Properties

Label 1183.2.c.j.337.8
Level $1183$
Weight $2$
Character 1183.337
Analytic conductor $9.446$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1183,2,Mod(337,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.c (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.8
Character \(\chi\) \(=\) 1183.337
Dual form 1183.2.c.j.337.17

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.10989i q^{2} +0.955760 q^{3} +0.768150 q^{4} -3.55862i q^{5} -1.06079i q^{6} -1.00000i q^{7} -3.07233i q^{8} -2.08652 q^{9} +O(q^{10})\) \(q-1.10989i q^{2} +0.955760 q^{3} +0.768150 q^{4} -3.55862i q^{5} -1.06079i q^{6} -1.00000i q^{7} -3.07233i q^{8} -2.08652 q^{9} -3.94966 q^{10} +4.00483i q^{11} +0.734167 q^{12} -1.10989 q^{14} -3.40118i q^{15} -1.87364 q^{16} -1.86471 q^{17} +2.31581i q^{18} -6.34660i q^{19} -2.73355i q^{20} -0.955760i q^{21} +4.44491 q^{22} -4.50206 q^{23} -2.93641i q^{24} -7.66376 q^{25} -4.86149 q^{27} -0.768150i q^{28} +8.63110 q^{29} -3.77493 q^{30} -3.22435i q^{31} -4.06514i q^{32} +3.82766i q^{33} +2.06962i q^{34} -3.55862 q^{35} -1.60276 q^{36} -1.83406i q^{37} -7.04401 q^{38} -10.9333 q^{40} +10.3590i q^{41} -1.06079 q^{42} +10.3318 q^{43} +3.07631i q^{44} +7.42514i q^{45} +4.99678i q^{46} +0.832988i q^{47} -1.79075 q^{48} -1.00000 q^{49} +8.50590i q^{50} -1.78221 q^{51} +6.53512 q^{53} +5.39571i q^{54} +14.2517 q^{55} -3.07233 q^{56} -6.06582i q^{57} -9.57955i q^{58} -5.20780i q^{59} -2.61262i q^{60} -1.83794 q^{61} -3.57866 q^{62} +2.08652i q^{63} -8.25913 q^{64} +4.24827 q^{66} +3.05008i q^{67} -1.43238 q^{68} -4.30289 q^{69} +3.94966i q^{70} +14.4391i q^{71} +6.41050i q^{72} -12.9055i q^{73} -2.03559 q^{74} -7.32471 q^{75} -4.87514i q^{76} +4.00483 q^{77} +0.0931919 q^{79} +6.66758i q^{80} +1.61315 q^{81} +11.4973 q^{82} +3.17863i q^{83} -0.734167i q^{84} +6.63578i q^{85} -11.4671i q^{86} +8.24926 q^{87} +12.3042 q^{88} -12.2290i q^{89} +8.24106 q^{90} -3.45826 q^{92} -3.08170i q^{93} +0.924522 q^{94} -22.5851 q^{95} -3.88529i q^{96} -13.1916i q^{97} +1.10989i q^{98} -8.35617i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 16 q^{3} - 30 q^{4} + 52 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 16 q^{3} - 30 q^{4} + 52 q^{9} + 12 q^{10} - 26 q^{12} + 6 q^{14} + 26 q^{16} - 62 q^{17} - 8 q^{22} - 36 q^{23} - 64 q^{25} + 64 q^{27} + 30 q^{29} + 20 q^{30} - 8 q^{35} - 98 q^{36} - 90 q^{38} - 40 q^{40} + 4 q^{42} + 44 q^{43} + 22 q^{48} - 24 q^{49} - 36 q^{51} + 106 q^{53} - 52 q^{55} - 24 q^{56} + 44 q^{61} - 38 q^{62} - 4 q^{64} - 68 q^{66} + 68 q^{68} - 6 q^{69} - 80 q^{74} - 30 q^{75} - 24 q^{77} + 4 q^{79} + 72 q^{81} + 64 q^{82} + 54 q^{87} + 96 q^{88} + 52 q^{90} + 104 q^{92} - 88 q^{94} - 58 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times\).

