Properties

Label 8020.2.a.e.1.5
Level $8020$
Weight $2$
Character 8020.1
Self dual yes
Analytic conductor $64.040$
Analytic rank $0$
Dimension $35$
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] = [8020,2,Mod(1,8020)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8020, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8020.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8020 = 2^{2} \cdot 5 \cdot 401 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8020.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: \(64.0400224211\)
Analytic rank: \(0\)
Dimension: \(35\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 8020.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.91827 q^{3} -1.00000 q^{5} -4.31238 q^{7} +5.51629 q^{9} +O(q^{10})\) \(q-2.91827 q^{3} -1.00000 q^{5} -4.31238 q^{7} +5.51629 q^{9} +3.32037 q^{11} +5.27795 q^{13} +2.91827 q^{15} +5.57087 q^{17} +2.74798 q^{19} +12.5847 q^{21} +1.03475 q^{23} +1.00000 q^{25} -7.34323 q^{27} -0.925860 q^{29} +8.47043 q^{31} -9.68973 q^{33} +4.31238 q^{35} +1.78281 q^{37} -15.4025 q^{39} +9.98377 q^{41} +0.674120 q^{43} -5.51629 q^{45} +1.41456 q^{47} +11.5966 q^{49} -16.2573 q^{51} +2.67960 q^{53} -3.32037 q^{55} -8.01933 q^{57} +12.9693 q^{59} +5.37423 q^{61} -23.7884 q^{63} -5.27795 q^{65} +9.52502 q^{67} -3.01968 q^{69} -3.09924 q^{71} -5.83066 q^{73} -2.91827 q^{75} -14.3187 q^{77} -12.9855 q^{79} +4.88062 q^{81} -9.48897 q^{83} -5.57087 q^{85} +2.70191 q^{87} +14.6686 q^{89} -22.7605 q^{91} -24.7190 q^{93} -2.74798 q^{95} -1.14852 q^{97} +18.3161 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 35 q - q^{3} - 35 q^{5} + 6 q^{7} + 52 q^{9} - 2 q^{11} - q^{13} + q^{15} + 18 q^{17} + 2 q^{19} + 12 q^{21} + 13 q^{23} + 35 q^{25} - 7 q^{27} + 25 q^{29} + 13 q^{31} + 14 q^{33} - 6 q^{35} - 19 q^{37} - 3 q^{39} + 24 q^{41} - 5 q^{43} - 52 q^{45} + 19 q^{47} + 55 q^{49} + 41 q^{53} + 2 q^{55} + 14 q^{57} + 3 q^{59} + 13 q^{61} + 70 q^{63} + q^{65} - 17 q^{67} + 64 q^{69} + 17 q^{71} - 63 q^{73} - q^{75} + 54 q^{77} + 11 q^{79} + 107 q^{81} - 8 q^{83} - 18 q^{85} + 36 q^{87} + 38 q^{89} - 27 q^{91} + q^{93} - 2 q^{95} - 54 q^{97} - 51 q^{99}+O(q^{100}) \) 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
\(3\) −2.91827 −1.68486 −0.842432 0.538803i \(-0.818877\pi\)
−0.842432 + 0.538803i \(0.818877\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.31238 −1.62993 −0.814963 0.579513i \(-0.803243\pi\)
−0.814963 + 0.579513i \(0.803243\pi\)
\(8\) 0 0
\(9\) 5.51629 1.83876
\(10\) 0 0
\(11\) 3.32037 1.00113 0.500564 0.865699i \(-0.333126\pi\)
0.500564 + 0.865699i \(0.333126\pi\)
\(12\) 0 0
\(13\) 5.27795 1.46384 0.731920 0.681391i \(-0.238625\pi\)
0.731920 + 0.681391i \(0.238625\pi\)
\(14\) 0 0
\(15\) 2.91827 0.753494
\(16\) 0 0
\(17\) 5.57087 1.35113 0.675567 0.737298i \(-0.263899\pi\)
0.675567 + 0.737298i \(0.263899\pi\)
\(18\) 0 0
\(19\) 2.74798 0.630429 0.315214 0.949020i \(-0.397924\pi\)
0.315214 + 0.949020i \(0.397924\pi\)
\(20\) 0 0
\(21\) 12.5847 2.74620
\(22\) 0 0
\(23\) 1.03475 0.215760 0.107880 0.994164i \(-0.465594\pi\)
0.107880 + 0.994164i \(0.465594\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −7.34323 −1.41320
\(28\) 0 0
\(29\) −0.925860 −0.171928 −0.0859639 0.996298i \(-0.527397\pi\)
−0.0859639 + 0.996298i \(0.527397\pi\)
\(30\) 0 0
\(31\) 8.47043 1.52133 0.760667 0.649142i \(-0.224872\pi\)
0.760667 + 0.649142i \(0.224872\pi\)
\(32\) 0 0
\(33\) −9.68973 −1.68677
\(34\) 0 0
\(35\) 4.31238 0.728925
\(36\) 0 0
\(37\) 1.78281 0.293093 0.146546 0.989204i \(-0.453184\pi\)
0.146546 + 0.989204i \(0.453184\pi\)
\(38\) 0 0
\(39\) −15.4025 −2.46637
\(40\) 0 0
\(41\) 9.98377 1.55920 0.779601 0.626276i \(-0.215422\pi\)
0.779601 + 0.626276i \(0.215422\pi\)
\(42\) 0 0
\(43\) 0.674120 0.102802 0.0514011 0.998678i \(-0.483631\pi\)
0.0514011 + 0.998678i \(0.483631\pi\)
\(44\) 0 0
\(45\) −5.51629 −0.822321
\(46\) 0 0
\(47\) 1.41456 0.206335 0.103167 0.994664i \(-0.467102\pi\)
0.103167 + 0.994664i \(0.467102\pi\)
\(48\) 0 0
\(49\) 11.5966 1.65666
\(50\) 0 0
\(51\) −16.2573 −2.27648
\(52\) 0 0
\(53\) 2.67960 0.368071 0.184036 0.982920i \(-0.441084\pi\)
0.184036 + 0.982920i \(0.441084\pi\)
\(54\) 0 0
