\(10\) 0 0
\(11\) 6.44787 1.94410 0.972052 0.234764i \(-0.0754316\pi\)
0.972052 + 0.234764i \(0.0754316\pi\)
\(12\) 0 0
\(13\) 3.76325 1.04374 0.521869 0.853025i \(-0.325235\pi\)
0.521869 + 0.853025i \(0.325235\pi\)
\(14\) 0 0
\(15\) 2.68461 0.693164
\(16\) 0 0
\(17\) 5.65501 1.37154 0.685771 0.727817i \(-0.259465\pi\)
0.685771 + 0.727817i \(0.259465\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −8.60995 −1.87884
\(22\) 0 0
\(23\) 7.20715 1.50279 0.751397 0.659850i \(-0.229380\pi\)
0.751397 + 0.659850i \(0.229380\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.24072 −0.623677
\(28\) 0 0
\(29\) −2.12851 −0.395254 −0.197627 0.980277i \(-0.563323\pi\)
−0.197627 + 0.980277i \(0.563323\pi\)
\(30\) 0 0
\(31\) 10.8957 1.95693 0.978466 0.206409i \(-0.0661778\pi\)
0.978466 + 0.206409i \(0.0661778\pi\)
\(32\) 0 0
\(33\) −17.3100 −3.01329
\(34\) 0 0
\(35\) −3.20715 −0.542107
\(36\) 0 0
\(37\) −1.89176 −0.311003 −0.155502 0.987836i \(-0.549699\pi\)
−0.155502 + 0.987836i \(0.549699\pi\)
\(38\) 0 0
\(39\) −10.1029 −1.61776
\(40\) 0 0
\(41\) −6.48144 −1.01223 −0.506116 0.862466i \(-0.668919\pi\)
−0.506116 + 0.862466i \(0.668919\pi\)
\(42\) 0 0
\(43\) −5.49293 −0.837665 −0.418832 0.908064i \(-0.637560\pi\)
−0.418832 + 0.908064i \(0.637560\pi\)
\(44\) 0 0
\(45\) −4.20715 −0.627164
\(46\) 0 0
\(47\) −2.86216 −0.417489 −0.208744 0.977970i \(-0.566938\pi\)
−0.208744 + 0.977970i \(0.566938\pi\)
\(48\) 0 0
\(49\) 3.28579 0.469398
\(50\) 0 0
\(51\) −15.1815 −2.12584
\(52\) 0 0
\(53\) −3.60597 −0.495318 −0.247659 0.968847i \(-0.579661\pi\)
−0.247659 + 0.968847i \(0.579661\pi\)
\(54\) 0 0
\(55\) −6.44787 −0.869430
\(56\) 0 0
\(57\) 2.68461 0.355586
\(58\) 0 0
\(59\) 10.7672 1.40177 0.700887 0.713272i \(-0.252788\pi\)
0.700887 + 0.713272i \(0.252788\pi\)
\(60\) 0 0
\(61\) −6.44787 −0.825565 −0.412782 0.910830i \(-0.635443\pi\)
−0.412782 + 0.910830i \(0.635443\pi\)
\(62\) 0 0
\(63\) 13.4929 1.69995
\(64\) 0 0
\(65\) −3.76325 −0.466774
\(66\) 0 0
\(67\) 6.62541 0.809423 0.404712 0.914444i \(-0.367372\pi\)
0.404712 + 0.914444i \(0.367372\pi\)
\(68\) 0 0
\(69\) −19.3484 −2.32927
\(70\) 0 0
\(71\) 4.41429 0.523880 0.261940 0.965084i \(-0.415638\pi\)
0.261940 + 0.965084i \(0.415638\pi\)
\(72\) 0 0
\(73\) 13.1815 1.54278 0.771390 0.636363i \(-0.219562\pi\)
0.771390 + 0.636363i \(0.219562\pi\)
\(74\) 0 0
\(75\) −2.68461 −0.309992
\(76\) 0 0
\(77\) 20.6793 2.35662
\(78\) 0 0
\(79\) −9.78352 −1.10073 −0.550366 0.834924i \(-0.685512\pi\)
−0.550366 + 0.834924i \(0.685512\pi\)
\(80\) 0 0
\(81\) −3.92136 −0.435707
\(82\) 0 0
\(83\) −1.49293 −0.163871 −0.0819354 0.996638i \(-0.526110\pi\)
−0.0819354 + 0.996638i \(0.526110\pi\)
\(84\) 0 0
\(85\) −5.65501 −0.613372
\(86\) 0 0
\(87\) 5.71421 0.612628
\(88\) 0 0
\(89\) −3.94080 −0.417724 −0.208862 0.977945i \(-0.566976\pi\)
−0.208862 + 0.977945i \(0.566976\pi\)
\(90\) 0 0
\(91\) 12.0693 1.26521
\(92\) 0 0
\(93\) −29.2508 −3.03317
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −5.09411 −0.517228 −0.258614 0.965981i \(-0.583266\pi\)
−0.258614 + 0.965981i \(0.583266\pi\)
\(98\) 0 0
\(99\) 27.1271 2.72638
\(100\) 0 0
\(101\) −3.49293 −0.347560 −0.173780 0.984785i \(-0.555598\pi\)
−0.173780 + 0.984785i \(0.555598\pi\)
\(102\) 0 0
\(103\) 2.07467 0.204423 0.102211 0.994763i \(-0.467408\pi\)
0.102211 + 0.994763i \(0.467408\pi\)
\(104\) 0 0
\(105\) 8.60995 0.840245
\(106\) 0 0
\(107\) −6.62541 −0.640503 −0.320251 0.947333i \(-0.603767\pi\)
−0.320251 + 0.947333i \(0.603767\pi\)
\(108\) 0 0
\(109\) −17.0242 −1.63063 −0.815313 0.579020i \(-0.803435\pi\)
−0.815313 + 0.579020i \(0.803435\pi\)
\(110\) 0 0
\(111\) 5.07864 0.482043
\(112\) 0 0
\(113\) −15.6753 −1.47461 −0.737303 0.675562i \(-0.763901\pi\)
−0.737303 + 0.675562i \(0.763901\pi\)
\(114\) 0 0
\(115\) −7.20715 −0.672070
\(116\) 0 0
\(117\) 15.8326 1.46372
\(118\) 0 0
\(119\) 18.1365 1.66257
\(120\) 0 0
\(121\) 30.5750 2.77954