\(n\) \(339\) \(1016\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.10989i − 0.784809i −0.919793 0.392404i \(-0.871643\pi\)
0.919793 0.392404i \(-0.128357\pi\)
\(3\) 0.955760 0.551808 0.275904 0.961185i \(-0.411023\pi\)
0.275904 + 0.961185i \(0.411023\pi\)
\(4\) 0.768150 0.384075
\(5\) − 3.55862i − 1.59146i −0.605650 0.795731i \(-0.707087\pi\)
0.605650 0.795731i \(-0.292913\pi\)
\(6\) − 1.06079i − 0.433064i
\(7\) − 1.00000i − 0.377964i
\(8\) − 3.07233i − 1.08623i
\(9\) −2.08652 −0.695508
\(10\) −3.94966 −1.24899
\(11\) 4.00483i 1.20750i 0.797173 + 0.603751i \(0.206328\pi\)
−0.797173 + 0.603751i \(0.793672\pi\)
\(12\) 0.734167 0.211936
\(13\) 0 0
\(14\) −1.10989 −0.296630
\(15\) − 3.40118i − 0.878182i
\(16\) −1.87364 −0.468411
\(17\) −1.86471 −0.452258 −0.226129 0.974097i \(-0.572607\pi\)
−0.226129 + 0.974097i \(0.572607\pi\)
\(18\) 2.31581i 0.545841i
\(19\) − 6.34660i − 1.45601i −0.685572 0.728005i \(-0.740448\pi\)
0.685572 0.728005i \(-0.259552\pi\)
\(20\) − 2.73355i − 0.611241i
\(21\) − 0.955760i − 0.208564i
\(22\) 4.44491 0.947658
\(23\) −4.50206 −0.938744 −0.469372 0.883001i \(-0.655520\pi\)
−0.469372 + 0.883001i \(0.655520\pi\)
\(24\) − 2.93641i − 0.599393i
\(25\) −7.66376 −1.53275
\(26\) 0 0
\(27\) −4.86149 −0.935595
\(28\) − 0.768150i − 0.145167i
\(29\) 8.63110 1.60275 0.801377 0.598159i \(-0.204101\pi\)
0.801377 + 0.598159i \(0.204101\pi\)
\(30\) −3.77493 −0.689205
\(31\) − 3.22435i − 0.579110i −0.957161 0.289555i \(-0.906493\pi\)
0.957161 0.289555i \(-0.0935073\pi\)
\(32\) − 4.06514i − 0.718621i
\(33\) 3.82766i 0.666309i
\(34\) 2.06962i 0.354936i
\(35\) −3.55862 −0.601516
\(36\) −1.60276 −0.267127
\(37\) − 1.83406i − 0.301517i −0.988571 0.150758i \(-0.951828\pi\)
0.988571 0.150758i \(-0.0481716\pi\)
\(38\) −7.04401 −1.14269
\(39\) 0 0
\(40\) −10.9333 −1.72870
\(41\) 10.3590i 1.61780i 0.587947 + 0.808899i \(0.299936\pi\)
−0.587947 + 0.808899i \(0.700064\pi\)
\(42\) −1.06079 −0.163683
\(43\) 10.3318 1.57559 0.787793 0.615941i \(-0.211224\pi\)
0.787793 + 0.615941i \(0.211224\pi\)
\(44\) 3.07631i 0.463771i
\(45\) 7.42514i 1.10687i
\(46\) 4.99678i 0.736735i
\(47\) 0.832988i 0.121504i 0.998153 + 0.0607519i \(0.0193498\pi\)
−0.998153 + 0.0607519i \(0.980650\pi\)
\(48\) −1.79075 −0.258473
\(49\) −1.00000 −0.142857
\(50\) 8.50590i 1.20292i
\(51\) −1.78221 −0.249560
\(52\) 0 0
\(53\) 6.53512 0.897667 0.448834 0.893615i \(-0.351840\pi\)
0.448834 + 0.893615i \(0.351840\pi\)