\(55\) −3.32037 −0.447718
\(56\) 0 0
\(57\) −8.01933 −1.06219
\(58\) 0 0
\(59\) 12.9693 1.68846 0.844232 0.535977i \(-0.180057\pi\)
0.844232 + 0.535977i \(0.180057\pi\)
\(60\) 0 0
\(61\) 5.37423 0.688100 0.344050 0.938951i \(-0.388201\pi\)
0.344050 + 0.938951i \(0.388201\pi\)
\(62\) 0 0
\(63\) −23.7884 −2.99705
\(64\) 0 0
\(65\) −5.27795 −0.654649
\(66\) 0 0
\(67\) 9.52502 1.16367 0.581833 0.813308i \(-0.302336\pi\)
0.581833 + 0.813308i \(0.302336\pi\)
\(68\) 0 0
\(69\) −3.01968 −0.363526
\(70\) 0 0
\(71\) −3.09924 −0.367812 −0.183906 0.982944i \(-0.558874\pi\)
−0.183906 + 0.982944i \(0.558874\pi\)
\(72\) 0 0
\(73\) −5.83066 −0.682427 −0.341214 0.939986i \(-0.610838\pi\)
−0.341214 + 0.939986i \(0.610838\pi\)
\(74\) 0 0
\(75\) −2.91827 −0.336973
\(76\) 0 0
\(77\) −14.3187 −1.63177
\(78\) 0 0
\(79\) −12.9855 −1.46098 −0.730490 0.682923i \(-0.760708\pi\)
−0.730490 + 0.682923i \(0.760708\pi\)
\(80\) 0 0
\(81\) 4.88062 0.542292
\(82\) 0 0
\(83\) −9.48897 −1.04155 −0.520775 0.853694i \(-0.674357\pi\)
−0.520775 + 0.853694i \(0.674357\pi\)
\(84\) 0 0
\(85\) −5.57087 −0.604246
\(86\) 0 0
\(87\) 2.70191 0.289675
\(88\) 0 0
\(89\) 14.6686 1.55487 0.777436 0.628962i \(-0.216520\pi\)
0.777436 + 0.628962i \(0.216520\pi\)
\(90\) 0 0
\(91\) −22.7605 −2.38595
\(92\) 0 0
\(93\) −24.7190 −2.56324
\(94\) 0 0
\(95\) −2.74798 −0.281936
\(96\) 0 0
\(97\) −1.14852 −0.116614 −0.0583072 0.998299i \(-0.518570\pi\)
−0.0583072 + 0.998299i \(0.518570\pi\)
\(98\) 0 0
\(99\) 18.3161 1.84084
\(100\) 0 0
\(101\) 12.4855 1.24235 0.621177 0.783670i \(-0.286655\pi\)
0.621177 + 0.783670i \(0.286655\pi\)
\(102\) 0 0
\(103\) 5.16985 0.509400 0.254700 0.967020i \(-0.418023\pi\)
0.254700 + 0.967020i \(0.418023\pi\)
\(104\) 0 0
\(105\) −12.5847 −1.22814
\(106\) 0 0
\(107\) −12.9665 −1.25352 −0.626758 0.779214i \(-0.715618\pi\)
−0.626758 + 0.779214i \(0.715618\pi\)
\(108\) 0 0
\(109\) 10.7096 1.02579 0.512896 0.858451i \(-0.328573\pi\)
0.512896 + 0.858451i \(0.328573\pi\)
\(110\) 0 0
\(111\) −5.20273 −0.493821
\(112\) 0 0
\(113\) −17.4369 −1.64033 −0.820163 0.572130i \(-0.806117\pi\)
−0.820163 + 0.572130i \(0.806117\pi\)
\(114\) 0 0
\(115\) −1.03475 −0.0964909
\(116\) 0 0
\(117\) 29.1147 2.69166
\(118\) 0 0
\(119\) −24.0237 −2.20225
\(120\) 0 0
\(121\) 0.0248507 0.00225915
\(122\) 0 0
\(123\) −29.1353 −2.62704
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 18.8509 1.67275 0.836373 0.548161i \(-0.184672\pi\)
0.836373 + 0.548161i \(0.184672\pi\)
\(128\) 0 0
\(129\) −1.96726 −0.173208
\(130\) 0 0
\(131\) −17.1817 −1.50118 −0.750588 0.660770i \(-0.770230\pi\)
−0.750588 + 0.660770i \(0.770230\pi\)
\(132\) 0 0
\(133\) −11.8503 −1.02755
\(134\) 0 0
\(135\) 7.34323 0.632004
\(136\) 0 0
\(137\) 3.91551 0.334524 0.167262 0.985912i \(-0.446507\pi\)
0.167262 + 0.985912i \(0.446507\pi\)
\(138\) 0 0
\(139\) −22.6985 −1.92527 −0.962633 0.270810i \(-0.912709\pi\)
−0.962633 + 0.270810i \(0.912709\pi\)
\(140\) 0 0
\(141\) −4.12806 −0.347645
\(142\) 0 0
\(143\) 17.5247 1.46549
\(144\) 0 0
\(145\) 0.925860 0.0768885
\(146\) 0 0
\(147\) −33.8420 −2.79124
\(148\) 0 0
\(149\) 1.04715 0.0857862 0.0428931 0.999080i \(-0.486343\pi\)
0.0428931 + 0.999080i \(0.486343\pi\)
\(150\) 0 0
\(151\) 5.12153 0.416784 0.208392 0.978045i \(-0.433177\pi\)
0.208392 + 0.978045i \(0.433177\pi\)
\(152\) 0 0
\(153\) 30.7306 2.48442
\(154\) 0 0
\(155\) −8.47043 −0.680362
\(156\) 0 0
\(157\) −3.28082 −0.261838 −0.130919 0.991393i \(-0.541793\pi\)
−0.130919 + 0.991393i \(0.541793\pi\)
\(158\) 0 0
\(159\) −7.81979 −0.620150
\(160\) 0 0
\(161\) −4.46223 −0.351673
\(162\) 0 0
\(163\) −12.6811 −0.993262 −0.496631 0.867962i \(-0.665430\pi\)
−0.496631 + 0.867962i \(0.665430\pi\)
\(164\) 0 0
\(165\) 9.68973 0.754345
\(166\) 0 0
\(167\) −9.21551 −0.713118 −0.356559 0.934273i \(-0.616050\pi\)
−0.356559 + 0.934273i \(0.616050\pi\)
\(168\) 0 0
\(169\) 14.8567 1.14283
\(170\) 0 0
\(171\) 15.1586 1.15921
\(172\) 0 0
\(173\) 10.6937 0.813027 0.406514 0.913645i \(-0.366744\pi\)
0.406514 + 0.913645i \(0.366744\pi\)
\(174\) 0 0
\(175\) −4.31238 −0.325985
\(176\) 0 0
\(177\) −37.8480 −2.84483
\(178\) 0 0
\(179\) −0.412909 −0.0308623 −0.0154311 0.999881i \(-0.504912\pi\)
−0.0154311 + 0.999881i \(0.504912\pi\)
