Properties

Label 8016.2.a.p.1.3
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $0$
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] = [8016,2,Mod(1,8016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8016, 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("8016.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.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: \(64.0080822603\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.36497.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 3x^{3} + 5x^{2} + x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 501)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.410375\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.19483 q^{5} +4.52759 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.19483 q^{5} +4.52759 q^{7} +1.00000 q^{9} +2.62592 q^{11} +4.50686 q^{13} +2.19483 q^{15} +2.14478 q^{17} +6.09764 q^{19} -4.52759 q^{21} +6.18843 q^{23} -0.182739 q^{25} -1.00000 q^{27} +1.84359 q^{29} -1.20401 q^{31} -2.62592 q^{33} -9.93726 q^{35} +10.7075 q^{37} -4.50686 q^{39} +0.353073 q^{41} +11.9236 q^{43} -2.19483 q^{45} -0.986349 q^{47} +13.4990 q^{49} -2.14478 q^{51} -7.36740 q^{53} -5.76345 q^{55} -6.09764 q^{57} +4.63958 q^{59} +11.9846 q^{61} +4.52759 q^{63} -9.89178 q^{65} +0.652622 q^{67} -6.18843 q^{69} +13.2454 q^{71} -7.23916 q^{73} +0.182739 q^{75} +11.8891 q^{77} -7.65804 q^{79} +1.00000 q^{81} +6.48636 q^{83} -4.70742 q^{85} -1.84359 q^{87} -6.00336 q^{89} +20.4052 q^{91} +1.20401 q^{93} -13.3833 q^{95} -8.29839 q^{97} +2.62592 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9} + 15 q^{11} + 9 q^{15} - 11 q^{17} + 16 q^{19} - 4 q^{21} + 9 q^{23} + 6 q^{25} - 5 q^{27} - q^{29} + 18 q^{31} - 15 q^{33} + 4 q^{35} + 7 q^{37} - 10 q^{41} - 6 q^{43} - 9 q^{45} + 7 q^{47} + 11 q^{49} + 11 q^{51} - 9 q^{53} - 17 q^{55} - 16 q^{57} + 37 q^{59} - 2 q^{61} + 4 q^{63} - 16 q^{65} - 9 q^{69} - 13 q^{71} - 6 q^{73} - 6 q^{75} - 2 q^{77} - 14 q^{79} + 5 q^{81} + 5 q^{83} + 29 q^{85} + q^{87} - 30 q^{89} + 33 q^{91} - 18 q^{93} - 43 q^{95} - 9 q^{97} + 15 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −2.19483 −0.981556 −0.490778 0.871285i \(-0.663287\pi\)
−0.490778 + 0.871285i \(0.663287\pi\)
\(6\) 0 0
\(7\) 4.52759 1.71127 0.855633 0.517582i \(-0.173168\pi\)
0.855633 + 0.517582i \(0.173168\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.62592 0.791746 0.395873 0.918305i \(-0.370442\pi\)
0.395873 + 0.918305i \(0.370442\pi\)
\(12\) 0 0
\(13\) 4.50686 1.24998 0.624989 0.780633i \(-0.285103\pi\)
0.624989 + 0.780633i \(0.285103\pi\)
\(14\) 0 0
\(15\) 2.19483 0.566702
\(16\) 0 0
\(17\) 2.14478 0.520186 0.260093 0.965584i \(-0.416247\pi\)
0.260093 + 0.965584i \(0.416247\pi\)
\(18\) 0 0
\(19\) 6.09764 1.39889 0.699447 0.714684i \(-0.253430\pi\)
0.699447 + 0.714684i \(0.253430\pi\)
\(20\) 0 0
\(21\) −4.52759 −0.988000
\(22\) 0 0
\(23\) 6.18843 1.29038 0.645189 0.764023i \(-0.276779\pi\)
0.645189 + 0.764023i \(0.276779\pi\)
\(24\) 0 0
\(25\) −0.182739 −0.0365478
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.84359 0.342345 0.171173 0.985241i \(-0.445244\pi\)
0.171173 + 0.985241i \(0.445244\pi\)
\(30\) 0 0
\(31\) −1.20401 −0.216247 −0.108123 0.994137i \(-0.534484\pi\)
−0.108123 + 0.994137i \(0.534484\pi\)
\(32\) 0 0
\(33\) −2.62592 −0.457115
\(34\) 0 0
\(35\) −9.93726 −1.67970
\(36\) 0 0
\(37\) 10.7075 1.76030 0.880152 0.474691i \(-0.157440\pi\)
0.880152 + 0.474691i \(0.157440\pi\)
\(38\) 0 0
\(39\) −4.50686 −0.721676
\(40\) 0 0
\(41\) 0.353073 0.0551407 0.0275703 0.999620i \(-0.491223\pi\)
0.0275703 + 0.999620i \(0.491223\pi\)
\(42\) 0 0
\(43\) 11.9236 1.81833 0.909167 0.416432i \(-0.136720\pi\)
0.909167 + 0.416432i \(0.136720\pi\)
\(44\) 0 0
\(45\) −2.19483 −0.327185
\(46\) 0 0
\(47\) −0.986349 −0.143874 −0.0719369 0.997409i \(-0.522918\pi\)
−0.0719369 + 0.997409i \(0.522918\pi\)
\(48\) 0 0
\(49\) 13.4990 1.92843
\(50\) 0 0
\(51\) −2.14478 −0.300330
\(52\) 0 0
\(53\) −7.36740 −1.01199 −0.505995 0.862536i \(-0.668875\pi\)
−0.505995 + 0.862536i \(0.668875\pi\)
\(54\) 0 0
\(55\) −5.76345 −0.777143
\(56\) 0 0
\(57\) −6.09764 −0.807652
\(58\) 0 0
\(59\) 4.63958 0.604021 0.302011 0.953305i \(-0.402342\pi\)
