Properties

Label 3040.2.a.e.1.1
Level $3040$
Weight $2$
Character 3040.1
Self dual yes
Analytic conductor $24.275$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) 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)
 
\(f(q)\) \(=\) \(q-2.56155 q^{3} -1.00000 q^{5} +4.56155 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q-2.56155 q^{3} -1.00000 q^{5} +4.56155 q^{7} +3.56155 q^{9} -4.00000 q^{11} +2.56155 q^{13} +2.56155 q^{15} -0.561553 q^{17} -1.00000 q^{19} -11.6847 q^{21} -5.68466 q^{23} +1.00000 q^{25} -1.43845 q^{27} -3.43845 q^{29} -5.12311 q^{31} +10.2462 q^{33} -4.56155 q^{35} -1.12311 q^{37} -6.56155 q^{39} +8.24621 q^{41} -2.00000 q^{43} -3.56155 q^{45} +3.12311 q^{47} +13.8078 q^{49} +1.43845 q^{51} +6.56155 q^{53} +4.00000 q^{55} +2.56155 q^{57} +12.8078 q^{59} -4.87689 q^{61} +16.2462 q^{63} -2.56155 q^{65} -10.5616 q^{67} +14.5616 q^{69} -8.00000 q^{71} +14.8078 q^{73} -2.56155 q^{75} -18.2462 q^{77} +5.12311 q^{79} -7.00000 q^{81} -14.0000 q^{83} +0.561553 q^{85} +8.80776 q^{87} -10.0000 q^{89} +11.6847 q^{91} +13.1231 q^{93} +1.00000 q^{95} -1.12311 q^{97} -14.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} - 2 q^{5} + 5 q^{7} + 3 q^{9} - 8 q^{11} + q^{13} + q^{15} + 3 q^{17} - 2 q^{19} - 11 q^{21} + q^{23} + 2 q^{25} - 7 q^{27} - 11 q^{29} - 2 q^{31} + 4 q^{33} - 5 q^{35} + 6 q^{37} - 9 q^{39} - 4 q^{43} - 3 q^{45} - 2 q^{47} + 7 q^{49} + 7 q^{51} + 9 q^{53} + 8 q^{55} + q^{57} + 5 q^{59} - 18 q^{61} + 16 q^{63} - q^{65} - 17 q^{67} + 25 q^{69} - 16 q^{71} + 9 q^{73} - q^{75} - 20 q^{77} + 2 q^{79} - 14 q^{81} - 28 q^{83} - 3 q^{85} - 3 q^{87} - 20 q^{89} + 11 q^{91} + 18 q^{93} + 2 q^{95} + 6 q^{97} - 12 q^{99}+O(q^{100}) \) 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
\(3\) −2.56155 −1.47891 −0.739457 0.673204i \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 4.56155 1.72410 0.862052 0.506819i \(-0.169179\pi\)
0.862052 + 0.506819i \(0.169179\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) −4.00000 −1.20605 −0.603023 0.797724i \(-0.706037\pi\)
−0.603023 + 0.797724i \(0.706037\pi\)
\(12\) 0 0
\(13\) 2.56155 0.710447 0.355223 0.934781i \(-0.384405\pi\)
0.355223 + 0.934781i \(0.384405\pi\)
\(14\) 0 0
\(15\) 2.56155 0.661390
\(16\) 0 0
\(17\) −0.561553 −0.136197 −0.0680983 0.997679i \(-0.521693\pi\)
−0.0680983 + 0.997679i \(0.521693\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −11.6847 −2.54980
\(22\) 0 0
\(23\) −5.68466 −1.18533 −0.592667 0.805448i \(-0.701925\pi\)
−0.592667 + 0.805448i \(0.701925\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) −3.43845 −0.638504 −0.319252 0.947670i \(-0.603432\pi\)
−0.319252 + 0.947670i \(0.603432\pi\)
\(30\) 0 0
\(31\) −5.12311 −0.920137 −0.460068 0.887883i \(-0.652175\pi\)
−0.460068 + 0.887883i \(0.652175\pi\)
\(32\) 0 0
\(33\) 10.2462 1.78364
\(34\) 0 0
\(35\) −4.56155 −0.771043
\(36\) 0 0
\(37\) −1.12311 −0.184637 −0.0923187 0.995730i \(-0.529428\pi\)
−0.0923187 + 0.995730i \(0.529428\pi\)
\(38\) 0 0
\(39\) −6.56155 −1.05069
\(40\) 0 0
\(41\) 8.24621 1.28784 0.643921 0.765092i \(-0.277307\pi\)
0.643921 + 0.765092i \(0.277307\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) −3.56155 −0.530925
\(46\) 0 0
\(47\) 3.12311 0.455552 0.227776 0.973714i \(-0.426855\pi\)
0.227776 + 0.973714i \(0.426855\pi\)
\(48\) 0 0
\(49\) 13.8078 1.97254
\(50\) 0 0
\(51\) 1.43845 0.201423
\(52\) 0 0
\(53\) 6.56155 0.901299 0.450649 0.892701i \(-0.351193\pi\)
0.450649 + 0.892701i \(0.351193\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 2.56155 0.339286
\(58\) 0 0
\(59\) 12.8078 1.66743 0.833714 0.552196i \(-0.186210\pi\)
0.833714 + 0.552196i \(0.186210\pi\)
\(60\) 0 0
\(61\) −4.87689 −0.624422 −0.312211 0.950013i \(-0.601070\pi\)
−0.312211 + 0.950013i \(0.601070\pi\)
\(62\) 0 0
\(63\) 16.2462 2.04683
\(64\) 0 0
\(65\) −2.56155 −0.317722
\(66\) 0 0
\(67\) −10.5616 −1.29030 −0.645150 0.764056i \(-0.723205\pi\)
−0.645150 + 0.764056i \(0.723205\pi\)
\(68\) 0 0
\(69\) 14.5616 1.75300
\(70\) 0 0
\(71\) −8.00000 −0.949425 −0.474713 0.880141i \(-0.657448\pi\)
−0.474713 + 0.880141i \(0.657448\pi\)