\(122\) 0 0
\(123\) 17.4002 1.56892
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 14.8803 1.32041 0.660205 0.751085i \(-0.270469\pi\)
0.660205 + 0.751085i \(0.270469\pi\)
\(128\) 0 0
\(129\) 14.7464 1.29835
\(130\) 0 0
\(131\) −7.85067 −0.685916 −0.342958 0.939351i \(-0.611429\pi\)
−0.342958 + 0.939351i \(0.611429\pi\)
\(132\) 0 0
\(133\) −3.20715 −0.278095
\(134\) 0 0
\(135\) 3.24072 0.278917
\(136\) 0 0
\(137\) 11.4977 0.982317 0.491159 0.871070i \(-0.336574\pi\)
0.491159 + 0.871070i \(0.336574\pi\)
\(138\) 0 0
\(139\) −7.27645 −0.617181 −0.308590 0.951195i \(-0.599857\pi\)
−0.308590 + 0.951195i \(0.599857\pi\)
\(140\) 0 0
\(141\) 7.68379 0.647092
\(142\) 0 0
\(143\) 24.2650 2.02914
\(144\) 0 0
\(145\) 2.12851 0.176763
\(146\) 0 0
\(147\) −8.82107 −0.727549
\(148\) 0 0
\(149\) −12.3577 −1.01238 −0.506192 0.862421i \(-0.668947\pi\)
−0.506192 + 0.862421i \(0.668947\pi\)
\(150\) 0 0
\(151\) −16.3551 −1.33096 −0.665479 0.746416i \(-0.731773\pi\)
−0.665479 + 0.746416i \(0.731773\pi\)
\(152\) 0 0
\(153\) 23.7915 1.92343
\(154\) 0 0
\(155\) −10.8957 −0.875166
\(156\) 0 0
\(157\) 23.4769 1.87366 0.936830 0.349784i \(-0.113745\pi\)
0.936830 + 0.349784i \(0.113745\pi\)
\(158\) 0 0
\(159\) 9.68064 0.767725
\(160\) 0 0
\(161\) 23.1144 1.82167
\(162\) 0 0
\(163\) 3.40280 0.266528 0.133264 0.991081i \(-0.457454\pi\)
0.133264 + 0.991081i \(0.457454\pi\)
\(164\) 0 0
\(165\) 17.3100 1.34758
\(166\) 0 0
\(167\) 18.0826 1.39927 0.699637 0.714498i \(-0.253345\pi\)
0.699637 + 0.714498i \(0.253345\pi\)
\(168\) 0 0
\(169\) 1.16208 0.0893907
\(170\) 0 0
\(171\) −4.20715 −0.321729
\(172\) 0 0
\(173\) 8.14877 0.619539 0.309770 0.950812i \(-0.399748\pi\)
0.309770 + 0.950812i \(0.399748\pi\)
\(174\) 0 0
\(175\) 3.20715 0.242437
\(176\) 0 0
\(177\) −28.9058 −2.17270
\(178\) 0 0
\(179\) −8.26496 −0.617752 −0.308876 0.951102i \(-0.599953\pi\)
−0.308876 + 0.951102i \(0.599953\pi\)
\(180\) 0 0
\(181\) −0.0901336 −0.00669958 −0.00334979 0.999994i \(-0.501066\pi\)
−0.00334979 + 0.999994i \(0.501066\pi\)
\(182\) 0 0
\(183\) 17.3100 1.27959
\(184\) 0 0
\(185\) 1.89176 0.139085
\(186\) 0 0
\(187\) 36.4628 2.66642
\(188\) 0 0
\(189\) −10.3935 −0.756013
\(190\) 0 0
\(191\) 0.916561 0.0663201 0.0331600 0.999450i \(-0.489443\pi\)
0.0331600 + 0.999450i \(0.489443\pi\)
\(192\) 0 0
\(193\) −17.7424 −1.27713 −0.638564 0.769569i \(-0.720471\pi\)
−0.638564 + 0.769569i \(0.720471\pi\)
\(194\) 0 0
\(195\) 10.1029 0.723482
\(196\) 0 0
\(197\) 26.6121 1.89603 0.948017 0.318220i \(-0.103085\pi\)
0.948017 + 0.318220i \(0.103085\pi\)
\(198\) 0 0
\(199\) 3.58786 0.254337 0.127168 0.991881i \(-0.459411\pi\)
0.127168 + 0.991881i \(0.459411\pi\)
\(200\) 0 0
\(201\) −17.7867 −1.25457
\(202\) 0 0
\(203\) −6.82643 −0.479121
\(204\) 0 0
\(205\) 6.48144 0.452683
\(206\) 0 0
\(207\) 30.3215 2.10749
\(208\) 0 0
\(209\) −6.44787 −0.446008
\(210\) 0 0
\(211\) −8.45266 −0.581905 −0.290953 0.956737i \(-0.593972\pi\)
−0.290953 + 0.956737i \(0.593972\pi\)
\(212\) 0 0
\(213\) −11.8507 −0.811994
\(214\) 0 0
\(215\) 5.49293 0.374615
\(216\) 0 0
\(217\) 34.9442 2.37217
\(218\) 0 0
\(219\) −35.3873 −2.39125
\(220\) 0 0
\(221\) 21.2812 1.43153
\(222\) 0 0
\(223\) −28.9191 −1.93657 −0.968285 0.249849i \(-0.919619\pi\)
−0.968285 + 0.249849i \(0.919619\pi\)
\(224\) 0 0
\(225\) 4.20715 0.280476
\(226\) 0 0
\(227\) 12.7518 0.846364 0.423182 0.906045i \(-0.360913\pi\)
0.423182 + 0.906045i \(0.360913\pi\)
\(228\) 0 0
\(229\) −6.98851 −0.461814 −0.230907 0.972976i \(-0.574169\pi\)
−0.230907 + 0.972976i \(0.574169\pi\)
\(230\) 0 0
\(231\) −55.5158 −3.65267
\(232\) 0 0
\(233\) −7.04507 −0.461538 −0.230769 0.973009i \(-0.574124\pi\)
−0.230769 + 0.973009i \(0.574124\pi\)
\(234\) 0 0
\(235\) 2.86216 0.186707
\(236\) 0 0
\(237\) 26.2650 1.70609
\(238\) 0 0
\(239\) −13.3388 −0.862815 −0.431408 0.902157i \(-0.641983\pi\)
−0.431408 + 0.902157i \(0.641983\pi\)
\(240\) 0 0