0.302011 + 0.953305i \(0.402342\pi\)
\(60\) 0 0
\(61\) 11.9846 1.53447 0.767237 0.641363i \(-0.221631\pi\)
0.767237 + 0.641363i \(0.221631\pi\)
\(62\) 0 0
\(63\) 4.52759 0.570422
\(64\) 0 0
\(65\) −9.89178 −1.22692
\(66\) 0 0
\(67\) 0.652622 0.0797304 0.0398652 0.999205i \(-0.487307\pi\)
0.0398652 + 0.999205i \(0.487307\pi\)
\(68\) 0 0
\(69\) −6.18843 −0.745000
\(70\) 0 0
\(71\) 13.2454 1.57194 0.785972 0.618262i \(-0.212163\pi\)
0.785972 + 0.618262i \(0.212163\pi\)
\(72\) 0 0
\(73\) −7.23916 −0.847279 −0.423640 0.905831i \(-0.639248\pi\)
−0.423640 + 0.905831i \(0.639248\pi\)
\(74\) 0 0
\(75\) 0.182739 0.0211009
\(76\) 0 0
\(77\) 11.8891 1.35489
\(78\) 0 0
\(79\) −7.65804 −0.861596 −0.430798 0.902448i \(-0.641768\pi\)
−0.430798 + 0.902448i \(0.641768\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.48636 0.711971 0.355985 0.934491i \(-0.384145\pi\)
0.355985 + 0.934491i \(0.384145\pi\)
\(84\) 0 0
\(85\) −4.70742 −0.510592
\(86\) 0 0
\(87\) −1.84359 −0.197653
\(88\) 0 0
\(89\) −6.00336 −0.636355 −0.318177 0.948031i \(-0.603071\pi\)
−0.318177 + 0.948031i \(0.603071\pi\)
\(90\) 0 0
\(91\) 20.4052 2.13905
\(92\) 0 0
\(93\) 1.20401 0.124850
\(94\) 0 0
\(95\) −13.3833 −1.37309
\(96\) 0 0
\(97\) −8.29839 −0.842573 −0.421287 0.906928i \(-0.638421\pi\)
−0.421287 + 0.906928i \(0.638421\pi\)
\(98\) 0 0
\(99\) 2.62592 0.263915
\(100\) 0 0
\(101\) −12.2104 −1.21498 −0.607489 0.794328i \(-0.707823\pi\)
−0.607489 + 0.794328i \(0.707823\pi\)
\(102\) 0 0
\(103\) −14.8965 −1.46780 −0.733898 0.679260i \(-0.762301\pi\)
−0.733898 + 0.679260i \(0.762301\pi\)
\(104\) 0 0
\(105\) 9.93726 0.969778
\(106\) 0 0
\(107\) −10.0613 −0.972666 −0.486333 0.873773i \(-0.661666\pi\)
−0.486333 + 0.873773i \(0.661666\pi\)
\(108\) 0 0
\(109\) −10.9585 −1.04964 −0.524819 0.851214i \(-0.675867\pi\)
−0.524819 + 0.851214i \(0.675867\pi\)
\(110\) 0 0
\(111\) −10.7075 −1.01631
\(112\) 0 0
\(113\) −12.0220 −1.13093 −0.565465 0.824772i \(-0.691303\pi\)
−0.565465 + 0.824772i \(0.691303\pi\)
\(114\) 0 0
\(115\) −13.5825 −1.26658
\(116\) 0 0
\(117\) 4.50686 0.416660
\(118\) 0 0
\(119\) 9.71069 0.890177
\(120\) 0 0
\(121\) −4.10452 −0.373138
\(122\) 0 0
\(123\) −0.353073 −0.0318355
\(124\) 0 0
\(125\) 11.3752 1.01743
\(126\) 0 0
\(127\) 2.03046 0.180174 0.0900869 0.995934i \(-0.471286\pi\)
0.0900869 + 0.995934i \(0.471286\pi\)
\(128\) 0 0
\(129\) −11.9236 −1.04982
\(130\) 0 0
\(131\) −7.73416 −0.675737 −0.337868 0.941193i \(-0.609706\pi\)
−0.337868 + 0.941193i \(0.609706\pi\)
\(132\) 0 0
\(133\) 27.6076 2.39388
\(134\) 0 0
\(135\) 2.19483 0.188901
\(136\) 0 0
\(137\) −19.1222 −1.63372 −0.816862 0.576833i \(-0.804288\pi\)
−0.816862 + 0.576833i \(0.804288\pi\)
\(138\) 0 0
\(139\) 2.90796 0.246650 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(140\) 0 0
\(141\) 0.986349 0.0830655
\(142\) 0 0
\(143\) 11.8347 0.989666
\(144\) 0 0
\(145\) −4.04635 −0.336031
\(146\) 0 0
\(147\) −13.4990 −1.11338
\(148\) 0 0
\(149\) −13.5856 −1.11298 −0.556488 0.830855i \(-0.687852\pi\)
−0.556488 + 0.830855i \(0.687852\pi\)
\(150\) 0 0
\(151\) −17.3253 −1.40991 −0.704955 0.709252i \(-0.749033\pi\)
−0.704955 + 0.709252i \(0.749033\pi\)
\(152\) 0 0
\(153\) 2.14478 0.173395
\(154\) 0 0
\(155\) 2.64259 0.212258
\(156\) 0 0
\(157\) 14.9268 1.19129 0.595646 0.803247i \(-0.296896\pi\)
0.595646 + 0.803247i \(0.296896\pi\)
\(158\) 0 0
\(159\) 7.36740 0.584273
\(160\) 0 0
\(161\) 28.0187 2.20818
\(162\) 0 0
\(163\) −18.0795 −1.41609 −0.708046 0.706166i \(-0.750423\pi\)
−0.708046 + 0.706166i \(0.750423\pi\)
\(164\) 0 0
\(165\) 5.76345 0.448684
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 7.31181 0.562447
\(170\) 0 0
\(171\) 6.09764 0.466298
\(172\) 0 0
\(173\) −17.2480 −1.31134 −0.655671 0.755047i \(-0.727614\pi\)
−0.655671 + 0.755047i \(0.727614\pi\)
\(174\) 0 0
\(175\) −0.827368 −0.0625431
\(176\) 0 0
\(177\) −4.63958 −0.348732
\(178\) 0 0
\(179\) 13.8251 1.03334 0.516670 0.856185i \(-0.327171\pi\)
0.516670 + 0.856185i \(0.327171\pi\)
\(180\) 0 0
\(181\) 0.887785 0.0659886 0.0329943 0.999456i \(-0.489496\pi\)
0.0329943 + 0.999456i \(0.489496\pi\)
\(182\) 0 0
\(183\) −11.9846 −0.885929
\(184\) 0 0
\(185\) −23.5011 −1.72784
\(186\) 0 0