\(72\) 0 0
\(73\) 14.8078 1.73312 0.866559 0.499075i \(-0.166327\pi\)
0.866559 + 0.499075i \(0.166327\pi\)
\(74\) 0 0
\(75\) −2.56155 −0.295783
\(76\) 0 0
\(77\) −18.2462 −2.07935
\(78\) 0 0
\(79\) 5.12311 0.576394 0.288197 0.957571i \(-0.406944\pi\)
0.288197 + 0.957571i \(0.406944\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −14.0000 −1.53670 −0.768350 0.640030i \(-0.778922\pi\)
−0.768350 + 0.640030i \(0.778922\pi\)
\(84\) 0 0
\(85\) 0.561553 0.0609090
\(86\) 0 0
\(87\) 8.80776 0.944291
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 11.6847 1.22489
\(92\) 0 0
\(93\) 13.1231 1.36080
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −1.12311 −0.114034 −0.0570170 0.998373i \(-0.518159\pi\)
−0.0570170 + 0.998373i \(0.518159\pi\)
\(98\) 0 0
\(99\) −14.2462 −1.43180
\(100\) 0 0
\(101\) −3.12311 −0.310761 −0.155380 0.987855i \(-0.549660\pi\)
−0.155380 + 0.987855i \(0.549660\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 11.6847 1.14031
\(106\) 0 0
\(107\) −11.6847 −1.12960 −0.564799 0.825228i \(-0.691046\pi\)
−0.564799 + 0.825228i \(0.691046\pi\)
\(108\) 0 0
\(109\) 11.4384 1.09560 0.547802 0.836608i \(-0.315465\pi\)
0.547802 + 0.836608i \(0.315465\pi\)
\(110\) 0 0
\(111\) 2.87689 0.273063
\(112\) 0 0
\(113\) −6.24621 −0.587594 −0.293797 0.955868i \(-0.594919\pi\)
−0.293797 + 0.955868i \(0.594919\pi\)
\(114\) 0 0
\(115\) 5.68466 0.530097
\(116\) 0 0
\(117\) 9.12311 0.843431
\(118\) 0 0
\(119\) −2.56155 −0.234817
\(120\) 0 0
\(121\) 5.00000 0.454545
\(122\) 0 0
\(123\) −21.1231 −1.90461
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −15.3693 −1.36381 −0.681903 0.731442i \(-0.738847\pi\)
−0.681903 + 0.731442i \(0.738847\pi\)
\(128\) 0 0
\(129\) 5.12311 0.451064
\(130\) 0 0
\(131\) −16.0000 −1.39793 −0.698963 0.715158i \(-0.746355\pi\)
−0.698963 + 0.715158i \(0.746355\pi\)
\(132\) 0 0
\(133\) −4.56155 −0.395537
\(134\) 0 0
\(135\) 1.43845 0.123802
\(136\) 0 0
\(137\) −2.80776 −0.239883 −0.119942 0.992781i \(-0.538271\pi\)
−0.119942 + 0.992781i \(0.538271\pi\)
\(138\) 0 0
\(139\) 2.24621 0.190521 0.0952606 0.995452i \(-0.469632\pi\)
0.0952606 + 0.995452i \(0.469632\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) −10.2462 −0.856831
\(144\) 0 0
\(145\) 3.43845 0.285547
\(146\) 0 0
\(147\) −35.3693 −2.91721
\(148\) 0 0
\(149\) 3.12311 0.255855 0.127927 0.991784i \(-0.459168\pi\)
0.127927 + 0.991784i \(0.459168\pi\)
\(150\) 0 0
\(151\) 13.1231 1.06794 0.533972 0.845502i \(-0.320699\pi\)
0.533972 + 0.845502i \(0.320699\pi\)
\(152\) 0 0
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) 5.12311 0.411498
\(156\) 0 0
\(157\) 6.00000 0.478852 0.239426 0.970915i \(-0.423041\pi\)
0.239426 + 0.970915i \(0.423041\pi\)
\(158\) 0 0
\(159\) −16.8078 −1.33294
\(160\) 0 0
\(161\) −25.9309 −2.04364
\(162\) 0 0
\(163\) 3.75379 0.294019 0.147010 0.989135i \(-0.453035\pi\)
0.147010 + 0.989135i \(0.453035\pi\)
\(164\) 0 0
\(165\) −10.2462 −0.797666
\(166\) 0 0
\(167\) −19.3693 −1.49884 −0.749421 0.662093i \(-0.769668\pi\)
−0.749421 + 0.662093i \(0.769668\pi\)
\(168\) 0 0
\(169\) −6.43845 −0.495265
\(170\) 0 0
\(171\) −3.56155 −0.272359
\(172\) 0 0
\(173\) 2.87689 0.218726 0.109363 0.994002i \(-0.465119\pi\)
0.109363 + 0.994002i \(0.465119\pi\)
\(174\) 0 0
\(175\) 4.56155 0.344821
\(176\) 0 0
\(177\) −32.8078 −2.46598
\(178\) 0 0
\(179\) 14.2462 1.06481 0.532406 0.846489i \(-0.321288\pi\)
0.532406 + 0.846489i \(0.321288\pi\)
\(180\) 0 0
\(181\) 7.75379 0.576335 0.288167 0.957580i \(-0.406954\pi\)
0.288167 + 0.957580i \(0.406954\pi\)
\(182\) 0 0
\(183\) 12.4924 0.923466
\(184\) 0 0
\(185\) 1.12311 0.0825724
\(186\) 0 0
\(187\) 2.24621 0.164259
\(188\) 0 0
\(189\) −6.56155 −0.477283
\(190\) 0 0
\(191\) 4.31534 0.312247 0.156124 0.987738i \(-0.450100\pi\)
0.156124 + 0.987738i \(0.450100\pi\)
\(192\) 0 0
\(193\) 9.12311 0.656696 0.328348 0.944557i \(-0.393508\pi\)
0.328348 + 0.944557i \(0.393508\pi\)
\(194\) 0 0
\(195\) 6.56155 0.469883
\(196\) 0 0
\(197\) −24.2462 −1.72747 −0.863736 0.503945i \(-0.831881\pi\)
−0.863736 + 0.503945i \(0.831881\pi\)
\(198\) 0 0