\(180\) 0 0
\(181\) 8.46717 0.629360 0.314680 0.949198i \(-0.398103\pi\)
0.314680 + 0.949198i \(0.398103\pi\)
\(182\) 0 0
\(183\) −15.6835 −1.15935
\(184\) 0 0
\(185\) −1.78281 −0.131075
\(186\) 0 0
\(187\) 18.4973 1.35266
\(188\) 0 0
\(189\) 31.6668 2.30342
\(190\) 0 0
\(191\) 10.9445 0.791917 0.395958 0.918269i \(-0.370412\pi\)
0.395958 + 0.918269i \(0.370412\pi\)
\(192\) 0 0
\(193\) 12.1178 0.872260 0.436130 0.899884i \(-0.356349\pi\)
0.436130 + 0.899884i \(0.356349\pi\)
\(194\) 0 0
\(195\) 15.4025 1.10299
\(196\) 0 0
\(197\) −2.82132 −0.201010 −0.100505 0.994937i \(-0.532046\pi\)
−0.100505 + 0.994937i \(0.532046\pi\)
\(198\) 0 0
\(199\) −6.85412 −0.485875 −0.242938 0.970042i \(-0.578111\pi\)
−0.242938 + 0.970042i \(0.578111\pi\)
\(200\) 0 0
\(201\) −27.7966 −1.96062
\(202\) 0 0
\(203\) 3.99266 0.280230
\(204\) 0 0
\(205\) −9.98377 −0.697297
\(206\) 0 0
\(207\) 5.70798 0.396732
\(208\) 0 0
\(209\) 9.12429 0.631140
\(210\) 0 0
\(211\) 25.5752 1.76067 0.880336 0.474350i \(-0.157317\pi\)
0.880336 + 0.474350i \(0.157317\pi\)
\(212\) 0 0
\(213\) 9.04442 0.619714
\(214\) 0 0
\(215\) −0.674120 −0.0459746
\(216\) 0 0
\(217\) −36.5277 −2.47966
\(218\) 0 0
\(219\) 17.0154 1.14980
\(220\) 0 0
\(221\) 29.4028 1.97784
\(222\) 0 0
\(223\) 2.59665 0.173885 0.0869424 0.996213i \(-0.472290\pi\)
0.0869424 + 0.996213i \(0.472290\pi\)
\(224\) 0 0
\(225\) 5.51629 0.367753
\(226\) 0 0
\(227\) 10.8232 0.718360 0.359180 0.933268i \(-0.383056\pi\)
0.359180 + 0.933268i \(0.383056\pi\)
\(228\) 0 0
\(229\) 14.9657 0.988962 0.494481 0.869188i \(-0.335358\pi\)
0.494481 + 0.869188i \(0.335358\pi\)
\(230\) 0 0
\(231\) 41.7858 2.74930
\(232\) 0 0
\(233\) −5.54068 −0.362982 −0.181491 0.983393i \(-0.558092\pi\)
−0.181491 + 0.983393i \(0.558092\pi\)
\(234\) 0 0
\(235\) −1.41456 −0.0922756
\(236\) 0 0
\(237\) 37.8951 2.46155
\(238\) 0 0
\(239\) −6.12190 −0.395993 −0.197996 0.980203i \(-0.563443\pi\)
−0.197996 + 0.980203i \(0.563443\pi\)
\(240\) 0 0
\(241\) −16.6578 −1.07303 −0.536513 0.843892i \(-0.680259\pi\)
−0.536513 + 0.843892i \(0.680259\pi\)
\(242\) 0 0
\(243\) 7.78670 0.499517
\(244\) 0 0
\(245\) −11.5966 −0.740880
\(246\) 0 0
\(247\) 14.5037 0.922847
\(248\) 0 0
\(249\) 27.6914 1.75487
\(250\) 0 0
\(251\) −2.31022 −0.145820 −0.0729098 0.997339i \(-0.523229\pi\)
−0.0729098 + 0.997339i \(0.523229\pi\)
\(252\) 0 0
\(253\) 3.43575 0.216004
\(254\) 0 0
\(255\) 16.2573 1.01807
\(256\) 0 0
\(257\) 3.78628 0.236182 0.118091 0.993003i \(-0.462323\pi\)
0.118091 + 0.993003i \(0.462323\pi\)
\(258\) 0 0
\(259\) −7.68816 −0.477719
\(260\) 0 0
\(261\) −5.10732 −0.316135
\(262\) 0 0
\(263\) 0.183960 0.0113434 0.00567172 0.999984i \(-0.498195\pi\)
0.00567172 + 0.999984i \(0.498195\pi\)
\(264\) 0 0
\(265\) −2.67960 −0.164606
\(266\) 0 0
\(267\) −42.8070 −2.61975
\(268\) 0 0
\(269\) −9.72726 −0.593081 −0.296541 0.955020i \(-0.595833\pi\)
−0.296541 + 0.955020i \(0.595833\pi\)
\(270\) 0 0
\(271\) −6.31224 −0.383441 −0.191721 0.981450i \(-0.561407\pi\)
−0.191721 + 0.981450i \(0.561407\pi\)
\(272\) 0 0
\(273\) 66.4213 4.02000
\(274\) 0 0
\(275\) 3.32037 0.200226
\(276\) 0 0
\(277\) 13.3757 0.803666 0.401833 0.915713i \(-0.368373\pi\)
0.401833 + 0.915713i \(0.368373\pi\)
\(278\) 0 0
\(279\) 46.7254 2.79738
\(280\) 0 0
\(281\) 29.0312 1.73186 0.865928 0.500168i \(-0.166729\pi\)
0.865928 + 0.500168i \(0.166729\pi\)
\(282\) 0 0
\(283\) 13.3119 0.791308 0.395654 0.918400i \(-0.370518\pi\)
0.395654 + 0.918400i \(0.370518\pi\)
\(284\) 0 0
\(285\) 8.01933 0.475024
\(286\) 0 0
\(287\) −43.0538 −2.54139
\(288\) 0 0
\(289\) 14.0346 0.825564
\(290\) 0 0
\(291\) 3.35168 0.196479
\(292\) 0 0
\(293\) −0.298727 −0.0174518 −0.00872591 0.999962i \(-0.502778\pi\)
−0.00872591 + 0.999962i \(0.502778\pi\)
\(294\) 0 0
\(295\) −12.9693 −0.755104
\(296\) 0 0
\(297\) −24.3822 −1.41480
\(298\) 0 0
\(299\) 5.46135 0.315838
\(300\) 0 0
\(301\) −2.90706 −0.167560
\(302\) 0 0
\(303\) −36.4360 −2.09320
\(304\) 0 0
\(305\) −5.37423 −0.307728
\(306\) 0 0
\(307\) −17.9705 −1.02563 −0.512814 0.858500i \(-0.671397\pi\)
−0.512814 + 0.858500i \(0.671397\pi\)
\(308\) 0 0
\(309\) −15.0870 −0.858270
\(310\) 0 0
\(311\) −8.09599 −0.459081 −0.229541 0.973299i \(-0.573722\pi\)