\(187\) 5.63204 0.411855
\(188\) 0 0
\(189\) −4.52759 −0.329333
\(190\) 0 0
\(191\) −12.8833 −0.932200 −0.466100 0.884732i \(-0.654341\pi\)
−0.466100 + 0.884732i \(0.654341\pi\)
\(192\) 0 0
\(193\) 19.6292 1.41294 0.706470 0.707743i \(-0.250286\pi\)
0.706470 + 0.707743i \(0.250286\pi\)
\(194\) 0 0
\(195\) 9.89178 0.708365
\(196\) 0 0
\(197\) −4.98382 −0.355082 −0.177541 0.984113i \(-0.556814\pi\)
−0.177541 + 0.984113i \(0.556814\pi\)
\(198\) 0 0
\(199\) 14.6453 1.03818 0.519089 0.854720i \(-0.326271\pi\)
0.519089 + 0.854720i \(0.326271\pi\)
\(200\) 0 0
\(201\) −0.652622 −0.0460324
\(202\) 0 0
\(203\) 8.34700 0.585844
\(204\) 0 0
\(205\) −0.774933 −0.0541236
\(206\) 0 0
\(207\) 6.18843 0.430126
\(208\) 0 0
\(209\) 16.0119 1.10757
\(210\) 0 0
\(211\) −3.01322 −0.207438 −0.103719 0.994607i \(-0.533074\pi\)
−0.103719 + 0.994607i \(0.533074\pi\)
\(212\) 0 0
\(213\) −13.2454 −0.907563
\(214\) 0 0
\(215\) −26.1703 −1.78480
\(216\) 0 0
\(217\) −5.45126 −0.370056
\(218\) 0 0
\(219\) 7.23916 0.489177
\(220\) 0 0
\(221\) 9.66624 0.650222
\(222\) 0 0
\(223\) 9.11072 0.610099 0.305050 0.952336i \(-0.401327\pi\)
0.305050 + 0.952336i \(0.401327\pi\)
\(224\) 0 0
\(225\) −0.182739 −0.0121826
\(226\) 0 0
\(227\) 18.3539 1.21819 0.609097 0.793096i \(-0.291532\pi\)
0.609097 + 0.793096i \(0.291532\pi\)
\(228\) 0 0
\(229\) 5.54404 0.366360 0.183180 0.983079i \(-0.441361\pi\)
0.183180 + 0.983079i \(0.441361\pi\)
\(230\) 0 0
\(231\) −11.8891 −0.782245
\(232\) 0 0
\(233\) 2.85999 0.187364 0.0936821 0.995602i \(-0.470136\pi\)
0.0936821 + 0.995602i \(0.470136\pi\)
\(234\) 0 0
\(235\) 2.16486 0.141220
\(236\) 0 0
\(237\) 7.65804 0.497443
\(238\) 0 0
\(239\) −25.0961 −1.62333 −0.811666 0.584122i \(-0.801439\pi\)
−0.811666 + 0.584122i \(0.801439\pi\)
\(240\) 0 0
\(241\) −18.3872 −1.18442 −0.592211 0.805783i \(-0.701745\pi\)
−0.592211 + 0.805783i \(0.701745\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −29.6280 −1.89287
\(246\) 0 0
\(247\) 27.4812 1.74859
\(248\) 0 0
\(249\) −6.48636 −0.411057
\(250\) 0 0
\(251\) −15.0744 −0.951491 −0.475745 0.879583i \(-0.657822\pi\)
−0.475745 + 0.879583i \(0.657822\pi\)
\(252\) 0 0
\(253\) 16.2504 1.02165
\(254\) 0 0
\(255\) 4.70742 0.294790
\(256\) 0 0
\(257\) 26.1018 1.62819 0.814094 0.580733i \(-0.197234\pi\)
0.814094 + 0.580733i \(0.197234\pi\)
\(258\) 0 0
\(259\) 48.4792 3.01235
\(260\) 0 0
\(261\) 1.84359 0.114115
\(262\) 0 0
\(263\) 17.5960 1.08502 0.542508 0.840051i \(-0.317475\pi\)
0.542508 + 0.840051i \(0.317475\pi\)
\(264\) 0 0
\(265\) 16.1702 0.993326
\(266\) 0 0
\(267\) 6.00336 0.367400
\(268\) 0 0
\(269\) −17.8079 −1.08577 −0.542885 0.839807i \(-0.682668\pi\)
−0.542885 + 0.839807i \(0.682668\pi\)
\(270\) 0 0
\(271\) 20.5804 1.25017 0.625085 0.780557i \(-0.285064\pi\)
0.625085 + 0.780557i \(0.285064\pi\)
\(272\) 0 0
\(273\) −20.4052 −1.23498
\(274\) 0 0
\(275\) −0.479860 −0.0289366
\(276\) 0 0
\(277\) 4.86009 0.292014 0.146007 0.989284i \(-0.453358\pi\)
0.146007 + 0.989284i \(0.453358\pi\)
\(278\) 0 0
\(279\) −1.20401 −0.0720822
\(280\) 0 0
\(281\) 0.340342 0.0203031 0.0101516 0.999948i \(-0.496769\pi\)
0.0101516 + 0.999948i \(0.496769\pi\)
\(282\) 0 0
\(283\) 10.8080 0.642469 0.321235 0.947000i \(-0.395902\pi\)
0.321235 + 0.947000i \(0.395902\pi\)
\(284\) 0 0
\(285\) 13.3833 0.792756
\(286\) 0 0
\(287\) 1.59857 0.0943604
\(288\) 0 0
\(289\) −12.3999 −0.729406
\(290\) 0 0
\(291\) 8.29839 0.486460
\(292\) 0 0
\(293\) 13.0134 0.760254 0.380127 0.924934i \(-0.375880\pi\)
0.380127 + 0.924934i \(0.375880\pi\)
\(294\) 0 0
\(295\) −10.1831 −0.592881
\(296\) 0 0
\(297\) −2.62592 −0.152372
\(298\) 0 0
\(299\) 27.8904 1.61294
\(300\) 0 0
\(301\) 53.9852 3.11165
\(302\) 0 0
\(303\) 12.2104 0.701468
\(304\) 0 0
\(305\) −26.3042 −1.50617
\(306\) 0 0
\(307\) −10.6004 −0.605000 −0.302500 0.953149i \(-0.597821\pi\)
−0.302500 + 0.953149i \(0.597821\pi\)
\(308\) 0 0
\(309\) 14.8965 0.847432
\(310\) 0 0
\(311\) −23.2712 −1.31959 −0.659795 0.751446i \(-0.729357\pi\)
−0.659795 + 0.751446i \(0.729357\pi\)
\(312\) 0 0
\(313\) −6.59154 −0.372576 −0.186288 0.982495i \(-0.559646\pi\)
−0.186288 + 0.982495i \(0.559646\pi\)
\(314\) 0 0
\(315\) −9.93726 −0.559901
\(316\) 0 0