\(54\) 5.39571i 0.734263i
\(55\) 14.2517 1.92169
\(56\) −3.07233 −0.410558
\(57\) − 6.06582i − 0.803438i
\(58\) − 9.57955i − 1.25786i
\(59\) − 5.20780i − 0.677997i −0.940787 0.338999i \(-0.889912\pi\)
0.940787 0.338999i \(-0.110088\pi\)
\(60\) − 2.61262i − 0.337288i
\(61\) −1.83794 −0.235324 −0.117662 0.993054i \(-0.537540\pi\)
−0.117662 + 0.993054i \(0.537540\pi\)
\(62\) −3.57866 −0.454491
\(63\) 2.08652i 0.262877i
\(64\) −8.25913 −1.03239
\(65\) 0 0
\(66\) 4.24827 0.522925
\(67\) 3.05008i 0.372626i 0.982490 + 0.186313i \(0.0596539\pi\)
−0.982490 + 0.186313i \(0.940346\pi\)
\(68\) −1.43238 −0.173701
\(69\) −4.30289 −0.518007
\(70\) 3.94966i 0.472075i
\(71\) 14.4391i 1.71361i 0.515640 + 0.856805i \(0.327554\pi\)
−0.515640 + 0.856805i \(0.672446\pi\)
\(72\) 6.41050i 0.755484i
\(73\) − 12.9055i − 1.51047i −0.655454 0.755235i \(-0.727523\pi\)
0.655454 0.755235i \(-0.272477\pi\)
\(74\) −2.03559 −0.236633
\(75\) −7.32471 −0.845785
\(76\) − 4.87514i − 0.559217i
\(77\) 4.00483 0.456393
\(78\) 0 0
\(79\) 0.0931919 0.0104849 0.00524245 0.999986i \(-0.498331\pi\)
0.00524245 + 0.999986i \(0.498331\pi\)
\(80\) 6.66758i 0.745458i
\(81\) 1.61315 0.179239
\(82\) 11.4973 1.26966
\(83\) 3.17863i 0.348900i 0.984666 + 0.174450i \(0.0558147\pi\)
−0.984666 + 0.174450i \(0.944185\pi\)
\(84\) − 0.734167i − 0.0801042i
\(85\) 6.63578i 0.719752i
\(86\) − 11.4671i − 1.23653i
\(87\) 8.24926 0.884413
\(88\) 12.3042 1.31163
\(89\) − 12.2290i − 1.29627i −0.761524 0.648137i \(-0.775549\pi\)
0.761524 0.648137i \(-0.224451\pi\)
\(90\) 8.24106 0.868684
\(91\) 0 0
\(92\) −3.45826 −0.360548
\(93\) − 3.08170i − 0.319558i
\(94\) 0.924522 0.0953572
\(95\) −22.5851 −2.31718
\(96\) − 3.88529i − 0.396541i
\(97\) − 13.1916i − 1.33941i −0.742628 0.669704i \(-0.766421\pi\)
0.742628 0.669704i \(-0.233579\pi\)
\(98\) 1.10989i 0.112116i
\(99\) − 8.35617i − 0.839827i
\(100\) −5.88692 −0.588692
\(101\) 7.73025 0.769188 0.384594 0.923086i \(-0.374341\pi\)
0.384594 + 0.923086i \(0.374341\pi\)
\(102\) 1.97806i 0.195857i
\(103\) 0.359154 0.0353885 0.0176943 0.999843i \(-0.494367\pi\)
0.0176943 + 0.999843i \(0.494367\pi\)
\(104\) 0 0
\(105\) −3.40118 −0.331922
\(106\) − 7.25324i − 0.704497i
\(107\) −3.99044 −0.385770 −0.192885 0.981221i \(-0.561784\pi\)
−0.192885 + 0.981221i \(0.561784\pi\)
\(108\) −3.73436 −0.359339
\(109\) − 9.65395i − 0.924681i −0.886702 0.462341i \(-0.847010\pi\)
0.886702 0.462341i \(-0.152990\pi\)
\(110\) − 15.8177i − 1.50816i
