Properties

Label 3040.2.a.s.1.4
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(4\)
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 1.4
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 \(-0.217759\pi\)
0.774981 + 0.631985i \(0.217759\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.20715 −1.21219 −0.606094 0.795393i \(-0.707264\pi\)
−0.606094 + 0.795393i \(0.707264\pi\)
\(8\) 0 0
\(9\) 4.20715 1.40238
\(10\) 0 0
\(11\) −6.44787 −1.94410 −0.972052 0.234764i \(-0.924568\pi\)
−0.972052 + 0.234764i \(0.924568\pi\)
\(12\) 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 \(-0.770620\pi\)
−0.751397 + 0.659850i \(0.770620\pi\)
\(24\) 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 \(-0.933822\pi\)
−0.978466 + 0.206409i \(0.933822\pi\)
\(32\) 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 \(-0.362440\pi\)
0.418832 + 0.908064i \(0.362440\pi\)
\(44\) 0 0
\(45\) −4.20715 −0.627164
\(46\) 0 0
\(47\) 2.86216 0.417489 0.208744 0.977970i \(-0.433062\pi\)
0.208744 + 0.977970i \(0.433062\pi\)
\(48\) 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 \(-0.747212\pi\)
−0.700887 + 0.713272i \(0.747212\pi\)
\(60\) 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 \(-0.632628\pi\)
−0.404712 + 0.914444i \(0.632628\pi\)
\(68\) 0 0
\(69\) −19.3484 −2.32927
\(70\) 0 0
\(71\) −4.41429 −0.523880 −0.261940 0.965084i \(-0.584362\pi\)
−0.261940 + 0.965084i \(0.584362\pi\)
\(72\) 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 \(-0.314488\pi\)
0.550366 + 0.834924i \(0.314488\pi\)
\(80\) 0 0
\(81\) −3.92136 −0.435707
\(82\) 0 0
\(83\) 1.49293 0.163871 0.0819354 0.996638i \(-0.473890\pi\)
0.0819354 + 0.996638i \(0.473890\pi\)
\(84\) 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 \(-0.532592\pi\)
−0.102211 + 0.994763i \(0.532592\pi\)
\(104\) 0 0
\(105\) 8.60995 0.840245
\(106\) 0 0
\(107\) 6.62541 0.640503 0.320251 0.947333i \(-0.396233\pi\)
0.320251 + 0.947333i \(0.396233\pi\)
\(108\) 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 \(-0.729531\pi\)
−0.660205 + 0.751085i \(0.729531\pi\)
\(128\) 0 0
\(129\) 14.7464 1.29835
\(130\) 0 0
\(131\) 7.85067 0.685916 0.342958 0.939351i \(-0.388571\pi\)
0.342958 + 0.939351i \(0.388571\pi\)
\(132\) 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 \(-0.400143\pi\)
0.308590 + 0.951195i \(0.400143\pi\)
\(140\) 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 \(-0.268227\pi\)
0.665479 + 0.746416i \(0.268227\pi\)
\(152\) 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 \(-0.542546\pi\)
−0.133264 + 0.991081i \(0.542546\pi\)
\(164\) 0 0
\(165\) 17.3100 1.34758
\(166\) 0 0
\(167\) −18.0826 −1.39927 −0.699637 0.714498i \(-0.746655\pi\)
−0.699637 + 0.714498i \(0.746655\pi\)
\(168\) 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 \(-0.400047\pi\)
0.308876 + 0.951102i \(0.400047\pi\)
\(180\) 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 \(-0.510557\pi\)
−0.0331600 + 0.999450i \(0.510557\pi\)
\(192\) 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 \(-0.540589\pi\)
−0.127168 + 0.991881i \(0.540589\pi\)
\(200\) 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 \(-0.406028\pi\)
0.290953 + 0.956737i \(0.406028\pi\)
\(212\) 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 \(-0.0803811\pi\)
0.968285 + 0.249849i \(0.0803811\pi\)
\(224\) 0 0
\(225\) 4.20715 0.280476
\(226\) 0 0
\(227\) −12.7518 −0.846364 −0.423182 0.906045i \(-0.639087\pi\)
−0.423182 + 0.906045i \(0.639087\pi\)
\(228\) 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 \(-0.358017\pi\)
0.431408 + 0.902157i \(0.358017\pi\)
\(240\) 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 \(-0.546183\pi\)
−0.144580 + 0.989493i \(0.546183\pi\)
\(252\) 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 \(-0.371884\pi\)