\(317\) 8.90860 0.500357 0.250178 0.968200i \(-0.419511\pi\)
0.250178 + 0.968200i \(0.419511\pi\)
\(318\) 0 0
\(319\) 4.84112 0.271051
\(320\) 0 0
\(321\) 10.0613 0.561569
\(322\) 0 0
\(323\) 13.0781 0.727686
\(324\) 0 0
\(325\) −0.823581 −0.0456840
\(326\) 0 0
\(327\) 10.9585 0.606008
\(328\) 0 0
\(329\) −4.46578 −0.246206
\(330\) 0 0
\(331\) 11.2953 0.620845 0.310422 0.950599i \(-0.399530\pi\)
0.310422 + 0.950599i \(0.399530\pi\)
\(332\) 0 0
\(333\) 10.7075 0.586768
\(334\) 0 0
\(335\) −1.43239 −0.0782599
\(336\) 0 0
\(337\) 21.4635 1.16919 0.584596 0.811325i \(-0.301253\pi\)
0.584596 + 0.811325i \(0.301253\pi\)
\(338\) 0 0
\(339\) 12.0220 0.652943
\(340\) 0 0
\(341\) −3.16164 −0.171212
\(342\) 0 0
\(343\) 29.4250 1.58880
\(344\) 0 0
\(345\) 13.5825 0.731259
\(346\) 0 0
\(347\) −23.3110 −1.25140 −0.625699 0.780065i \(-0.715186\pi\)
−0.625699 + 0.780065i \(0.715186\pi\)
\(348\) 0 0
\(349\) −22.1588 −1.18613 −0.593066 0.805154i \(-0.702083\pi\)
−0.593066 + 0.805154i \(0.702083\pi\)
\(350\) 0 0
\(351\) −4.50686 −0.240559
\(352\) 0 0
\(353\) −11.5024 −0.612209 −0.306105 0.951998i \(-0.599026\pi\)
−0.306105 + 0.951998i \(0.599026\pi\)
\(354\) 0 0
\(355\) −29.0714 −1.54295
\(356\) 0 0
\(357\) −9.71069 −0.513944
\(358\) 0 0
\(359\) −33.4833 −1.76718 −0.883590 0.468262i \(-0.844880\pi\)
−0.883590 + 0.468262i \(0.844880\pi\)
\(360\) 0 0
\(361\) 18.1812 0.956906
\(362\) 0 0
\(363\) 4.10452 0.215431
\(364\) 0 0
\(365\) 15.8887 0.831652
\(366\) 0 0
\(367\) 2.74783 0.143436 0.0717179 0.997425i \(-0.477152\pi\)
0.0717179 + 0.997425i \(0.477152\pi\)
\(368\) 0 0
\(369\) 0.353073 0.0183802
\(370\) 0 0
\(371\) −33.3566 −1.73179
\(372\) 0 0
\(373\) 8.10471 0.419646 0.209823 0.977739i \(-0.432711\pi\)
0.209823 + 0.977739i \(0.432711\pi\)
\(374\) 0 0
\(375\) −11.3752 −0.587413
\(376\) 0 0
\(377\) 8.30879 0.427925
\(378\) 0 0
\(379\) 4.29718 0.220731 0.110366 0.993891i \(-0.464798\pi\)
0.110366 + 0.993891i \(0.464798\pi\)
\(380\) 0 0
\(381\) −2.03046 −0.104023
\(382\) 0 0
\(383\) 27.4329 1.40176 0.700878 0.713281i \(-0.252792\pi\)
0.700878 + 0.713281i \(0.252792\pi\)
\(384\) 0 0
\(385\) −26.0945 −1.32990
\(386\) 0 0
\(387\) 11.9236 0.606111
\(388\) 0 0
\(389\) 1.77324 0.0899067 0.0449534 0.998989i \(-0.485686\pi\)
0.0449534 + 0.998989i \(0.485686\pi\)
\(390\) 0 0
\(391\) 13.2728 0.671236
\(392\) 0 0
\(393\) 7.73416 0.390137
\(394\) 0 0
\(395\) 16.8081 0.845705
\(396\) 0 0
\(397\) −5.70497 −0.286324 −0.143162 0.989699i \(-0.545727\pi\)
−0.143162 + 0.989699i \(0.545727\pi\)
\(398\) 0 0
\(399\) −27.6076 −1.38211
\(400\) 0 0
\(401\) −22.7358 −1.13537 −0.567686 0.823245i \(-0.692161\pi\)
−0.567686 + 0.823245i \(0.692161\pi\)
\(402\) 0 0
\(403\) −5.42631 −0.270304
\(404\) 0 0
\(405\) −2.19483 −0.109062
\(406\) 0 0
\(407\) 28.1171 1.39371
\(408\) 0 0
\(409\) −22.6282 −1.11889 −0.559445 0.828867i \(-0.688986\pi\)
−0.559445 + 0.828867i \(0.688986\pi\)
\(410\) 0 0
\(411\) 19.1222 0.943231
\(412\) 0 0
\(413\) 21.0061 1.03364
\(414\) 0 0
\(415\) −14.2364 −0.698839
\(416\) 0 0
\(417\) −2.90796 −0.142403
\(418\) 0 0
\(419\) 24.0907 1.17691 0.588453 0.808531i \(-0.299737\pi\)
0.588453 + 0.808531i \(0.299737\pi\)
\(420\) 0 0
\(421\) −5.42990 −0.264637 −0.132319 0.991207i \(-0.542242\pi\)
−0.132319 + 0.991207i \(0.542242\pi\)
\(422\) 0 0
\(423\) −0.986349 −0.0479579
\(424\) 0 0
\(425\) −0.391936 −0.0190117
\(426\) 0 0
\(427\) 54.2614 2.62590
\(428\) 0 0
\(429\) −11.8347 −0.571384
\(430\) 0 0
\(431\) 10.8167 0.521024 0.260512 0.965471i \(-0.416109\pi\)
0.260512 + 0.965471i \(0.416109\pi\)
\(432\) 0 0
\(433\) −17.6776 −0.849530 −0.424765 0.905304i \(-0.639643\pi\)
−0.424765 + 0.905304i \(0.639643\pi\)
\(434\) 0 0
\(435\) 4.04635 0.194008
\(436\) 0 0
\(437\) 37.7348 1.80510
\(438\) 0 0
\(439\) −13.9547 −0.666023 −0.333011 0.942923i \(-0.608065\pi\)
−0.333011 + 0.942923i \(0.608065\pi\)
\(440\) 0 0
\(441\) 13.4990 0.642811
\(442\) 0 0
\(443\) −41.1782 −1.95643 −0.978217 0.207584i \(-0.933440\pi\)
−0.978217 + 0.207584i \(0.933440\pi\)
\(444\) 0 0
\(445\) 13.1763 0.624618
\(446\) 0 0
\(447\) 13.5856 0.642577
\(448\) 0 0
\(449\) 33.9009 1.59988 0.799942 0.600077i \(-0.204863\pi\)
0.799942 + 0.600077i \(0.204863\pi\)
\(450\) 0 0
\(451\) 0.927142 0.0436574