\(111\) − 1.75292i − 0.166379i
\(112\) 1.87364i 0.177043i
\(113\) 2.08248 0.195904 0.0979518 0.995191i \(-0.468771\pi\)
0.0979518 + 0.995191i \(0.468771\pi\)
\(114\) −6.73238 −0.630545
\(115\) 16.0211i 1.49398i
\(116\) 6.62998 0.615578
\(117\) 0 0
\(118\) −5.78007 −0.532098
\(119\) 1.86471i 0.170937i
\(120\) −10.4496 −0.953911
\(121\) −5.03866 −0.458060
\(122\) 2.03991i 0.184685i
\(123\) 9.90068i 0.892714i
\(124\) − 2.47678i − 0.222422i
\(125\) 9.47929i 0.847853i
\(126\) 2.31581 0.206308
\(127\) −7.18287 −0.637376 −0.318688 0.947860i \(-0.603242\pi\)
−0.318688 + 0.947860i \(0.603242\pi\)
\(128\) 1.03643i 0.0916084i
\(129\) 9.87472 0.869421
\(130\) 0 0
\(131\) −1.48085 −0.129383 −0.0646913 0.997905i \(-0.520606\pi\)
−0.0646913 + 0.997905i \(0.520606\pi\)
\(132\) 2.94022i 0.255913i
\(133\) −6.34660 −0.550320
\(134\) 3.38524 0.292440
\(135\) 17.3002i 1.48896i
\(136\) 5.72901i 0.491258i
\(137\) 4.43864i 0.379219i 0.981860 + 0.189609i \(0.0607222\pi\)
−0.981860 + 0.189609i \(0.939278\pi\)
\(138\) 4.77572i 0.406536i
\(139\) 17.3529 1.47186 0.735929 0.677059i \(-0.236746\pi\)
0.735929 + 0.677059i \(0.236746\pi\)
\(140\) −2.73355 −0.231027
\(141\) 0.796136i 0.0670468i
\(142\) 16.0258 1.34486
\(143\) 0 0
\(144\) 3.90940 0.325783
\(145\) − 30.7148i − 2.55072i
\(146\) −14.3236 −1.18543
\(147\) −0.955760 −0.0788297
\(148\) − 1.40883i − 0.115805i
\(149\) 23.1381i 1.89554i 0.318950 + 0.947771i \(0.396670\pi\)
−0.318950 + 0.947771i \(0.603330\pi\)
\(150\) 8.12960i 0.663779i
\(151\) − 10.8846i − 0.885778i −0.896576 0.442889i \(-0.853953\pi\)
0.896576 0.442889i \(-0.146047\pi\)
\(152\) −19.4989 −1.58157
\(153\) 3.89076 0.314549
\(154\) − 4.44491i − 0.358181i
\(155\) −11.4742 −0.921632
\(156\) 0 0
\(157\) 14.2555 1.13771 0.568855 0.822438i \(-0.307387\pi\)
0.568855 + 0.822438i \(0.307387\pi\)
\(158\) − 0.103432i − 0.00822865i
\(159\) 6.24600 0.495340
\(160\) −14.4663 −1.14366
\(161\) 4.50206i 0.354812i
\(162\) − 1.79041i − 0.140668i
\(163\) 2.01430i 0.157772i 0.996884 + 0.0788861i \(0.0251364\pi\)
−0.996884 + 0.0788861i \(0.974864\pi\)
\(164\) 7.95724i 0.621356i
\(165\) 13.6212 1.06041
\(166\) 3.52792 0.273820
\(167\) 18.8978i 1.46236i 0.682187 + 0.731178i \(0.261029\pi\)
−0.682187 + 0.731178i \(0.738971\pi\)
\(168\) −2.93641 −0.226549
\(169\) 0 0
\(170\) 7.36497 0.564867
\(171\) 13.2423i 1.01267i
\(172\) 7.93638 0.605143
\(173\) 12.4146 0.943868 0.471934 0.881634i \(-0.343556\pi\)
0.471934 + 0.881634i \(0.343556\pi\)