\(199\) −5.43845 −0.385521 −0.192761 0.981246i \(-0.561744\pi\)
−0.192761 + 0.981246i \(0.561744\pi\)
\(200\) 0 0
\(201\) 27.0540 1.90824
\(202\) 0 0
\(203\) −15.6847 −1.10085
\(204\) 0 0
\(205\) −8.24621 −0.575940
\(206\) 0 0
\(207\) −20.2462 −1.40721
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) 23.0540 1.58710 0.793551 0.608504i \(-0.208230\pi\)
0.793551 + 0.608504i \(0.208230\pi\)
\(212\) 0 0
\(213\) 20.4924 1.40412
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) −23.3693 −1.58641
\(218\) 0 0
\(219\) −37.9309 −2.56313
\(220\) 0 0
\(221\) −1.43845 −0.0967604
\(222\) 0 0
\(223\) 15.3693 1.02921 0.514603 0.857429i \(-0.327939\pi\)
0.514603 + 0.857429i \(0.327939\pi\)
\(224\) 0 0
\(225\) 3.56155 0.237437
\(226\) 0 0
\(227\) −4.31534 −0.286419 −0.143210 0.989692i \(-0.545742\pi\)
−0.143210 + 0.989692i \(0.545742\pi\)
\(228\) 0 0
\(229\) −28.7386 −1.89910 −0.949551 0.313612i \(-0.898461\pi\)
−0.949551 + 0.313612i \(0.898461\pi\)
\(230\) 0 0
\(231\) 46.7386 3.07518
\(232\) 0 0
\(233\) −26.4924 −1.73558 −0.867788 0.496934i \(-0.834459\pi\)
−0.867788 + 0.496934i \(0.834459\pi\)
\(234\) 0 0
\(235\) −3.12311 −0.203729
\(236\) 0 0
\(237\) −13.1231 −0.852437
\(238\) 0 0
\(239\) −3.19224 −0.206489 −0.103244 0.994656i \(-0.532922\pi\)
−0.103244 + 0.994656i \(0.532922\pi\)
\(240\) 0 0
\(241\) −29.8617 −1.92356 −0.961782 0.273817i \(-0.911714\pi\)
−0.961782 + 0.273817i \(0.911714\pi\)
\(242\) 0 0
\(243\) 22.2462 1.42710
\(244\) 0 0
\(245\) −13.8078 −0.882146
\(246\) 0 0
\(247\) −2.56155 −0.162988
\(248\) 0 0
\(249\) 35.8617 2.27265
\(250\) 0 0
\(251\) 5.75379 0.363176 0.181588 0.983375i \(-0.441876\pi\)
0.181588 + 0.983375i \(0.441876\pi\)
\(252\) 0 0
\(253\) 22.7386 1.42957
\(254\) 0 0
\(255\) −1.43845 −0.0900791
\(256\) 0 0
\(257\) −31.3693 −1.95676 −0.978382 0.206805i \(-0.933693\pi\)
−0.978382 + 0.206805i \(0.933693\pi\)
\(258\) 0 0
\(259\) −5.12311 −0.318334
\(260\) 0 0
\(261\) −12.2462 −0.758021
\(262\) 0 0
\(263\) −9.36932 −0.577737 −0.288868 0.957369i \(-0.593279\pi\)
−0.288868 + 0.957369i \(0.593279\pi\)
\(264\) 0 0
\(265\) −6.56155 −0.403073
\(266\) 0 0
\(267\) 25.6155 1.56764
\(268\) 0 0
\(269\) −24.2462 −1.47832 −0.739159 0.673531i \(-0.764777\pi\)
−0.739159 + 0.673531i \(0.764777\pi\)
\(270\) 0 0
\(271\) −20.1771 −1.22567 −0.612835 0.790211i \(-0.709971\pi\)
−0.612835 + 0.790211i \(0.709971\pi\)
\(272\) 0 0
\(273\) −29.9309 −1.81150
\(274\) 0 0
\(275\) −4.00000 −0.241209
\(276\) 0 0
\(277\) −19.6155 −1.17858 −0.589291 0.807921i \(-0.700593\pi\)
−0.589291 + 0.807921i \(0.700593\pi\)
\(278\) 0 0
\(279\) −18.2462 −1.09237
\(280\) 0 0
\(281\) 24.2462 1.44641 0.723204 0.690635i \(-0.242669\pi\)
0.723204 + 0.690635i \(0.242669\pi\)
\(282\) 0 0
\(283\) −6.00000 −0.356663 −0.178331 0.983970i \(-0.557070\pi\)
−0.178331 + 0.983970i \(0.557070\pi\)
\(284\) 0 0
\(285\) −2.56155 −0.151733
\(286\) 0 0
\(287\) 37.6155 2.22037
\(288\) 0 0
\(289\) −16.6847 −0.981450
\(290\) 0 0
\(291\) 2.87689 0.168647
\(292\) 0 0
\(293\) 5.93087 0.346485 0.173243 0.984879i \(-0.444576\pi\)
0.173243 + 0.984879i \(0.444576\pi\)
\(294\) 0 0
\(295\) −12.8078 −0.745697
\(296\) 0 0
\(297\) 5.75379 0.333869
\(298\) 0 0
\(299\) −14.5616 −0.842116
\(300\) 0 0
\(301\) −9.12311 −0.525847
\(302\) 0 0
\(303\) 8.00000 0.459588
\(304\) 0 0
\(305\) 4.87689 0.279250
\(306\) 0 0
\(307\) −5.75379 −0.328386 −0.164193 0.986428i \(-0.552502\pi\)
−0.164193 + 0.986428i \(0.552502\pi\)
\(308\) 0 0
\(309\) 40.9848 2.33155
\(310\) 0 0
\(311\) −5.43845 −0.308386 −0.154193 0.988041i \(-0.549278\pi\)
−0.154193 + 0.988041i \(0.549278\pi\)
\(312\) 0 0
\(313\) −30.1771 −1.70571 −0.852855 0.522148i \(-0.825131\pi\)
−0.852855 + 0.522148i \(0.825131\pi\)
\(314\) 0 0
\(315\) −16.2462 −0.915370
\(316\) 0 0
\(317\) 5.43845 0.305454 0.152727 0.988268i \(-0.451195\pi\)
0.152727 + 0.988268i \(0.451195\pi\)
\(318\) 0 0
\(319\) 13.7538 0.770064
\(320\) 0 0
\(321\) 29.9309 1.67058
\(322\) 0 0
\(323\) 0.561553 0.0312456
\(324\) 0 0
\(325\) 2.56155 0.142089
\(326\) 0 0
\(327\) −29.3002 −1.62030
\(328\) 0 0
\(329\) 14.2462 0.785419
\(330\) 0 0
