Properties

Label 3040.2.a.x.1.4
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 1.4
Root \(-1.50372\) 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)
 
\(f(q)\) \(=\) \(q+1.76491 q^{3} -1.00000 q^{5} +3.89255 q^{7} +0.114895 q^{9} +O(q^{10})\) \(q+1.76491 q^{3} -1.00000 q^{5} +3.89255 q^{7} +0.114895 q^{9} -4.80767 q^{11} +0.0353128 q^{13} -1.76491 q^{15} -4.39275 q^{17} -1.00000 q^{19} +6.86999 q^{21} -9.12195 q^{23} +1.00000 q^{25} -5.09194 q^{27} -6.21514 q^{29} -9.71448 q^{31} -8.48508 q^{33} -3.89255 q^{35} -0.749158 q^{37} +0.0623238 q^{39} -3.52981 q^{41} +11.4625 q^{43} -0.114895 q^{45} -5.57827 q^{47} +8.15196 q^{49} -7.75279 q^{51} +8.67877 q^{53} +4.80767 q^{55} -1.76491 q^{57} +2.56252 q^{59} +6.50725 q^{61} +0.447233 q^{63} -0.0353128 q^{65} +13.2496 q^{67} -16.0994 q^{69} -1.67106 q^{71} -11.3217 q^{73} +1.76491 q^{75} -18.7141 q^{77} -11.4141 q^{79} -9.33148 q^{81} +5.64889 q^{83} +4.39275 q^{85} -10.9691 q^{87} -1.24079 q^{89} +0.137457 q^{91} -17.1451 q^{93} +1.00000 q^{95} +9.31976 q^{97} -0.552375 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 4 q^{3} - 5 q^{5} - 4 q^{7} + 7 q^{9}+O(q^{10}) \) 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
\(3\) 1.76491 1.01897 0.509485 0.860480i \(-0.329836\pi\)
0.509485 + 0.860480i \(0.329836\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 3.89255 1.47125 0.735623 0.677391i \(-0.236889\pi\)
0.735623 + 0.677391i \(0.236889\pi\)
\(8\) 0 0
\(9\) 0.114895 0.0382982
\(10\) 0 0
\(11\) −4.80767 −1.44957 −0.724783 0.688977i \(-0.758060\pi\)
−0.724783 + 0.688977i \(0.758060\pi\)
\(12\) 0 0
\(13\) 0.0353128 0.00979401 0.00489701 0.999988i \(-0.498441\pi\)
0.00489701 + 0.999988i \(0.498441\pi\)
\(14\) 0 0
\(15\) −1.76491 −0.455697
\(16\) 0 0
\(17\) −4.39275 −1.06540 −0.532699 0.846305i \(-0.678822\pi\)
−0.532699 + 0.846305i \(0.678822\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 6.86999 1.49915
\(22\) 0 0
\(23\) −9.12195 −1.90206 −0.951029 0.309102i \(-0.899972\pi\)
−0.951029 + 0.309102i \(0.899972\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.09194 −0.979944
\(28\) 0 0
\(29\) −6.21514 −1.15412 −0.577061 0.816701i \(-0.695801\pi\)
−0.577061 + 0.816701i \(0.695801\pi\)
\(30\) 0 0
\(31\) −9.71448 −1.74477 −0.872386 0.488818i \(-0.837428\pi\)
−0.872386 + 0.488818i \(0.837428\pi\)
\(32\) 0 0
\(33\) −8.48508 −1.47706
\(34\) 0 0
\(35\) −3.89255 −0.657961
\(36\) 0 0
\(37\) −0.749158 −0.123161 −0.0615804 0.998102i \(-0.519614\pi\)
−0.0615804 + 0.998102i \(0.519614\pi\)
\(38\) 0 0
\(39\) 0.0623238 0.00997980
\(40\) 0 0
\(41\) −3.52981 −0.551264 −0.275632 0.961263i \(-0.588887\pi\)
−0.275632 + 0.961263i \(0.588887\pi\)
\(42\) 0 0
\(43\) 11.4625 1.74802 0.874009 0.485910i \(-0.161512\pi\)
0.874009 + 0.485910i \(0.161512\pi\)
\(44\) 0 0
\(45\) −0.114895 −0.0171275
\(46\) 0 0
\(47\) −5.57827 −0.813674 −0.406837 0.913501i \(-0.633368\pi\)
−0.406837 + 0.913501i \(0.633368\pi\)
\(48\) 0 0
\(49\) 8.15196 1.16457
\(50\) 0 0
\(51\) −7.75279 −1.08561
\(52\) 0 0
\(53\) 8.67877 1.19212 0.596061 0.802939i \(-0.296732\pi\)
0.596061 + 0.802939i \(0.296732\pi\)
\(54\) 0 0
\(55\) 4.80767 0.648265
\(56\) 0 0
\(57\) −1.76491 −0.233768
\(58\) 0 0
\(59\) 2.56252 0.333612 0.166806 0.985990i \(-0.446655\pi\)
0.166806 + 0.985990i \(0.446655\pi\)
\(60\) 0 0
\(61\) 6.50725 0.833168 0.416584 0.909097i \(-0.363227\pi\)
0.416584 + 0.909097i \(0.363227\pi\)
\(62\) 0 0
\(63\) 0.447233 0.0563461
\(64\) 0 0
\(65\) −0.0353128 −0.00438002
\(66\) 0 0
\(67\) 13.2496 1.61870 0.809348 0.587330i \(-0.199821\pi\)
0.809348 + 0.587330i \(0.199821\pi\)
\(68\) 0 0
\(69\) −16.0994 −1.93814
\(70\) 0 0
\(71\) −1.67106 −0.198319 −0.0991594 0.995072i \(-0.531615\pi\)
−0.0991594 + 0.995072i \(0.531615\pi\)
\(72\) 0 0
\(73\) −11.3217 −1.32511 −0.662554 0.749014i \(-0.730528\pi\)
−0.662554 + 0.749014i \(0.730528\pi\)
\(74\) 0 0
\(75\) 1.76491 0.203794
\(76\) 0 0
\(77\) −18.7141 −2.13267
\(78\) 0 0
\(79\) −11.4141 −1.28418 −0.642091 0.766628i \(-0.721933\pi\)
−0.642091 + 0.766628i \(0.721933\pi\)
\(80\) 0 0
\(81\) −9.33148 −1.03683
\(82\) 0 0
\(83\) 5.64889 0.620047 0.310023 0.950729i \(-0.399663\pi\)