0.391708 + 0.920090i \(0.371884\pi\)
\(264\) 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 \(-0.464917\pi\)
0.109993 + 0.993932i \(0.464917\pi\)
\(272\) 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 \(-0.217764\pi\)
0.774972 + 0.631995i \(0.217764\pi\)
\(284\) 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 \(-0.124886\pi\)
0.924017 + 0.382353i \(0.124886\pi\)
\(308\) 0 0
\(309\) −5.56968 −0.316848
\(310\) 0 0
\(311\) −14.6840 −0.832656 −0.416328 0.909214i \(-0.636683\pi\)
−0.416328 + 0.909214i \(0.636683\pi\)
\(312\) 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 \(-0.218735\pi\)
0.773041 + 0.634356i \(0.218735\pi\)
\(332\) 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 \(-0.680529\pi\)
−0.537228 + 0.843437i \(0.680529\pi\)
\(348\) 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 \(-0.541131\pi\)
−0.128857 + 0.991663i \(0.541131\pi\)
\(360\) 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 \(-0.328987\pi\)
0.511779 + 0.859117i \(0.328987\pi\)
\(368\) 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 \(-0.443345\pi\)
0.177050 + 0.984202i \(0.443345\pi\)
\(380\) 0 0
\(381\) −39.9478 −2.04659
\(382\) 0 0
\(383\) −18.2399 −0.932015 −0.466008 0.884781i \(-0.654308\pi\)
−0.466008 + 0.884781i \(0.654308\pi\)
\(384\) 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 \(-0.307958\pi\)
0.567377 + 0.823458i \(0.307958\pi\)
\(420\) 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 \(-0.558095\pi\)
−0.181501 + 0.983391i \(0.558095\pi\)
\(432\) 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 \(-0.594473\pi\)
−0.292458 + 0.956278i \(0.594473\pi\)
\(440\) 0 0
\(441\) 13.8238 0.658276
\(442\) 0 0
\(443\) −17.9823 −0.854366 −0.427183 0.904165i \(-0.640494\pi\)
−0.427183 + 0.904165i \(0.640494\pi\)
\(444\) 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 \(-0.585672\pi\)
−0.265908 + 0.963998i \(0.585672\pi\)
\(464\) 0 0
\(465\) 29.2508 1.35647
\(466\) 0 0
\(467\) 37.6165 1.74068 0.870342 0.492447i \(-0.163898\pi\)
0.870342 + 0.492447i \(0.163898\pi\)
\(468\) 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 \(-0.714771\pi\)
−0.624680 + 0.780880i \(0.714771\pi\)
\(480\) 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 \(-0.490720\pi\)
0.0291509 + 0.999575i \(0.490720\pi\)
\(488\) 0 0
\(489\) −9.13520 −0.413108
\(490\) 0 0
\(491\) −26.6296 −1.20178 −0.600890 0.799332i \(-0.705187\pi\)
−0.600890 + 0.799332i \(0.705187\pi\)
\(492\) 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 \(-0.637505\pi\)
−0.418673 + 0.908137i \(0.637505\pi\)
\(500\) 0 0
\(501\) −48.5448 −2.16882
\(502\) 0 0
\(503\) 18.8989 0.842660 0.421330 0.906907i \(-0.361563\pi\)
0.421330 + 0.906907i \(0.361563\pi\)
\(504\) 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 \(-0.839456\pi\)
−0.875482 + 0.483251i \(0.839456\pi\)
\(524\) 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 \(-0.617385\pi\)
−0.360475 + 0.932769i \(0.617385\pi\)
\(548\) 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 \(-0.809834\pi\)
−0.826787 + 0.562516i \(0.809834\pi\)
\(564\) 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 \(-0.432057\pi\)
0.211832 + 0.977306i \(0.432057\pi\)
\(572\) 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 \(-0.405273\pi\)
0.293222 + 0.956044i \(0.405273\pi\)
\(588\) 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 \(-0.584010\pi\)
−0.260871 + 0.965374i \(0.584010\pi\)
\(600\) 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 \(-0.735366\pi\)
−0.673863 + 0.738856i \(0.735366\pi\)
\(608\) 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 \(-0.141347\pi\)
0.903017 + 0.429604i \(0.141347\pi\)
\(620\) 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 \(-0.579513\pi\)
