Properties

Label 8016.2.a.p.1.4
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 1.4
Root \(1.33419\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.19539 q^{5} -1.77240 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.19539 q^{5} -1.77240 q^{7} +1.00000 q^{9} +5.47298 q^{11} -4.71582 q^{13} +1.19539 q^{15} +4.92365 q^{17} +8.56490 q^{19} +1.77240 q^{21} +4.04298 q^{23} -3.57105 q^{25} -1.00000 q^{27} +9.38643 q^{29} +0.0932685 q^{31} -5.47298 q^{33} +2.11871 q^{35} -1.65649 q^{37} +4.71582 q^{39} +0.208144 q^{41} -6.12542 q^{43} -1.19539 q^{45} +5.00671 q^{47} -3.85859 q^{49} -4.92365 q^{51} -8.99915 q^{53} -6.54233 q^{55} -8.56490 q^{57} +13.4797 q^{59} -9.91008 q^{61} -1.77240 q^{63} +5.63722 q^{65} -4.73622 q^{67} -4.04298 q^{69} +6.89736 q^{71} +2.90100 q^{73} +3.57105 q^{75} -9.70033 q^{77} -2.36024 q^{79} +1.00000 q^{81} -15.8094 q^{83} -5.88566 q^{85} -9.38643 q^{87} -7.84740 q^{89} +8.35833 q^{91} -0.0932685 q^{93} -10.2384 q^{95} +5.91792 q^{97} +5.47298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} - 9 q^{5} + 4 q^{7} + 5 q^{9}+O(q^{10}) \) 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
\(5\) −1.19539 −0.534593 −0.267296 0.963614i \(-0.586130\pi\)
−0.267296 + 0.963614i \(0.586130\pi\)
\(6\) 0 0
\(7\) −1.77240 −0.669905 −0.334953 0.942235i \(-0.608720\pi\)
−0.334953 + 0.942235i \(0.608720\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.47298 1.65017 0.825083 0.565011i \(-0.191128\pi\)
0.825083 + 0.565011i \(0.191128\pi\)
\(12\) 0 0
\(13\) −4.71582 −1.30793 −0.653966 0.756524i \(-0.726896\pi\)
−0.653966 + 0.756524i \(0.726896\pi\)
\(14\) 0 0
\(15\) 1.19539 0.308647
\(16\) 0 0
\(17\) 4.92365 1.19416 0.597080 0.802182i \(-0.296327\pi\)
0.597080 + 0.802182i \(0.296327\pi\)
\(18\) 0 0
\(19\) 8.56490 1.96492 0.982461 0.186466i \(-0.0597035\pi\)
0.982461 + 0.186466i \(0.0597035\pi\)
\(20\) 0 0
\(21\) 1.77240 0.386770
\(22\) 0 0
\(23\) 4.04298 0.843019 0.421509 0.906824i \(-0.361500\pi\)
0.421509 + 0.906824i \(0.361500\pi\)
\(24\) 0 0
\(25\) −3.57105 −0.714210
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 9.38643 1.74302 0.871508 0.490381i \(-0.163142\pi\)
0.871508 + 0.490381i \(0.163142\pi\)
\(30\) 0 0
\(31\) 0.0932685 0.0167515 0.00837576 0.999965i \(-0.497334\pi\)
0.00837576 + 0.999965i \(0.497334\pi\)
\(32\) 0 0
\(33\) −5.47298 −0.952724
\(34\) 0 0
\(35\) 2.11871 0.358127
\(36\) 0 0
\(37\) −1.65649 −0.272325 −0.136162 0.990687i \(-0.543477\pi\)
−0.136162 + 0.990687i \(0.543477\pi\)
\(38\) 0 0
\(39\) 4.71582 0.755135
\(40\) 0 0
\(41\) 0.208144 0.0325066 0.0162533 0.999868i \(-0.494826\pi\)
0.0162533 + 0.999868i \(0.494826\pi\)
\(42\) 0 0
\(43\) −6.12542 −0.934118 −0.467059 0.884226i \(-0.654686\pi\)
−0.467059 + 0.884226i \(0.654686\pi\)
\(44\) 0 0
\(45\) −1.19539 −0.178198
\(46\) 0 0
\(47\) 5.00671 0.730304 0.365152 0.930948i \(-0.381017\pi\)
0.365152 + 0.930948i \(0.381017\pi\)
\(48\) 0 0
\(49\) −3.85859 −0.551227
\(50\) 0 0
\(51\) −4.92365 −0.689449
\(52\) 0 0
\(53\) −8.99915 −1.23613 −0.618064 0.786128i \(-0.712083\pi\)
−0.618064 + 0.786128i \(0.712083\pi\)
\(54\) 0 0
\(55\) −6.54233 −0.882167
\(56\) 0 0
\(57\) −8.56490 −1.13445
\(58\) 0 0
\(59\) 13.4797 1.75491 0.877454 0.479661i \(-0.159240\pi\)
0.877454 + 0.479661i \(0.159240\pi\)
\(60\) 0 0
\(61\) −9.91008 −1.26886 −0.634428 0.772982i \(-0.718764\pi\)
−0.634428 + 0.772982i \(0.718764\pi\)
\(62\) 0 0
\(63\) −1.77240 −0.223302
\(64\) 0 0
\(65\) 5.63722 0.699211
\(66\) 0 0
\(67\) −4.73622 −0.578621 −0.289311 0.957235i \(-0.593426\pi\)
−0.289311 + 0.957235i \(0.593426\pi\)
\(68\) 0 0
\(69\) −4.04298 −0.486717
\(70\) 0 0
\(71\) 6.89736 0.818566 0.409283 0.912408i \(-0.365779\pi\)
0.409283 + 0.912408i \(0.365779\pi\)
\(72\) 0 0
\(73\) 2.90100 0.339536 0.169768 0.985484i \(-0.445698\pi\)
0.169768 + 0.985484i \(0.445698\pi\)
\(74\) 0 0
\(75\) 3.57105 0.412350
\(76\) 0 0
\(77\) −9.70033 −1.10546
\(78\) 0 0
\(79\) −2.36024 −0.265548 −0.132774 0.991146i \(-0.542388\pi\)
−0.132774 + 0.991146i \(0.542388\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.8094 −1.73530 −0.867651 0.497174i \(-0.834371\pi\)