\(241\) −11.6342 −0.749424 −0.374712 0.927141i \(-0.622258\pi\)
−0.374712 + 0.927141i \(0.622258\pi\)
\(242\) 0 0
\(243\) 20.2495 1.29901
\(244\) 0 0
\(245\) −3.28579 −0.209921
\(246\) 0 0
\(247\) −3.76325 −0.239450
\(248\) 0 0
\(249\) 4.00795 0.253993
\(250\) 0 0
\(251\) 4.58117 0.289161 0.144580 0.989493i \(-0.453817\pi\)
0.144580 + 0.989493i \(0.453817\pi\)
\(252\) 0 0
\(253\) 46.4707 2.92159
\(254\) 0 0
\(255\) 15.1815 0.950704
\(256\) 0 0
\(257\) 5.67528 0.354014 0.177007 0.984210i \(-0.443358\pi\)
0.177007 + 0.984210i \(0.443358\pi\)
\(258\) 0 0
\(259\) −6.06715 −0.376994
\(260\) 0 0
\(261\) −8.95493 −0.554296
\(262\) 0 0
\(263\) −12.7049 −0.783416 −0.391708 0.920090i \(-0.628116\pi\)
−0.391708 + 0.920090i \(0.628116\pi\)
\(264\) 0 0
\(265\) 3.60597 0.221513
\(266\) 0 0
\(267\) 10.5795 0.647456
\(268\) 0 0
\(269\) 4.54064 0.276848 0.138424 0.990373i \(-0.455796\pi\)
0.138424 + 0.990373i \(0.455796\pi\)
\(270\) 0 0
\(271\) −3.62144 −0.219987 −0.109993 0.993932i \(-0.535083\pi\)
−0.109993 + 0.993932i \(0.535083\pi\)
\(272\) 0 0
\(273\) −32.4014 −1.96102
\(274\) 0 0
\(275\) 6.44787 0.388821
\(276\) 0 0
\(277\) −18.5716 −1.11586 −0.557929 0.829889i \(-0.688404\pi\)
−0.557929 + 0.829889i \(0.688404\pi\)
\(278\) 0 0
\(279\) 45.8399 2.74437
\(280\) 0 0
\(281\) −23.3100 −1.39056 −0.695280 0.718739i \(-0.744720\pi\)
−0.695280 + 0.718739i \(0.744720\pi\)
\(282\) 0 0
\(283\) −26.0741 −1.54994 −0.774972 0.631995i \(-0.782236\pi\)
−0.774972 + 0.631995i \(0.782236\pi\)
\(284\) 0 0
\(285\) −2.68461 −0.159023
\(286\) 0 0
\(287\) −20.7869 −1.22701
\(288\) 0 0
\(289\) 14.9792 0.881128
\(290\) 0 0
\(291\) 13.6757 0.801684
\(292\) 0 0
\(293\) 14.6281 0.854580 0.427290 0.904115i \(-0.359468\pi\)
0.427290 + 0.904115i \(0.359468\pi\)
\(294\) 0 0
\(295\) −10.7672 −0.626892
\(296\) 0 0
\(297\) −20.8957 −1.21249
\(298\) 0 0
\(299\) 27.1223 1.56852
\(300\) 0 0
\(301\) −17.6166 −1.01541
\(302\) 0 0
\(303\) 9.37717 0.538705
\(304\) 0 0
\(305\) 6.44787 0.369204
\(306\) 0 0
\(307\) −32.3802 −1.84803 −0.924017 0.382353i \(-0.875114\pi\)
−0.924017 + 0.382353i \(0.875114\pi\)
\(308\) 0 0
\(309\) −5.56968 −0.316848
\(310\) 0 0
\(311\) 14.6840 0.832656 0.416328 0.909214i \(-0.363317\pi\)
0.416328 + 0.909214i \(0.363317\pi\)
\(312\) 0 0
\(313\) −5.50568 −0.311199 −0.155600 0.987820i \(-0.549731\pi\)
−0.155600 + 0.987820i \(0.549731\pi\)
\(314\) 0 0
\(315\) −13.4929 −0.760241
\(316\) 0 0
\(317\) 12.6014 0.707767 0.353884 0.935290i \(-0.384861\pi\)
0.353884 + 0.935290i \(0.384861\pi\)
\(318\) 0 0
\(319\) −13.7243 −0.768414
\(320\) 0 0
\(321\) 17.7867 0.992755
\(322\) 0 0
\(323\) −5.65501 −0.314653
\(324\) 0 0
\(325\) 3.76325 0.208748
\(326\) 0 0
\(327\) 45.7035 2.52741
\(328\) 0 0
\(329\) −9.17936 −0.506075
\(330\) 0 0
\(331\) −28.1285 −1.54608 −0.773041 0.634356i \(-0.781265\pi\)
−0.773041 + 0.634356i \(0.781265\pi\)
\(332\) 0 0
\(333\) −7.95891 −0.436145
\(334\) 0 0
\(335\) −6.62541 −0.361985
\(336\) 0 0
\(337\) −12.2981 −0.669920 −0.334960 0.942232i \(-0.608723\pi\)
−0.334960 + 0.942232i \(0.608723\pi\)
\(338\) 0 0
\(339\) 42.0821 2.28558
\(340\) 0 0
\(341\) 70.2542 3.80448
\(342\) 0 0
\(343\) −11.9120 −0.643189
\(344\) 0 0
\(345\) 19.3484 1.04168
\(346\) 0 0
\(347\) 20.0149 1.07446 0.537228 0.843437i \(-0.319471\pi\)
0.537228 + 0.843437i \(0.319471\pi\)
\(348\) 0 0
\(349\) 26.6121 1.42451 0.712257 0.701919i \(-0.247673\pi\)
0.712257 + 0.701919i \(0.247673\pi\)
\(350\) 0 0
\(351\) −12.1957 −0.650956
\(352\) 0 0
\(353\) 20.4510 1.08850 0.544249 0.838924i \(-0.316815\pi\)
0.544249 + 0.838924i \(0.316815\pi\)
\(354\) 0 0
\(355\) −4.41429 −0.234286
\(356\) 0 0
\(357\) −48.6894 −2.57691
\(358\) 0 0
\(359\) 4.88299 0.257714 0.128857 0.991663i \(-0.458869\pi\)
0.128857 + 0.991663i \(0.458869\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −82.0820 −4.30819
\(364\) 0 0
\(365\) −13.1815 −0.689952
\(366\) 0 0
\(367\) −19.6086 −1.02356 −0.511779 0.859117i \(-0.671013\pi\)