\(331\) −12.8078 −0.703978 −0.351989 0.936004i \(-0.614495\pi\)
−0.351989 + 0.936004i \(0.614495\pi\)
\(332\) 0 0
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) 10.5616 0.577039
\(336\) 0 0
\(337\) 14.8769 0.810396 0.405198 0.914229i \(-0.367203\pi\)
0.405198 + 0.914229i \(0.367203\pi\)
\(338\) 0 0
\(339\) 16.0000 0.869001
\(340\) 0 0
\(341\) 20.4924 1.10973
\(342\) 0 0
\(343\) 31.0540 1.67676
\(344\) 0 0
\(345\) −14.5616 −0.783968
\(346\) 0 0
\(347\) −18.0000 −0.966291 −0.483145 0.875540i \(-0.660506\pi\)
−0.483145 + 0.875540i \(0.660506\pi\)
\(348\) 0 0
\(349\) −2.00000 −0.107058 −0.0535288 0.998566i \(-0.517047\pi\)
−0.0535288 + 0.998566i \(0.517047\pi\)
\(350\) 0 0
\(351\) −3.68466 −0.196673
\(352\) 0 0
\(353\) −36.4233 −1.93862 −0.969308 0.245849i \(-0.920933\pi\)
−0.969308 + 0.245849i \(0.920933\pi\)
\(354\) 0 0
\(355\) 8.00000 0.424596
\(356\) 0 0
\(357\) 6.56155 0.347274
\(358\) 0 0
\(359\) 5.93087 0.313019 0.156510 0.987676i \(-0.449976\pi\)
0.156510 + 0.987676i \(0.449976\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −12.8078 −0.672233
\(364\) 0 0
\(365\) −14.8078 −0.775074
\(366\) 0 0
\(367\) 35.1231 1.83341 0.916706 0.399563i \(-0.130838\pi\)
0.916706 + 0.399563i \(0.130838\pi\)
\(368\) 0 0
\(369\) 29.3693 1.52891
\(370\) 0 0
\(371\) 29.9309 1.55393
\(372\) 0 0
\(373\) −24.8078 −1.28450 −0.642249 0.766496i \(-0.721998\pi\)
−0.642249 + 0.766496i \(0.721998\pi\)
\(374\) 0 0
\(375\) 2.56155 0.132278
\(376\) 0 0
\(377\) −8.80776 −0.453623
\(378\) 0 0
\(379\) 8.31534 0.427130 0.213565 0.976929i \(-0.431492\pi\)
0.213565 + 0.976929i \(0.431492\pi\)
\(380\) 0 0
\(381\) 39.3693 2.01695
\(382\) 0 0
\(383\) 26.7386 1.36628 0.683140 0.730287i \(-0.260614\pi\)
0.683140 + 0.730287i \(0.260614\pi\)
\(384\) 0 0
\(385\) 18.2462 0.929913
\(386\) 0 0
\(387\) −7.12311 −0.362088
\(388\) 0 0
\(389\) 7.12311 0.361156 0.180578 0.983561i \(-0.442203\pi\)
0.180578 + 0.983561i \(0.442203\pi\)
\(390\) 0 0
\(391\) 3.19224 0.161438
\(392\) 0 0
\(393\) 40.9848 2.06741
\(394\) 0 0
\(395\) −5.12311 −0.257771
\(396\) 0 0
\(397\) 25.8617 1.29796 0.648982 0.760804i \(-0.275195\pi\)
0.648982 + 0.760804i \(0.275195\pi\)
\(398\) 0 0
\(399\) 11.6847 0.584965
\(400\) 0 0
\(401\) −20.7386 −1.03564 −0.517819 0.855490i \(-0.673256\pi\)
−0.517819 + 0.855490i \(0.673256\pi\)
\(402\) 0 0
\(403\) −13.1231 −0.653708
\(404\) 0 0
\(405\) 7.00000 0.347833
\(406\) 0 0
\(407\) 4.49242 0.222681
\(408\) 0 0
\(409\) 24.7386 1.22325 0.611623 0.791149i \(-0.290517\pi\)
0.611623 + 0.791149i \(0.290517\pi\)
\(410\) 0 0
\(411\) 7.19224 0.354767
\(412\) 0 0
\(413\) 58.4233 2.87482
\(414\) 0 0
\(415\) 14.0000 0.687233
\(416\) 0 0
\(417\) −5.75379 −0.281764
\(418\) 0 0
\(419\) −22.8769 −1.11761 −0.558805 0.829299i \(-0.688740\pi\)
−0.558805 + 0.829299i \(0.688740\pi\)
\(420\) 0 0
\(421\) 19.3002 0.940634 0.470317 0.882498i \(-0.344140\pi\)
0.470317 + 0.882498i \(0.344140\pi\)
\(422\) 0 0
\(423\) 11.1231 0.540824
\(424\) 0 0
\(425\) −0.561553 −0.0272393
\(426\) 0 0
\(427\) −22.2462 −1.07657
\(428\) 0 0
\(429\) 26.2462 1.26718
\(430\) 0 0
\(431\) 20.4924 0.987085 0.493543 0.869722i \(-0.335702\pi\)
0.493543 + 0.869722i \(0.335702\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 0 0
\(435\) −8.80776 −0.422300
\(436\) 0 0
\(437\) 5.68466 0.271934
\(438\) 0 0
\(439\) 31.3693 1.49718 0.748588 0.663036i \(-0.230732\pi\)
0.748588 + 0.663036i \(0.230732\pi\)
\(440\) 0 0
\(441\) 49.1771 2.34177
\(442\) 0 0
\(443\) 11.6155 0.551870 0.275935 0.961176i \(-0.411012\pi\)
0.275935 + 0.961176i \(0.411012\pi\)
\(444\) 0 0
\(445\) 10.0000 0.474045
\(446\) 0 0
\(447\) −8.00000 −0.378387
\(448\) 0 0
\(449\) 16.7386 0.789945 0.394972 0.918693i \(-0.370754\pi\)
0.394972 + 0.918693i \(0.370754\pi\)
\(450\) 0 0
\(451\) −32.9848 −1.55320
\(452\) 0 0
\(453\) −33.6155 −1.57940
\(454\) 0 0
\(455\) −11.6847 −0.547785
\(456\) 0 0
\(457\) 16.5616 0.774717 0.387358 0.921929i \(-0.373388\pi\)
0.387358 + 0.921929i \(0.373388\pi\)
\(458\) 0 0
\(459\) 0.807764 0.0377032
\(460\) 0 0
\(461\) 24.2462 1.12926 0.564629 0.825345i \(-0.309019\pi\)