0.310023 + 0.950729i \(0.399663\pi\)
\(84\) 0 0
\(85\) 4.39275 0.476460
\(86\) 0 0
\(87\) −10.9691 −1.17601
\(88\) 0 0
\(89\) −1.24079 −0.131523 −0.0657617 0.997835i \(-0.520948\pi\)
−0.0657617 + 0.997835i \(0.520948\pi\)
\(90\) 0 0
\(91\) 0.137457 0.0144094
\(92\) 0 0
\(93\) −17.1451 −1.77787
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 9.31976 0.946278 0.473139 0.880988i \(-0.343121\pi\)
0.473139 + 0.880988i \(0.343121\pi\)
\(98\) 0 0
\(99\) −0.552375 −0.0555158
\(100\) 0 0
\(101\) 0.807666 0.0803657 0.0401829 0.999192i \(-0.487206\pi\)
0.0401829 + 0.999192i \(0.487206\pi\)
\(102\) 0 0
\(103\) 17.0564 1.68062 0.840309 0.542108i \(-0.182374\pi\)
0.840309 + 0.542108i \(0.182374\pi\)
\(104\) 0 0
\(105\) −6.86999 −0.670442
\(106\) 0 0
\(107\) 8.41008 0.813033 0.406516 0.913643i \(-0.366743\pi\)
0.406516 + 0.913643i \(0.366743\pi\)
\(108\) 0 0
\(109\) −16.1309 −1.54506 −0.772529 0.634979i \(-0.781009\pi\)
−0.772529 + 0.634979i \(0.781009\pi\)
\(110\) 0 0
\(111\) −1.32219 −0.125497
\(112\) 0 0
\(113\) 13.0794 1.23040 0.615201 0.788370i \(-0.289075\pi\)
0.615201 + 0.788370i \(0.289075\pi\)
\(114\) 0 0
\(115\) 9.12195 0.850626
\(116\) 0 0
\(117\) 0.00405725 0.000375093 0
\(118\) 0 0
\(119\) −17.0990 −1.56746
\(120\) 0 0
\(121\) 12.1136 1.10124
\(122\) 0 0
\(123\) −6.22979 −0.561721
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 15.0024 1.33125 0.665623 0.746288i \(-0.268166\pi\)
0.665623 + 0.746288i \(0.268166\pi\)
\(128\) 0 0
\(129\) 20.2303 1.78118
\(130\) 0 0
\(131\) −4.11404 −0.359445 −0.179723 0.983717i \(-0.557520\pi\)
−0.179723 + 0.983717i \(0.557520\pi\)
\(132\) 0 0
\(133\) −3.89255 −0.337527
\(134\) 0 0
\(135\) 5.09194 0.438244
\(136\) 0 0
\(137\) −2.77742 −0.237291 −0.118645 0.992937i \(-0.537855\pi\)
−0.118645 + 0.992937i \(0.537855\pi\)
\(138\) 0 0
\(139\) 5.40850 0.458743 0.229371 0.973339i \(-0.426333\pi\)
0.229371 + 0.973339i \(0.426333\pi\)
\(140\) 0 0
\(141\) −9.84512 −0.829109
\(142\) 0 0
\(143\) −0.169772 −0.0141971
\(144\) 0 0
\(145\) 6.21514 0.516139
\(146\) 0 0
\(147\) 14.3874 1.18666
\(148\) 0 0
\(149\) −0.291090 −0.0238470 −0.0119235 0.999929i \(-0.503795\pi\)
−0.0119235 + 0.999929i \(0.503795\pi\)
\(150\) 0 0
\(151\) −15.0259 −1.22279 −0.611395 0.791326i \(-0.709391\pi\)
−0.611395 + 0.791326i \(0.709391\pi\)
\(152\) 0 0
\(153\) −0.504703 −0.0408028
\(154\) 0 0
\(155\) 9.71448 0.780286
\(156\) 0 0
\(157\) −9.60044 −0.766198 −0.383099 0.923707i \(-0.625143\pi\)
−0.383099 + 0.923707i \(0.625143\pi\)
\(158\) 0 0
\(159\) 15.3172 1.21473
\(160\) 0 0
\(161\) −35.5077 −2.79840
\(162\) 0 0
\(163\) 4.90681 0.384331 0.192166 0.981363i \(-0.438449\pi\)
0.192166 + 0.981363i \(0.438449\pi\)
\(164\) 0 0
\(165\) 8.48508 0.660563
\(166\) 0 0
\(167\) 6.54582 0.506531 0.253266 0.967397i \(-0.418495\pi\)
0.253266 + 0.967397i \(0.418495\pi\)
\(168\) 0 0
\(169\) −12.9988 −0.999904
\(170\) 0 0
\(171\) −0.114895 −0.00878621
\(172\) 0 0
\(173\) −23.9483 −1.82076 −0.910379 0.413776i \(-0.864210\pi\)
−0.910379 + 0.413776i \(0.864210\pi\)
\(174\) 0 0
\(175\) 3.89255 0.294249
\(176\) 0 0
\(177\) 4.52261 0.339940
\(178\) 0 0
\(179\) −9.98637 −0.746416 −0.373208 0.927748i \(-0.621742\pi\)
−0.373208 + 0.927748i \(0.621742\pi\)
\(180\) 0 0
\(181\) −7.41406 −0.551083 −0.275541 0.961289i \(-0.588857\pi\)
−0.275541 + 0.961289i \(0.588857\pi\)
\(182\) 0 0
\(183\) 11.4847 0.848973
\(184\) 0 0
\(185\) 0.749158 0.0550792
\(186\) 0 0
\(187\) 21.1189 1.54436
\(188\) 0 0
\(189\) −19.8206 −1.44174
\(190\) 0 0
\(191\) 22.3007 1.61362 0.806809 0.590812i \(-0.201192\pi\)
0.806809 + 0.590812i \(0.201192\pi\)
\(192\) 0 0
\(193\) −3.15393 −0.227025 −0.113512 0.993537i \(-0.536210\pi\)
−0.113512 + 0.993537i \(0.536210\pi\)
\(194\) 0 0
\(195\) −0.0623238 −0.00446310
\(196\) 0 0
\(197\) 5.21801 0.371768 0.185884 0.982572i \(-0.440485\pi\)
0.185884 + 0.982572i \(0.440485\pi\)
\(198\) 0 0
\(199\) −22.2841 −1.57967 −0.789837 0.613316i \(-0.789835\pi\)
−0.789837 + 0.613316i \(0.789835\pi\)
\(200\) 0 0
\(201\) 23.3843 1.64940
\(202\) 0 0
\(203\) −24.1927 −1.69800
\(204\) 0 0
\(205\) 3.52981 0.246533