−0.511779 + 0.859117i \(0.671013\pi\)
\(368\) 0 0
\(369\) −27.2684 −1.41953
\(370\) 0 0
\(371\) −11.5649 −0.600419
\(372\) 0 0
\(373\) 4.71819 0.244298 0.122149 0.992512i \(-0.461021\pi\)
0.122149 + 0.992512i \(0.461021\pi\)
\(374\) 0 0
\(375\) 2.68461 0.138633
\(376\) 0 0
\(377\) −8.01011 −0.412541
\(378\) 0 0
\(379\) −6.89358 −0.354099 −0.177050 0.984202i \(-0.556655\pi\)
−0.177050 + 0.984202i \(0.556655\pi\)
\(380\) 0 0
\(381\) −39.9478 −2.04659
\(382\) 0 0
\(383\) 18.2399 0.932015 0.466008 0.884781i \(-0.345692\pi\)
0.466008 + 0.884781i \(0.345692\pi\)
\(384\) 0 0
\(385\) −20.6793 −1.05391
\(386\) 0 0
\(387\) −23.1096 −1.17473
\(388\) 0 0
\(389\) −6.04053 −0.306267 −0.153133 0.988206i \(-0.548936\pi\)
−0.153133 + 0.988206i \(0.548936\pi\)
\(390\) 0 0
\(391\) 40.7565 2.06115
\(392\) 0 0
\(393\) 21.0760 1.06314
\(394\) 0 0
\(395\) 9.78352 0.492262
\(396\) 0 0
\(397\) −11.6166 −0.583023 −0.291511 0.956567i \(-0.594158\pi\)
−0.291511 + 0.956567i \(0.594158\pi\)
\(398\) 0 0
\(399\) 8.60995 0.431036
\(400\) 0 0
\(401\) 7.21030 0.360065 0.180033 0.983661i \(-0.442380\pi\)
0.180033 + 0.983661i \(0.442380\pi\)
\(402\) 0 0
\(403\) 41.0034 2.04253
\(404\) 0 0
\(405\) 3.92136 0.194854
\(406\) 0 0
\(407\) −12.1978 −0.604623
\(408\) 0 0
\(409\) −27.9893 −1.38398 −0.691990 0.721907i \(-0.743266\pi\)
−0.691990 + 0.721907i \(0.743266\pi\)
\(410\) 0 0
\(411\) −30.8670 −1.52255
\(412\) 0 0
\(413\) 34.5321 1.69921
\(414\) 0 0
\(415\) 1.49293 0.0732852
\(416\) 0 0
\(417\) 19.5345 0.956606
\(418\) 0 0
\(419\) −23.2278 −1.13475 −0.567377 0.823458i \(-0.692042\pi\)
−0.567377 + 0.823458i \(0.692042\pi\)
\(420\) 0 0
\(421\) −9.74515 −0.474949 −0.237475 0.971394i \(-0.576320\pi\)
−0.237475 + 0.971394i \(0.576320\pi\)
\(422\) 0 0
\(423\) −12.0415 −0.585479
\(424\) 0 0
\(425\) 5.65501 0.274308
\(426\) 0 0
\(427\) −20.6793 −1.00074
\(428\) 0 0
\(429\) −65.1420 −3.14509
\(430\) 0 0
\(431\) 7.53610 0.363001 0.181501 0.983391i \(-0.441905\pi\)
0.181501 + 0.983391i \(0.441905\pi\)
\(432\) 0 0
\(433\) −38.5480 −1.85250 −0.926250 0.376910i \(-0.876987\pi\)
−0.926250 + 0.376910i \(0.876987\pi\)
\(434\) 0 0
\(435\) −5.71421 −0.273976
\(436\) 0 0
\(437\) −7.20715 −0.344765
\(438\) 0 0
\(439\) 12.2554 0.584917 0.292458 0.956278i \(-0.405527\pi\)
0.292458 + 0.956278i \(0.405527\pi\)
\(440\) 0 0
\(441\) 13.8238 0.658276
\(442\) 0 0
\(443\) 17.9823 0.854366 0.427183 0.904165i \(-0.359506\pi\)
0.427183 + 0.904165i \(0.359506\pi\)
\(444\) 0 0
\(445\) 3.94080 0.186812
\(446\) 0 0
\(447\) 33.1757 1.56916
\(448\) 0 0
\(449\) −15.5265 −0.732741 −0.366371 0.930469i \(-0.619400\pi\)
−0.366371 + 0.930469i \(0.619400\pi\)
\(450\) 0 0
\(451\) −41.7915 −1.96788
\(452\) 0 0
\(453\) 43.9071 2.06294
\(454\) 0 0
\(455\) −12.0693 −0.565818
\(456\) 0 0
\(457\) −35.5136 −1.66126 −0.830629 0.556827i \(-0.812019\pi\)
−0.830629 + 0.556827i \(0.812019\pi\)
\(458\) 0 0
\(459\) −18.3263 −0.855399
\(460\) 0 0
\(461\) 40.7543 1.89812 0.949060 0.315097i \(-0.102037\pi\)
0.949060 + 0.315097i \(0.102037\pi\)
\(462\) 0 0
\(463\) 11.4433 0.531817 0.265908 0.963998i \(-0.414328\pi\)
0.265908 + 0.963998i \(0.414328\pi\)
\(464\) 0 0
\(465\) 29.2508 1.35647
\(466\) 0 0
\(467\) −37.6165 −1.74068 −0.870342 0.492447i \(-0.836102\pi\)
−0.870342 + 0.492447i \(0.836102\pi\)
\(468\) 0 0
\(469\) 21.2487 0.981172
\(470\) 0 0
\(471\) −63.0264 −2.90410
\(472\) 0 0
\(473\) −35.4177 −1.62851
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −15.1709 −0.694626
\(478\) 0 0
\(479\) 27.3436 1.24936 0.624680 0.780880i \(-0.285229\pi\)
0.624680 + 0.780880i \(0.285229\pi\)
\(480\) 0 0
\(481\) −7.11917 −0.324606
\(482\) 0 0
\(483\) −62.0531 −2.82352
\(484\) 0 0
\(485\) 5.09411 0.231312
\(486\) 0 0
\(487\) −1.28661 −0.0583019 −0.0291509 0.999575i \(-0.509280\pi\)
−0.0291509 + 0.999575i \(0.509280\pi\)
\(488\) 0 0
\(489\) −9.13520 −0.413108
\(490\) 0 0