−0.867651 + 0.497174i \(0.834371\pi\)
\(84\) 0 0
\(85\) −5.88566 −0.638390
\(86\) 0 0
\(87\) −9.38643 −1.00633
\(88\) 0 0
\(89\) −7.84740 −0.831823 −0.415911 0.909405i \(-0.636537\pi\)
−0.415911 + 0.909405i \(0.636537\pi\)
\(90\) 0 0
\(91\) 8.35833 0.876191
\(92\) 0 0
\(93\) −0.0932685 −0.00967149
\(94\) 0 0
\(95\) −10.2384 −1.05043
\(96\) 0 0
\(97\) 5.91792 0.600873 0.300437 0.953802i \(-0.402868\pi\)
0.300437 + 0.953802i \(0.402868\pi\)
\(98\) 0 0
\(99\) 5.47298 0.550056
\(100\) 0 0
\(101\) −1.16294 −0.115717 −0.0578585 0.998325i \(-0.518427\pi\)
−0.0578585 + 0.998325i \(0.518427\pi\)
\(102\) 0 0
\(103\) 7.22124 0.711530 0.355765 0.934575i \(-0.384220\pi\)
0.355765 + 0.934575i \(0.384220\pi\)
\(104\) 0 0
\(105\) −2.11871 −0.206765
\(106\) 0 0
\(107\) −8.75773 −0.846642 −0.423321 0.905980i \(-0.639136\pi\)
−0.423321 + 0.905980i \(0.639136\pi\)
\(108\) 0 0
\(109\) 5.78303 0.553914 0.276957 0.960882i \(-0.410674\pi\)
0.276957 + 0.960882i \(0.410674\pi\)
\(110\) 0 0
\(111\) 1.65649 0.157227
\(112\) 0 0
\(113\) −3.11997 −0.293502 −0.146751 0.989173i \(-0.546882\pi\)
−0.146751 + 0.989173i \(0.546882\pi\)
\(114\) 0 0
\(115\) −4.83292 −0.450672
\(116\) 0 0
\(117\) −4.71582 −0.435977
\(118\) 0 0
\(119\) −8.72669 −0.799974
\(120\) 0 0
\(121\) 18.9536 1.72305
\(122\) 0 0
\(123\) −0.208144 −0.0187677
\(124\) 0 0
\(125\) 10.2457 0.916405
\(126\) 0 0
\(127\) −2.95298 −0.262034 −0.131017 0.991380i \(-0.541824\pi\)
−0.131017 + 0.991380i \(0.541824\pi\)
\(128\) 0 0
\(129\) 6.12542 0.539313
\(130\) 0 0
\(131\) 17.3913 1.51948 0.759741 0.650225i \(-0.225326\pi\)
0.759741 + 0.650225i \(0.225326\pi\)
\(132\) 0 0
\(133\) −15.1805 −1.31631
\(134\) 0 0
\(135\) 1.19539 0.102882
\(136\) 0 0
\(137\) 18.3829 1.57056 0.785278 0.619143i \(-0.212520\pi\)
0.785278 + 0.619143i \(0.212520\pi\)
\(138\) 0 0
\(139\) 8.44917 0.716649 0.358325 0.933597i \(-0.383348\pi\)
0.358325 + 0.933597i \(0.383348\pi\)
\(140\) 0 0
\(141\) −5.00671 −0.421641
\(142\) 0 0
\(143\) −25.8096 −2.15831
\(144\) 0 0
\(145\) −11.2204 −0.931804
\(146\) 0 0
\(147\) 3.85859 0.318251
\(148\) 0 0
\(149\) −12.3049 −1.00805 −0.504027 0.863688i \(-0.668149\pi\)
−0.504027 + 0.863688i \(0.668149\pi\)
\(150\) 0 0
\(151\) −6.43183 −0.523415 −0.261708 0.965147i \(-0.584286\pi\)
−0.261708 + 0.965147i \(0.584286\pi\)
\(152\) 0 0
\(153\) 4.92365 0.398053
\(154\) 0 0
\(155\) −0.111492 −0.00895524
\(156\) 0 0
\(157\) 4.09719 0.326992 0.163496 0.986544i \(-0.447723\pi\)
0.163496 + 0.986544i \(0.447723\pi\)
\(158\) 0 0
\(159\) 8.99915 0.713679
\(160\) 0 0
\(161\) −7.16578 −0.564743
\(162\) 0 0
\(163\) 10.8003 0.845946 0.422973 0.906142i \(-0.360986\pi\)
0.422973 + 0.906142i \(0.360986\pi\)
\(164\) 0 0
\(165\) 6.54233 0.509320
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 9.23892 0.710686
\(170\) 0 0
\(171\) 8.56490 0.654974
\(172\) 0 0
\(173\) 6.62825 0.503936 0.251968 0.967736i \(-0.418922\pi\)
0.251968 + 0.967736i \(0.418922\pi\)
\(174\) 0 0
\(175\) 6.32934 0.478453
\(176\) 0 0
\(177\) −13.4797 −1.01320
\(178\) 0 0
\(179\) 6.54195 0.488969 0.244484 0.969653i \(-0.421381\pi\)
0.244484 + 0.969653i \(0.421381\pi\)
\(180\) 0 0
\(181\) −4.10198 −0.304898 −0.152449 0.988311i \(-0.548716\pi\)
−0.152449 + 0.988311i \(0.548716\pi\)
\(182\) 0 0
\(183\) 9.91008 0.732574
\(184\) 0 0
\(185\) 1.98014 0.145583
\(186\) 0 0
\(187\) 26.9471 1.97056
\(188\) 0 0
\(189\) 1.77240 0.128923
\(190\) 0 0
\(191\) 2.92293 0.211496 0.105748 0.994393i \(-0.466276\pi\)
0.105748 + 0.994393i \(0.466276\pi\)
\(192\) 0 0
\(193\) −5.63895 −0.405900 −0.202950 0.979189i \(-0.565053\pi\)
−0.202950 + 0.979189i \(0.565053\pi\)
\(194\) 0 0
\(195\) −5.63722 −0.403690
\(196\) 0 0
\(197\) 16.0864 1.14611 0.573054 0.819518i \(-0.305759\pi\)
0.573054 + 0.819518i \(0.305759\pi\)
\(198\) 0 0
\(199\) −11.0123 −0.780642 −0.390321 0.920679i \(-0.627636\pi\)
−0.390321 + 0.920679i \(0.627636\pi\)
\(200\) 0 0
\(201\) 4.73622 0.334067
\(202\) 0 0
\(203\) −16.6365 −1.16766
\(204\) 0 0
\(205\) −0.248812 −0.0173778
\(206\) 0 0
\(207\) 4.04298 0.281006
\(208\) 0 0