\(452\) 0 0
\(453\) 17.3253 0.814012
\(454\) 0 0
\(455\) −44.7859 −2.09959
\(456\) 0 0
\(457\) −1.56208 −0.0730708 −0.0365354 0.999332i \(-0.511632\pi\)
−0.0365354 + 0.999332i \(0.511632\pi\)
\(458\) 0 0
\(459\) −2.14478 −0.100110
\(460\) 0 0
\(461\) 6.34383 0.295462 0.147731 0.989028i \(-0.452803\pi\)
0.147731 + 0.989028i \(0.452803\pi\)
\(462\) 0 0
\(463\) −15.7171 −0.730437 −0.365218 0.930922i \(-0.619006\pi\)
−0.365218 + 0.930922i \(0.619006\pi\)
\(464\) 0 0
\(465\) −2.64259 −0.122547
\(466\) 0 0
\(467\) 41.4281 1.91706 0.958532 0.284987i \(-0.0919891\pi\)
0.958532 + 0.284987i \(0.0919891\pi\)
\(468\) 0 0
\(469\) 2.95480 0.136440
\(470\) 0 0
\(471\) −14.9268 −0.687792
\(472\) 0 0
\(473\) 31.3105 1.43966
\(474\) 0 0
\(475\) −1.11428 −0.0511266
\(476\) 0 0
\(477\) −7.36740 −0.337330
\(478\) 0 0
\(479\) −21.7486 −0.993720 −0.496860 0.867831i \(-0.665514\pi\)
−0.496860 + 0.867831i \(0.665514\pi\)
\(480\) 0 0
\(481\) 48.2573 2.20034
\(482\) 0 0
\(483\) −28.0187 −1.27489
\(484\) 0 0
\(485\) 18.2135 0.827033
\(486\) 0 0
\(487\) 4.22677 0.191533 0.0957667 0.995404i \(-0.469470\pi\)
0.0957667 + 0.995404i \(0.469470\pi\)
\(488\) 0 0
\(489\) 18.0795 0.817581
\(490\) 0 0
\(491\) 11.2226 0.506469 0.253235 0.967405i \(-0.418505\pi\)
0.253235 + 0.967405i \(0.418505\pi\)
\(492\) 0 0
\(493\) 3.95409 0.178083
\(494\) 0 0
\(495\) −5.76345 −0.259048
\(496\) 0 0
\(497\) 59.9699 2.69002
\(498\) 0 0
\(499\) −8.02996 −0.359470 −0.179735 0.983715i \(-0.557524\pi\)
−0.179735 + 0.983715i \(0.557524\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) −35.0823 −1.56424 −0.782121 0.623127i \(-0.785862\pi\)
−0.782121 + 0.623127i \(0.785862\pi\)
\(504\) 0 0
\(505\) 26.7997 1.19257
\(506\) 0 0
\(507\) −7.31181 −0.324729
\(508\) 0 0
\(509\) −0.620513 −0.0275038 −0.0137519 0.999905i \(-0.504377\pi\)
−0.0137519 + 0.999905i \(0.504377\pi\)
\(510\) 0 0
\(511\) −32.7759 −1.44992
\(512\) 0 0
\(513\) −6.09764 −0.269217
\(514\) 0 0
\(515\) 32.6952 1.44072
\(516\) 0 0
\(517\) −2.59008 −0.113911
\(518\) 0 0
\(519\) 17.2480 0.757103
\(520\) 0 0
\(521\) 17.2556 0.755983 0.377992 0.925809i \(-0.376615\pi\)
0.377992 + 0.925809i \(0.376615\pi\)
\(522\) 0 0
\(523\) −27.4817 −1.20169 −0.600845 0.799366i \(-0.705169\pi\)
−0.600845 + 0.799366i \(0.705169\pi\)
\(524\) 0 0
\(525\) 0.827368 0.0361093
\(526\) 0 0
\(527\) −2.58234 −0.112488
\(528\) 0 0
\(529\) 15.2967 0.665074
\(530\) 0 0
\(531\) 4.63958 0.201340
\(532\) 0 0
\(533\) 1.59125 0.0689247
\(534\) 0 0
\(535\) 22.0829 0.954727
\(536\) 0 0
\(537\) −13.8251 −0.596599
\(538\) 0 0
\(539\) 35.4475 1.52683
\(540\) 0 0
\(541\) 14.2636 0.613239 0.306620 0.951832i \(-0.400802\pi\)
0.306620 + 0.951832i \(0.400802\pi\)
\(542\) 0 0
\(543\) −0.887785 −0.0380985
\(544\) 0 0
\(545\) 24.0521 1.03028
\(546\) 0 0
\(547\) −26.1563 −1.11836 −0.559182 0.829045i \(-0.688885\pi\)
−0.559182 + 0.829045i \(0.688885\pi\)
\(548\) 0 0
\(549\) 11.9846 0.511492
\(550\) 0 0
\(551\) 11.2415 0.478905
\(552\) 0 0
\(553\) −34.6724 −1.47442
\(554\) 0 0
\(555\) 23.5011 0.997568
\(556\) 0 0
\(557\) −22.0575 −0.934608 −0.467304 0.884097i \(-0.654775\pi\)
−0.467304 + 0.884097i \(0.654775\pi\)
\(558\) 0 0
\(559\) 53.7381 2.27288
\(560\) 0 0
\(561\) −5.63204 −0.237785
\(562\) 0 0
\(563\) −4.77511 −0.201247 −0.100623 0.994925i \(-0.532084\pi\)
−0.100623 + 0.994925i \(0.532084\pi\)
\(564\) 0 0
\(565\) 26.3861 1.11007
\(566\) 0 0
\(567\) 4.52759 0.190141
\(568\) 0 0
\(569\) −39.2560 −1.64570 −0.822850 0.568259i \(-0.807617\pi\)
−0.822850 + 0.568259i \(0.807617\pi\)
\(570\) 0 0
\(571\) −27.3668 −1.14527 −0.572633 0.819812i \(-0.694078\pi\)
−0.572633 + 0.819812i \(0.694078\pi\)
\(572\) 0 0
\(573\) 12.8833 0.538206
\(574\) 0 0
\(575\) −1.13087 −0.0471605
\(576\) 0 0
\(577\) 42.7549 1.77991 0.889954 0.456050i \(-0.150736\pi\)
0.889954 + 0.456050i \(0.150736\pi\)
\(578\) 0 0
\(579\) −19.6292 −0.815762
\(580\) 0 0
\(581\) 29.3676 1.21837
\(582\) 0 0
\(583\) −19.3462 −0.801240
\(584\) 0 0
\(585\) −9.89178 −0.408975
\(586\) 0 0
\(587\) 21.8718 0.902747 0.451374 0.892335i \(-0.350934\pi\)
0.451374 + 0.892335i \(0.350934\pi\)
\(588\) 0 0
\(589\) −7.34162 −0.302506
\(590\) 0 0
\(591\) 4.98382 0.205007
\(592\) 0 0