\(206\) 0 0
\(207\) −1.04806 −0.0728454
\(208\) 0 0
\(209\) 4.80767 0.332553
\(210\) 0 0
\(211\) 2.27871 0.156873 0.0784364 0.996919i \(-0.475007\pi\)
0.0784364 + 0.996919i \(0.475007\pi\)
\(212\) 0 0
\(213\) −2.94927 −0.202081
\(214\) 0 0
\(215\) −11.4625 −0.781737
\(216\) 0 0
\(217\) −37.8141 −2.56699
\(218\) 0 0
\(219\) −19.9818 −1.35024
\(220\) 0 0
\(221\) −0.155120 −0.0104345
\(222\) 0 0
\(223\) −20.1108 −1.34672 −0.673360 0.739315i \(-0.735150\pi\)
−0.673360 + 0.739315i \(0.735150\pi\)
\(224\) 0 0
\(225\) 0.114895 0.00765964
\(226\) 0 0
\(227\) −1.16184 −0.0771140 −0.0385570 0.999256i \(-0.512276\pi\)
−0.0385570 + 0.999256i \(0.512276\pi\)
\(228\) 0 0
\(229\) 2.77617 0.183454 0.0917272 0.995784i \(-0.470761\pi\)
0.0917272 + 0.995784i \(0.470761\pi\)
\(230\) 0 0
\(231\) −33.0286 −2.17312
\(232\) 0 0
\(233\) 25.2847 1.65645 0.828227 0.560392i \(-0.189350\pi\)
0.828227 + 0.560392i \(0.189350\pi\)
\(234\) 0 0
\(235\) 5.57827 0.363886
\(236\) 0 0
\(237\) −20.1448 −1.30854
\(238\) 0 0
\(239\) 0.228766 0.0147976 0.00739881 0.999973i \(-0.497645\pi\)
0.00739881 + 0.999973i \(0.497645\pi\)
\(240\) 0 0
\(241\) −3.21056 −0.206810 −0.103405 0.994639i \(-0.532974\pi\)
−0.103405 + 0.994639i \(0.532974\pi\)
\(242\) 0 0
\(243\) −1.19337 −0.0765548
\(244\) 0 0
\(245\) −8.15196 −0.520809
\(246\) 0 0
\(247\) −0.0353128 −0.00224690
\(248\) 0 0
\(249\) 9.96977 0.631808
\(250\) 0 0
\(251\) 26.5329 1.67474 0.837372 0.546634i \(-0.184091\pi\)
0.837372 + 0.546634i \(0.184091\pi\)
\(252\) 0 0
\(253\) 43.8553 2.75716
\(254\) 0 0
\(255\) 7.75279 0.485498
\(256\) 0 0
\(257\) −28.5807 −1.78281 −0.891406 0.453206i \(-0.850280\pi\)
−0.891406 + 0.453206i \(0.850280\pi\)
\(258\) 0 0
\(259\) −2.91614 −0.181200
\(260\) 0 0
\(261\) −0.714086 −0.0442008
\(262\) 0 0
\(263\) −18.4959 −1.14050 −0.570252 0.821470i \(-0.693154\pi\)
−0.570252 + 0.821470i \(0.693154\pi\)
\(264\) 0 0
\(265\) −8.67877 −0.533133
\(266\) 0 0
\(267\) −2.18988 −0.134018
\(268\) 0 0
\(269\) −15.6882 −0.956526 −0.478263 0.878217i \(-0.658733\pi\)
−0.478263 + 0.878217i \(0.658733\pi\)
\(270\) 0 0
\(271\) 27.7803 1.68753 0.843767 0.536710i \(-0.180333\pi\)
0.843767 + 0.536710i \(0.180333\pi\)
\(272\) 0 0
\(273\) 0.242599 0.0146827
\(274\) 0 0
\(275\) −4.80767 −0.289913
\(276\) 0 0
\(277\) 19.9044 1.19594 0.597968 0.801520i \(-0.295975\pi\)
0.597968 + 0.801520i \(0.295975\pi\)
\(278\) 0 0
\(279\) −1.11614 −0.0668217
\(280\) 0 0
\(281\) 10.5220 0.627688 0.313844 0.949475i \(-0.398383\pi\)
0.313844 + 0.949475i \(0.398383\pi\)
\(282\) 0 0
\(283\) 4.26164 0.253328 0.126664 0.991946i \(-0.459573\pi\)
0.126664 + 0.991946i \(0.459573\pi\)
\(284\) 0 0
\(285\) 1.76491 0.104544
\(286\) 0 0
\(287\) −13.7400 −0.811045
\(288\) 0 0
\(289\) 2.29623 0.135072
\(290\) 0 0
\(291\) 16.4485 0.964228
\(292\) 0 0
\(293\) −2.16596 −0.126536 −0.0632682 0.997997i \(-0.520152\pi\)
−0.0632682 + 0.997997i \(0.520152\pi\)
\(294\) 0 0
\(295\) −2.56252 −0.149196
\(296\) 0 0
\(297\) 24.4803 1.42049
\(298\) 0 0
\(299\) −0.322122 −0.0186288
\(300\) 0 0
\(301\) 44.6184 2.57176
\(302\) 0 0
\(303\) 1.42545 0.0818902
\(304\) 0 0
\(305\) −6.50725 −0.372604
\(306\) 0 0
\(307\) 15.1287 0.863443 0.431721 0.902007i \(-0.357906\pi\)
0.431721 + 0.902007i \(0.357906\pi\)
\(308\) 0 0
\(309\) 30.1030 1.71250
\(310\) 0 0
\(311\) −12.0180 −0.681477 −0.340739 0.940158i \(-0.610677\pi\)
−0.340739 + 0.940158i \(0.610677\pi\)
\(312\) 0 0
\(313\) −23.1532 −1.30870 −0.654349 0.756193i \(-0.727057\pi\)
−0.654349 + 0.756193i \(0.727057\pi\)
\(314\) 0 0
\(315\) −0.447233 −0.0251987
\(316\) 0 0
\(317\) 6.88004 0.386422 0.193211 0.981157i \(-0.438110\pi\)
0.193211 + 0.981157i \(0.438110\pi\)
\(318\) 0 0
\(319\) 29.8803 1.67298
\(320\) 0 0
\(321\) 14.8430 0.828456
\(322\) 0 0
\(323\) 4.39275 0.244419
\(324\) 0 0
\(325\) 0.0353128 0.00195880
\(326\) 0 0
\(327\) −28.4695 −1.57437
\(328\) 0 0
\(329\) −21.7137 −1.19711
\(330\) 0 0
\(331\) −10.6924 −0.587706 −0.293853 0.955851i \(-0.594938\pi\)
−0.293853 + 0.955851i \(0.594938\pi\)
\(332\) 0 0
\(333\) −0.0860743 −0.00471684
\(334\) 0 0
\(335\) −13.2496 −0.723903
\(336\) 0 0