\(209\) 46.8756 3.24245
\(210\) 0 0
\(211\) 25.8916 1.78245 0.891227 0.453558i \(-0.149846\pi\)
0.891227 + 0.453558i \(0.149846\pi\)
\(212\) 0 0
\(213\) −6.89736 −0.472599
\(214\) 0 0
\(215\) 7.32224 0.499373
\(216\) 0 0
\(217\) −0.165309 −0.0112219
\(218\) 0 0
\(219\) −2.90100 −0.196031
\(220\) 0 0
\(221\) −23.2190 −1.56188
\(222\) 0 0
\(223\) −20.7188 −1.38743 −0.693715 0.720249i \(-0.744027\pi\)
−0.693715 + 0.720249i \(0.744027\pi\)
\(224\) 0 0
\(225\) −3.57105 −0.238070
\(226\) 0 0
\(227\) 21.9100 1.45422 0.727110 0.686521i \(-0.240863\pi\)
0.727110 + 0.686521i \(0.240863\pi\)
\(228\) 0 0
\(229\) 18.3308 1.21133 0.605667 0.795719i \(-0.292907\pi\)
0.605667 + 0.795719i \(0.292907\pi\)
\(230\) 0 0
\(231\) 9.70033 0.638235
\(232\) 0 0
\(233\) −1.98209 −0.129851 −0.0649255 0.997890i \(-0.520681\pi\)
−0.0649255 + 0.997890i \(0.520681\pi\)
\(234\) 0 0
\(235\) −5.98496 −0.390416
\(236\) 0 0
\(237\) 2.36024 0.153314
\(238\) 0 0
\(239\) −23.2205 −1.50201 −0.751004 0.660297i \(-0.770430\pi\)
−0.751004 + 0.660297i \(0.770430\pi\)
\(240\) 0 0
\(241\) 10.7187 0.690450 0.345225 0.938520i \(-0.387803\pi\)
0.345225 + 0.938520i \(0.387803\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 4.61250 0.294682
\(246\) 0 0
\(247\) −40.3905 −2.56999
\(248\) 0 0
\(249\) 15.8094 1.00188
\(250\) 0 0
\(251\) 12.4880 0.788233 0.394116 0.919061i \(-0.371051\pi\)
0.394116 + 0.919061i \(0.371051\pi\)
\(252\) 0 0
\(253\) 22.1271 1.39112
\(254\) 0 0
\(255\) 5.88566 0.368574
\(256\) 0 0
\(257\) −11.7839 −0.735060 −0.367530 0.930012i \(-0.619796\pi\)
−0.367530 + 0.930012i \(0.619796\pi\)
\(258\) 0 0
\(259\) 2.93596 0.182432
\(260\) 0 0
\(261\) 9.38643 0.581005
\(262\) 0 0
\(263\) 6.92203 0.426830 0.213415 0.976962i \(-0.431541\pi\)
0.213415 + 0.976962i \(0.431541\pi\)
\(264\) 0 0
\(265\) 10.7575 0.660825
\(266\) 0 0
\(267\) 7.84740 0.480253
\(268\) 0 0
\(269\) 19.7673 1.20523 0.602617 0.798030i \(-0.294125\pi\)
0.602617 + 0.798030i \(0.294125\pi\)
\(270\) 0 0
\(271\) −28.0726 −1.70529 −0.852644 0.522492i \(-0.825003\pi\)
−0.852644 + 0.522492i \(0.825003\pi\)
\(272\) 0 0
\(273\) −8.35833 −0.505869
\(274\) 0 0
\(275\) −19.5443 −1.17857
\(276\) 0 0
\(277\) 9.78055 0.587656 0.293828 0.955858i \(-0.405071\pi\)
0.293828 + 0.955858i \(0.405071\pi\)
\(278\) 0 0
\(279\) 0.0932685 0.00558384
\(280\) 0 0
\(281\) 2.42891 0.144896 0.0724482 0.997372i \(-0.476919\pi\)
0.0724482 + 0.997372i \(0.476919\pi\)
\(282\) 0 0
\(283\) −7.70418 −0.457966 −0.228983 0.973430i \(-0.573540\pi\)
−0.228983 + 0.973430i \(0.573540\pi\)
\(284\) 0 0
\(285\) 10.2384 0.606468
\(286\) 0 0
\(287\) −0.368915 −0.0217763
\(288\) 0 0
\(289\) 7.24232 0.426019
\(290\) 0 0
\(291\) −5.91792 −0.346914
\(292\) 0 0
\(293\) 0.192909 0.0112698 0.00563492 0.999984i \(-0.498206\pi\)
0.00563492 + 0.999984i \(0.498206\pi\)
\(294\) 0 0
\(295\) −16.1134 −0.938161
\(296\) 0 0
\(297\) −5.47298 −0.317575
\(298\) 0 0
\(299\) −19.0659 −1.10261
\(300\) 0 0
\(301\) 10.8567 0.625771
\(302\) 0 0
\(303\) 1.16294 0.0668093
\(304\) 0 0
\(305\) 11.8464 0.678321
\(306\) 0 0
\(307\) 4.67881 0.267034 0.133517 0.991047i \(-0.457373\pi\)
0.133517 + 0.991047i \(0.457373\pi\)
\(308\) 0 0
\(309\) −7.22124 −0.410802
\(310\) 0 0
\(311\) −0.460674 −0.0261225 −0.0130612 0.999915i \(-0.504158\pi\)
−0.0130612 + 0.999915i \(0.504158\pi\)
\(312\) 0 0
\(313\) 14.0147 0.792157 0.396079 0.918217i \(-0.370371\pi\)
0.396079 + 0.918217i \(0.370371\pi\)
\(314\) 0 0
\(315\) 2.11871 0.119376
\(316\) 0 0
\(317\) −6.70071 −0.376349 −0.188175 0.982136i \(-0.560257\pi\)
−0.188175 + 0.982136i \(0.560257\pi\)
\(318\) 0 0
\(319\) 51.3718 2.87627
\(320\) 0 0
\(321\) 8.75773 0.488809
\(322\) 0 0
\(323\) 42.1706 2.34643
\(324\) 0 0
\(325\) 16.8404 0.934139
\(326\) 0 0
\(327\) −5.78303 −0.319802
\(328\) 0 0
\(329\) −8.87392 −0.489235
\(330\) 0 0
\(331\) 22.2154 1.22107 0.610535 0.791989i \(-0.290954\pi\)
0.610535 + 0.791989i \(0.290954\pi\)
\(332\) 0 0
\(333\) −1.65649 −0.0907749
\(334\) 0 0
\(335\) 5.66161 0.309327
\(336\) 0 0
\(337\) −17.4236 −0.949125 −0.474562 0.880222i \(-0.657394\pi\)
−0.474562 + 0.880222i \(0.657394\pi\)