\(491\) 26.6296 1.20178 0.600890 0.799332i \(-0.294813\pi\)
0.600890 + 0.799332i \(0.294813\pi\)
\(492\) 0 0
\(493\) −12.0367 −0.542107
\(494\) 0 0
\(495\) −27.1271 −1.21927
\(496\) 0 0
\(497\) 14.1573 0.635041
\(498\) 0 0
\(499\) 18.7049 0.837345 0.418673 0.908137i \(-0.362495\pi\)
0.418673 + 0.908137i \(0.362495\pi\)
\(500\) 0 0
\(501\) −48.5448 −2.16882
\(502\) 0 0
\(503\) −18.8989 −0.842660 −0.421330 0.906907i \(-0.638437\pi\)
−0.421330 + 0.906907i \(0.638437\pi\)
\(504\) 0 0
\(505\) 3.49293 0.155434
\(506\) 0 0
\(507\) −3.11973 −0.138552
\(508\) 0 0
\(509\) 1.91781 0.0850057 0.0425028 0.999096i \(-0.486467\pi\)
0.0425028 + 0.999096i \(0.486467\pi\)
\(510\) 0 0
\(511\) 42.2751 1.87014
\(512\) 0 0
\(513\) 3.24072 0.143081
\(514\) 0 0
\(515\) −2.07467 −0.0914207
\(516\) 0 0
\(517\) −18.4548 −0.811642
\(518\) 0 0
\(519\) −21.8763 −0.960263
\(520\) 0 0
\(521\) −11.3330 −0.496508 −0.248254 0.968695i \(-0.579857\pi\)
−0.248254 + 0.968695i \(0.579857\pi\)
\(522\) 0 0
\(523\) 40.0431 1.75096 0.875482 0.483251i \(-0.160544\pi\)
0.875482 + 0.483251i \(0.160544\pi\)
\(524\) 0 0
\(525\) −8.60995 −0.375769
\(526\) 0 0
\(527\) 61.6155 2.68401
\(528\) 0 0
\(529\) 28.9430 1.25839
\(530\) 0 0
\(531\) 45.2993 1.96582
\(532\) 0 0
\(533\) −24.3913 −1.05650
\(534\) 0 0
\(535\) 6.62541 0.286442
\(536\) 0 0
\(537\) 22.1882 0.957492
\(538\) 0 0
\(539\) 21.1863 0.912559
\(540\) 0 0
\(541\) −14.6131 −0.628266 −0.314133 0.949379i \(-0.601714\pi\)
−0.314133 + 0.949379i \(0.601714\pi\)
\(542\) 0 0
\(543\) 0.241974 0.0103841
\(544\) 0 0
\(545\) 17.0242 0.729238
\(546\) 0 0
\(547\) 16.8616 0.720950 0.360475 0.932769i \(-0.382615\pi\)
0.360475 + 0.932769i \(0.382615\pi\)
\(548\) 0 0
\(549\) −27.1271 −1.15776
\(550\) 0 0
\(551\) 2.12851 0.0906774
\(552\) 0 0
\(553\) −31.3772 −1.33429
\(554\) 0 0
\(555\) −5.07864 −0.215576
\(556\) 0 0
\(557\) 15.5186 0.657542 0.328771 0.944410i \(-0.393365\pi\)
0.328771 + 0.944410i \(0.393365\pi\)
\(558\) 0 0
\(559\) −20.6713 −0.874303
\(560\) 0 0
\(561\) −97.8884 −4.13285
\(562\) 0 0
\(563\) 39.2354 1.65357 0.826787 0.562516i \(-0.190166\pi\)
0.826787 + 0.562516i \(0.190166\pi\)
\(564\) 0 0
\(565\) 15.6753 0.659464
\(566\) 0 0
\(567\) −12.5764 −0.528158
\(568\) 0 0
\(569\) 11.7323 0.491842 0.245921 0.969290i \(-0.420910\pi\)
0.245921 + 0.969290i \(0.420910\pi\)
\(570\) 0 0
\(571\) −10.1237 −0.423664 −0.211832 0.977306i \(-0.567943\pi\)
−0.211832 + 0.977306i \(0.567943\pi\)
\(572\) 0 0
\(573\) −2.46061 −0.102794
\(574\) 0 0
\(575\) 7.20715 0.300559
\(576\) 0 0
\(577\) 31.1628 1.29733 0.648663 0.761076i \(-0.275328\pi\)
0.648663 + 0.761076i \(0.275328\pi\)
\(578\) 0 0
\(579\) 47.6315 1.97950
\(580\) 0 0
\(581\) −4.78806 −0.198642
\(582\) 0 0
\(583\) −23.2508 −0.962951
\(584\) 0 0
\(585\) −15.8326 −0.654596
\(586\) 0 0
\(587\) −14.2084 −0.586443 −0.293222 0.956044i \(-0.594727\pi\)
−0.293222 + 0.956044i \(0.594727\pi\)
\(588\) 0 0
\(589\) −10.8957 −0.448951
\(590\) 0 0
\(591\) −71.4432 −2.93878
\(592\) 0 0
\(593\) −40.3055 −1.65515 −0.827574 0.561357i \(-0.810280\pi\)
−0.827574 + 0.561357i \(0.810280\pi\)
\(594\) 0 0
\(595\) −18.1365 −0.743522
\(596\) 0 0
\(597\) −9.63203 −0.394213
\(598\) 0 0
\(599\) 12.7694 0.521743 0.260871 0.965374i \(-0.415990\pi\)
0.260871 + 0.965374i \(0.415990\pi\)
\(600\) 0 0
\(601\) 44.6014 1.81933 0.909664 0.415345i \(-0.136339\pi\)
0.909664 + 0.415345i \(0.136339\pi\)
\(602\) 0 0
\(603\) 27.8741 1.13512
\(604\) 0 0
\(605\) −30.5750 −1.24305
\(606\) 0 0
\(607\) 33.2044 1.34773 0.673863 0.738856i \(-0.264634\pi\)
0.673863 + 0.738856i \(0.264634\pi\)
\(608\) 0 0
\(609\) 18.3263 0.742620
\(610\) 0 0
\(611\) −10.7710 −0.435749
\(612\) 0 0
\(613\) −30.3038 −1.22396 −0.611980 0.790873i \(-0.709627\pi\)
−0.611980 + 0.790873i \(0.709627\pi\)
\(614\) 0 0
\(615\) −17.4002 −0.701642
\(616\) 0 0
\(617\) −23.5078 −0.946390 −0.473195 0.880958i \(-0.656899\pi\)
−0.473195 + 0.880958i \(0.656899\pi\)