\(174\) − 9.15575i − 0.694095i
\(175\) 7.66376i 0.579325i
\(176\) − 7.50363i − 0.565607i
\(177\) − 4.97740i − 0.374125i
\(178\) −13.5728 −1.01733
\(179\) 9.37393 0.700641 0.350320 0.936630i \(-0.386073\pi\)
0.350320 + 0.936630i \(0.386073\pi\)
\(180\) 5.70362i 0.425123i
\(181\) −5.83547 −0.433747 −0.216873 0.976200i \(-0.569586\pi\)
−0.216873 + 0.976200i \(0.569586\pi\)
\(182\) 0 0
\(183\) −1.75663 −0.129854
\(184\) 13.8318i 1.01970i
\(185\) −6.52670 −0.479852
\(186\) −3.42034 −0.250792
\(187\) − 7.46784i − 0.546102i
\(188\) 0.639860i 0.0466666i
\(189\) 4.86149i 0.353622i
\(190\) 25.0669i 1.81855i
\(191\) −16.3051 −1.17980 −0.589899 0.807477i \(-0.700833\pi\)
−0.589899 + 0.807477i \(0.700833\pi\)
\(192\) −7.89375 −0.569682
\(193\) − 21.2251i − 1.52782i −0.645323 0.763910i \(-0.723277\pi\)
0.645323 0.763910i \(-0.276723\pi\)
\(194\) −14.6412 −1.05118
\(195\) 0 0
\(196\) −0.768150 −0.0548679
\(197\) − 16.0798i − 1.14564i −0.819683 0.572818i \(-0.805850\pi\)
0.819683 0.572818i \(-0.194150\pi\)
\(198\) −9.27441 −0.659103
\(199\) 15.4383 1.09440 0.547198 0.837003i \(-0.315695\pi\)
0.547198 + 0.837003i \(0.315695\pi\)
\(200\) 23.5456i 1.66493i
\(201\) 2.91514i 0.205618i
\(202\) − 8.57970i − 0.603666i
\(203\) − 8.63110i − 0.605784i
\(204\) −1.36901 −0.0958497
\(205\) 36.8636 2.57466
\(206\) − 0.398621i − 0.0277732i
\(207\) 9.39365 0.652904
\(208\) 0 0
\(209\) 25.4170 1.75813
\(210\) 3.77493i 0.260495i
\(211\) 3.45377 0.237767 0.118884 0.992908i \(-0.462068\pi\)
0.118884 + 0.992908i \(0.462068\pi\)
\(212\) 5.01995 0.344772
\(213\) 13.8004i 0.945584i
\(214\) 4.42893i 0.302756i
\(215\) − 36.7669i − 2.50748i
\(216\) 14.9361i 1.01628i
\(217\) −3.22435 −0.218883
\(218\) −10.7148 −0.725698
\(219\) − 12.3345i − 0.833490i
\(220\) 10.9474 0.738075
\(221\) 0 0
\(222\) −1.94554 −0.130576
\(223\) 15.6638i 1.04893i 0.851433 + 0.524464i \(0.175734\pi\)
−0.851433 + 0.524464i \(0.824266\pi\)
\(224\) −4.06514 −0.271613
\(225\) 15.9906 1.06604
\(226\) − 2.31132i − 0.153747i
\(227\) − 8.05474i − 0.534612i −0.963612 0.267306i \(-0.913867\pi\)
0.963612 0.267306i \(-0.0861335\pi\)
\(228\) − 4.65947i − 0.308581i
\(229\) 19.9670i 1.31946i 0.751503 + 0.659730i \(0.229329\pi\)
−0.751503 + 0.659730i \(0.770671\pi\)
\(230\) 17.7816 1.17248
\(231\) 3.82766 0.251841
\(232\) − 26.5176i − 1.74097i
\(233\) −12.1790 −0.797871 −0.398935 0.916979i \(-0.630620\pi\)
−0.398935 + 0.916979i \(0.630620\pi\)
\(234\) 0 0