\(338\) 0 0
\(339\) 3.11997 0.169453
\(340\) 0 0
\(341\) 0.510457 0.0276428
\(342\) 0 0
\(343\) 19.2458 1.03918
\(344\) 0 0
\(345\) 4.83292 0.260195
\(346\) 0 0
\(347\) 21.9645 1.17911 0.589557 0.807727i \(-0.299302\pi\)
0.589557 + 0.807727i \(0.299302\pi\)
\(348\) 0 0
\(349\) −4.50882 −0.241352 −0.120676 0.992692i \(-0.538506\pi\)
−0.120676 + 0.992692i \(0.538506\pi\)
\(350\) 0 0
\(351\) 4.71582 0.251712
\(352\) 0 0
\(353\) −28.5941 −1.52191 −0.760955 0.648805i \(-0.775269\pi\)
−0.760955 + 0.648805i \(0.775269\pi\)
\(354\) 0 0
\(355\) −8.24500 −0.437599
\(356\) 0 0
\(357\) 8.72669 0.461865
\(358\) 0 0
\(359\) −22.6985 −1.19798 −0.598991 0.800756i \(-0.704431\pi\)
−0.598991 + 0.800756i \(0.704431\pi\)
\(360\) 0 0
\(361\) 54.3575 2.86092
\(362\) 0 0
\(363\) −18.9536 −0.994804
\(364\) 0 0
\(365\) −3.46781 −0.181514
\(366\) 0 0
\(367\) 20.7463 1.08295 0.541474 0.840717i \(-0.317866\pi\)
0.541474 + 0.840717i \(0.317866\pi\)
\(368\) 0 0
\(369\) 0.208144 0.0108355
\(370\) 0 0
\(371\) 15.9501 0.828089
\(372\) 0 0
\(373\) −18.7331 −0.969961 −0.484981 0.874525i \(-0.661173\pi\)
−0.484981 + 0.874525i \(0.661173\pi\)
\(374\) 0 0
\(375\) −10.2457 −0.529087
\(376\) 0 0
\(377\) −44.2647 −2.27975
\(378\) 0 0
\(379\) 12.8017 0.657581 0.328790 0.944403i \(-0.393359\pi\)
0.328790 + 0.944403i \(0.393359\pi\)
\(380\) 0 0
\(381\) 2.95298 0.151286
\(382\) 0 0
\(383\) −20.7589 −1.06073 −0.530364 0.847770i \(-0.677945\pi\)
−0.530364 + 0.847770i \(0.677945\pi\)
\(384\) 0 0
\(385\) 11.5956 0.590969
\(386\) 0 0
\(387\) −6.12542 −0.311373
\(388\) 0 0
\(389\) 22.4763 1.13959 0.569797 0.821785i \(-0.307022\pi\)
0.569797 + 0.821785i \(0.307022\pi\)
\(390\) 0 0
\(391\) 19.9062 1.00670
\(392\) 0 0
\(393\) −17.3913 −0.877274
\(394\) 0 0
\(395\) 2.82140 0.141960
\(396\) 0 0
\(397\) 18.0948 0.908153 0.454076 0.890963i \(-0.349969\pi\)
0.454076 + 0.890963i \(0.349969\pi\)
\(398\) 0 0
\(399\) 15.1805 0.759973
\(400\) 0 0
\(401\) 1.90969 0.0953655 0.0476828 0.998863i \(-0.484816\pi\)
0.0476828 + 0.998863i \(0.484816\pi\)
\(402\) 0 0
\(403\) −0.439837 −0.0219099
\(404\) 0 0
\(405\) −1.19539 −0.0593992
\(406\) 0 0
\(407\) −9.06592 −0.449381
\(408\) 0 0
\(409\) −11.9762 −0.592186 −0.296093 0.955159i \(-0.595684\pi\)
−0.296093 + 0.955159i \(0.595684\pi\)
\(410\) 0 0
\(411\) −18.3829 −0.906761
\(412\) 0 0
\(413\) −23.8915 −1.17562
\(414\) 0 0
\(415\) 18.8983 0.927680
\(416\) 0 0
\(417\) −8.44917 −0.413758
\(418\) 0 0
\(419\) −27.4154 −1.33933 −0.669666 0.742663i \(-0.733563\pi\)
−0.669666 + 0.742663i \(0.733563\pi\)
\(420\) 0 0
\(421\) −18.3718 −0.895386 −0.447693 0.894187i \(-0.647754\pi\)
−0.447693 + 0.894187i \(0.647754\pi\)
\(422\) 0 0
\(423\) 5.00671 0.243435
\(424\) 0 0
\(425\) −17.5826 −0.852882
\(426\) 0 0
\(427\) 17.5647 0.850014
\(428\) 0 0
\(429\) 25.8096 1.24610
\(430\) 0 0
\(431\) 8.66607 0.417430 0.208715 0.977977i \(-0.433072\pi\)
0.208715 + 0.977977i \(0.433072\pi\)
\(432\) 0 0
\(433\) 4.56610 0.219433 0.109716 0.993963i \(-0.465006\pi\)
0.109716 + 0.993963i \(0.465006\pi\)
\(434\) 0 0
\(435\) 11.2204 0.537977
\(436\) 0 0
\(437\) 34.6277 1.65647
\(438\) 0 0
\(439\) 10.5648 0.504232 0.252116 0.967697i \(-0.418873\pi\)
0.252116 + 0.967697i \(0.418873\pi\)
\(440\) 0 0
\(441\) −3.85859 −0.183742
\(442\) 0 0
\(443\) −11.9577 −0.568127 −0.284064 0.958805i \(-0.591683\pi\)
−0.284064 + 0.958805i \(0.591683\pi\)
\(444\) 0 0
\(445\) 9.38067 0.444687
\(446\) 0 0
\(447\) 12.3049 0.582000
\(448\) 0 0
\(449\) 7.17792 0.338747 0.169374 0.985552i \(-0.445826\pi\)
0.169374 + 0.985552i \(0.445826\pi\)
\(450\) 0 0
\(451\) 1.13917 0.0536413
\(452\) 0 0
\(453\) 6.43183 0.302194
\(454\) 0 0
\(455\) −9.99143 −0.468405
\(456\) 0 0
\(457\) 3.32645 0.155605 0.0778025 0.996969i \(-0.475210\pi\)
0.0778025 + 0.996969i \(0.475210\pi\)
\(458\) 0 0
\(459\) −4.92365 −0.229816
\(460\) 0 0
\(461\) −21.3336 −0.993603 −0.496802 0.867864i \(-0.665492\pi\)
−0.496802 + 0.867864i \(0.665492\pi\)
\(462\) 0 0
\(463\) −41.5946 −1.93306 −0.966531 0.256548i \(-0.917415\pi\)
−0.966531 + 0.256548i \(0.917415\pi\)
\(464\) 0 0