−0.229541 + 0.973299i \(0.573722\pi\)
\(312\) 0 0
\(313\) 14.0964 0.796773 0.398386 0.917218i \(-0.369570\pi\)
0.398386 + 0.917218i \(0.369570\pi\)
\(314\) 0 0
\(315\) 23.7884 1.34032
\(316\) 0 0
\(317\) 33.7475 1.89545 0.947723 0.319093i \(-0.103378\pi\)
0.947723 + 0.319093i \(0.103378\pi\)
\(318\) 0 0
\(319\) −3.07420 −0.172122
\(320\) 0 0
\(321\) 37.8396 2.11200
\(322\) 0 0
\(323\) 15.3086 0.851794
\(324\) 0 0
\(325\) 5.27795 0.292768
\(326\) 0 0
\(327\) −31.2534 −1.72832
\(328\) 0 0
\(329\) −6.10011 −0.336310
\(330\) 0 0
\(331\) 32.5670 1.79005 0.895023 0.446021i \(-0.147159\pi\)
0.895023 + 0.446021i \(0.147159\pi\)
\(332\) 0 0
\(333\) 9.83452 0.538928
\(334\) 0 0
\(335\) −9.52502 −0.520408
\(336\) 0 0
\(337\) −20.2884 −1.10518 −0.552591 0.833453i \(-0.686361\pi\)
−0.552591 + 0.833453i \(0.686361\pi\)
\(338\) 0 0
\(339\) 50.8856 2.76372
\(340\) 0 0
\(341\) 28.1250 1.52305
\(342\) 0 0
\(343\) −19.8223 −1.07030
\(344\) 0 0
\(345\) 3.01968 0.162574
\(346\) 0 0
\(347\) −11.5164 −0.618234 −0.309117 0.951024i \(-0.600033\pi\)
−0.309117 + 0.951024i \(0.600033\pi\)
\(348\) 0 0
\(349\) −11.0146 −0.589599 −0.294800 0.955559i \(-0.595253\pi\)
−0.294800 + 0.955559i \(0.595253\pi\)
\(350\) 0 0
\(351\) −38.7572 −2.06870
\(352\) 0 0
\(353\) −14.1217 −0.751624 −0.375812 0.926696i \(-0.622636\pi\)
−0.375812 + 0.926696i \(0.622636\pi\)
\(354\) 0 0
\(355\) 3.09924 0.164491
\(356\) 0 0
\(357\) 70.1076 3.71049
\(358\) 0 0
\(359\) 30.6604 1.61819 0.809096 0.587676i \(-0.199957\pi\)
0.809096 + 0.587676i \(0.199957\pi\)
\(360\) 0 0
\(361\) −11.4486 −0.602560
\(362\) 0 0
\(363\) −0.0725209 −0.00380636
\(364\) 0 0
\(365\) 5.83066 0.305191
\(366\) 0 0
\(367\) −8.65178 −0.451619 −0.225810 0.974171i \(-0.572503\pi\)
−0.225810 + 0.974171i \(0.572503\pi\)
\(368\) 0 0
\(369\) 55.0734 2.86701
\(370\) 0 0
\(371\) −11.5554 −0.599929
\(372\) 0 0
\(373\) −12.6273 −0.653818 −0.326909 0.945056i \(-0.606007\pi\)
−0.326909 + 0.945056i \(0.606007\pi\)
\(374\) 0 0
\(375\) 2.91827 0.150699
\(376\) 0 0
\(377\) −4.88664 −0.251675
\(378\) 0 0
\(379\) −16.4358 −0.844252 −0.422126 0.906537i \(-0.638716\pi\)
−0.422126 + 0.906537i \(0.638716\pi\)
\(380\) 0 0
\(381\) −55.0120 −2.81835
\(382\) 0 0
\(383\) −18.2167 −0.930832 −0.465416 0.885092i \(-0.654095\pi\)
−0.465416 + 0.885092i \(0.654095\pi\)
\(384\) 0 0
\(385\) 14.3187 0.729748
\(386\) 0 0
\(387\) 3.71864 0.189029
\(388\) 0 0
\(389\) 1.38413 0.0701784 0.0350892 0.999384i \(-0.488828\pi\)
0.0350892 + 0.999384i \(0.488828\pi\)
\(390\) 0 0
\(391\) 5.76445 0.291521
\(392\) 0 0
\(393\) 50.1410 2.52928
\(394\) 0 0
\(395\) 12.9855 0.653370
\(396\) 0 0
\(397\) 0.235691 0.0118290 0.00591450 0.999983i \(-0.498117\pi\)
0.00591450 + 0.999983i \(0.498117\pi\)
\(398\) 0 0
\(399\) 34.5824 1.73129
\(400\) 0 0
\(401\) −1.00000 −0.0499376
\(402\) 0 0
\(403\) 44.7065 2.22699
\(404\) 0 0
\(405\) −4.88062 −0.242520
\(406\) 0 0
\(407\) 5.91959 0.293423
\(408\) 0 0
\(409\) −3.86214 −0.190970 −0.0954852 0.995431i \(-0.530440\pi\)
−0.0954852 + 0.995431i \(0.530440\pi\)
\(410\) 0 0
\(411\) −11.4265 −0.563628
\(412\) 0 0
\(413\) −55.9287 −2.75207
\(414\) 0 0
\(415\) 9.48897 0.465795
\(416\) 0 0
\(417\) 66.2405 3.24381
\(418\) 0 0
\(419\) −25.9451 −1.26750 −0.633751 0.773537i \(-0.718486\pi\)
−0.633751 + 0.773537i \(0.718486\pi\)
\(420\) 0 0
\(421\) 32.5329 1.58556 0.792778 0.609510i \(-0.208634\pi\)
0.792778 + 0.609510i \(0.208634\pi\)
\(422\) 0 0
\(423\) 7.80312 0.379401
\(424\) 0 0
\(425\) 5.57087 0.270227
\(426\) 0 0
\(427\) −23.1757 −1.12155
\(428\) 0 0
\(429\) −51.1419 −2.46915
\(430\) 0 0
\(431\) −9.18644 −0.442495 −0.221248 0.975218i \(-0.571013\pi\)
−0.221248 + 0.975218i \(0.571013\pi\)
\(432\) 0 0
\(433\) −23.7717 −1.14239 −0.571197 0.820813i \(-0.693521\pi\)
−0.571197 + 0.820813i \(0.693521\pi\)
\(434\) 0 0
\(435\) −2.70191 −0.129547
\(436\) 0 0
\(437\) 2.84347 0.136021
\(438\) 0 0
\(439\) 24.2561 1.15768 0.578840 0.815441i \(-0.303506\pi\)
0.578840 + 0.815441i \(0.303506\pi\)
\(440\) 0 0
\(441\) 63.9703 3.04621
\(442\) 0 0
\(443\) 8.10423 0.385044 0.192522 0.981293i \(-0.438333\pi\)
0.192522 + 0.981293i \(0.438333\pi\)
\(444\) 0 0
\(445\) −14.6686 −0.695360
\(446\) 0 0