\(235\) 2.96428 0.193369
\(236\) − 4.00037i − 0.260402i
\(237\) 0.0890691 0.00578566
\(238\) 2.06962 0.134153
\(239\) 15.5704i 1.00717i 0.863946 + 0.503584i \(0.167986\pi\)
−0.863946 + 0.503584i \(0.832014\pi\)
\(240\) 6.37261i 0.411350i
\(241\) − 1.30581i − 0.0841148i −0.999115 0.0420574i \(-0.986609\pi\)
0.999115 0.0420574i \(-0.0133912\pi\)
\(242\) 5.59235i 0.359490i
\(243\) 16.1263 1.03450
\(244\) −1.41182 −0.0903822
\(245\) 3.55862i 0.227352i
\(246\) 10.9886 0.700610
\(247\) 0 0
\(248\) −9.90628 −0.629049
\(249\) 3.03801i 0.192526i
\(250\) 10.5209 0.665402
\(251\) 20.2852 1.28039 0.640196 0.768212i \(-0.278853\pi\)
0.640196 + 0.768212i \(0.278853\pi\)
\(252\) 1.60276i 0.100965i
\(253\) − 18.0300i − 1.13353i
\(254\) 7.97217i 0.500219i
\(255\) 6.34221i 0.397165i
\(256\) −15.3679 −0.960496
\(257\) −17.3450 −1.08195 −0.540977 0.841038i \(-0.681945\pi\)
−0.540977 + 0.841038i \(0.681945\pi\)
\(258\) − 10.9598i − 0.682329i
\(259\) −1.83406 −0.113963
\(260\) 0 0
\(261\) −18.0090 −1.11473
\(262\) 1.64358i 0.101541i
\(263\) −10.8891 −0.671448 −0.335724 0.941960i \(-0.608981\pi\)
−0.335724 + 0.941960i \(0.608981\pi\)
\(264\) 11.7598 0.723768
\(265\) − 23.2560i − 1.42860i
\(266\) 7.04401i 0.431896i
\(267\) − 11.6880i − 0.715294i
\(268\) 2.34292i 0.143117i
\(269\) 8.99414 0.548382 0.274191 0.961675i \(-0.411590\pi\)
0.274191 + 0.961675i \(0.411590\pi\)
\(270\) 19.2013 1.16855
\(271\) − 4.94282i − 0.300255i −0.988667 0.150127i \(-0.952032\pi\)
0.988667 0.150127i \(-0.0479684\pi\)
\(272\) 3.49380 0.211843
\(273\) 0 0
\(274\) 4.92639 0.297614
\(275\) − 30.6920i − 1.85080i
\(276\) −3.30526 −0.198954
\(277\) −19.4610 −1.16930 −0.584650 0.811286i \(-0.698768\pi\)
−0.584650 + 0.811286i \(0.698768\pi\)
\(278\) − 19.2598i − 1.15513i
\(279\) 6.72768i 0.402776i
\(280\) 10.9333i 0.653387i
\(281\) 20.1468i 1.20185i 0.799304 + 0.600927i \(0.205202\pi\)
−0.799304 + 0.600927i \(0.794798\pi\)
\(282\) 0.883621 0.0526189
\(283\) 22.8282 1.35699 0.678496 0.734604i \(-0.262632\pi\)
0.678496 + 0.734604i \(0.262632\pi\)
\(284\) 11.0914i 0.658155i
\(285\) −21.5859 −1.27864
\(286\) 0 0
\(287\) 10.3590 0.611470
\(288\) 8.48200i 0.499807i
\(289\) −13.5229 −0.795463
\(290\) −34.0899 −2.00183
\(291\) − 12.6080i − 0.739096i
\(292\) − 9.91334i − 0.580134i
\(293\) − 26.5717i − 1.55233i −0.630528 0.776166i \(-0.717162\pi\)
0.630528 0.776166i \(-0.282838\pi\)
\(294\) 1.06079i 0.0618663i
\(295\) −18.5326 −1.07901
\(296\) −5.63483 −0.327518