\(465\) 0.111492 0.00517031
\(466\) 0 0
\(467\) 28.7058 1.32834 0.664172 0.747579i \(-0.268784\pi\)
0.664172 + 0.747579i \(0.268784\pi\)
\(468\) 0 0
\(469\) 8.39449 0.387622
\(470\) 0 0
\(471\) −4.09719 −0.188789
\(472\) 0 0
\(473\) −33.5243 −1.54145
\(474\) 0 0
\(475\) −30.5857 −1.40337
\(476\) 0 0
\(477\) −8.99915 −0.412043
\(478\) 0 0
\(479\) 21.5130 0.982956 0.491478 0.870890i \(-0.336457\pi\)
0.491478 + 0.870890i \(0.336457\pi\)
\(480\) 0 0
\(481\) 7.81168 0.356182
\(482\) 0 0
\(483\) 7.16578 0.326054
\(484\) 0 0
\(485\) −7.07420 −0.321223
\(486\) 0 0
\(487\) 33.0631 1.49823 0.749117 0.662438i \(-0.230478\pi\)
0.749117 + 0.662438i \(0.230478\pi\)
\(488\) 0 0
\(489\) −10.8003 −0.488407
\(490\) 0 0
\(491\) −4.46205 −0.201370 −0.100685 0.994918i \(-0.532103\pi\)
−0.100685 + 0.994918i \(0.532103\pi\)
\(492\) 0 0
\(493\) 46.2155 2.08144
\(494\) 0 0
\(495\) −6.54233 −0.294056
\(496\) 0 0
\(497\) −12.2249 −0.548362
\(498\) 0 0
\(499\) −15.1803 −0.679565 −0.339783 0.940504i \(-0.610354\pi\)
−0.339783 + 0.940504i \(0.610354\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) 27.8467 1.24162 0.620811 0.783961i \(-0.286804\pi\)
0.620811 + 0.783961i \(0.286804\pi\)
\(504\) 0 0
\(505\) 1.39016 0.0618615
\(506\) 0 0
\(507\) −9.23892 −0.410315
\(508\) 0 0
\(509\) 5.39072 0.238939 0.119470 0.992838i \(-0.461881\pi\)
0.119470 + 0.992838i \(0.461881\pi\)
\(510\) 0 0
\(511\) −5.14174 −0.227457
\(512\) 0 0
\(513\) −8.56490 −0.378150
\(514\) 0 0
\(515\) −8.63218 −0.380379
\(516\) 0 0
\(517\) 27.4017 1.20512
\(518\) 0 0
\(519\) −6.62825 −0.290948
\(520\) 0 0
\(521\) −6.40308 −0.280524 −0.140262 0.990114i \(-0.544795\pi\)
−0.140262 + 0.990114i \(0.544795\pi\)
\(522\) 0 0
\(523\) 30.8147 1.34743 0.673717 0.738989i \(-0.264697\pi\)
0.673717 + 0.738989i \(0.264697\pi\)
\(524\) 0 0
\(525\) −6.32934 −0.276235
\(526\) 0 0
\(527\) 0.459221 0.0200040
\(528\) 0 0
\(529\) −6.65435 −0.289320
\(530\) 0 0
\(531\) 13.4797 0.584969
\(532\) 0 0
\(533\) −0.981568 −0.0425164
\(534\) 0 0
\(535\) 10.4689 0.452609
\(536\) 0 0
\(537\) −6.54195 −0.282306
\(538\) 0 0
\(539\) −21.1180 −0.909616
\(540\) 0 0
\(541\) 22.3284 0.959973 0.479987 0.877276i \(-0.340642\pi\)
0.479987 + 0.877276i \(0.340642\pi\)
\(542\) 0 0
\(543\) 4.10198 0.176033
\(544\) 0 0
\(545\) −6.91295 −0.296118
\(546\) 0 0
\(547\) −17.6182 −0.753298 −0.376649 0.926356i \(-0.622924\pi\)
−0.376649 + 0.926356i \(0.622924\pi\)
\(548\) 0 0
\(549\) −9.91008 −0.422952
\(550\) 0 0
\(551\) 80.3938 3.42489
\(552\) 0 0
\(553\) 4.18330 0.177892
\(554\) 0 0
\(555\) −1.98014 −0.0840523
\(556\) 0 0
\(557\) 39.2928 1.66489 0.832445 0.554108i \(-0.186941\pi\)
0.832445 + 0.554108i \(0.186941\pi\)
\(558\) 0 0
\(559\) 28.8864 1.22176
\(560\) 0 0
\(561\) −26.9471 −1.13771
\(562\) 0 0
\(563\) 2.37113 0.0999313 0.0499656 0.998751i \(-0.484089\pi\)
0.0499656 + 0.998751i \(0.484089\pi\)
\(564\) 0 0
\(565\) 3.72956 0.156904
\(566\) 0 0
\(567\) −1.77240 −0.0744339
\(568\) 0 0
\(569\) 41.0333 1.72020 0.860102 0.510121i \(-0.170400\pi\)
0.860102 + 0.510121i \(0.170400\pi\)
\(570\) 0 0
\(571\) −35.0112 −1.46517 −0.732586 0.680674i \(-0.761687\pi\)
−0.732586 + 0.680674i \(0.761687\pi\)
\(572\) 0 0
\(573\) −2.92293 −0.122107
\(574\) 0 0
\(575\) −14.4377 −0.602093
\(576\) 0 0
\(577\) −3.96953 −0.165254 −0.0826269 0.996581i \(-0.526331\pi\)
−0.0826269 + 0.996581i \(0.526331\pi\)
\(578\) 0 0
\(579\) 5.63895 0.234347
\(580\) 0 0
\(581\) 28.0206 1.16249
\(582\) 0 0
\(583\) −49.2522 −2.03982
\(584\) 0 0
\(585\) 5.63722 0.233070
\(586\) 0 0
\(587\) 20.6634 0.852870 0.426435 0.904518i \(-0.359769\pi\)
0.426435 + 0.904518i \(0.359769\pi\)
\(588\) 0 0
\(589\) 0.798835 0.0329154
\(590\) 0 0
\(591\) −16.0864 −0.661706
\(592\) 0 0
\(593\) 26.9249 1.10567 0.552836 0.833290i \(-0.313546\pi\)
0.552836 + 0.833290i \(0.313546\pi\)
\(594\) 0 0
\(595\) 10.4318 0.427661
\(596\) 0 0
\(597\) 11.0123 0.450704
\(598\) 0 0
\(599\) 39.1015 1.59764 0.798822 0.601568i \(-0.205457\pi\)
0.798822 + 0.601568i \(0.205457\pi\)
\(600\) 0 0
\(601\) −22.7034 −0.926093 −0.463046 0.886334i \(-0.653244\pi\)