\(447\) −3.05588 −0.144538
\(448\) 0 0
\(449\) −17.0286 −0.803629 −0.401814 0.915721i \(-0.631620\pi\)
−0.401814 + 0.915721i \(0.631620\pi\)
\(450\) 0 0
\(451\) 33.1498 1.56096
\(452\) 0 0
\(453\) −14.9460 −0.702225
\(454\) 0 0
\(455\) 22.7605 1.06703
\(456\) 0 0
\(457\) −18.9279 −0.885412 −0.442706 0.896667i \(-0.645981\pi\)
−0.442706 + 0.896667i \(0.645981\pi\)
\(458\) 0 0
\(459\) −40.9081 −1.90943
\(460\) 0 0
\(461\) −37.2886 −1.73671 −0.868353 0.495947i \(-0.834821\pi\)
−0.868353 + 0.495947i \(0.834821\pi\)
\(462\) 0 0
\(463\) 31.0510 1.44306 0.721531 0.692382i \(-0.243439\pi\)
0.721531 + 0.692382i \(0.243439\pi\)
\(464\) 0 0
\(465\) 24.7190 1.14632
\(466\) 0 0
\(467\) −2.65892 −0.123040 −0.0615201 0.998106i \(-0.519595\pi\)
−0.0615201 + 0.998106i \(0.519595\pi\)
\(468\) 0 0
\(469\) −41.0755 −1.89669
\(470\) 0 0
\(471\) 9.57433 0.441162
\(472\) 0 0
\(473\) 2.23833 0.102918
\(474\) 0 0
\(475\) 2.74798 0.126086
\(476\) 0 0
\(477\) 14.7815 0.676796
\(478\) 0 0
\(479\) 26.3789 1.20528 0.602642 0.798012i \(-0.294115\pi\)
0.602642 + 0.798012i \(0.294115\pi\)
\(480\) 0 0
\(481\) 9.40959 0.429040
\(482\) 0 0
\(483\) 13.0220 0.592521
\(484\) 0 0
\(485\) 1.14852 0.0521515
\(486\) 0 0
\(487\) 10.2087 0.462600 0.231300 0.972882i \(-0.425702\pi\)
0.231300 + 0.972882i \(0.425702\pi\)
\(488\) 0 0
\(489\) 37.0069 1.67351
\(490\) 0 0
\(491\) −38.9047 −1.75574 −0.877872 0.478896i \(-0.841037\pi\)
−0.877872 + 0.478896i \(0.841037\pi\)
\(492\) 0 0
\(493\) −5.15784 −0.232298
\(494\) 0 0
\(495\) −18.3161 −0.823249
\(496\) 0 0
\(497\) 13.3651 0.599507
\(498\) 0 0
\(499\) −24.7854 −1.10955 −0.554774 0.832001i \(-0.687195\pi\)
−0.554774 + 0.832001i \(0.687195\pi\)
\(500\) 0 0
\(501\) 26.8934 1.20151
\(502\) 0 0
\(503\) 31.3530 1.39796 0.698982 0.715140i \(-0.253637\pi\)
0.698982 + 0.715140i \(0.253637\pi\)
\(504\) 0 0
\(505\) −12.4855 −0.555597
\(506\) 0 0
\(507\) −43.3560 −1.92551
\(508\) 0 0
\(509\) −43.4976 −1.92800 −0.963999 0.265905i \(-0.914329\pi\)
−0.963999 + 0.265905i \(0.914329\pi\)
\(510\) 0 0
\(511\) 25.1440 1.11231
\(512\) 0 0
\(513\) −20.1790 −0.890925
\(514\) 0 0
\(515\) −5.16985 −0.227811
\(516\) 0 0
\(517\) 4.69686 0.206567
\(518\) 0 0
\(519\) −31.2071 −1.36984
\(520\) 0 0
\(521\) 18.5220 0.811462 0.405731 0.913993i \(-0.367017\pi\)
0.405731 + 0.913993i \(0.367017\pi\)
\(522\) 0 0
\(523\) −18.6845 −0.817017 −0.408509 0.912754i \(-0.633951\pi\)
−0.408509 + 0.912754i \(0.633951\pi\)
\(524\) 0 0
\(525\) 12.5847 0.549241
\(526\) 0 0
\(527\) 47.1877 2.05553
\(528\) 0 0
\(529\) −21.9293 −0.953448
\(530\) 0 0
\(531\) 71.5427 3.10469
\(532\) 0 0
\(533\) 52.6938 2.28242
\(534\) 0 0
\(535\) 12.9665 0.560589
\(536\) 0 0
\(537\) 1.20498 0.0519988
\(538\) 0 0
\(539\) 38.5050 1.65853
\(540\) 0 0
\(541\) 20.9363 0.900123 0.450062 0.892997i \(-0.351402\pi\)
0.450062 + 0.892997i \(0.351402\pi\)
\(542\) 0 0
\(543\) −24.7095 −1.06039
\(544\) 0 0
\(545\) −10.7096 −0.458748
\(546\) 0 0
\(547\) −22.8226 −0.975825 −0.487912 0.872893i \(-0.662241\pi\)
−0.487912 + 0.872893i \(0.662241\pi\)
\(548\) 0 0
\(549\) 29.6458 1.26525
\(550\) 0 0
\(551\) −2.54424 −0.108388
\(552\) 0 0
\(553\) 55.9983 2.38129
\(554\) 0 0
\(555\) 5.20273 0.220843
\(556\) 0 0
\(557\) −36.8697 −1.56222 −0.781111 0.624393i \(-0.785346\pi\)
−0.781111 + 0.624393i \(0.785346\pi\)
\(558\) 0 0
\(559\) 3.55797 0.150486
\(560\) 0 0
\(561\) −53.9802 −2.27905
\(562\) 0 0
\(563\) −32.1487 −1.35490 −0.677452 0.735567i \(-0.736916\pi\)
−0.677452 + 0.735567i \(0.736916\pi\)
\(564\) 0 0
\(565\) 17.4369 0.733576
\(566\) 0 0
\(567\) −21.0471 −0.883895
\(568\) 0 0
\(569\) 19.8745 0.833184 0.416592 0.909094i \(-0.363224\pi\)
0.416592 + 0.909094i \(0.363224\pi\)
\(570\) 0 0
\(571\) 34.0334 1.42425 0.712126 0.702051i \(-0.247732\pi\)
0.712126 + 0.702051i \(0.247732\pi\)
\(572\) 0 0
\(573\) −31.9390 −1.33427
\(574\) 0 0
\(575\) 1.03475 0.0431520
\(576\) 0 0
\(577\) −45.1983 −1.88163 −0.940815 0.338921i \(-0.889938\pi\)
−0.940815 + 0.338921i \(0.889938\pi\)
\(578\) 0 0
\(579\) −35.3631 −1.46964
\(580\) 0 0
\(581\) 40.9200 1.69765
\(582\) 0 0
\(583\) 8.89726 0.368487
\(584\) 0 0
\(585\) −29.1147 −1.20375
\(586\) 0 0
\(587\) −6.16135 −0.254306 −0.127153 0.991883i \(-0.540584\pi\)