\(337\) −29.7049 −1.61813 −0.809065 0.587720i \(-0.800026\pi\)
−0.809065 + 0.587720i \(0.800026\pi\)
\(338\) 0 0
\(339\) 23.0838 1.25374
\(340\) 0 0
\(341\) 46.7040 2.52916
\(342\) 0 0
\(343\) 4.48406 0.242116
\(344\) 0 0
\(345\) 16.0994 0.866762
\(346\) 0 0
\(347\) 3.25157 0.174553 0.0872767 0.996184i \(-0.472184\pi\)
0.0872767 + 0.996184i \(0.472184\pi\)
\(348\) 0 0
\(349\) −31.4979 −1.68604 −0.843021 0.537880i \(-0.819225\pi\)
−0.843021 + 0.537880i \(0.819225\pi\)
\(350\) 0 0
\(351\) −0.179811 −0.00959759
\(352\) 0 0
\(353\) −23.0362 −1.22609 −0.613047 0.790047i \(-0.710056\pi\)
−0.613047 + 0.790047i \(0.710056\pi\)
\(354\) 0 0
\(355\) 1.67106 0.0886909
\(356\) 0 0
\(357\) −30.1781 −1.59720
\(358\) 0 0
\(359\) 13.0768 0.690168 0.345084 0.938572i \(-0.387850\pi\)
0.345084 + 0.938572i \(0.387850\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 21.3795 1.12213
\(364\) 0 0
\(365\) 11.3217 0.592606
\(366\) 0 0
\(367\) −32.4936 −1.69615 −0.848077 0.529874i \(-0.822239\pi\)
−0.848077 + 0.529874i \(0.822239\pi\)
\(368\) 0 0
\(369\) −0.405557 −0.0211124
\(370\) 0 0
\(371\) 33.7826 1.75390
\(372\) 0 0
\(373\) 6.75164 0.349586 0.174793 0.984605i \(-0.444074\pi\)
0.174793 + 0.984605i \(0.444074\pi\)
\(374\) 0 0
\(375\) −1.76491 −0.0911394
\(376\) 0 0
\(377\) −0.219474 −0.0113035
\(378\) 0 0
\(379\) −34.7328 −1.78410 −0.892051 0.451934i \(-0.850734\pi\)
−0.892051 + 0.451934i \(0.850734\pi\)
\(380\) 0 0
\(381\) 26.4778 1.35650
\(382\) 0 0
\(383\) 0.230516 0.0117788 0.00588942 0.999983i \(-0.498125\pi\)
0.00588942 + 0.999983i \(0.498125\pi\)
\(384\) 0 0
\(385\) 18.7141 0.953758
\(386\) 0 0
\(387\) 1.31698 0.0669460
\(388\) 0 0
\(389\) −26.1973 −1.32826 −0.664128 0.747619i \(-0.731197\pi\)
−0.664128 + 0.747619i \(0.731197\pi\)
\(390\) 0 0
\(391\) 40.0704 2.02645
\(392\) 0 0
\(393\) −7.26089 −0.366264
\(394\) 0 0
\(395\) 11.4141 0.574304
\(396\) 0 0
\(397\) −24.7005 −1.23968 −0.619840 0.784728i \(-0.712803\pi\)
−0.619840 + 0.784728i \(0.712803\pi\)
\(398\) 0 0
\(399\) −6.86999 −0.343930
\(400\) 0 0
\(401\) 2.43645 0.121671 0.0608354 0.998148i \(-0.480624\pi\)
0.0608354 + 0.998148i \(0.480624\pi\)
\(402\) 0 0
\(403\) −0.343046 −0.0170883
\(404\) 0 0
\(405\) 9.33148 0.463685
\(406\) 0 0
\(407\) 3.60170 0.178530
\(408\) 0 0
\(409\) −22.2293 −1.09917 −0.549585 0.835438i \(-0.685214\pi\)
−0.549585 + 0.835438i \(0.685214\pi\)
\(410\) 0 0
\(411\) −4.90188 −0.241792
\(412\) 0 0
\(413\) 9.97474 0.490825
\(414\) 0 0
\(415\) −5.64889 −0.277293
\(416\) 0 0
\(417\) 9.54549 0.467445
\(418\) 0 0
\(419\) −4.34305 −0.212172 −0.106086 0.994357i \(-0.533832\pi\)
−0.106086 + 0.994357i \(0.533832\pi\)
\(420\) 0 0
\(421\) −0.903722 −0.0440447 −0.0220224 0.999757i \(-0.507011\pi\)
−0.0220224 + 0.999757i \(0.507011\pi\)
\(422\) 0 0
\(423\) −0.640913 −0.0311623
\(424\) 0 0
\(425\) −4.39275 −0.213080
\(426\) 0 0
\(427\) 25.3298 1.22580
\(428\) 0 0
\(429\) −0.299632 −0.0144664
\(430\) 0 0
\(431\) 27.6872 1.33365 0.666823 0.745216i \(-0.267654\pi\)
0.666823 + 0.745216i \(0.267654\pi\)
\(432\) 0 0
\(433\) 15.7540 0.757088 0.378544 0.925583i \(-0.376425\pi\)
0.378544 + 0.925583i \(0.376425\pi\)
\(434\) 0 0
\(435\) 10.9691 0.525930
\(436\) 0 0
\(437\) 9.12195 0.436362
\(438\) 0 0
\(439\) 2.54081 0.121266 0.0606332 0.998160i \(-0.480688\pi\)
0.0606332 + 0.998160i \(0.480688\pi\)
\(440\) 0 0
\(441\) 0.936616 0.0446008
\(442\) 0 0
\(443\) −20.3166 −0.965270 −0.482635 0.875821i \(-0.660320\pi\)
−0.482635 + 0.875821i \(0.660320\pi\)
\(444\) 0 0
\(445\) 1.24079 0.0588191
\(446\) 0 0
\(447\) −0.513746 −0.0242993
\(448\) 0 0
\(449\) −20.0132 −0.944480 −0.472240 0.881470i \(-0.656555\pi\)
−0.472240 + 0.881470i \(0.656555\pi\)
\(450\) 0 0
\(451\) 16.9702 0.799094
\(452\) 0 0
\(453\) −26.5193 −1.24599
\(454\) 0 0
\(455\) −0.137457 −0.00644408
\(456\) 0 0
\(457\) 28.9331 1.35343 0.676717 0.736243i \(-0.263402\pi\)
0.676717 + 0.736243i \(0.263402\pi\)
\(458\) 0 0
\(459\) 22.3676 1.04403
\(460\) 0 0
\(461\) 9.04395 0.421219 0.210609 0.977570i \(-0.432455\pi\)
0.210609 + 0.977570i \(0.432455\pi\)
\(462\) 0 0
\(463\) −33.2226 −1.54399 −0.771993 0.635632i \(-0.780740\pi\)