−0.247208 + 0.968962i \(0.579513\pi\)
\(632\) 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 \(-0.477264\pi\)
0.0713653 + 0.997450i \(0.477264\pi\)
\(644\) 0 0
\(645\) −14.7464 −0.580639
\(646\) 0 0
\(647\) 14.4810 0.569305 0.284653 0.958631i \(-0.408122\pi\)
0.284653 + 0.958631i \(0.408122\pi\)
\(648\) 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 \(-0.755571\pi\)
−0.719374 + 0.694623i \(0.755571\pi\)
\(660\) 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 \(-0.430108\pi\)
0.217812 + 0.975991i \(0.430108\pi\)
\(684\) 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 \(-0.559424\pi\)
−0.185604 + 0.982625i \(0.559424\pi\)
\(692\) 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 \(-0.329836\pi\)
0.509484 + 0.860480i \(0.329836\pi\)
\(720\) 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 \(-0.536453\pi\)
−0.114269 + 0.993450i \(0.536453\pi\)
\(728\) 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 \(-0.411872\pi\)
0.273338 + 0.961918i \(0.411872\pi\)
\(740\) 0 0
\(741\) 10.1029 0.371139
\(742\) 0 0
\(743\) 4.80517 0.176285 0.0881423 0.996108i \(-0.471907\pi\)
0.0881423 + 0.996108i \(0.471907\pi\)
\(744\) 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 \(-0.349650\pi\)
0.454969 + 0.890507i \(0.349650\pi\)
\(752\) 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 \(-0.548688\pi\)
−0.152364 + 0.988325i \(0.548688\pi\)
\(788\) 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 \(-0.462265\pi\)
0.118272 + 0.992981i \(0.462265\pi\)
\(812\) 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 \(-0.364830\pi\)
0.412002 + 0.911183i \(0.364830\pi\)
\(824\) 0 0
\(825\) −17.3100 −0.602658
\(826\) 0 0
\(827\) −10.4276 −0.362603 −0.181302 0.983428i \(-0.558031\pi\)
−0.181302 + 0.983428i \(0.558031\pi\)
\(828\) 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 \(-0.431717\pi\)
0.212876 + 0.977079i \(0.431717\pi\)
\(840\) 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 \(-0.151364\pi\)
0.889053 + 0.457804i \(0.151364\pi\)
\(860\) 0 0
\(861\) 55.8049 1.90182
\(862\) 0 0
\(863\) 0.166447 0.00566592 0.00283296 0.999996i \(-0.499098\pi\)
0.00283296 + 0.999996i \(0.499098\pi\)
\(864\) 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 \(-0.804748\pi\)
−0.817694 + 0.575653i \(0.804748\pi\)
\(884\) 0 0
\(885\) 28.9058 0.971659
\(886\) 0 0
\(887\) 12.7619 0.428502 0.214251 0.976779i \(-0.431269\pi\)
0.214251 + 0.976779i \(0.431269\pi\)
\(888\) 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 \(-0.421284\pi\)
0.244780 + 0.969579i \(0.421284\pi\)
\(908\) 0 0
\(909\) −14.6953 −0.487412
\(910\) 0 0
\(911\) −27.4177 −0.908389 −0.454195 0.890903i \(-0.650073\pi\)
−0.454195 + 0.890903i \(0.650073\pi\)
\(912\) 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 \(-0.764710\pi\)
−0.739017 + 0.673687i \(0.764710\pi\)
\(920\) 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 \(-0.630081\pi\)
−0.397380 + 0.917654i \(0.630081\pi\)
\(948\) 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 \(-0.268147\pi\)
0.665668 + 0.746248i \(0.268147\pi\)
\(968\) 0 0
\(969\) 15.1815 0.487701
\(970\) 0 0
\(971\) 21.5095 0.690272 0.345136 0.938553i \(-0.387833\pi\)
0.345136 + 0.938553i \(0.387833\pi\)
\(972\) 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 \(-0.875856\pi\)
−0.924905 + 0.380198i \(0.875856\pi\)
\(984\) 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 \(-0.412333\pi\)
0.271946 + 0.962312i \(0.412333\pi\)
\(992\) 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.s.1.4 yes 4
4.3 odd 2 3040.2.a.q.1.1 4
8.3 odd 2 6080.2.a.cg.1.4 4
8.5 even 2 6080.2.a.ce.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.q.1.1 4 4.3 odd 2
3040.2.a.s.1.4 yes 4 1.1 even 1 trivial
6080.2.a.ce.1.1 4 8.5 even 2
6080.2.a.cg.1.4 4 8.3 odd 2