−0.127153 + 0.991883i \(0.540584\pi\)
\(588\) 0 0
\(589\) 23.2765 0.959093
\(590\) 0 0
\(591\) 8.23336 0.338675
\(592\) 0 0
\(593\) 10.4969 0.431055 0.215527 0.976498i \(-0.430853\pi\)
0.215527 + 0.976498i \(0.430853\pi\)
\(594\) 0 0
\(595\) 24.0237 0.984876
\(596\) 0 0
\(597\) 20.0022 0.818634
\(598\) 0 0
\(599\) 7.53354 0.307812 0.153906 0.988085i \(-0.450815\pi\)
0.153906 + 0.988085i \(0.450815\pi\)
\(600\) 0 0
\(601\) 38.4629 1.56893 0.784466 0.620171i \(-0.212937\pi\)
0.784466 + 0.620171i \(0.212937\pi\)
\(602\) 0 0
\(603\) 52.5428 2.13971
\(604\) 0 0
\(605\) −0.0248507 −0.00101032
\(606\) 0 0
\(607\) 26.3296 1.06869 0.534343 0.845268i \(-0.320559\pi\)
0.534343 + 0.845268i \(0.320559\pi\)
\(608\) 0 0
\(609\) −11.6517 −0.472149
\(610\) 0 0
\(611\) 7.46597 0.302041
\(612\) 0 0
\(613\) −23.3413 −0.942746 −0.471373 0.881934i \(-0.656241\pi\)
−0.471373 + 0.881934i \(0.656241\pi\)
\(614\) 0 0
\(615\) 29.1353 1.17485
\(616\) 0 0
\(617\) 29.6407 1.19329 0.596646 0.802505i \(-0.296500\pi\)
0.596646 + 0.802505i \(0.296500\pi\)
\(618\) 0 0
\(619\) −44.2273 −1.77764 −0.888822 0.458253i \(-0.848476\pi\)
−0.888822 + 0.458253i \(0.848476\pi\)
\(620\) 0 0
\(621\) −7.59840 −0.304913
\(622\) 0 0
\(623\) −63.2567 −2.53433
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −26.6271 −1.06339
\(628\) 0 0
\(629\) 9.93181 0.396007
\(630\) 0 0
\(631\) 23.7717 0.946336 0.473168 0.880972i \(-0.343110\pi\)
0.473168 + 0.880972i \(0.343110\pi\)
\(632\) 0 0
\(633\) −74.6355 −2.96649
\(634\) 0 0
\(635\) −18.8509 −0.748075
\(636\) 0 0
\(637\) 61.2063 2.42508
\(638\) 0 0
\(639\) −17.0963 −0.676321
\(640\) 0 0
\(641\) −17.2078 −0.679666 −0.339833 0.940486i \(-0.610371\pi\)
−0.339833 + 0.940486i \(0.610371\pi\)
\(642\) 0 0
\(643\) −42.9441 −1.69355 −0.846776 0.531950i \(-0.821459\pi\)
−0.846776 + 0.531950i \(0.821459\pi\)
\(644\) 0 0
\(645\) 1.96726 0.0774609
\(646\) 0 0
\(647\) −5.04996 −0.198535 −0.0992673 0.995061i \(-0.531650\pi\)
−0.0992673 + 0.995061i \(0.531650\pi\)
\(648\) 0 0
\(649\) 43.0630 1.69037
\(650\) 0 0
\(651\) 106.598 4.17789
\(652\) 0 0
\(653\) 41.9871 1.64308 0.821542 0.570149i \(-0.193114\pi\)
0.821542 + 0.570149i \(0.193114\pi\)
\(654\) 0 0
\(655\) 17.1817 0.671346
\(656\) 0 0
\(657\) −32.1636 −1.25482
\(658\) 0 0
\(659\) 44.6790 1.74045 0.870223 0.492658i \(-0.163975\pi\)
0.870223 + 0.492658i \(0.163975\pi\)
\(660\) 0 0
\(661\) 33.7989 1.31463 0.657313 0.753618i \(-0.271693\pi\)
0.657313 + 0.753618i \(0.271693\pi\)
\(662\) 0 0
\(663\) −85.8052 −3.33240
\(664\) 0 0
\(665\) 11.8503 0.459535
\(666\) 0 0
\(667\) −0.958033 −0.0370952
\(668\) 0 0
\(669\) −7.57773 −0.292972
\(670\) 0 0
\(671\) 17.8444 0.688877
\(672\) 0 0
\(673\) −16.0709 −0.619489 −0.309745 0.950820i \(-0.600244\pi\)
−0.309745 + 0.950820i \(0.600244\pi\)
\(674\) 0 0
\(675\) −7.34323 −0.282641
\(676\) 0 0
\(677\) 12.4968 0.480289 0.240145 0.970737i \(-0.422805\pi\)
0.240145 + 0.970737i \(0.422805\pi\)
\(678\) 0 0
\(679\) 4.95284 0.190073
\(680\) 0 0
\(681\) −31.5849 −1.21034
\(682\) 0 0
\(683\) −18.1924 −0.696112 −0.348056 0.937474i \(-0.613158\pi\)
−0.348056 + 0.937474i \(0.613158\pi\)
\(684\) 0 0
\(685\) −3.91551 −0.149604
\(686\) 0 0
\(687\) −43.6740 −1.66627
\(688\) 0 0
\(689\) 14.1428 0.538797
\(690\) 0 0
\(691\) 25.4643 0.968706 0.484353 0.874873i \(-0.339055\pi\)
0.484353 + 0.874873i \(0.339055\pi\)
\(692\) 0 0
\(693\) −78.9861 −3.00043
\(694\) 0 0
\(695\) 22.6985 0.861005
\(696\) 0 0
\(697\) 55.6183 2.10669
\(698\) 0 0
\(699\) 16.1692 0.611576
\(700\) 0 0
\(701\) −31.6968 −1.19717 −0.598586 0.801058i \(-0.704271\pi\)
−0.598586 + 0.801058i \(0.704271\pi\)
\(702\) 0 0
\(703\) 4.89912 0.184774
\(704\) 0 0
\(705\) 4.12806 0.155472
\(706\) 0 0
\(707\) −53.8422 −2.02494
\(708\) 0 0
\(709\) −28.2265 −1.06007 −0.530034 0.847976i \(-0.677821\pi\)
−0.530034 + 0.847976i \(0.677821\pi\)
\(710\) 0 0
\(711\) −71.6317 −2.68640
\(712\) 0 0
\(713\) 8.76478 0.328243
\(714\) 0 0
\(715\) −17.5247 −0.655388
\(716\) 0 0
\(717\) 17.8654 0.667194
\(718\) 0 0
\(719\) −50.4890 −1.88292 −0.941461 0.337122i \(-0.890546\pi\)
−0.941461 + 0.337122i \(0.890546\pi\)
\(720\) 0 0
\(721\) −22.2943 −0.830285
\(722\) 0 0
\(723\) 48.6121 1.80790
\(724\) 0 0