0.564629 + 0.825345i \(0.309019\pi\)
\(462\) 0 0
\(463\) −21.8617 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(464\) 0 0
\(465\) −13.1231 −0.608569
\(466\) 0 0
\(467\) −29.8617 −1.38184 −0.690918 0.722933i \(-0.742794\pi\)
−0.690918 + 0.722933i \(0.742794\pi\)
\(468\) 0 0
\(469\) −48.1771 −2.22461
\(470\) 0 0
\(471\) −15.3693 −0.708181
\(472\) 0 0
\(473\) 8.00000 0.367840
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) 23.3693 1.07001
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) −2.87689 −0.131175
\(482\) 0 0
\(483\) 66.4233 3.02236
\(484\) 0 0
\(485\) 1.12311 0.0509976
\(486\) 0 0
\(487\) 27.3693 1.24022 0.620111 0.784514i \(-0.287088\pi\)
0.620111 + 0.784514i \(0.287088\pi\)
\(488\) 0 0
\(489\) −9.61553 −0.434829
\(490\) 0 0
\(491\) −2.87689 −0.129832 −0.0649162 0.997891i \(-0.520678\pi\)
−0.0649162 + 0.997891i \(0.520678\pi\)
\(492\) 0 0
\(493\) 1.93087 0.0869620
\(494\) 0 0
\(495\) 14.2462 0.640320
\(496\) 0 0
\(497\) −36.4924 −1.63691
\(498\) 0 0
\(499\) 29.1231 1.30373 0.651865 0.758335i \(-0.273987\pi\)
0.651865 + 0.758335i \(0.273987\pi\)
\(500\) 0 0
\(501\) 49.6155 2.21666
\(502\) 0 0
\(503\) −36.4233 −1.62403 −0.812017 0.583634i \(-0.801630\pi\)
−0.812017 + 0.583634i \(0.801630\pi\)
\(504\) 0 0
\(505\) 3.12311 0.138976
\(506\) 0 0
\(507\) 16.4924 0.732454
\(508\) 0 0
\(509\) −2.00000 −0.0886484 −0.0443242 0.999017i \(-0.514113\pi\)
−0.0443242 + 0.999017i \(0.514113\pi\)
\(510\) 0 0
\(511\) 67.5464 2.98808
\(512\) 0 0
\(513\) 1.43845 0.0635090
\(514\) 0 0
\(515\) 16.0000 0.705044
\(516\) 0 0
\(517\) −12.4924 −0.549416
\(518\) 0 0
\(519\) −7.36932 −0.323477
\(520\) 0 0
\(521\) 7.12311 0.312069 0.156034 0.987752i \(-0.450129\pi\)
0.156034 + 0.987752i \(0.450129\pi\)
\(522\) 0 0
\(523\) −13.9309 −0.609154 −0.304577 0.952488i \(-0.598515\pi\)
−0.304577 + 0.952488i \(0.598515\pi\)
\(524\) 0 0
\(525\) −11.6847 −0.509960
\(526\) 0 0
\(527\) 2.87689 0.125319
\(528\) 0 0
\(529\) 9.31534 0.405015
\(530\) 0 0
\(531\) 45.6155 1.97955
\(532\) 0 0
\(533\) 21.1231 0.914943
\(534\) 0 0
\(535\) 11.6847 0.505172
\(536\) 0 0
\(537\) −36.4924 −1.57476
\(538\) 0 0
\(539\) −55.2311 −2.37897
\(540\) 0 0
\(541\) −16.8769 −0.725594 −0.362797 0.931868i \(-0.618178\pi\)
−0.362797 + 0.931868i \(0.618178\pi\)
\(542\) 0 0
\(543\) −19.8617 −0.852349
\(544\) 0 0
\(545\) −11.4384 −0.489969
\(546\) 0 0
\(547\) −21.7538 −0.930125 −0.465062 0.885278i \(-0.653968\pi\)
−0.465062 + 0.885278i \(0.653968\pi\)
\(548\) 0 0
\(549\) −17.3693 −0.741304
\(550\) 0 0
\(551\) 3.43845 0.146483
\(552\) 0 0
\(553\) 23.3693 0.993764
\(554\) 0 0
\(555\) −2.87689 −0.122117
\(556\) 0 0
\(557\) 29.3693 1.24442 0.622209 0.782851i \(-0.286235\pi\)
0.622209 + 0.782851i \(0.286235\pi\)
\(558\) 0 0
\(559\) −5.12311 −0.216684
\(560\) 0 0
\(561\) −5.75379 −0.242925
\(562\) 0 0
\(563\) 30.7386 1.29548 0.647739 0.761862i \(-0.275715\pi\)
0.647739 + 0.761862i \(0.275715\pi\)
\(564\) 0 0
\(565\) 6.24621 0.262780
\(566\) 0 0
\(567\) −31.9309 −1.34097
\(568\) 0 0
\(569\) −0.876894 −0.0367613 −0.0183807 0.999831i \(-0.505851\pi\)
−0.0183807 + 0.999831i \(0.505851\pi\)
\(570\) 0 0
\(571\) 27.3693 1.14537 0.572685 0.819775i \(-0.305902\pi\)
0.572685 + 0.819775i \(0.305902\pi\)
\(572\) 0 0
\(573\) −11.0540 −0.461786
\(574\) 0 0
\(575\) −5.68466 −0.237067
\(576\) 0 0
\(577\) 28.4233 1.18328 0.591639 0.806203i \(-0.298481\pi\)
0.591639 + 0.806203i \(0.298481\pi\)
\(578\) 0 0
\(579\) −23.3693 −0.971196
\(580\) 0 0
\(581\) −63.8617 −2.64943
\(582\) 0 0
\(583\) −26.2462 −1.08701
\(584\) 0 0
\(585\) −9.12311 −0.377194
\(586\) 0 0
\(587\) −15.7538 −0.650228 −0.325114 0.945675i \(-0.605403\pi\)
−0.325114 + 0.945675i \(0.605403\pi\)
\(588\) 0 0
\(589\) 5.12311 0.211094
\(590\) 0 0
\(591\) 62.1080 2.55478
\(592\) 0 0
\(593\) −8.24621 −0.338631 −0.169316 0.985562i \(-0.554156\pi\)
−0.169316 + 0.985562i \(0.554156\pi\)
\(594\) 0 0
\(595\) 2.56155 0.105013
\(596\) 0 0
\(597\) 13.9309 0.570153
\(598\) 0 0
\(599\) 9.61553 0.392880 0.196440 0.980516i \(-0.437062\pi\)
0.196440 + 0.980516i \(0.437062\pi\)
\(600\) 0 0
\(601\) −48.7386 −1.98809 −0.994045 0.108969i \(-0.965245\pi\)