\(297\) − 19.4695i − 1.12973i
\(298\) 25.6806 1.48764
\(299\) 0 0
\(300\) −5.62648 −0.324845
\(301\) − 10.3318i − 0.595515i
\(302\) −12.0807 −0.695167
\(303\) 7.38826 0.424444
\(304\) 11.8913i 0.682011i
\(305\) 6.54053i 0.374510i
\(306\) − 4.31830i − 0.246861i
\(307\) 7.16833i 0.409118i 0.978854 + 0.204559i \(0.0655761\pi\)
−0.978854 + 0.204559i \(0.934424\pi\)
\(308\) 3.07631 0.175289
\(309\) 0.343265 0.0195277
\(310\) 12.7351i 0.723305i
\(311\) 13.6640 0.774817 0.387408 0.921908i \(-0.373370\pi\)
0.387408 + 0.921908i \(0.373370\pi\)
\(312\) 0 0
\(313\) 10.5821 0.598137 0.299068 0.954232i \(-0.403324\pi\)
0.299068 + 0.954232i \(0.403324\pi\)
\(314\) − 15.8219i − 0.892884i
\(315\) 7.42514 0.418359
\(316\) 0.0715854 0.00402699
\(317\) − 3.65041i − 0.205027i −0.994732 0.102514i \(-0.967311\pi\)
0.994732 0.102514i \(-0.0326886\pi\)
\(318\) − 6.93236i − 0.388747i
\(319\) 34.5661i 1.93533i
\(320\) 29.3911i 1.64301i
\(321\) −3.81390 −0.212871
\(322\) 4.99678 0.278459
\(323\) 11.8346i 0.658492i
\(324\) 1.23914 0.0688411
\(325\) 0 0
\(326\) 2.23565 0.123821
\(327\) − 9.22686i − 0.510247i
\(328\) 31.8262 1.75731
\(329\) 0.832988 0.0459241
\(330\) − 15.1180i − 0.832216i
\(331\) 21.7576i 1.19590i 0.801532 + 0.597952i \(0.204019\pi\)
−0.801532 + 0.597952i \(0.795981\pi\)
\(332\) 2.44167i 0.134004i
\(333\) 3.82680i 0.209707i
\(334\) 20.9744 1.14767
\(335\) 10.8541 0.593021
\(336\) 1.79075i 0.0976936i
\(337\) 17.8877 0.974405 0.487202 0.873289i \(-0.338017\pi\)
0.487202 + 0.873289i \(0.338017\pi\)
\(338\) 0 0
\(339\) 1.99036 0.108101
\(340\) 5.09728i 0.276439i
\(341\) 12.9130 0.699276
\(342\) 14.6975 0.794749
\(343\) 1.00000i 0.0539949i
\(344\) − 31.7428i − 1.71145i
\(345\) 15.3123i 0.824388i
\(346\) − 13.7789i − 0.740756i
\(347\) 11.2388 0.603331 0.301666 0.953414i \(-0.402457\pi\)
0.301666 + 0.953414i \(0.402457\pi\)
\(348\) 6.33667 0.339681
\(349\) 24.4183i 1.30708i 0.756891 + 0.653542i \(0.226718\pi\)
−0.756891 + 0.653542i \(0.773282\pi\)
\(350\) 8.50590 0.454660
\(351\) 0 0
\(352\) 16.2802 0.867736
\(353\) − 15.9072i − 0.846655i −0.905977 0.423327i \(-0.860862\pi\)
0.905977 0.423327i \(-0.139138\pi\)
\(354\) −5.52436 −0.293616
\(355\) 51.3834 2.72715
\(356\) − 9.39372i − 0.497866i
\(357\) 1.78221i 0.0943247i
\(358\) − 10.4040i − 0.549869i
\(359\) − 12.4456i − 0.656853i −0.944529 0.328427i \(-0.893482\pi\)
0.944529 0.328427i \(-0.106518\pi\)
\(360\) 22.8125 1.20232
\(361\) −21.2793 −1.11996