−0.771993 + 0.635632i \(0.780740\pi\)
\(464\) 0 0
\(465\) 17.1451 0.795087
\(466\) 0 0
\(467\) 34.0888 1.57744 0.788722 0.614750i \(-0.210743\pi\)
0.788722 + 0.614750i \(0.210743\pi\)
\(468\) 0 0
\(469\) 51.5747 2.38150
\(470\) 0 0
\(471\) −16.9439 −0.780733
\(472\) 0 0
\(473\) −55.1079 −2.53387
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 0.997145 0.0456561
\(478\) 0 0
\(479\) 28.2735 1.29185 0.645925 0.763401i \(-0.276472\pi\)
0.645925 + 0.763401i \(0.276472\pi\)
\(480\) 0 0
\(481\) −0.0264549 −0.00120624
\(482\) 0 0
\(483\) −62.6677 −2.85148
\(484\) 0 0
\(485\) −9.31976 −0.423188
\(486\) 0 0
\(487\) 25.5836 1.15930 0.579652 0.814864i \(-0.303189\pi\)
0.579652 + 0.814864i \(0.303189\pi\)
\(488\) 0 0
\(489\) 8.66006 0.391622
\(490\) 0 0
\(491\) 3.53949 0.159735 0.0798675 0.996805i \(-0.474550\pi\)
0.0798675 + 0.996805i \(0.474550\pi\)
\(492\) 0 0
\(493\) 27.3015 1.22960
\(494\) 0 0
\(495\) 0.552375 0.0248274
\(496\) 0 0
\(497\) −6.50470 −0.291776
\(498\) 0 0
\(499\) −26.7046 −1.19546 −0.597730 0.801697i \(-0.703931\pi\)
−0.597730 + 0.801697i \(0.703931\pi\)
\(500\) 0 0
\(501\) 11.5528 0.516140
\(502\) 0 0
\(503\) 24.7956 1.10558 0.552790 0.833320i \(-0.313563\pi\)
0.552790 + 0.833320i \(0.313563\pi\)
\(504\) 0 0
\(505\) −0.807666 −0.0359406
\(506\) 0 0
\(507\) −22.9416 −1.01887
\(508\) 0 0
\(509\) 17.4860 0.775051 0.387526 0.921859i \(-0.373330\pi\)
0.387526 + 0.921859i \(0.373330\pi\)
\(510\) 0 0
\(511\) −44.0704 −1.94956
\(512\) 0 0
\(513\) 5.09194 0.224815
\(514\) 0 0
\(515\) −17.0564 −0.751595
\(516\) 0 0
\(517\) 26.8184 1.17947
\(518\) 0 0
\(519\) −42.2666 −1.85530
\(520\) 0 0
\(521\) −1.37888 −0.0604099 −0.0302049 0.999544i \(-0.509616\pi\)
−0.0302049 + 0.999544i \(0.509616\pi\)
\(522\) 0 0
\(523\) −38.5666 −1.68640 −0.843199 0.537602i \(-0.819330\pi\)
−0.843199 + 0.537602i \(0.819330\pi\)
\(524\) 0 0
\(525\) 6.86999 0.299831
\(526\) 0 0
\(527\) 42.6732 1.85888
\(528\) 0 0
\(529\) 60.2100 2.61782
\(530\) 0 0
\(531\) 0.294420 0.0127767
\(532\) 0 0
\(533\) −0.124648 −0.00539909
\(534\) 0 0
\(535\) −8.41008 −0.363599
\(536\) 0 0
\(537\) −17.6250 −0.760575
\(538\) 0 0
\(539\) −39.1919 −1.68811
\(540\) 0 0
\(541\) 8.73354 0.375484 0.187742 0.982218i \(-0.439883\pi\)
0.187742 + 0.982218i \(0.439883\pi\)
\(542\) 0 0
\(543\) −13.0851 −0.561537
\(544\) 0 0
\(545\) 16.1309 0.690971
\(546\) 0 0
\(547\) −31.8117 −1.36017 −0.680084 0.733134i \(-0.738057\pi\)
−0.680084 + 0.733134i \(0.738057\pi\)
\(548\) 0 0
\(549\) 0.747648 0.0319089
\(550\) 0 0
\(551\) 6.21514 0.264774
\(552\) 0 0
\(553\) −44.4298 −1.88935
\(554\) 0 0
\(555\) 1.32219 0.0561240
\(556\) 0 0
\(557\) −31.9044 −1.35183 −0.675915 0.736979i \(-0.736252\pi\)
−0.675915 + 0.736979i \(0.736252\pi\)
\(558\) 0 0
\(559\) 0.404774 0.0171201
\(560\) 0 0
\(561\) 37.2728 1.57366
\(562\) 0 0
\(563\) 40.7223 1.71624 0.858119 0.513450i \(-0.171633\pi\)
0.858119 + 0.513450i \(0.171633\pi\)
\(564\) 0 0
\(565\) −13.0794 −0.550253
\(566\) 0 0
\(567\) −36.3233 −1.52543
\(568\) 0 0
\(569\) 12.9176 0.541534 0.270767 0.962645i \(-0.412723\pi\)
0.270767 + 0.962645i \(0.412723\pi\)
\(570\) 0 0
\(571\) 14.3129 0.598976 0.299488 0.954100i \(-0.403184\pi\)
0.299488 + 0.954100i \(0.403184\pi\)
\(572\) 0 0
\(573\) 39.3586 1.64423
\(574\) 0 0
\(575\) −9.12195 −0.380412
\(576\) 0 0
\(577\) 36.0033 1.49884 0.749419 0.662095i \(-0.230333\pi\)
0.749419 + 0.662095i \(0.230333\pi\)
\(578\) 0 0
\(579\) −5.56639 −0.231331
\(580\) 0 0
\(581\) 21.9886 0.912241
\(582\) 0 0
\(583\) −41.7246 −1.72806
\(584\) 0 0
\(585\) −0.00405725 −0.000167747 0
\(586\) 0 0
\(587\) 6.30974 0.260431 0.130215 0.991486i \(-0.458433\pi\)
0.130215 + 0.991486i \(0.458433\pi\)
\(588\) 0 0
\(589\) 9.71448 0.400278
\(590\) 0 0
\(591\) 9.20929 0.378820
\(592\) 0 0
\(593\) 6.17104 0.253414 0.126707 0.991940i \(-0.459559\pi\)
0.126707 + 0.991940i \(0.459559\pi\)
\(594\) 0 0
\(595\) 17.0990 0.700990
\(596\) 0 0
\(597\) −39.3293 −1.60964
\(598\) 0 0
\(599\) −23.3127 −0.952530 −0.476265 0.879302i \(-0.658010\pi\)
−0.476265 + 0.879302i \(0.658010\pi\)
\(600\) 0 0
\(601\) −20.0776 −0.818984 −0.409492 0.912314i \(-0.634294\pi\)