−0.994045 + 0.108969i \(0.965245\pi\)
\(602\) 0 0
\(603\) −37.6155 −1.53182
\(604\) 0 0
\(605\) −5.00000 −0.203279
\(606\) 0 0
\(607\) −9.12311 −0.370295 −0.185148 0.982711i \(-0.559276\pi\)
−0.185148 + 0.982711i \(0.559276\pi\)
\(608\) 0 0
\(609\) 40.1771 1.62806
\(610\) 0 0
\(611\) 8.00000 0.323645
\(612\) 0 0
\(613\) −41.3693 −1.67089 −0.835445 0.549573i \(-0.814790\pi\)
−0.835445 + 0.549573i \(0.814790\pi\)
\(614\) 0 0
\(615\) 21.1231 0.851766
\(616\) 0 0
\(617\) 22.9848 0.925335 0.462668 0.886532i \(-0.346892\pi\)
0.462668 + 0.886532i \(0.346892\pi\)
\(618\) 0 0
\(619\) −2.24621 −0.0902829 −0.0451414 0.998981i \(-0.514374\pi\)
−0.0451414 + 0.998981i \(0.514374\pi\)
\(620\) 0 0
\(621\) 8.17708 0.328135
\(622\) 0 0
\(623\) −45.6155 −1.82755
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −10.2462 −0.409194
\(628\) 0 0
\(629\) 0.630683 0.0251470
\(630\) 0 0
\(631\) −26.7386 −1.06445 −0.532224 0.846604i \(-0.678644\pi\)
−0.532224 + 0.846604i \(0.678644\pi\)
\(632\) 0 0
\(633\) −59.0540 −2.34718
\(634\) 0 0
\(635\) 15.3693 0.609913
\(636\) 0 0
\(637\) 35.3693 1.40138
\(638\) 0 0
\(639\) −28.4924 −1.12714
\(640\) 0 0
\(641\) −7.12311 −0.281346 −0.140673 0.990056i \(-0.544927\pi\)
−0.140673 + 0.990056i \(0.544927\pi\)
\(642\) 0 0
\(643\) 14.4924 0.571525 0.285763 0.958300i \(-0.407753\pi\)
0.285763 + 0.958300i \(0.407753\pi\)
\(644\) 0 0
\(645\) −5.12311 −0.201722
\(646\) 0 0
\(647\) −17.6847 −0.695256 −0.347628 0.937633i \(-0.613013\pi\)
−0.347628 + 0.937633i \(0.613013\pi\)
\(648\) 0 0
\(649\) −51.2311 −2.01099
\(650\) 0 0
\(651\) 59.8617 2.34617
\(652\) 0 0
\(653\) 45.3693 1.77544 0.887719 0.460385i \(-0.152289\pi\)
0.887719 + 0.460385i \(0.152289\pi\)
\(654\) 0 0
\(655\) 16.0000 0.625172
\(656\) 0 0
\(657\) 52.7386 2.05753
\(658\) 0 0
\(659\) 27.1922 1.05926 0.529630 0.848229i \(-0.322331\pi\)
0.529630 + 0.848229i \(0.322331\pi\)
\(660\) 0 0
\(661\) 19.9309 0.775221 0.387610 0.921823i \(-0.373301\pi\)
0.387610 + 0.921823i \(0.373301\pi\)
\(662\) 0 0
\(663\) 3.68466 0.143100
\(664\) 0 0
\(665\) 4.56155 0.176889
\(666\) 0 0
\(667\) 19.5464 0.756840
\(668\) 0 0
\(669\) −39.3693 −1.52211
\(670\) 0 0
\(671\) 19.5076 0.753082
\(672\) 0 0
\(673\) −36.4924 −1.40668 −0.703340 0.710854i \(-0.748309\pi\)
−0.703340 + 0.710854i \(0.748309\pi\)
\(674\) 0 0
\(675\) −1.43845 −0.0553659
\(676\) 0 0
\(677\) 15.1922 0.583885 0.291943 0.956436i \(-0.405698\pi\)
0.291943 + 0.956436i \(0.405698\pi\)
\(678\) 0 0
\(679\) −5.12311 −0.196607
\(680\) 0 0
\(681\) 11.0540 0.423589
\(682\) 0 0
\(683\) 34.2462 1.31039 0.655197 0.755458i \(-0.272585\pi\)
0.655197 + 0.755458i \(0.272585\pi\)
\(684\) 0 0
\(685\) 2.80776 0.107279
\(686\) 0 0
\(687\) 73.6155 2.80861
\(688\) 0 0
\(689\) 16.8078 0.640325
\(690\) 0 0
\(691\) −34.1080 −1.29753 −0.648764 0.760990i \(-0.724714\pi\)
−0.648764 + 0.760990i \(0.724714\pi\)
\(692\) 0 0
\(693\) −64.9848 −2.46857
\(694\) 0 0
\(695\) −2.24621 −0.0852036
\(696\) 0 0
\(697\) −4.63068 −0.175400
\(698\) 0 0
\(699\) 67.8617 2.56677
\(700\) 0 0
\(701\) 11.1231 0.420114 0.210057 0.977689i \(-0.432635\pi\)
0.210057 + 0.977689i \(0.432635\pi\)
\(702\) 0 0
\(703\) 1.12311 0.0423587
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 0 0
\(707\) −14.2462 −0.535784
\(708\) 0 0
\(709\) −26.0000 −0.976450 −0.488225 0.872718i \(-0.662356\pi\)
−0.488225 + 0.872718i \(0.662356\pi\)
\(710\) 0 0
\(711\) 18.2462 0.684286
\(712\) 0 0
\(713\) 29.1231 1.09067
\(714\) 0 0
\(715\) 10.2462 0.383187
\(716\) 0 0
\(717\) 8.17708 0.305379
\(718\) 0 0
\(719\) −34.4233 −1.28377 −0.641886 0.766800i \(-0.721848\pi\)
−0.641886 + 0.766800i \(0.721848\pi\)
\(720\) 0 0
\(721\) −72.9848 −2.71810
\(722\) 0 0
\(723\) 76.4924 2.84478
\(724\) 0 0
\(725\) −3.43845 −0.127701
\(726\) 0 0
\(727\) −13.6847 −0.507536 −0.253768 0.967265i \(-0.581670\pi\)
−0.253768 + 0.967265i \(0.581670\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) 1.12311 0.0415396
\(732\) 0 0
\(733\) −4.38447 −0.161944 −0.0809721 0.996716i \(-0.525802\pi\)
−0.0809721 + 0.996716i \(0.525802\pi\)
\(734\) 0 0