−0.463046 + 0.886334i \(0.653244\pi\)
\(602\) 0 0
\(603\) −4.73622 −0.192874
\(604\) 0 0
\(605\) −22.6568 −0.921131
\(606\) 0 0
\(607\) 33.0805 1.34270 0.671348 0.741142i \(-0.265716\pi\)
0.671348 + 0.741142i \(0.265716\pi\)
\(608\) 0 0
\(609\) 16.6365 0.674147
\(610\) 0 0
\(611\) −23.6107 −0.955189
\(612\) 0 0
\(613\) −3.34105 −0.134944 −0.0674719 0.997721i \(-0.521493\pi\)
−0.0674719 + 0.997721i \(0.521493\pi\)
\(614\) 0 0
\(615\) 0.248812 0.0100331
\(616\) 0 0
\(617\) −14.4996 −0.583731 −0.291866 0.956459i \(-0.594276\pi\)
−0.291866 + 0.956459i \(0.594276\pi\)
\(618\) 0 0
\(619\) 41.7308 1.67730 0.838652 0.544668i \(-0.183344\pi\)
0.838652 + 0.544668i \(0.183344\pi\)
\(620\) 0 0
\(621\) −4.04298 −0.162239
\(622\) 0 0
\(623\) 13.9088 0.557243
\(624\) 0 0
\(625\) 5.60768 0.224307
\(626\) 0 0
\(627\) −46.8756 −1.87203
\(628\) 0 0
\(629\) −8.15596 −0.325199
\(630\) 0 0
\(631\) 24.8326 0.988569 0.494285 0.869300i \(-0.335430\pi\)
0.494285 + 0.869300i \(0.335430\pi\)
\(632\) 0 0
\(633\) −25.8916 −1.02910
\(634\) 0 0
\(635\) 3.52995 0.140082
\(636\) 0 0
\(637\) 18.1964 0.720967
\(638\) 0 0
\(639\) 6.89736 0.272855
\(640\) 0 0
\(641\) −26.2163 −1.03548 −0.517742 0.855537i \(-0.673227\pi\)
−0.517742 + 0.855537i \(0.673227\pi\)
\(642\) 0 0
\(643\) 23.5950 0.930495 0.465247 0.885181i \(-0.345965\pi\)
0.465247 + 0.885181i \(0.345965\pi\)
\(644\) 0 0
\(645\) −7.32224 −0.288313
\(646\) 0 0
\(647\) −10.6851 −0.420074 −0.210037 0.977693i \(-0.567358\pi\)
−0.210037 + 0.977693i \(0.567358\pi\)
\(648\) 0 0
\(649\) 73.7742 2.89589
\(650\) 0 0
\(651\) 0.165309 0.00647899
\(652\) 0 0
\(653\) −20.4734 −0.801188 −0.400594 0.916256i \(-0.631196\pi\)
−0.400594 + 0.916256i \(0.631196\pi\)
\(654\) 0 0
\(655\) −20.7893 −0.812305
\(656\) 0 0
\(657\) 2.90100 0.113179
\(658\) 0 0
\(659\) −7.85694 −0.306063 −0.153031 0.988221i \(-0.548904\pi\)
−0.153031 + 0.988221i \(0.548904\pi\)
\(660\) 0 0
\(661\) 12.7399 0.495525 0.247762 0.968821i \(-0.420305\pi\)
0.247762 + 0.968821i \(0.420305\pi\)
\(662\) 0 0
\(663\) 23.2190 0.901752
\(664\) 0 0
\(665\) 18.1465 0.703691
\(666\) 0 0
\(667\) 37.9491 1.46940
\(668\) 0 0
\(669\) 20.7188 0.801033
\(670\) 0 0
\(671\) −54.2377 −2.09382
\(672\) 0 0
\(673\) 10.3912 0.400552 0.200276 0.979740i \(-0.435816\pi\)
0.200276 + 0.979740i \(0.435816\pi\)
\(674\) 0 0
\(675\) 3.57105 0.137450
\(676\) 0 0
\(677\) 6.74331 0.259166 0.129583 0.991569i \(-0.458636\pi\)
0.129583 + 0.991569i \(0.458636\pi\)
\(678\) 0 0
\(679\) −10.4889 −0.402528
\(680\) 0 0
\(681\) −21.9100 −0.839594
\(682\) 0 0
\(683\) 27.1693 1.03960 0.519802 0.854287i \(-0.326006\pi\)
0.519802 + 0.854287i \(0.326006\pi\)
\(684\) 0 0
\(685\) −21.9746 −0.839608
\(686\) 0 0
\(687\) −18.3308 −0.699363
\(688\) 0 0
\(689\) 42.4383 1.61677
\(690\) 0 0
\(691\) 15.0429 0.572258 0.286129 0.958191i \(-0.407631\pi\)
0.286129 + 0.958191i \(0.407631\pi\)
\(692\) 0 0
\(693\) −9.70033 −0.368485
\(694\) 0 0
\(695\) −10.1000 −0.383116
\(696\) 0 0
\(697\) 1.02483 0.0388181
\(698\) 0 0
\(699\) 1.98209 0.0749695
\(700\) 0 0
\(701\) −44.8197 −1.69282 −0.846409 0.532534i \(-0.821240\pi\)
−0.846409 + 0.532534i \(0.821240\pi\)
\(702\) 0 0
\(703\) −14.1876 −0.535097
\(704\) 0 0
\(705\) 5.98496 0.225407
\(706\) 0 0
\(707\) 2.06120 0.0775195
\(708\) 0 0
\(709\) −11.2478 −0.422421 −0.211211 0.977441i \(-0.567741\pi\)
−0.211211 + 0.977441i \(0.567741\pi\)
\(710\) 0 0
\(711\) −2.36024 −0.0885160
\(712\) 0 0
\(713\) 0.377082 0.0141218
\(714\) 0 0
\(715\) 30.8524 1.15382
\(716\) 0 0
\(717\) 23.2205 0.867185
\(718\) 0 0
\(719\) −21.9546 −0.818769 −0.409385 0.912362i \(-0.634257\pi\)
−0.409385 + 0.912362i \(0.634257\pi\)
\(720\) 0 0
\(721\) −12.7990 −0.476658
\(722\) 0 0
\(723\) −10.7187 −0.398632
\(724\) 0 0
\(725\) −33.5194 −1.24488
\(726\) 0 0
\(727\) −45.2750 −1.67916 −0.839579 0.543238i \(-0.817198\pi\)
−0.839579 + 0.543238i \(0.817198\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −30.1594 −1.11549
\(732\) 0 0
\(733\) −4.84016 −0.178775 −0.0893876 0.995997i \(-0.528491\pi\)
−0.0893876 + 0.995997i \(0.528491\pi\)
\(734\) 0 0
\(735\) −4.61250 −0.170135