\(618\) 0 0
\(619\) −44.9336 −1.80603 −0.903017 0.429604i \(-0.858653\pi\)
−0.903017 + 0.429604i \(0.858653\pi\)
\(620\) 0 0
\(621\) −23.3563 −0.937258
\(622\) 0 0
\(623\) −12.6387 −0.506360
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 17.3100 0.691296
\(628\) 0 0
\(629\) −10.6979 −0.426554
\(630\) 0 0
\(631\) 12.4196 0.494417 0.247208 0.968962i \(-0.420487\pi\)
0.247208 + 0.968962i \(0.420487\pi\)
\(632\) 0 0
\(633\) 22.6921 0.901931
\(634\) 0 0
\(635\) −14.8803 −0.590505
\(636\) 0 0
\(637\) 12.3653 0.489929
\(638\) 0 0
\(639\) 18.5716 0.734680
\(640\) 0 0
\(641\) −50.4345 −1.99204 −0.996022 0.0891124i \(-0.971597\pi\)
−0.996022 + 0.0891124i \(0.971597\pi\)
\(642\) 0 0
\(643\) −3.61928 −0.142731 −0.0713653 0.997450i \(-0.522736\pi\)
−0.0713653 + 0.997450i \(0.522736\pi\)
\(644\) 0 0
\(645\) −14.7464 −0.580639
\(646\) 0 0
\(647\) −14.4810 −0.569305 −0.284653 0.958631i \(-0.591878\pi\)
−0.284653 + 0.958631i \(0.591878\pi\)
\(648\) 0 0
\(649\) 69.4257 2.72520
\(650\) 0 0
\(651\) −93.8117 −3.67677
\(652\) 0 0
\(653\) 1.57776 0.0617425 0.0308712 0.999523i \(-0.490172\pi\)
0.0308712 + 0.999523i \(0.490172\pi\)
\(654\) 0 0
\(655\) 7.85067 0.306751
\(656\) 0 0
\(657\) 55.4566 2.16357
\(658\) 0 0
\(659\) 36.9341 1.43875 0.719374 0.694623i \(-0.244429\pi\)
0.719374 + 0.694623i \(0.244429\pi\)
\(660\) 0 0
\(661\) 10.5737 0.411270 0.205635 0.978629i \(-0.434074\pi\)
0.205635 + 0.978629i \(0.434074\pi\)
\(662\) 0 0
\(663\) −57.1319 −2.21882
\(664\) 0 0
\(665\) 3.20715 0.124368
\(666\) 0 0
\(667\) −15.3404 −0.593985
\(668\) 0 0
\(669\) 77.6367 3.00161
\(670\) 0 0
\(671\) −41.5750 −1.60498
\(672\) 0 0
\(673\) −2.69736 −0.103976 −0.0519878 0.998648i \(-0.516556\pi\)
−0.0519878 + 0.998648i \(0.516556\pi\)
\(674\) 0 0
\(675\) −3.24072 −0.124735
\(676\) 0 0
\(677\) 16.7341 0.643143 0.321572 0.946885i \(-0.395789\pi\)
0.321572 + 0.946885i \(0.395789\pi\)
\(678\) 0 0
\(679\) −16.3375 −0.626978
\(680\) 0 0
\(681\) −34.2335 −1.31183
\(682\) 0 0
\(683\) −11.3847 −0.435623 −0.217812 0.975991i \(-0.569892\pi\)
−0.217812 + 0.975991i \(0.569892\pi\)
\(684\) 0 0
\(685\) −11.4977 −0.439306
\(686\) 0 0
\(687\) 18.7614 0.715793
\(688\) 0 0
\(689\) −13.5702 −0.516983
\(690\) 0 0
\(691\) 9.75789 0.371208 0.185604 0.982625i \(-0.440576\pi\)
0.185604 + 0.982625i \(0.440576\pi\)
\(692\) 0 0
\(693\) 87.0006 3.30488
\(694\) 0 0
\(695\) 7.27645 0.276012
\(696\) 0 0
\(697\) −36.6526 −1.38832
\(698\) 0 0
\(699\) 18.9133 0.715366
\(700\) 0 0
\(701\) 13.0520 0.492968 0.246484 0.969147i \(-0.420725\pi\)
0.246484 + 0.969147i \(0.420725\pi\)
\(702\) 0 0
\(703\) 1.89176 0.0713490
\(704\) 0 0
\(705\) −7.68379 −0.289388
\(706\) 0 0
\(707\) −11.2023 −0.421308
\(708\) 0 0
\(709\) −9.46895 −0.355614 −0.177807 0.984065i \(-0.556900\pi\)
−0.177807 + 0.984065i \(0.556900\pi\)
\(710\) 0 0
\(711\) −41.1607 −1.54365
\(712\) 0 0
\(713\) 78.5271 2.94086
\(714\) 0 0
\(715\) −24.2650 −0.907458
\(716\) 0 0
\(717\) 35.8095 1.33733
\(718\) 0 0
\(719\) −27.3228 −1.01897 −0.509484 0.860480i \(-0.670164\pi\)
−0.509484 + 0.860480i \(0.670164\pi\)
\(720\) 0 0
\(721\) 6.65376 0.247799
\(722\) 0 0
\(723\) 31.2333 1.16158
\(724\) 0 0
\(725\) −2.12851 −0.0790507
\(726\) 0 0
\(727\) 6.16208 0.228539 0.114269 0.993450i \(-0.463547\pi\)
0.114269 + 0.993450i \(0.463547\pi\)
\(728\) 0 0
\(729\) −42.5980 −1.57770
\(730\) 0 0
\(731\) −31.0626 −1.14889
\(732\) 0 0
\(733\) 40.9804 1.51365 0.756823 0.653620i \(-0.226750\pi\)
0.756823 + 0.653620i \(0.226750\pi\)
\(734\) 0 0
\(735\) 8.82107 0.325370
\(736\) 0 0
\(737\) 42.7198 1.57360
\(738\) 0 0
\(739\) −14.8612 −0.546677 −0.273338 0.961918i \(-0.588128\pi\)
−0.273338 + 0.961918i \(0.588128\pi\)
\(740\) 0 0
\(741\) 10.1029 0.371139
\(742\) 0 0
\(743\) −4.80517 −0.176285 −0.0881423 0.996108i \(-0.528093\pi\)
−0.0881423 + 0.996108i \(0.528093\pi\)
\(744\) 0 0
\(745\) 12.3577 0.452752
\(746\) 0 0
\(747\) −6.28099 −0.229809
\(748\) 0 0