\(593\) −0.763853 −0.0313677 −0.0156839 0.999877i \(-0.504993\pi\)
−0.0156839 + 0.999877i \(0.504993\pi\)
\(594\) 0 0
\(595\) −21.3133 −0.873759
\(596\) 0 0
\(597\) −14.6453 −0.599392
\(598\) 0 0
\(599\) −3.83490 −0.156690 −0.0783450 0.996926i \(-0.524964\pi\)
−0.0783450 + 0.996926i \(0.524964\pi\)
\(600\) 0 0
\(601\) −38.9975 −1.59074 −0.795370 0.606124i \(-0.792723\pi\)
−0.795370 + 0.606124i \(0.792723\pi\)
\(602\) 0 0
\(603\) 0.652622 0.0265768
\(604\) 0 0
\(605\) 9.00870 0.366256
\(606\) 0 0
\(607\) −25.1931 −1.02256 −0.511279 0.859415i \(-0.670828\pi\)
−0.511279 + 0.859415i \(0.670828\pi\)
\(608\) 0 0
\(609\) −8.34700 −0.338237
\(610\) 0 0
\(611\) −4.44534 −0.179839
\(612\) 0 0
\(613\) −44.1016 −1.78125 −0.890623 0.454742i \(-0.849731\pi\)
−0.890623 + 0.454742i \(0.849731\pi\)
\(614\) 0 0
\(615\) 0.774933 0.0312483
\(616\) 0 0
\(617\) 3.77330 0.151908 0.0759538 0.997111i \(-0.475800\pi\)
0.0759538 + 0.997111i \(0.475800\pi\)
\(618\) 0 0
\(619\) 35.1811 1.41405 0.707024 0.707190i \(-0.250037\pi\)
0.707024 + 0.707190i \(0.250037\pi\)
\(620\) 0 0
\(621\) −6.18843 −0.248333
\(622\) 0 0
\(623\) −27.1807 −1.08897
\(624\) 0 0
\(625\) −24.0529 −0.962116
\(626\) 0 0
\(627\) −16.0119 −0.639456
\(628\) 0 0
\(629\) 22.9653 0.915686
\(630\) 0 0
\(631\) 15.6390 0.622579 0.311289 0.950315i \(-0.399239\pi\)
0.311289 + 0.950315i \(0.399239\pi\)
\(632\) 0 0
\(633\) 3.01322 0.119765
\(634\) 0 0
\(635\) −4.45650 −0.176851
\(636\) 0 0
\(637\) 60.8383 2.41050
\(638\) 0 0
\(639\) 13.2454 0.523981
\(640\) 0 0
\(641\) 24.1866 0.955313 0.477657 0.878547i \(-0.341486\pi\)
0.477657 + 0.878547i \(0.341486\pi\)
\(642\) 0 0
\(643\) 24.7104 0.974483 0.487242 0.873267i \(-0.338003\pi\)
0.487242 + 0.873267i \(0.338003\pi\)
\(644\) 0 0
\(645\) 26.1703 1.03045
\(646\) 0 0
\(647\) −5.00060 −0.196594 −0.0982969 0.995157i \(-0.531339\pi\)
−0.0982969 + 0.995157i \(0.531339\pi\)
\(648\) 0 0
\(649\) 12.1832 0.478232
\(650\) 0 0
\(651\) 5.45126 0.213652
\(652\) 0 0
\(653\) 14.9748 0.586009 0.293005 0.956111i \(-0.405345\pi\)
0.293005 + 0.956111i \(0.405345\pi\)
\(654\) 0 0
\(655\) 16.9751 0.663273
\(656\) 0 0
\(657\) −7.23916 −0.282426
\(658\) 0 0
\(659\) 33.2094 1.29366 0.646828 0.762636i \(-0.276095\pi\)
0.646828 + 0.762636i \(0.276095\pi\)
\(660\) 0 0
\(661\) −5.56439 −0.216430 −0.108215 0.994128i \(-0.534513\pi\)
−0.108215 + 0.994128i \(0.534513\pi\)
\(662\) 0 0
\(663\) −9.66624 −0.375406
\(664\) 0 0
\(665\) −60.5939 −2.34973
\(666\) 0 0
\(667\) 11.4089 0.441755
\(668\) 0 0
\(669\) −9.11072 −0.352241
\(670\) 0 0
\(671\) 31.4707 1.21491
\(672\) 0 0
\(673\) 42.9558 1.65582 0.827911 0.560859i \(-0.189529\pi\)
0.827911 + 0.560859i \(0.189529\pi\)
\(674\) 0 0
\(675\) 0.182739 0.00703364
\(676\) 0 0
\(677\) −12.7007 −0.488126 −0.244063 0.969759i \(-0.578480\pi\)
−0.244063 + 0.969759i \(0.578480\pi\)
\(678\) 0 0
\(679\) −37.5717 −1.44187
\(680\) 0 0
\(681\) −18.3539 −0.703324
\(682\) 0 0
\(683\) 29.4729 1.12775 0.563874 0.825861i \(-0.309310\pi\)
0.563874 + 0.825861i \(0.309310\pi\)
\(684\) 0 0
\(685\) 41.9700 1.60359
\(686\) 0 0
\(687\) −5.54404 −0.211518
\(688\) 0 0
\(689\) −33.2039 −1.26497
\(690\) 0 0
\(691\) 13.4746 0.512599 0.256300 0.966597i \(-0.417497\pi\)
0.256300 + 0.966597i \(0.417497\pi\)
\(692\) 0 0
\(693\) 11.8891 0.451630
\(694\) 0 0
\(695\) −6.38247 −0.242101
\(696\) 0 0
\(697\) 0.757264 0.0286834
\(698\) 0 0
\(699\) −2.85999 −0.108175
\(700\) 0 0
\(701\) −32.7670 −1.23759 −0.618796 0.785552i \(-0.712379\pi\)
−0.618796 + 0.785552i \(0.712379\pi\)
\(702\) 0 0
\(703\) 65.2906 2.46248
\(704\) 0 0
\(705\) −2.16486 −0.0815335
\(706\) 0 0
\(707\) −55.2836 −2.07915
\(708\) 0 0
\(709\) −34.9670 −1.31321 −0.656607 0.754233i \(-0.728009\pi\)
−0.656607 + 0.754233i \(0.728009\pi\)
\(710\) 0 0
\(711\) −7.65804 −0.287199
\(712\) 0 0
\(713\) −7.45094 −0.279040
\(714\) 0 0
\(715\) −25.9751 −0.971413
\(716\) 0 0
\(717\) 25.0961 0.937231
\(718\) 0 0
\(719\) 9.48429 0.353704 0.176852 0.984237i \(-0.443409\pi\)
0.176852 + 0.984237i \(0.443409\pi\)
\(720\) 0 0
\(721\) −67.4452 −2.51179
\(722\) 0 0
\(723\) 18.3872 0.683827
\(724\) 0 0
\(725\) −0.336896 −0.0125120
\(726\) 0 0
\(727\) 14.6705 0.544098 0.272049 0.962283i \(-0.412299\pi\)