\(736\) 0 0
\(737\) −25.9213 −0.954822
\(738\) 0 0
\(739\) −21.1540 −0.778163 −0.389081 0.921203i \(-0.627208\pi\)
−0.389081 + 0.921203i \(0.627208\pi\)
\(740\) 0 0
\(741\) 40.3905 1.48378
\(742\) 0 0
\(743\) −20.9429 −0.768320 −0.384160 0.923267i \(-0.625509\pi\)
−0.384160 + 0.923267i \(0.625509\pi\)
\(744\) 0 0
\(745\) 14.7091 0.538898
\(746\) 0 0
\(747\) −15.8094 −0.578434
\(748\) 0 0
\(749\) 15.5222 0.567170
\(750\) 0 0
\(751\) 21.2708 0.776184 0.388092 0.921621i \(-0.373134\pi\)
0.388092 + 0.921621i \(0.373134\pi\)
\(752\) 0 0
\(753\) −12.4880 −0.455086
\(754\) 0 0
\(755\) 7.68852 0.279814
\(756\) 0 0
\(757\) −34.7271 −1.26218 −0.631089 0.775710i \(-0.717392\pi\)
−0.631089 + 0.775710i \(0.717392\pi\)
\(758\) 0 0
\(759\) −22.1271 −0.803164
\(760\) 0 0
\(761\) −35.2988 −1.27958 −0.639789 0.768550i \(-0.720978\pi\)
−0.639789 + 0.768550i \(0.720978\pi\)
\(762\) 0 0
\(763\) −10.2499 −0.371070
\(764\) 0 0
\(765\) −5.88566 −0.212797
\(766\) 0 0
\(767\) −63.5678 −2.29530
\(768\) 0 0
\(769\) −48.1317 −1.73567 −0.867837 0.496849i \(-0.834490\pi\)
−0.867837 + 0.496849i \(0.834490\pi\)
\(770\) 0 0
\(771\) 11.7839 0.424387
\(772\) 0 0
\(773\) −11.0122 −0.396080 −0.198040 0.980194i \(-0.563458\pi\)
−0.198040 + 0.980194i \(0.563458\pi\)
\(774\) 0 0
\(775\) −0.333067 −0.0119641
\(776\) 0 0
\(777\) −2.93596 −0.105327
\(778\) 0 0
\(779\) 1.78273 0.0638730
\(780\) 0 0
\(781\) 37.7491 1.35077
\(782\) 0 0
\(783\) −9.38643 −0.335444
\(784\) 0 0
\(785\) −4.89773 −0.174807
\(786\) 0 0
\(787\) −21.1662 −0.754495 −0.377248 0.926112i \(-0.623129\pi\)
−0.377248 + 0.926112i \(0.623129\pi\)
\(788\) 0 0
\(789\) −6.92203 −0.246431
\(790\) 0 0
\(791\) 5.52984 0.196618
\(792\) 0 0
\(793\) 46.7341 1.65958
\(794\) 0 0
\(795\) −10.7575 −0.381528
\(796\) 0 0
\(797\) 2.97801 0.105486 0.0527432 0.998608i \(-0.483204\pi\)
0.0527432 + 0.998608i \(0.483204\pi\)
\(798\) 0 0
\(799\) 24.6513 0.872100
\(800\) 0 0
\(801\) −7.84740 −0.277274
\(802\) 0 0
\(803\) 15.8771 0.560292
\(804\) 0 0
\(805\) 8.56588 0.301907
\(806\) 0 0
\(807\) −19.7673 −0.695842
\(808\) 0 0
\(809\) 1.95543 0.0687491 0.0343746 0.999409i \(-0.489056\pi\)
0.0343746 + 0.999409i \(0.489056\pi\)
\(810\) 0 0
\(811\) 15.0091 0.527040 0.263520 0.964654i \(-0.415116\pi\)
0.263520 + 0.964654i \(0.415116\pi\)
\(812\) 0 0
\(813\) 28.0726 0.984549
\(814\) 0 0
\(815\) −12.9105 −0.452237
\(816\) 0 0
\(817\) −52.4636 −1.83547
\(818\) 0 0
\(819\) 8.35833 0.292064
\(820\) 0 0
\(821\) −1.80545 −0.0630107 −0.0315053 0.999504i \(-0.510030\pi\)
−0.0315053 + 0.999504i \(0.510030\pi\)
\(822\) 0 0
\(823\) 47.0093 1.63864 0.819321 0.573335i \(-0.194351\pi\)
0.819321 + 0.573335i \(0.194351\pi\)
\(824\) 0 0
\(825\) 19.5443 0.680446
\(826\) 0 0
\(827\) 51.7029 1.79788 0.898942 0.438067i \(-0.144337\pi\)
0.898942 + 0.438067i \(0.144337\pi\)
\(828\) 0 0
\(829\) 21.5559 0.748669 0.374334 0.927294i \(-0.377871\pi\)
0.374334 + 0.927294i \(0.377871\pi\)
\(830\) 0 0
\(831\) −9.78055 −0.339283
\(832\) 0 0
\(833\) −18.9983 −0.658253
\(834\) 0 0
\(835\) 1.19539 0.0413680
\(836\) 0 0
\(837\) −0.0932685 −0.00322383
\(838\) 0 0
\(839\) 31.1997 1.07713 0.538566 0.842583i \(-0.318966\pi\)
0.538566 + 0.842583i \(0.318966\pi\)
\(840\) 0 0
\(841\) 59.1051 2.03811
\(842\) 0 0
\(843\) −2.42891 −0.0836560
\(844\) 0 0
\(845\) −11.0441 −0.379928
\(846\) 0 0
\(847\) −33.5933 −1.15428
\(848\) 0 0
\(849\) 7.70418 0.264407
\(850\) 0 0
\(851\) −6.69713 −0.229575
\(852\) 0 0
\(853\) 33.4350 1.14479 0.572397 0.819977i \(-0.306014\pi\)
0.572397 + 0.819977i \(0.306014\pi\)
\(854\) 0 0
\(855\) −10.2384 −0.350145
\(856\) 0 0
\(857\) −22.7565 −0.777349 −0.388674 0.921375i \(-0.627067\pi\)
−0.388674 + 0.921375i \(0.627067\pi\)
\(858\) 0 0
\(859\) 46.6685 1.59231 0.796154 0.605094i \(-0.206864\pi\)
0.796154 + 0.605094i \(0.206864\pi\)
\(860\) 0 0
\(861\) 0.368915 0.0125726
\(862\) 0 0
\(863\) −1.56673 −0.0533321 −0.0266661 0.999644i \(-0.508489\pi\)
−0.0266661 + 0.999644i \(0.508489\pi\)
\(864\) 0 0
\(865\) −7.92331 −0.269401
\(866\) 0 0
\(867\) −7.24232 −0.245962
\(868\) 0 0
\(869\) −12.9176 −0.438199
\(870\) 0 0