\(749\) −21.2487 −0.776409
\(750\) 0 0
\(751\) −24.9363 −0.909937 −0.454969 0.890507i \(-0.650350\pi\)
−0.454969 + 0.890507i \(0.650350\pi\)
\(752\) 0 0
\(753\) −12.2987 −0.448188
\(754\) 0 0
\(755\) 16.3551 0.595223
\(756\) 0 0
\(757\) −22.3134 −0.810996 −0.405498 0.914096i \(-0.632902\pi\)
−0.405498 + 0.914096i \(0.632902\pi\)
\(758\) 0 0
\(759\) −124.756 −4.52835
\(760\) 0 0
\(761\) −43.6064 −1.58073 −0.790365 0.612636i \(-0.790109\pi\)
−0.790365 + 0.612636i \(0.790109\pi\)
\(762\) 0 0
\(763\) −54.5992 −1.97662
\(764\) 0 0
\(765\) −23.7915 −0.860182
\(766\) 0 0
\(767\) 40.5198 1.46309
\(768\) 0 0
\(769\) 7.09039 0.255686 0.127843 0.991794i \(-0.459195\pi\)
0.127843 + 0.991794i \(0.459195\pi\)
\(770\) 0 0
\(771\) −15.2359 −0.548708
\(772\) 0 0
\(773\) −9.38949 −0.337716 −0.168858 0.985640i \(-0.554008\pi\)
−0.168858 + 0.985640i \(0.554008\pi\)
\(774\) 0 0
\(775\) 10.8957 0.391386
\(776\) 0 0
\(777\) 16.2879 0.584327
\(778\) 0 0
\(779\) 6.48144 0.232222
\(780\) 0 0
\(781\) 28.4628 1.01848
\(782\) 0 0
\(783\) 6.89789 0.246511
\(784\) 0 0
\(785\) −23.4769 −0.837927
\(786\) 0 0
\(787\) 8.54867 0.304727 0.152364 0.988325i \(-0.451312\pi\)
0.152364 + 0.988325i \(0.451312\pi\)
\(788\) 0 0
\(789\) 34.1077 1.21427
\(790\) 0 0
\(791\) −50.2729 −1.78750
\(792\) 0 0
\(793\) −24.2650 −0.861674
\(794\) 0 0
\(795\) −9.68064 −0.343337
\(796\) 0 0
\(797\) −37.5777 −1.33107 −0.665535 0.746366i \(-0.731797\pi\)
−0.665535 + 0.746366i \(0.731797\pi\)
\(798\) 0 0
\(799\) −16.1855 −0.572604
\(800\) 0 0
\(801\) −16.5795 −0.585809
\(802\) 0 0
\(803\) 84.9927 2.99933
\(804\) 0 0
\(805\) −23.1144 −0.814675
\(806\) 0 0
\(807\) −12.1899 −0.429103
\(808\) 0 0
\(809\) −18.0773 −0.635562 −0.317781 0.948164i \(-0.602938\pi\)
−0.317781 + 0.948164i \(0.602938\pi\)
\(810\) 0 0
\(811\) −6.73629 −0.236543 −0.118272 0.992981i \(-0.537735\pi\)
−0.118272 + 0.992981i \(0.537735\pi\)
\(812\) 0 0
\(813\) 9.72216 0.340971
\(814\) 0 0
\(815\) −3.40280 −0.119195
\(816\) 0 0
\(817\) 5.49293 0.192173
\(818\) 0 0
\(819\) 50.7773 1.77430
\(820\) 0 0
\(821\) 40.4803 1.41277 0.706386 0.707826i \(-0.250324\pi\)
0.706386 + 0.707826i \(0.250324\pi\)
\(822\) 0 0
\(823\) −23.6390 −0.824003 −0.412002 0.911183i \(-0.635170\pi\)
−0.412002 + 0.911183i \(0.635170\pi\)
\(824\) 0 0
\(825\) −17.3100 −0.602658
\(826\) 0 0
\(827\) 10.4276 0.362603 0.181302 0.983428i \(-0.441969\pi\)
0.181302 + 0.983428i \(0.441969\pi\)
\(828\) 0 0
\(829\) 50.1257 1.74094 0.870469 0.492223i \(-0.163815\pi\)
0.870469 + 0.492223i \(0.163815\pi\)
\(830\) 0 0
\(831\) 49.8575 1.72954
\(832\) 0 0
\(833\) 18.5812 0.643799
\(834\) 0 0
\(835\) −18.0826 −0.625775
\(836\) 0 0
\(837\) −35.3100 −1.22049
\(838\) 0 0
\(839\) −12.3321 −0.425752 −0.212876 0.977079i \(-0.568283\pi\)
−0.212876 + 0.977079i \(0.568283\pi\)
\(840\) 0 0
\(841\) −24.4695 −0.843775
\(842\) 0 0
\(843\) 62.5784 2.15531
\(844\) 0 0
\(845\) −1.16208 −0.0399767
\(846\) 0 0
\(847\) 98.0584 3.36933
\(848\) 0 0
\(849\) 69.9989 2.40236
\(850\) 0 0
\(851\) −13.6342 −0.467374
\(852\) 0 0
\(853\) 43.9205 1.50381 0.751904 0.659272i \(-0.229136\pi\)
0.751904 + 0.659272i \(0.229136\pi\)
\(854\) 0 0
\(855\) 4.20715 0.143881
\(856\) 0 0
\(857\) −30.1488 −1.02986 −0.514931 0.857232i \(-0.672183\pi\)
−0.514931 + 0.857232i \(0.672183\pi\)
\(858\) 0 0
\(859\) −52.1140 −1.77811 −0.889053 0.457804i \(-0.848636\pi\)
−0.889053 + 0.457804i \(0.848636\pi\)
\(860\) 0 0
\(861\) 55.8049 1.90182
\(862\) 0 0
\(863\) −0.166447 −0.00566592 −0.00283296 0.999996i \(-0.500902\pi\)
−0.00283296 + 0.999996i \(0.500902\pi\)
\(864\) 0 0
\(865\) −8.14877 −0.277066
\(866\) 0 0
\(867\) −40.2133 −1.36571
\(868\) 0 0
\(869\) −63.0828 −2.13994
\(870\) 0 0
\(871\) 24.9331 0.844826
\(872\) 0 0
\(873\) −21.4317 −0.725352
\(874\) 0 0
\(875\) −3.20715 −0.108421
\(876\) 0 0
\(877\) 1.66517 0.0562289 0.0281144 0.999605i \(-0.491050\pi\)
0.0281144 + 0.999605i \(0.491050\pi\)