0.272049 + 0.962283i \(0.412299\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 25.5736 0.945872
\(732\) 0 0
\(733\) 1.04809 0.0387121 0.0193560 0.999813i \(-0.493838\pi\)
0.0193560 + 0.999813i \(0.493838\pi\)
\(734\) 0 0
\(735\) 29.6280 1.09285
\(736\) 0 0
\(737\) 1.71374 0.0631263
\(738\) 0 0
\(739\) −11.8911 −0.437422 −0.218711 0.975790i \(-0.570185\pi\)
−0.218711 + 0.975790i \(0.570185\pi\)
\(740\) 0 0
\(741\) −27.4812 −1.00955
\(742\) 0 0
\(743\) −36.4383 −1.33679 −0.668396 0.743806i \(-0.733019\pi\)
−0.668396 + 0.743806i \(0.733019\pi\)
\(744\) 0 0
\(745\) 29.8181 1.09245
\(746\) 0 0
\(747\) 6.48636 0.237324
\(748\) 0 0
\(749\) −45.5536 −1.66449
\(750\) 0 0
\(751\) 17.9405 0.654659 0.327330 0.944910i \(-0.393851\pi\)
0.327330 + 0.944910i \(0.393851\pi\)
\(752\) 0 0
\(753\) 15.0744 0.549343
\(754\) 0 0
\(755\) 38.0259 1.38391
\(756\) 0 0
\(757\) −12.8400 −0.466679 −0.233339 0.972395i \(-0.574965\pi\)
−0.233339 + 0.972395i \(0.574965\pi\)
\(758\) 0 0
\(759\) −16.2504 −0.589851
\(760\) 0 0
\(761\) −51.8862 −1.88087 −0.940437 0.339968i \(-0.889584\pi\)
−0.940437 + 0.339968i \(0.889584\pi\)
\(762\) 0 0
\(763\) −49.6157 −1.79621
\(764\) 0 0
\(765\) −4.70742 −0.170197
\(766\) 0 0
\(767\) 20.9099 0.755014
\(768\) 0 0
\(769\) −28.7814 −1.03789 −0.518943 0.854809i \(-0.673674\pi\)
−0.518943 + 0.854809i \(0.673674\pi\)
\(770\) 0 0
\(771\) −26.1018 −0.940035
\(772\) 0 0
\(773\) −6.52345 −0.234632 −0.117316 0.993095i \(-0.537429\pi\)
−0.117316 + 0.993095i \(0.537429\pi\)
\(774\) 0 0
\(775\) 0.220020 0.00790335
\(776\) 0 0
\(777\) −48.4792 −1.73918
\(778\) 0 0
\(779\) 2.15291 0.0771360
\(780\) 0 0
\(781\) 34.7815 1.24458
\(782\) 0 0
\(783\) −1.84359 −0.0658844
\(784\) 0 0
\(785\) −32.7618 −1.16932
\(786\) 0 0
\(787\) −27.3603 −0.975290 −0.487645 0.873042i \(-0.662144\pi\)
−0.487645 + 0.873042i \(0.662144\pi\)
\(788\) 0 0
\(789\) −17.5960 −0.626434
\(790\) 0 0
\(791\) −54.4304 −1.93532
\(792\) 0 0
\(793\) 54.0131 1.91806
\(794\) 0 0
\(795\) −16.1702 −0.573497
\(796\) 0 0
\(797\) 16.1698 0.572764 0.286382 0.958116i \(-0.407547\pi\)
0.286382 + 0.958116i \(0.407547\pi\)
\(798\) 0 0
\(799\) −2.11550 −0.0748411
\(800\) 0 0
\(801\) −6.00336 −0.212118
\(802\) 0 0
\(803\) −19.0095 −0.670830
\(804\) 0 0
\(805\) −61.4961 −2.16745
\(806\) 0 0
\(807\) 17.8079 0.626869
\(808\) 0 0
\(809\) −9.51312 −0.334464 −0.167232 0.985918i \(-0.553483\pi\)
−0.167232 + 0.985918i \(0.553483\pi\)
\(810\) 0 0
\(811\) 3.25453 0.114282 0.0571410 0.998366i \(-0.481802\pi\)
0.0571410 + 0.998366i \(0.481802\pi\)
\(812\) 0 0
\(813\) −20.5804 −0.721786
\(814\) 0 0
\(815\) 39.6813 1.38997
\(816\) 0 0
\(817\) 72.7059 2.54366
\(818\) 0 0
\(819\) 20.4052 0.713016
\(820\) 0 0
\(821\) 14.1377 0.493408 0.246704 0.969091i \(-0.420652\pi\)
0.246704 + 0.969091i \(0.420652\pi\)
\(822\) 0 0
\(823\) 4.99950 0.174272 0.0871358 0.996196i \(-0.472229\pi\)
0.0871358 + 0.996196i \(0.472229\pi\)
\(824\) 0 0
\(825\) 0.479860 0.0167066
\(826\) 0 0
\(827\) 30.7739 1.07011 0.535056 0.844816i \(-0.320290\pi\)
0.535056 + 0.844816i \(0.320290\pi\)
\(828\) 0 0
\(829\) 11.7810 0.409171 0.204586 0.978849i \(-0.434415\pi\)
0.204586 + 0.978849i \(0.434415\pi\)
\(830\) 0 0
\(831\) −4.86009 −0.168595
\(832\) 0 0
\(833\) 28.9525 1.00314
\(834\) 0 0
\(835\) 2.19483 0.0759551
\(836\) 0 0
\(837\) 1.20401 0.0416167
\(838\) 0 0
\(839\) 27.5577 0.951398 0.475699 0.879608i \(-0.342195\pi\)
0.475699 + 0.879608i \(0.342195\pi\)
\(840\) 0 0
\(841\) −25.6012 −0.882800
\(842\) 0 0
\(843\) −0.340342 −0.0117220
\(844\) 0 0
\(845\) −16.0482 −0.552073
\(846\) 0 0
\(847\) −18.5836 −0.638539
\(848\) 0 0
\(849\) −10.8080 −0.370930
\(850\) 0 0
\(851\) 66.2627 2.27146
\(852\) 0 0
\(853\) 11.9459 0.409020 0.204510 0.978864i \(-0.434440\pi\)
0.204510 + 0.978864i \(0.434440\pi\)
\(854\) 0 0
\(855\) −13.3833 −0.457698
\(856\) 0 0
\(857\) −41.7299 −1.42547 −0.712734 0.701435i \(-0.752543\pi\)
−0.712734 + 0.701435i \(0.752543\pi\)
\(858\) 0 0
\(859\) −24.4056 −0.832708 −0.416354 0.909203i \(-0.636692\pi\)
−0.416354 + 0.909203i \(0.636692\pi\)
\(860\) 0 0
\(861\) −1.59857 −0.0544790
\(862\) 0 0
\(863\) 6.92599 0.235763 0.117882 0.993028i \(-0.462390\pi\)
0.117882 + 0.993028i \(0.462390\pi\)
\(864\) 0 0