\(725\) −0.925860 −0.0343856
\(726\) 0 0
\(727\) 38.2197 1.41749 0.708744 0.705465i \(-0.249262\pi\)
0.708744 + 0.705465i \(0.249262\pi\)
\(728\) 0 0
\(729\) −37.3656 −1.38391
\(730\) 0 0
\(731\) 3.75543 0.138900
\(732\) 0 0
\(733\) −33.3158 −1.23055 −0.615273 0.788314i \(-0.710954\pi\)
−0.615273 + 0.788314i \(0.710954\pi\)
\(734\) 0 0
\(735\) 33.8420 1.24828
\(736\) 0 0
\(737\) 31.6266 1.16498
\(738\) 0 0
\(739\) 16.5781 0.609834 0.304917 0.952379i \(-0.401371\pi\)
0.304917 + 0.952379i \(0.401371\pi\)
\(740\) 0 0
\(741\) −42.3256 −1.55487
\(742\) 0 0
\(743\) 20.7424 0.760966 0.380483 0.924788i \(-0.375758\pi\)
0.380483 + 0.924788i \(0.375758\pi\)
\(744\) 0 0
\(745\) −1.04715 −0.0383648
\(746\) 0 0
\(747\) −52.3440 −1.91517
\(748\) 0 0
\(749\) 55.9163 2.04314
\(750\) 0 0
\(751\) 31.9330 1.16525 0.582625 0.812741i \(-0.302026\pi\)
0.582625 + 0.812741i \(0.302026\pi\)
\(752\) 0 0
\(753\) 6.74183 0.245686
\(754\) 0 0
\(755\) −5.12153 −0.186392
\(756\) 0 0
\(757\) −35.7381 −1.29892 −0.649461 0.760395i \(-0.725005\pi\)
−0.649461 + 0.760395i \(0.725005\pi\)
\(758\) 0 0
\(759\) −10.0264 −0.363937
\(760\) 0 0
\(761\) −17.3465 −0.628809 −0.314405 0.949289i \(-0.601805\pi\)
−0.314405 + 0.949289i \(0.601805\pi\)
\(762\) 0 0
\(763\) −46.1838 −1.67196
\(764\) 0 0
\(765\) −30.7306 −1.11107
\(766\) 0 0
\(767\) 68.4515 2.47164
\(768\) 0 0
\(769\) 10.0004 0.360625 0.180312 0.983609i \(-0.442289\pi\)
0.180312 + 0.983609i \(0.442289\pi\)
\(770\) 0 0
\(771\) −11.0494 −0.397934
\(772\) 0 0
\(773\) 3.54683 0.127571 0.0637853 0.997964i \(-0.479683\pi\)
0.0637853 + 0.997964i \(0.479683\pi\)
\(774\) 0 0
\(775\) 8.47043 0.304267
\(776\) 0 0
\(777\) 22.4361 0.804891
\(778\) 0 0
\(779\) 27.4351 0.982966
\(780\) 0 0
\(781\) −10.2906 −0.368228
\(782\) 0 0
\(783\) 6.79880 0.242969
\(784\) 0 0
\(785\) 3.28082 0.117098
\(786\) 0 0
\(787\) −8.52935 −0.304039 −0.152019 0.988378i \(-0.548578\pi\)
−0.152019 + 0.988378i \(0.548578\pi\)
\(788\) 0 0
\(789\) −0.536844 −0.0191122
\(790\) 0 0
\(791\) 75.1945 2.67361
\(792\) 0 0
\(793\) 28.3649 1.00727
\(794\) 0 0
\(795\) 7.81979 0.277339
\(796\) 0 0
\(797\) −35.0007 −1.23979 −0.619894 0.784686i \(-0.712824\pi\)
−0.619894 + 0.784686i \(0.712824\pi\)
\(798\) 0 0
\(799\) 7.88032 0.278786
\(800\) 0 0
\(801\) 80.9165 2.85905
\(802\) 0 0
\(803\) −19.3599 −0.683198
\(804\) 0 0
\(805\) 4.46223 0.157273
\(806\) 0 0
\(807\) 28.3868 0.999261
\(808\) 0 0
\(809\) −7.25919 −0.255220 −0.127610 0.991824i \(-0.540731\pi\)
−0.127610 + 0.991824i \(0.540731\pi\)
\(810\) 0 0
\(811\) −39.6997 −1.39404 −0.697022 0.717050i \(-0.745492\pi\)
−0.697022 + 0.717050i \(0.745492\pi\)
\(812\) 0 0
\(813\) 18.4208 0.646046
\(814\) 0 0
\(815\) 12.6811 0.444200
\(816\) 0 0
\(817\) 1.85246 0.0648095
\(818\) 0 0
\(819\) −125.554 −4.38720
\(820\) 0 0
\(821\) 8.50403 0.296793 0.148396 0.988928i \(-0.452589\pi\)
0.148396 + 0.988928i \(0.452589\pi\)
\(822\) 0 0
\(823\) −24.7797 −0.863765 −0.431882 0.901930i \(-0.642150\pi\)
−0.431882 + 0.901930i \(0.642150\pi\)
\(824\) 0 0
\(825\) −9.68973 −0.337353
\(826\) 0 0
\(827\) 32.9822 1.14690 0.573451 0.819240i \(-0.305604\pi\)
0.573451 + 0.819240i \(0.305604\pi\)
\(828\) 0 0
\(829\) −14.4214 −0.500876 −0.250438 0.968133i \(-0.580575\pi\)
−0.250438 + 0.968133i \(0.580575\pi\)
\(830\) 0 0
\(831\) −39.0338 −1.35407
\(832\) 0 0
\(833\) 64.6032 2.23837
\(834\) 0 0
\(835\) 9.21551 0.318916
\(836\) 0 0
\(837\) −62.2003 −2.14996
\(838\) 0 0
\(839\) −14.3853 −0.496637 −0.248319 0.968678i \(-0.579878\pi\)
−0.248319 + 0.968678i \(0.579878\pi\)
\(840\) 0 0
\(841\) −28.1428 −0.970441
\(842\) 0 0
\(843\) −84.7209 −2.91794
\(844\) 0 0
\(845\) −14.8567 −0.511088
\(846\) 0 0
\(847\) −0.107165 −0.00368225
\(848\) 0 0
\(849\) −38.8476 −1.33325
\(850\) 0 0
\(851\) 1.84476 0.0632377
\(852\) 0 0
\(853\) −11.8523 −0.405815 −0.202908 0.979198i \(-0.565039\pi\)
−0.202908 + 0.979198i \(0.565039\pi\)
\(854\) 0 0
\(855\) −15.1586 −0.518415
\(856\) 0 0
\(857\) 19.6244 0.670357 0.335178 0.942155i \(-0.391203\pi\)
0.335178 + 0.942155i \(0.391203\pi\)
\(858\) 0 0
\(859\) −23.6781 −0.807887 −0.403943 0.914784i \(-0.632361\pi\)
−0.403943 + 0.914784i \(0.632361\pi\)
\(860\) 0 0
\(861\) 125.643 4.28189
\(862\) 0 0