−0.409492 + 0.912314i \(0.634294\pi\)
\(602\) 0 0
\(603\) 1.52231 0.0619932
\(604\) 0 0
\(605\) −12.1136 −0.492490
\(606\) 0 0
\(607\) −5.12748 −0.208118 −0.104059 0.994571i \(-0.533183\pi\)
−0.104059 + 0.994571i \(0.533183\pi\)
\(608\) 0 0
\(609\) −42.6979 −1.73021
\(610\) 0 0
\(611\) −0.196984 −0.00796913
\(612\) 0 0
\(613\) 24.2342 0.978811 0.489405 0.872056i \(-0.337214\pi\)
0.489405 + 0.872056i \(0.337214\pi\)
\(614\) 0 0
\(615\) 6.22979 0.251209
\(616\) 0 0
\(617\) 10.8591 0.437171 0.218585 0.975818i \(-0.429856\pi\)
0.218585 + 0.975818i \(0.429856\pi\)
\(618\) 0 0
\(619\) −19.2362 −0.773169 −0.386585 0.922254i \(-0.626345\pi\)
−0.386585 + 0.922254i \(0.626345\pi\)
\(620\) 0 0
\(621\) 46.4484 1.86391
\(622\) 0 0
\(623\) −4.82984 −0.193503
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 8.48508 0.338861
\(628\) 0 0
\(629\) 3.29086 0.131215
\(630\) 0 0
\(631\) −46.0007 −1.83126 −0.915629 0.402025i \(-0.868306\pi\)
−0.915629 + 0.402025i \(0.868306\pi\)
\(632\) 0 0
\(633\) 4.02171 0.159848
\(634\) 0 0
\(635\) −15.0024 −0.595351
\(636\) 0 0
\(637\) 0.287869 0.0114058
\(638\) 0 0
\(639\) −0.191996 −0.00759526
\(640\) 0 0
\(641\) 7.97279 0.314906 0.157453 0.987526i \(-0.449672\pi\)
0.157453 + 0.987526i \(0.449672\pi\)
\(642\) 0 0
\(643\) 42.2951 1.66795 0.833977 0.551799i \(-0.186058\pi\)
0.833977 + 0.551799i \(0.186058\pi\)
\(644\) 0 0
\(645\) −20.2303 −0.796566
\(646\) 0 0
\(647\) −0.422756 −0.0166202 −0.00831012 0.999965i \(-0.502645\pi\)
−0.00831012 + 0.999965i \(0.502645\pi\)
\(648\) 0 0
\(649\) −12.3197 −0.483592
\(650\) 0 0
\(651\) −66.7384 −2.61568
\(652\) 0 0
\(653\) 9.74222 0.381242 0.190621 0.981664i \(-0.438950\pi\)
0.190621 + 0.981664i \(0.438950\pi\)
\(654\) 0 0
\(655\) 4.11404 0.160749
\(656\) 0 0
\(657\) −1.30081 −0.0507493
\(658\) 0 0
\(659\) 4.05689 0.158034 0.0790170 0.996873i \(-0.474822\pi\)
0.0790170 + 0.996873i \(0.474822\pi\)
\(660\) 0 0
\(661\) −21.5278 −0.837335 −0.418668 0.908140i \(-0.637503\pi\)
−0.418668 + 0.908140i \(0.637503\pi\)
\(662\) 0 0
\(663\) −0.273773 −0.0106325
\(664\) 0 0
\(665\) 3.89255 0.150947
\(666\) 0 0
\(667\) 56.6942 2.19521
\(668\) 0 0
\(669\) −35.4937 −1.37227
\(670\) 0 0
\(671\) −31.2847 −1.20773
\(672\) 0 0
\(673\) −30.3026 −1.16808 −0.584039 0.811725i \(-0.698529\pi\)
−0.584039 + 0.811725i \(0.698529\pi\)
\(674\) 0 0
\(675\) −5.09194 −0.195989
\(676\) 0 0
\(677\) −18.3642 −0.705795 −0.352898 0.935662i \(-0.614804\pi\)
−0.352898 + 0.935662i \(0.614804\pi\)
\(678\) 0 0
\(679\) 36.2776 1.39221
\(680\) 0 0
\(681\) −2.05054 −0.0785768
\(682\) 0 0
\(683\) 9.35629 0.358009 0.179004 0.983848i \(-0.442712\pi\)
0.179004 + 0.983848i \(0.442712\pi\)
\(684\) 0 0
\(685\) 2.77742 0.106120
\(686\) 0 0
\(687\) 4.89968 0.186934
\(688\) 0 0
\(689\) 0.306472 0.0116757
\(690\) 0 0
\(691\) 24.6498 0.937723 0.468861 0.883272i \(-0.344664\pi\)
0.468861 + 0.883272i \(0.344664\pi\)
\(692\) 0 0
\(693\) −2.15015 −0.0816774
\(694\) 0 0
\(695\) −5.40850 −0.205156
\(696\) 0 0
\(697\) 15.5056 0.587316
\(698\) 0 0
\(699\) 44.6251 1.68788
\(700\) 0 0
\(701\) −13.2467 −0.500321 −0.250160 0.968204i \(-0.580483\pi\)
−0.250160 + 0.968204i \(0.580483\pi\)
\(702\) 0 0
\(703\) 0.749158 0.0282550
\(704\) 0 0
\(705\) 9.84512 0.370789
\(706\) 0 0
\(707\) 3.14388 0.118238
\(708\) 0 0
\(709\) −30.0979 −1.13035 −0.565175 0.824971i \(-0.691192\pi\)
−0.565175 + 0.824971i \(0.691192\pi\)
\(710\) 0 0
\(711\) −1.31141 −0.0491819
\(712\) 0 0
\(713\) 88.6150 3.31866
\(714\) 0 0
\(715\) 0.169772 0.00634912
\(716\) 0 0
\(717\) 0.403750 0.0150783
\(718\) 0 0
\(719\) −25.5662 −0.953458 −0.476729 0.879050i \(-0.658178\pi\)
−0.476729 + 0.879050i \(0.658178\pi\)
\(720\) 0 0
\(721\) 66.3929 2.47260
\(722\) 0 0
\(723\) −5.66633 −0.210733
\(724\) 0 0
\(725\) −6.21514 −0.230824
\(726\) 0 0
\(727\) −46.2807 −1.71646 −0.858228 0.513268i \(-0.828435\pi\)
−0.858228 + 0.513268i \(0.828435\pi\)
\(728\) 0 0
\(729\) 25.8883 0.958824
\(730\) 0 0
\(731\) −50.3519 −1.86233
\(732\) 0 0
\(733\) 33.1328 1.22379 0.611894 0.790940i \(-0.290408\pi\)
0.611894 + 0.790940i \(0.290408\pi\)
\(734\) 0 0
\(735\) −14.3874 −0.530689