\(865\) 37.8564 1.28715
\(866\) 0 0
\(867\) 12.3999 0.421123
\(868\) 0 0
\(869\) −20.1094 −0.682166
\(870\) 0 0
\(871\) 2.94128 0.0996614
\(872\) 0 0
\(873\) −8.29839 −0.280858
\(874\) 0 0
\(875\) 51.5022 1.74109
\(876\) 0 0
\(877\) 28.7443 0.970625 0.485312 0.874341i \(-0.338706\pi\)
0.485312 + 0.874341i \(0.338706\pi\)
\(878\) 0 0
\(879\) −13.0134 −0.438933
\(880\) 0 0
\(881\) 39.8532 1.34269 0.671343 0.741146i \(-0.265718\pi\)
0.671343 + 0.741146i \(0.265718\pi\)
\(882\) 0 0
\(883\) 32.7450 1.10196 0.550979 0.834519i \(-0.314255\pi\)
0.550979 + 0.834519i \(0.314255\pi\)
\(884\) 0 0
\(885\) 10.1831 0.342300
\(886\) 0 0
\(887\) −34.6559 −1.16363 −0.581815 0.813321i \(-0.697657\pi\)
−0.581815 + 0.813321i \(0.697657\pi\)
\(888\) 0 0
\(889\) 9.19306 0.308325
\(890\) 0 0
\(891\) 2.62592 0.0879718
\(892\) 0 0
\(893\) −6.01440 −0.201264
\(894\) 0 0
\(895\) −30.3438 −1.01428
\(896\) 0 0
\(897\) −27.8904 −0.931234
\(898\) 0 0
\(899\) −2.21970 −0.0740310
\(900\) 0 0
\(901\) −15.8015 −0.526423
\(902\) 0 0
\(903\) −53.9852 −1.79651
\(904\) 0 0
\(905\) −1.94853 −0.0647715
\(906\) 0 0
\(907\) 12.3392 0.409717 0.204858 0.978792i \(-0.434327\pi\)
0.204858 + 0.978792i \(0.434327\pi\)
\(908\) 0 0
\(909\) −12.2104 −0.404993
\(910\) 0 0
\(911\) −52.4048 −1.73625 −0.868124 0.496348i \(-0.834674\pi\)
−0.868124 + 0.496348i \(0.834674\pi\)
\(912\) 0 0
\(913\) 17.0327 0.563700
\(914\) 0 0
\(915\) 26.3042 0.869589
\(916\) 0 0
\(917\) −35.0171 −1.15637
\(918\) 0 0
\(919\) 34.8078 1.14820 0.574101 0.818784i \(-0.305352\pi\)
0.574101 + 0.818784i \(0.305352\pi\)
\(920\) 0 0
\(921\) 10.6004 0.349297
\(922\) 0 0
\(923\) 59.6954 1.96490
\(924\) 0 0
\(925\) −1.95668 −0.0643354
\(926\) 0 0
\(927\) −14.8965 −0.489265
\(928\) 0 0
\(929\) 50.9235 1.67074 0.835372 0.549685i \(-0.185252\pi\)
0.835372 + 0.549685i \(0.185252\pi\)
\(930\) 0 0
\(931\) 82.3123 2.69768
\(932\) 0 0
\(933\) 23.2712 0.761865
\(934\) 0 0
\(935\) −12.3613 −0.404259
\(936\) 0 0
\(937\) 23.9406 0.782104 0.391052 0.920369i \(-0.372111\pi\)
0.391052 + 0.920369i \(0.372111\pi\)
\(938\) 0 0
\(939\) 6.59154 0.215107
\(940\) 0 0
\(941\) −58.0602 −1.89271 −0.946355 0.323130i \(-0.895265\pi\)
−0.946355 + 0.323130i \(0.895265\pi\)
\(942\) 0 0
\(943\) 2.18497 0.0711523
\(944\) 0 0
\(945\) 9.93726 0.323259
\(946\) 0 0
\(947\) −32.0239 −1.04064 −0.520319 0.853972i \(-0.674187\pi\)
−0.520319 + 0.853972i \(0.674187\pi\)
\(948\) 0 0
\(949\) −32.6259 −1.05908
\(950\) 0 0
\(951\) −8.90860 −0.288881
\(952\) 0 0
\(953\) 34.0370 1.10257 0.551283 0.834318i \(-0.314138\pi\)
0.551283 + 0.834318i \(0.314138\pi\)
\(954\) 0 0
\(955\) 28.2765 0.915007
\(956\) 0 0
\(957\) −4.84112 −0.156491
\(958\) 0 0
\(959\) −86.5776 −2.79574
\(960\) 0 0
\(961\) −29.5504 −0.953237
\(962\) 0 0
\(963\) −10.0613 −0.324222
\(964\) 0 0
\(965\) −43.0827 −1.38688
\(966\) 0 0
\(967\) −45.0634 −1.44914 −0.724571 0.689200i \(-0.757962\pi\)
−0.724571 + 0.689200i \(0.757962\pi\)
\(968\) 0 0
\(969\) −13.0781 −0.420129
\(970\) 0 0
\(971\) 47.3027 1.51802 0.759008 0.651082i \(-0.225684\pi\)
0.759008 + 0.651082i \(0.225684\pi\)
\(972\) 0 0
\(973\) 13.1660 0.422084
\(974\) 0 0
\(975\) 0.823581 0.0263757
\(976\) 0 0
\(977\) −16.0859 −0.514635 −0.257317 0.966327i \(-0.582839\pi\)
−0.257317 + 0.966327i \(0.582839\pi\)
\(978\) 0 0
\(979\) −15.7644 −0.503831
\(980\) 0 0
\(981\) −10.9585 −0.349879
\(982\) 0 0
\(983\) 5.84308 0.186365 0.0931826 0.995649i \(-0.470296\pi\)
0.0931826 + 0.995649i \(0.470296\pi\)
\(984\) 0 0
\(985\) 10.9386 0.348533
\(986\) 0 0
\(987\) 4.46578 0.142147
\(988\) 0 0
\(989\) 73.7885 2.34634
\(990\) 0 0
\(991\) −14.7445 −0.468376 −0.234188 0.972191i \(-0.575243\pi\)
−0.234188 + 0.972191i \(0.575243\pi\)
\(992\) 0 0
\(993\) −11.2953 −0.358445
\(994\) 0 0
\(995\) −32.1439 −1.01903
\(996\) 0 0
\(997\) −41.7942 −1.32364 −0.661818 0.749664i \(-0.730215\pi\)
−0.661818 + 0.749664i \(0.730215\pi\)
\(998\) 0 0
\(999\) −10.7075 −0.338771
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 8016.2.a.p.1.3 5
4.3 odd 2 501.2.a.b.1.5 5
12.11 even 2 1503.2.a.d.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
501.2.a.b.1.5 5 4.3 odd 2
1503.2.a.d.1.1 5 12.11 even 2
8016.2.a.p.1.3 5 1.1 even 1 trivial