\(863\) −27.5258 −0.936989 −0.468494 0.883467i \(-0.655203\pi\)
−0.468494 + 0.883467i \(0.655203\pi\)
\(864\) 0 0
\(865\) −10.6937 −0.363597
\(866\) 0 0
\(867\) −40.9567 −1.39096
\(868\) 0 0
\(869\) −43.1166 −1.46263
\(870\) 0 0
\(871\) 50.2726 1.70342
\(872\) 0 0
\(873\) −6.33556 −0.214426
\(874\) 0 0
\(875\) 4.31238 0.145785
\(876\) 0 0
\(877\) 42.1666 1.42387 0.711933 0.702247i \(-0.247820\pi\)
0.711933 + 0.702247i \(0.247820\pi\)
\(878\) 0 0
\(879\) 0.871766 0.0294039
\(880\) 0 0
\(881\) −30.1450 −1.01561 −0.507805 0.861472i \(-0.669543\pi\)
−0.507805 + 0.861472i \(0.669543\pi\)
\(882\) 0 0
\(883\) −48.5409 −1.63353 −0.816765 0.576970i \(-0.804235\pi\)
−0.816765 + 0.576970i \(0.804235\pi\)
\(884\) 0 0
\(885\) 37.8480 1.27225
\(886\) 0 0
\(887\) −18.8258 −0.632107 −0.316054 0.948741i \(-0.602358\pi\)
−0.316054 + 0.948741i \(0.602358\pi\)
\(888\) 0 0
\(889\) −81.2922 −2.72645
\(890\) 0 0
\(891\) 16.2055 0.542904
\(892\) 0 0
\(893\) 3.88717 0.130079
\(894\) 0 0
\(895\) 0.412909 0.0138020
\(896\) 0 0
\(897\) −15.9377 −0.532144
\(898\) 0 0
\(899\) −7.84244 −0.261560
\(900\) 0 0
\(901\) 14.9277 0.497314
\(902\) 0 0
\(903\) 8.48358 0.282316
\(904\) 0 0
\(905\) −8.46717 −0.281458
\(906\) 0 0
\(907\) −40.7334 −1.35253 −0.676266 0.736658i \(-0.736403\pi\)
−0.676266 + 0.736658i \(0.736403\pi\)
\(908\) 0 0
\(909\) 68.8737 2.28440
\(910\) 0 0
\(911\) 42.1347 1.39598 0.697992 0.716106i \(-0.254077\pi\)
0.697992 + 0.716106i \(0.254077\pi\)
\(912\) 0 0
\(913\) −31.5069 −1.04273
\(914\) 0 0
\(915\) 15.6835 0.518479
\(916\) 0 0
\(917\) 74.0942 2.44681
\(918\) 0 0
\(919\) 38.2283 1.26104 0.630518 0.776175i \(-0.282842\pi\)
0.630518 + 0.776175i \(0.282842\pi\)
\(920\) 0 0
\(921\) 52.4426 1.72804
\(922\) 0 0
\(923\) −16.3576 −0.538418
\(924\) 0 0
\(925\) 1.78281 0.0586185
\(926\) 0 0
\(927\) 28.5184 0.936667
\(928\) 0 0
\(929\) −27.1946 −0.892226 −0.446113 0.894977i \(-0.647192\pi\)
−0.446113 + 0.894977i \(0.647192\pi\)
\(930\) 0 0
\(931\) 31.8672 1.04441
\(932\) 0 0
\(933\) 23.6263 0.773489
\(934\) 0 0
\(935\) −18.4973 −0.604928
\(936\) 0 0
\(937\) −13.2139 −0.431678 −0.215839 0.976429i \(-0.569249\pi\)
−0.215839 + 0.976429i \(0.569249\pi\)
\(938\) 0 0
\(939\) −41.1370 −1.34245
\(940\) 0 0
\(941\) 7.66673 0.249928 0.124964 0.992161i \(-0.460118\pi\)
0.124964 + 0.992161i \(0.460118\pi\)
\(942\) 0 0
\(943\) 10.3307 0.336414
\(944\) 0 0
\(945\) −31.6668 −1.03012
\(946\) 0 0
\(947\) 35.8576 1.16522 0.582608 0.812753i \(-0.302032\pi\)
0.582608 + 0.812753i \(0.302032\pi\)
\(948\) 0 0
\(949\) −30.7739 −0.998964
\(950\) 0 0
\(951\) −98.4842 −3.19357
\(952\) 0 0
\(953\) 31.7551 1.02865 0.514324 0.857596i \(-0.328043\pi\)
0.514324 + 0.857596i \(0.328043\pi\)
\(954\) 0 0
\(955\) −10.9445 −0.354156
\(956\) 0 0
\(957\) 8.97133 0.290002
\(958\) 0 0
\(959\) −16.8851 −0.545250
\(960\) 0 0
\(961\) 40.7483 1.31446
\(962\) 0 0
\(963\) −71.5268 −2.30492
\(964\) 0 0
\(965\) −12.1178 −0.390087
\(966\) 0 0
\(967\) 3.74949 0.120576 0.0602878 0.998181i \(-0.480798\pi\)
0.0602878 + 0.998181i \(0.480798\pi\)
\(968\) 0 0
\(969\) −44.6746 −1.43516
\(970\) 0 0
\(971\) −23.4254 −0.751755 −0.375878 0.926669i \(-0.622659\pi\)
−0.375878 + 0.926669i \(0.622659\pi\)
\(972\) 0 0
\(973\) 97.8847 3.13804
\(974\) 0 0
\(975\) −15.4025 −0.493274
\(976\) 0 0
\(977\) 12.6290 0.404038 0.202019 0.979382i \(-0.435250\pi\)
0.202019 + 0.979382i \(0.435250\pi\)
\(978\) 0 0
\(979\) 48.7053 1.55663
\(980\) 0 0
\(981\) 59.0772 1.88619
\(982\) 0 0
\(983\) 41.3313 1.31826 0.659131 0.752028i \(-0.270924\pi\)
0.659131 + 0.752028i \(0.270924\pi\)
\(984\) 0 0
\(985\) 2.82132 0.0898946
\(986\) 0 0
\(987\) 17.8018 0.566636
\(988\) 0 0
\(989\) 0.697545 0.0221806
\(990\) 0 0
\(991\) −16.6134 −0.527742 −0.263871 0.964558i \(-0.584999\pi\)
−0.263871 + 0.964558i \(0.584999\pi\)
\(992\) 0 0
\(993\) −95.0393 −3.01598
\(994\) 0 0
\(995\) 6.85412 0.217290
\(996\) 0 0
\(997\) −30.0754 −0.952496 −0.476248 0.879311i \(-0.658004\pi\)
−0.476248 + 0.879311i \(0.658004\pi\)
\(998\) 0 0
\(999\) −13.0916 −0.414200
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 8020.2.a.e.1.5 35
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8020.2.a.e.1.5 35 1.1 even 1 trivial