\(735\) 35.3693 1.30462
\(736\) 0 0
\(737\) 42.2462 1.55616
\(738\) 0 0
\(739\) 17.1231 0.629884 0.314942 0.949111i \(-0.398015\pi\)
0.314942 + 0.949111i \(0.398015\pi\)
\(740\) 0 0
\(741\) 6.56155 0.241045
\(742\) 0 0
\(743\) −35.3693 −1.29757 −0.648787 0.760970i \(-0.724723\pi\)
−0.648787 + 0.760970i \(0.724723\pi\)
\(744\) 0 0
\(745\) −3.12311 −0.114422
\(746\) 0 0
\(747\) −49.8617 −1.82435
\(748\) 0 0
\(749\) −53.3002 −1.94755
\(750\) 0 0
\(751\) 31.3693 1.14468 0.572341 0.820015i \(-0.306035\pi\)
0.572341 + 0.820015i \(0.306035\pi\)
\(752\) 0 0
\(753\) −14.7386 −0.537106
\(754\) 0 0
\(755\) −13.1231 −0.477599
\(756\) 0 0
\(757\) −12.7386 −0.462994 −0.231497 0.972836i \(-0.574362\pi\)
−0.231497 + 0.972836i \(0.574362\pi\)
\(758\) 0 0
\(759\) −58.2462 −2.11420
\(760\) 0 0
\(761\) 18.3153 0.663931 0.331965 0.943292i \(-0.392288\pi\)
0.331965 + 0.943292i \(0.392288\pi\)
\(762\) 0 0
\(763\) 52.1771 1.88894
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) 32.8078 1.18462
\(768\) 0 0
\(769\) 38.1771 1.37670 0.688350 0.725378i \(-0.258335\pi\)
0.688350 + 0.725378i \(0.258335\pi\)
\(770\) 0 0
\(771\) 80.3542 2.89388
\(772\) 0 0
\(773\) 20.1771 0.725719 0.362860 0.931844i \(-0.381800\pi\)
0.362860 + 0.931844i \(0.381800\pi\)
\(774\) 0 0
\(775\) −5.12311 −0.184027
\(776\) 0 0
\(777\) 13.1231 0.470789
\(778\) 0 0
\(779\) −8.24621 −0.295451
\(780\) 0 0
\(781\) 32.0000 1.14505
\(782\) 0 0
\(783\) 4.94602 0.176757
\(784\) 0 0
\(785\) −6.00000 −0.214149
\(786\) 0 0
\(787\) −35.0540 −1.24954 −0.624770 0.780809i \(-0.714807\pi\)
−0.624770 + 0.780809i \(0.714807\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) −28.4924 −1.01307
\(792\) 0 0
\(793\) −12.4924 −0.443619
\(794\) 0 0
\(795\) 16.8078 0.596110
\(796\) 0 0
\(797\) 24.8078 0.878736 0.439368 0.898307i \(-0.355202\pi\)
0.439368 + 0.898307i \(0.355202\pi\)
\(798\) 0 0
\(799\) −1.75379 −0.0620446
\(800\) 0 0
\(801\) −35.6155 −1.25841
\(802\) 0 0
\(803\) −59.2311 −2.09022
\(804\) 0 0
\(805\) 25.9309 0.913943
\(806\) 0 0
\(807\) 62.1080 2.18630
\(808\) 0 0
\(809\) 39.9309 1.40389 0.701947 0.712229i \(-0.252314\pi\)
0.701947 + 0.712229i \(0.252314\pi\)
\(810\) 0 0
\(811\) −48.0388 −1.68687 −0.843436 0.537230i \(-0.819471\pi\)
−0.843436 + 0.537230i \(0.819471\pi\)
\(812\) 0 0
\(813\) 51.6847 1.81266
\(814\) 0 0
\(815\) −3.75379 −0.131489
\(816\) 0 0
\(817\) 2.00000 0.0699711
\(818\) 0 0
\(819\) 41.6155 1.45416
\(820\) 0 0
\(821\) 4.87689 0.170205 0.0851024 0.996372i \(-0.472878\pi\)
0.0851024 + 0.996372i \(0.472878\pi\)
\(822\) 0 0
\(823\) 20.4233 0.711911 0.355956 0.934503i \(-0.384155\pi\)
0.355956 + 0.934503i \(0.384155\pi\)
\(824\) 0 0
\(825\) 10.2462 0.356727
\(826\) 0 0
\(827\) −39.0540 −1.35804 −0.679020 0.734120i \(-0.737595\pi\)
−0.679020 + 0.734120i \(0.737595\pi\)
\(828\) 0 0
\(829\) 43.4384 1.50868 0.754340 0.656484i \(-0.227957\pi\)
0.754340 + 0.656484i \(0.227957\pi\)
\(830\) 0 0
\(831\) 50.2462 1.74302
\(832\) 0 0
\(833\) −7.75379 −0.268653
\(834\) 0 0
\(835\) 19.3693 0.670303
\(836\) 0 0
\(837\) 7.36932 0.254721
\(838\) 0 0
\(839\) 35.2311 1.21631 0.608156 0.793818i \(-0.291910\pi\)
0.608156 + 0.793818i \(0.291910\pi\)
\(840\) 0 0
\(841\) −17.1771 −0.592313
\(842\) 0 0
\(843\) −62.1080 −2.13911
\(844\) 0 0
\(845\) 6.43845 0.221489
\(846\) 0 0
\(847\) 22.8078 0.783684
\(848\) 0 0
\(849\) 15.3693 0.527474
\(850\) 0 0
\(851\) 6.38447 0.218857
\(852\) 0 0
\(853\) −44.7386 −1.53182 −0.765911 0.642947i \(-0.777712\pi\)
−0.765911 + 0.642947i \(0.777712\pi\)
\(854\) 0 0
\(855\) 3.56155 0.121803
\(856\) 0 0
\(857\) −24.9848 −0.853466 −0.426733 0.904378i \(-0.640336\pi\)
−0.426733 + 0.904378i \(0.640336\pi\)
\(858\) 0 0
\(859\) −5.26137 −0.179515 −0.0897577 0.995964i \(-0.528609\pi\)
−0.0897577 + 0.995964i \(0.528609\pi\)
\(860\) 0 0
\(861\) −96.3542 −3.28374
\(862\) 0 0
\(863\) 49.4773 1.68423 0.842113 0.539301i \(-0.181312\pi\)
0.842113 + 0.539301i \(0.181312\pi\)
\(864\) 0 0
\(865\) −2.87689 −0.0978173
\(866\) 0 0
\(867\) 42.7386 1.45148
\(868\) 0 0
\(869\) −20.4924 −0.695158
\(870\) 0 0
\(871\) −27.0540 −0.916689
\(872\) 0 0