\(736\) 0 0
\(737\) −63.6996 −2.34641
\(738\) 0 0
\(739\) −2.75493 −0.101342 −0.0506708 0.998715i \(-0.516136\pi\)
−0.0506708 + 0.998715i \(0.516136\pi\)
\(740\) 0 0
\(741\) −0.0623238 −0.00228952
\(742\) 0 0
\(743\) 9.01027 0.330555 0.165277 0.986247i \(-0.447148\pi\)
0.165277 + 0.986247i \(0.447148\pi\)
\(744\) 0 0
\(745\) 0.291090 0.0106647
\(746\) 0 0
\(747\) 0.649028 0.0237467
\(748\) 0 0
\(749\) 32.7367 1.19617
\(750\) 0 0
\(751\) 10.7150 0.390996 0.195498 0.980704i \(-0.437368\pi\)
0.195498 + 0.980704i \(0.437368\pi\)
\(752\) 0 0
\(753\) 46.8281 1.70651
\(754\) 0 0
\(755\) 15.0259 0.546848
\(756\) 0 0
\(757\) 22.4197 0.814857 0.407428 0.913237i \(-0.366426\pi\)
0.407428 + 0.913237i \(0.366426\pi\)
\(758\) 0 0
\(759\) 77.4005 2.80946
\(760\) 0 0
\(761\) −7.21311 −0.261475 −0.130738 0.991417i \(-0.541735\pi\)
−0.130738 + 0.991417i \(0.541735\pi\)
\(762\) 0 0
\(763\) −62.7903 −2.27316
\(764\) 0 0
\(765\) 0.504703 0.0182476
\(766\) 0 0
\(767\) 0.0904898 0.00326740
\(768\) 0 0
\(769\) 6.14157 0.221471 0.110735 0.993850i \(-0.464679\pi\)
0.110735 + 0.993850i \(0.464679\pi\)
\(770\) 0 0
\(771\) −50.4422 −1.81663
\(772\) 0 0
\(773\) −51.0612 −1.83655 −0.918273 0.395949i \(-0.870416\pi\)
−0.918273 + 0.395949i \(0.870416\pi\)
\(774\) 0 0
\(775\) −9.71448 −0.348954
\(776\) 0 0
\(777\) −5.14671 −0.184637
\(778\) 0 0
\(779\) 3.52981 0.126469
\(780\) 0 0
\(781\) 8.03392 0.287476
\(782\) 0 0
\(783\) 31.6471 1.13098
\(784\) 0 0
\(785\) 9.60044 0.342654
\(786\) 0 0
\(787\) 38.6448 1.37754 0.688769 0.724980i \(-0.258151\pi\)
0.688769 + 0.724980i \(0.258151\pi\)
\(788\) 0 0
\(789\) −32.6435 −1.16214
\(790\) 0 0
\(791\) 50.9121 1.81023
\(792\) 0 0
\(793\) 0.229789 0.00816006
\(794\) 0 0
\(795\) −15.3172 −0.543246
\(796\) 0 0
\(797\) −8.38007 −0.296837 −0.148419 0.988925i \(-0.547418\pi\)
−0.148419 + 0.988925i \(0.547418\pi\)
\(798\) 0 0
\(799\) 24.5039 0.866886
\(800\) 0 0
\(801\) −0.142560 −0.00503711
\(802\) 0 0
\(803\) 54.4311 1.92083
\(804\) 0 0
\(805\) 35.5077 1.25148
\(806\) 0 0
\(807\) −27.6882 −0.974671
\(808\) 0 0
\(809\) −20.2642 −0.712450 −0.356225 0.934400i \(-0.615936\pi\)
−0.356225 + 0.934400i \(0.615936\pi\)
\(810\) 0 0
\(811\) −15.1391 −0.531606 −0.265803 0.964027i \(-0.585637\pi\)
−0.265803 + 0.964027i \(0.585637\pi\)
\(812\) 0 0
\(813\) 49.0296 1.71954
\(814\) 0 0
\(815\) −4.90681 −0.171878
\(816\) 0 0
\(817\) −11.4625 −0.401023
\(818\) 0 0
\(819\) 0.0157931 0.000551855 0
\(820\) 0 0
\(821\) 11.6562 0.406803 0.203402 0.979095i \(-0.434800\pi\)
0.203402 + 0.979095i \(0.434800\pi\)
\(822\) 0 0
\(823\) −37.6062 −1.31087 −0.655436 0.755251i \(-0.727515\pi\)
−0.655436 + 0.755251i \(0.727515\pi\)
\(824\) 0 0
\(825\) −8.48508 −0.295413
\(826\) 0 0
\(827\) −50.0978 −1.74207 −0.871036 0.491219i \(-0.836551\pi\)
−0.871036 + 0.491219i \(0.836551\pi\)
\(828\) 0 0
\(829\) −18.7309 −0.650552 −0.325276 0.945619i \(-0.605457\pi\)
−0.325276 + 0.945619i \(0.605457\pi\)
\(830\) 0 0
\(831\) 35.1293 1.21862
\(832\) 0 0
\(833\) −35.8095 −1.24073
\(834\) 0 0
\(835\) −6.54582 −0.226528
\(836\) 0 0
\(837\) 49.4655 1.70978
\(838\) 0 0
\(839\) 23.7615 0.820339 0.410170 0.912009i \(-0.365469\pi\)
0.410170 + 0.912009i \(0.365469\pi\)
\(840\) 0 0
\(841\) 9.62793 0.331998
\(842\) 0 0
\(843\) 18.5703 0.639595
\(844\) 0 0
\(845\) 12.9988 0.447171
\(846\) 0 0
\(847\) 47.1530 1.62020
\(848\) 0 0
\(849\) 7.52140 0.258134
\(850\) 0 0
\(851\) 6.83378 0.234259
\(852\) 0 0
\(853\) −3.61980 −0.123940 −0.0619699 0.998078i \(-0.519738\pi\)
−0.0619699 + 0.998078i \(0.519738\pi\)
\(854\) 0 0
\(855\) 0.114895 0.00392931
\(856\) 0 0
\(857\) −36.1820 −1.23595 −0.617977 0.786196i \(-0.712048\pi\)
−0.617977 + 0.786196i \(0.712048\pi\)
\(858\) 0 0
\(859\) 52.3890 1.78749 0.893745 0.448576i \(-0.148069\pi\)
0.893745 + 0.448576i \(0.148069\pi\)
\(860\) 0 0
\(861\) −24.2498 −0.826430
\(862\) 0 0
\(863\) 48.7594 1.65979 0.829895 0.557919i \(-0.188400\pi\)
0.829895 + 0.557919i \(0.188400\pi\)
\(864\) 0 0
\(865\) 23.9483 0.814267
\(866\) 0 0
\(867\) 4.05263 0.137635
\(868\) 0 0
\(869\) 54.8750 1.86151
\(870\) 0 0
\(871\) 0.467880 0.0158535
\(872\) 0 0