\(878\) 0 0
\(879\) −39.2707 −1.32457
\(880\) 0 0
\(881\) −37.2419 −1.25471 −0.627355 0.778733i \(-0.715863\pi\)
−0.627355 + 0.778733i \(0.715863\pi\)
\(882\) 0 0
\(883\) 48.5961 1.63539 0.817694 0.575653i \(-0.195252\pi\)
0.817694 + 0.575653i \(0.195252\pi\)
\(884\) 0 0
\(885\) 28.9058 0.971659
\(886\) 0 0
\(887\) −12.7619 −0.428502 −0.214251 0.976779i \(-0.568731\pi\)
−0.214251 + 0.976779i \(0.568731\pi\)
\(888\) 0 0
\(889\) 47.7232 1.60058
\(890\) 0 0
\(891\) −25.2844 −0.847059
\(892\) 0 0
\(893\) 2.86216 0.0957785
\(894\) 0 0
\(895\) 8.26496 0.276267
\(896\) 0 0
\(897\) −72.8129 −2.43115
\(898\) 0 0
\(899\) −23.1916 −0.773484
\(900\) 0 0
\(901\) −20.3918 −0.679350
\(902\) 0 0
\(903\) 47.2939 1.57384
\(904\) 0 0
\(905\) 0.0901336 0.00299614
\(906\) 0 0
\(907\) −14.7438 −0.489560 −0.244780 0.969579i \(-0.578716\pi\)
−0.244780 + 0.969579i \(0.578716\pi\)
\(908\) 0 0
\(909\) −14.6953 −0.487412
\(910\) 0 0
\(911\) 27.4177 0.908389 0.454195 0.890903i \(-0.349927\pi\)
0.454195 + 0.890903i \(0.349927\pi\)
\(912\) 0 0
\(913\) −9.62624 −0.318582
\(914\) 0 0
\(915\) −17.3100 −0.572252
\(916\) 0 0
\(917\) −25.1782 −0.831459
\(918\) 0 0
\(919\) 44.8066 1.47803 0.739017 0.673687i \(-0.235290\pi\)
0.739017 + 0.673687i \(0.235290\pi\)
\(920\) 0 0
\(921\) 86.9282 2.86438
\(922\) 0 0
\(923\) 16.6121 0.546794
\(924\) 0 0
\(925\) −1.89176 −0.0622007
\(926\) 0 0
\(927\) 8.72843 0.286679
\(928\) 0 0
\(929\) 41.9375 1.37593 0.687963 0.725746i \(-0.258505\pi\)
0.687963 + 0.725746i \(0.258505\pi\)
\(930\) 0 0
\(931\) −3.28579 −0.107687
\(932\) 0 0
\(933\) −39.4210 −1.29059
\(934\) 0 0
\(935\) −36.4628 −1.19246
\(936\) 0 0
\(937\) 51.1073 1.66960 0.834801 0.550552i \(-0.185583\pi\)
0.834801 + 0.550552i \(0.185583\pi\)
\(938\) 0 0
\(939\) 14.7806 0.482347
\(940\) 0 0
\(941\) 12.8264 0.418130 0.209065 0.977902i \(-0.432958\pi\)
0.209065 + 0.977902i \(0.432958\pi\)
\(942\) 0 0
\(943\) −46.7127 −1.52117
\(944\) 0 0
\(945\) 10.3935 0.338099
\(946\) 0 0
\(947\) 24.4575 0.794761 0.397380 0.917654i \(-0.369919\pi\)
0.397380 + 0.917654i \(0.369919\pi\)
\(948\) 0 0
\(949\) 49.6054 1.61026
\(950\) 0 0
\(951\) −33.8300 −1.09701
\(952\) 0 0
\(953\) 21.4854 0.695981 0.347990 0.937498i \(-0.386864\pi\)
0.347990 + 0.937498i \(0.386864\pi\)
\(954\) 0 0
\(955\) −0.916561 −0.0296592
\(956\) 0 0
\(957\) 36.8445 1.19101
\(958\) 0 0
\(959\) 36.8749 1.19075
\(960\) 0 0
\(961\) 87.7170 2.82958
\(962\) 0 0
\(963\) −27.8741 −0.898230
\(964\) 0 0
\(965\) 17.7424 0.571149
\(966\) 0 0
\(967\) −41.4000 −1.33134 −0.665668 0.746248i \(-0.731853\pi\)
−0.665668 + 0.746248i \(0.731853\pi\)
\(968\) 0 0
\(969\) 15.1815 0.487701
\(970\) 0 0
\(971\) −21.5095 −0.690272 −0.345136 0.938553i \(-0.612167\pi\)
−0.345136 + 0.938553i \(0.612167\pi\)
\(972\) 0 0
\(973\) −23.3366 −0.748139
\(974\) 0 0
\(975\) −10.1029 −0.323551
\(976\) 0 0
\(977\) 42.0896 1.34656 0.673282 0.739385i \(-0.264884\pi\)
0.673282 + 0.739385i \(0.264884\pi\)
\(978\) 0 0
\(979\) −25.4098 −0.812099
\(980\) 0 0
\(981\) −71.6235 −2.28676
\(982\) 0 0
\(983\) 57.9968 1.84981 0.924905 0.380198i \(-0.124144\pi\)
0.924905 + 0.380198i \(0.124144\pi\)
\(984\) 0 0
\(985\) −26.6121 −0.847932
\(986\) 0 0
\(987\) 24.6430 0.784397
\(988\) 0 0
\(989\) −39.5884 −1.25884
\(990\) 0 0
\(991\) −17.1218 −0.543892 −0.271946 0.962312i \(-0.587667\pi\)
−0.271946 + 0.962312i \(0.587667\pi\)
\(992\) 0 0
\(993\) 75.5141 2.39637
\(994\) 0 0
\(995\) −3.58786 −0.113743
\(996\) 0 0
\(997\) −21.7526 −0.688911 −0.344456 0.938803i \(-0.611936\pi\)
−0.344456 + 0.938803i \(0.611936\pi\)
\(998\) 0 0
\(999\) 6.13066 0.193966
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.q.1.1 4
4.3 odd 2 3040.2.a.s.1.4 yes 4
8.3 odd 2 6080.2.a.ce.1.1 4
8.5 even 2 6080.2.a.cg.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.q.1.1 4 1.1 even 1 trivial
3040.2.a.s.1.4 yes 4 4.3 odd 2
6080.2.a.ce.1.1 4 8.3 odd 2
6080.2.a.cg.1.4 4 8.5 even 2