\(871\) 22.3351 0.756798
\(872\) 0 0
\(873\) 5.91792 0.200291
\(874\) 0 0
\(875\) −18.1595 −0.613904
\(876\) 0 0
\(877\) −49.6890 −1.67788 −0.838939 0.544225i \(-0.816824\pi\)
−0.838939 + 0.544225i \(0.816824\pi\)
\(878\) 0 0
\(879\) −0.192909 −0.00650665
\(880\) 0 0
\(881\) −15.5704 −0.524580 −0.262290 0.964989i \(-0.584478\pi\)
−0.262290 + 0.964989i \(0.584478\pi\)
\(882\) 0 0
\(883\) −48.1084 −1.61898 −0.809488 0.587136i \(-0.800255\pi\)
−0.809488 + 0.587136i \(0.800255\pi\)
\(884\) 0 0
\(885\) 16.1134 0.541647
\(886\) 0 0
\(887\) 9.74435 0.327183 0.163592 0.986528i \(-0.447692\pi\)
0.163592 + 0.986528i \(0.447692\pi\)
\(888\) 0 0
\(889\) 5.23386 0.175538
\(890\) 0 0
\(891\) 5.47298 0.183352
\(892\) 0 0
\(893\) 42.8820 1.43499
\(894\) 0 0
\(895\) −7.82016 −0.261399
\(896\) 0 0
\(897\) 19.0659 0.636593
\(898\) 0 0
\(899\) 0.875458 0.0291982
\(900\) 0 0
\(901\) −44.3086 −1.47613
\(902\) 0 0
\(903\) −10.8567 −0.361289
\(904\) 0 0
\(905\) 4.90345 0.162996
\(906\) 0 0
\(907\) −0.915262 −0.0303908 −0.0151954 0.999885i \(-0.504837\pi\)
−0.0151954 + 0.999885i \(0.504837\pi\)
\(908\) 0 0
\(909\) −1.16294 −0.0385723
\(910\) 0 0
\(911\) 25.6329 0.849256 0.424628 0.905368i \(-0.360405\pi\)
0.424628 + 0.905368i \(0.360405\pi\)
\(912\) 0 0
\(913\) −86.5244 −2.86354
\(914\) 0 0
\(915\) −11.8464 −0.391629
\(916\) 0 0
\(917\) −30.8244 −1.01791
\(918\) 0 0
\(919\) 6.78251 0.223734 0.111867 0.993723i \(-0.464317\pi\)
0.111867 + 0.993723i \(0.464317\pi\)
\(920\) 0 0
\(921\) −4.67881 −0.154172
\(922\) 0 0
\(923\) −32.5267 −1.07063
\(924\) 0 0
\(925\) 5.91540 0.194497
\(926\) 0 0
\(927\) 7.22124 0.237177
\(928\) 0 0
\(929\) 53.5883 1.75817 0.879087 0.476662i \(-0.158153\pi\)
0.879087 + 0.476662i \(0.158153\pi\)
\(930\) 0 0
\(931\) −33.0484 −1.08312
\(932\) 0 0
\(933\) 0.460674 0.0150818
\(934\) 0 0
\(935\) −32.2121 −1.05345
\(936\) 0 0
\(937\) 9.20311 0.300653 0.150326 0.988636i \(-0.451968\pi\)
0.150326 + 0.988636i \(0.451968\pi\)
\(938\) 0 0
\(939\) −14.0147 −0.457352
\(940\) 0 0
\(941\) 28.3226 0.923290 0.461645 0.887065i \(-0.347259\pi\)
0.461645 + 0.887065i \(0.347259\pi\)
\(942\) 0 0
\(943\) 0.841520 0.0274037
\(944\) 0 0
\(945\) −2.11871 −0.0689215
\(946\) 0 0
\(947\) 19.4865 0.633226 0.316613 0.948555i \(-0.397454\pi\)
0.316613 + 0.948555i \(0.397454\pi\)
\(948\) 0 0
\(949\) −13.6806 −0.444091
\(950\) 0 0
\(951\) 6.70071 0.217285
\(952\) 0 0
\(953\) 54.1670 1.75464 0.877320 0.479905i \(-0.159329\pi\)
0.877320 + 0.479905i \(0.159329\pi\)
\(954\) 0 0
\(955\) −3.49403 −0.113064
\(956\) 0 0
\(957\) −51.3718 −1.66061
\(958\) 0 0
\(959\) −32.5819 −1.05212
\(960\) 0 0
\(961\) −30.9913 −0.999719
\(962\) 0 0
\(963\) −8.75773 −0.282214
\(964\) 0 0
\(965\) 6.74072 0.216991
\(966\) 0 0
\(967\) 41.3548 1.32988 0.664941 0.746896i \(-0.268457\pi\)
0.664941 + 0.746896i \(0.268457\pi\)
\(968\) 0 0
\(969\) −42.1706 −1.35471
\(970\) 0 0
\(971\) 37.4978 1.20336 0.601681 0.798736i \(-0.294498\pi\)
0.601681 + 0.798736i \(0.294498\pi\)
\(972\) 0 0
\(973\) −14.9753 −0.480087
\(974\) 0 0
\(975\) −16.8404 −0.539325
\(976\) 0 0
\(977\) −49.6720 −1.58915 −0.794574 0.607168i \(-0.792306\pi\)
−0.794574 + 0.607168i \(0.792306\pi\)
\(978\) 0 0
\(979\) −42.9487 −1.37265
\(980\) 0 0
\(981\) 5.78303 0.184638
\(982\) 0 0
\(983\) 2.92643 0.0933387 0.0466694 0.998910i \(-0.485139\pi\)
0.0466694 + 0.998910i \(0.485139\pi\)
\(984\) 0 0
\(985\) −19.2294 −0.612701
\(986\) 0 0
\(987\) 8.87392 0.282460
\(988\) 0 0
\(989\) −24.7649 −0.787479
\(990\) 0 0
\(991\) −45.1038 −1.43277 −0.716385 0.697705i \(-0.754204\pi\)
−0.716385 + 0.697705i \(0.754204\pi\)
\(992\) 0 0
\(993\) −22.2154 −0.704985
\(994\) 0 0
\(995\) 13.1640 0.417326
\(996\) 0 0
\(997\) 29.6044 0.937581 0.468790 0.883309i \(-0.344690\pi\)
0.468790 + 0.883309i \(0.344690\pi\)
\(998\) 0 0
\(999\) 1.65649 0.0524089
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.1.4 5
4.3 odd 2 501.2.a.b.1.1 5
12.11 even 2 1503.2.a.d.1.5 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
501.2.a.b.1.1 5 4.3 odd 2
1503.2.a.d.1.5 5 12.11 even 2
8016.2.a.p.1.4 5 1.1 even 1 trivial