\(873\) 1.07079 0.0362408
\(874\) 0 0
\(875\) −3.89255 −0.131592
\(876\) 0 0
\(877\) −9.17301 −0.309751 −0.154875 0.987934i \(-0.549498\pi\)
−0.154875 + 0.987934i \(0.549498\pi\)
\(878\) 0 0
\(879\) −3.82271 −0.128937
\(880\) 0 0
\(881\) 44.7985 1.50930 0.754649 0.656128i \(-0.227807\pi\)
0.754649 + 0.656128i \(0.227807\pi\)
\(882\) 0 0
\(883\) −40.4371 −1.36082 −0.680408 0.732833i \(-0.738198\pi\)
−0.680408 + 0.732833i \(0.738198\pi\)
\(884\) 0 0
\(885\) −4.52261 −0.152026
\(886\) 0 0
\(887\) −11.6569 −0.391400 −0.195700 0.980664i \(-0.562698\pi\)
−0.195700 + 0.980664i \(0.562698\pi\)
\(888\) 0 0
\(889\) 58.3975 1.95859
\(890\) 0 0
\(891\) 44.8627 1.50296
\(892\) 0 0
\(893\) 5.57827 0.186670
\(894\) 0 0
\(895\) 9.98637 0.333808
\(896\) 0 0
\(897\) −0.568515 −0.0189822
\(898\) 0 0
\(899\) 60.3768 2.01368
\(900\) 0 0
\(901\) −38.1237 −1.27008
\(902\) 0 0
\(903\) 78.7474 2.62055
\(904\) 0 0
\(905\) 7.41406 0.246452
\(906\) 0 0
\(907\) 29.7723 0.988575 0.494287 0.869299i \(-0.335429\pi\)
0.494287 + 0.869299i \(0.335429\pi\)
\(908\) 0 0
\(909\) 0.0927965 0.00307786
\(910\) 0 0
\(911\) 7.86284 0.260507 0.130254 0.991481i \(-0.458421\pi\)
0.130254 + 0.991481i \(0.458421\pi\)
\(912\) 0 0
\(913\) −27.1580 −0.898798
\(914\) 0 0
\(915\) −11.4847 −0.379672
\(916\) 0 0
\(917\) −16.0141 −0.528832
\(918\) 0 0
\(919\) 11.2607 0.371455 0.185727 0.982601i \(-0.440536\pi\)
0.185727 + 0.982601i \(0.440536\pi\)
\(920\) 0 0
\(921\) 26.7008 0.879822
\(922\) 0 0
\(923\) −0.0590100 −0.00194234
\(924\) 0 0
\(925\) −0.749158 −0.0246322
\(926\) 0 0
\(927\) 1.95969 0.0643647
\(928\) 0 0
\(929\) −6.12699 −0.201020 −0.100510 0.994936i \(-0.532047\pi\)
−0.100510 + 0.994936i \(0.532047\pi\)
\(930\) 0 0
\(931\) −8.15196 −0.267170
\(932\) 0 0
\(933\) −21.2106 −0.694404
\(934\) 0 0
\(935\) −21.1189 −0.690661
\(936\) 0 0
\(937\) 20.2530 0.661635 0.330818 0.943695i \(-0.392676\pi\)
0.330818 + 0.943695i \(0.392676\pi\)
\(938\) 0 0
\(939\) −40.8633 −1.33352
\(940\) 0 0
\(941\) 0.565626 0.0184389 0.00921945 0.999958i \(-0.497065\pi\)
0.00921945 + 0.999958i \(0.497065\pi\)
\(942\) 0 0
\(943\) 32.1988 1.04854
\(944\) 0 0
\(945\) 19.8206 0.644766
\(946\) 0 0
\(947\) −6.16157 −0.200224 −0.100112 0.994976i \(-0.531920\pi\)
−0.100112 + 0.994976i \(0.531920\pi\)
\(948\) 0 0
\(949\) −0.399802 −0.0129781
\(950\) 0 0
\(951\) 12.1426 0.393752
\(952\) 0 0
\(953\) 15.1913 0.492095 0.246047 0.969258i \(-0.420868\pi\)
0.246047 + 0.969258i \(0.420868\pi\)
\(954\) 0 0
\(955\) −22.3007 −0.721632
\(956\) 0 0
\(957\) 52.7359 1.70471
\(958\) 0 0
\(959\) −10.8112 −0.349113
\(960\) 0 0
\(961\) 63.3711 2.04423
\(962\) 0 0
\(963\) 0.966273 0.0311377
\(964\) 0 0
\(965\) 3.15393 0.101529
\(966\) 0 0
\(967\) −26.5387 −0.853429 −0.426714 0.904386i \(-0.640329\pi\)
−0.426714 + 0.904386i \(0.640329\pi\)
\(968\) 0 0
\(969\) 7.75279 0.249055
\(970\) 0 0
\(971\) 59.8610 1.92103 0.960516 0.278226i \(-0.0897463\pi\)
0.960516 + 0.278226i \(0.0897463\pi\)
\(972\) 0 0
\(973\) 21.0528 0.674923
\(974\) 0 0
\(975\) 0.0623238 0.00199596
\(976\) 0 0
\(977\) 22.5670 0.721983 0.360991 0.932569i \(-0.382438\pi\)
0.360991 + 0.932569i \(0.382438\pi\)
\(978\) 0 0
\(979\) 5.96530 0.190652
\(980\) 0 0
\(981\) −1.85335 −0.0591730
\(982\) 0 0
\(983\) 5.97094 0.190443 0.0952217 0.995456i \(-0.469644\pi\)
0.0952217 + 0.995456i \(0.469644\pi\)
\(984\) 0 0
\(985\) −5.21801 −0.166260
\(986\) 0 0
\(987\) −38.3226 −1.21982
\(988\) 0 0
\(989\) −104.561 −3.32483
\(990\) 0 0
\(991\) −12.8794 −0.409127 −0.204564 0.978853i \(-0.565578\pi\)
−0.204564 + 0.978853i \(0.565578\pi\)
\(992\) 0 0
\(993\) −18.8710 −0.598855
\(994\) 0 0
\(995\) 22.2841 0.706452
\(996\) 0 0
\(997\) 0.948160 0.0300285 0.0150143 0.999887i \(-0.495221\pi\)
0.0150143 + 0.999887i \(0.495221\pi\)
\(998\) 0 0
\(999\) 3.81467 0.120691
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.1.4 yes 5
4.3 odd 2 3040.2.a.u.1.2 5
8.3 odd 2 6080.2.a.cl.1.4 5
8.5 even 2 6080.2.a.ci.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.u.1.2 5 4.3 odd 2
3040.2.a.x.1.4 yes 5 1.1 even 1 trivial
6080.2.a.ci.1.2 5 8.5 even 2
6080.2.a.cl.1.4 5 8.3 odd 2