\(873\) −4.00000 −0.135379
\(874\) 0 0
\(875\) −4.56155 −0.154209
\(876\) 0 0
\(877\) −24.8078 −0.837699 −0.418849 0.908056i \(-0.637566\pi\)
−0.418849 + 0.908056i \(0.637566\pi\)
\(878\) 0 0
\(879\) −15.1922 −0.512421
\(880\) 0 0
\(881\) −26.4924 −0.892552 −0.446276 0.894895i \(-0.647250\pi\)
−0.446276 + 0.894895i \(0.647250\pi\)
\(882\) 0 0
\(883\) 19.7538 0.664768 0.332384 0.943144i \(-0.392147\pi\)
0.332384 + 0.943144i \(0.392147\pi\)
\(884\) 0 0
\(885\) 32.8078 1.10282
\(886\) 0 0
\(887\) −24.0000 −0.805841 −0.402921 0.915235i \(-0.632005\pi\)
−0.402921 + 0.915235i \(0.632005\pi\)
\(888\) 0 0
\(889\) −70.1080 −2.35135
\(890\) 0 0
\(891\) 28.0000 0.938035
\(892\) 0 0
\(893\) −3.12311 −0.104511
\(894\) 0 0
\(895\) −14.2462 −0.476198
\(896\) 0 0
\(897\) 37.3002 1.24542
\(898\) 0 0
\(899\) 17.6155 0.587511
\(900\) 0 0
\(901\) −3.68466 −0.122754
\(902\) 0 0
\(903\) 23.3693 0.777682
\(904\) 0 0
\(905\) −7.75379 −0.257745
\(906\) 0 0
\(907\) −42.4233 −1.40864 −0.704321 0.709881i \(-0.748748\pi\)
−0.704321 + 0.709881i \(0.748748\pi\)
\(908\) 0 0
\(909\) −11.1231 −0.368930
\(910\) 0 0
\(911\) −22.7386 −0.753365 −0.376682 0.926343i \(-0.622935\pi\)
−0.376682 + 0.926343i \(0.622935\pi\)
\(912\) 0 0
\(913\) 56.0000 1.85333
\(914\) 0 0
\(915\) −12.4924 −0.412987
\(916\) 0 0
\(917\) −72.9848 −2.41017
\(918\) 0 0
\(919\) −20.8078 −0.686385 −0.343192 0.939265i \(-0.611508\pi\)
−0.343192 + 0.939265i \(0.611508\pi\)
\(920\) 0 0
\(921\) 14.7386 0.485654
\(922\) 0 0
\(923\) −20.4924 −0.674516
\(924\) 0 0
\(925\) −1.12311 −0.0369275
\(926\) 0 0
\(927\) −56.9848 −1.87163
\(928\) 0 0
\(929\) −5.05398 −0.165816 −0.0829078 0.996557i \(-0.526421\pi\)
−0.0829078 + 0.996557i \(0.526421\pi\)
\(930\) 0 0
\(931\) −13.8078 −0.452531
\(932\) 0 0
\(933\) 13.9309 0.456076
\(934\) 0 0
\(935\) −2.24621 −0.0734590
\(936\) 0 0
\(937\) 17.0540 0.557129 0.278565 0.960418i \(-0.410141\pi\)
0.278565 + 0.960418i \(0.410141\pi\)
\(938\) 0 0
\(939\) 77.3002 2.52260
\(940\) 0 0
\(941\) 14.3153 0.466667 0.233333 0.972397i \(-0.425037\pi\)
0.233333 + 0.972397i \(0.425037\pi\)
\(942\) 0 0
\(943\) −46.8769 −1.52652
\(944\) 0 0
\(945\) 6.56155 0.213447
\(946\) 0 0
\(947\) −33.3693 −1.08436 −0.542179 0.840263i \(-0.682400\pi\)
−0.542179 + 0.840263i \(0.682400\pi\)
\(948\) 0 0
\(949\) 37.9309 1.23129
\(950\) 0 0
\(951\) −13.9309 −0.451739
\(952\) 0 0
\(953\) 51.2311 1.65954 0.829768 0.558108i \(-0.188473\pi\)
0.829768 + 0.558108i \(0.188473\pi\)
\(954\) 0 0
\(955\) −4.31534 −0.139641
\(956\) 0 0
\(957\) −35.2311 −1.13886
\(958\) 0 0
\(959\) −12.8078 −0.413584
\(960\) 0 0
\(961\) −4.75379 −0.153348
\(962\) 0 0
\(963\) −41.6155 −1.34104
\(964\) 0 0
\(965\) −9.12311 −0.293683
\(966\) 0 0
\(967\) 39.1231 1.25811 0.629057 0.777359i \(-0.283441\pi\)
0.629057 + 0.777359i \(0.283441\pi\)
\(968\) 0 0
\(969\) −1.43845 −0.0462096
\(970\) 0 0
\(971\) 12.9848 0.416704 0.208352 0.978054i \(-0.433190\pi\)
0.208352 + 0.978054i \(0.433190\pi\)
\(972\) 0 0
\(973\) 10.2462 0.328478
\(974\) 0 0
\(975\) −6.56155 −0.210138
\(976\) 0 0
\(977\) 5.12311 0.163903 0.0819513 0.996636i \(-0.473885\pi\)
0.0819513 + 0.996636i \(0.473885\pi\)
\(978\) 0 0
\(979\) 40.0000 1.27841
\(980\) 0 0
\(981\) 40.7386 1.30068
\(982\) 0 0
\(983\) 29.6155 0.944589 0.472294 0.881441i \(-0.343426\pi\)
0.472294 + 0.881441i \(0.343426\pi\)
\(984\) 0 0
\(985\) 24.2462 0.772549
\(986\) 0 0
\(987\) −36.4924 −1.16157
\(988\) 0 0
\(989\) 11.3693 0.361523
\(990\) 0 0
\(991\) 21.1231 0.670998 0.335499 0.942041i \(-0.391095\pi\)
0.335499 + 0.942041i \(0.391095\pi\)
\(992\) 0 0
\(993\) 32.8078 1.04112
\(994\) 0 0
\(995\) 5.43845 0.172410
\(996\) 0 0
\(997\) 8.24621 0.261160 0.130580 0.991438i \(-0.458316\pi\)
0.130580 + 0.991438i \(0.458316\pi\)
\(998\) 0 0
\(999\) 1.61553 0.0511130
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.e.1.1 2
4.3 odd 2 3040.2.a.h.1.2 yes 2
8.3 odd 2 6080.2.a.bc.1.1 2
8.5 even 2 6080.2.a.bi.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.e.1.1 2 1.1 even 1 trivial
3040.2.a.h.1.2 yes 2 4.3 odd 2
6080.2.a.bc.1.1 2 8.3 odd 2
6080.2.a.bi.1.2 2 8.5 even 2