\(362\) 6.47671i 0.340408i
\(363\) −4.81575 −0.252761
\(364\) 0 0
\(365\) −45.9256 −2.40386
\(366\) 1.94966i 0.101910i
\(367\) −18.2038 −0.950231 −0.475115 0.879923i \(-0.657594\pi\)
−0.475115 + 0.879923i \(0.657594\pi\)
\(368\) 8.43525 0.439718
\(369\) − 21.6142i − 1.12519i
\(370\) 7.24390i 0.376592i
\(371\) − 6.53512i − 0.339286i
\(372\) − 2.36721i − 0.122734i
\(373\) 32.6828 1.69225 0.846126 0.532983i \(-0.178929\pi\)
0.846126 + 0.532983i \(0.178929\pi\)
\(374\) −8.28846 −0.428586
\(375\) 9.05992i 0.467852i
\(376\) 2.55922 0.131982
\(377\) 0 0
\(378\) 5.39571 0.277525
\(379\) 8.81792i 0.452946i 0.974017 + 0.226473i \(0.0727195\pi\)
−0.974017 + 0.226473i \(0.927280\pi\)
\(380\) −17.3488 −0.889973
\(381\) −6.86510 −0.351710
\(382\) 18.0969i 0.925916i
\(383\) − 2.54364i − 0.129974i −0.997886 0.0649870i \(-0.979299\pi\)
0.997886 0.0649870i \(-0.0207006\pi\)
\(384\) 0.990579i 0.0505502i
\(385\) − 14.2517i − 0.726332i
\(386\) −23.5575 −1.19905
\(387\) −21.5575 −1.09583
\(388\) − 10.1332i − 0.514433i
\(389\) 0.516707 0.0261981 0.0130990 0.999914i \(-0.495830\pi\)
0.0130990 + 0.999914i \(0.495830\pi\)
\(390\) 0 0
\(391\) 8.39502 0.424555
\(392\) 3.07233i 0.155176i
\(393\) −1.41534 −0.0713944
\(394\) −17.8467 −0.899104
\(395\) − 0.331634i − 0.0166863i
\(396\) − 6.41880i − 0.322557i
\(397\) 25.8956i 1.29967i 0.760077 + 0.649833i \(0.225161\pi\)
−0.760077 + 0.649833i \(0.774839\pi\)
\(398\) − 17.1348i − 0.858891i
\(399\) −6.06582 −0.303671
\(400\) 14.3591 0.717957
\(401\) 10.2493i 0.511828i 0.966700 + 0.255914i \(0.0823764\pi\)
−0.966700 + 0.255914i \(0.917624\pi\)
\(402\) 3.23548 0.161371
\(403\) 0 0
\(404\) 5.93799 0.295426
\(405\) − 5.74057i − 0.285251i
\(406\) −9.57955 −0.475425
\(407\) 7.34508 0.364082
\(408\) 5.47556i 0.271080i
\(409\) 20.5100i 1.01416i 0.861900 + 0.507078i \(0.169274\pi\)
−0.861900 + 0.507078i \(0.830726\pi\)
\(410\) − 40.9144i − 2.02062i
\(411\) 4.24228i 0.209256i
\(412\) 0.275884 0.0135918
\(413\) −5.20780 −0.256259
\(414\) − 10.4259i − 0.512405i
\(415\) 11.3115 0.555261
\(416\) 0 0
\(417\) 16.5852 0.812183
\(418\) − 28.2101i − 1.37980i
\(419\) 15.2380 0.744426 0.372213 0.928147i \(-0.378599\pi\)
0.372213 + 0.928147i \(0.378599\pi\)
\(420\) −2.61262 −0.127483
\(421\) − 26.3846i − 1.28590i −0.765906 0.642952i \(-0.777709\pi\)
0.765906 0.642952i \(-0.222291\pi\)
\(422\) − 3.83329i − 0.186602i
\(423\) − 1.73805i − 0.0845068i
\(424\) − 20.0781i − 0.975077i
\(425\) 14.2907 0.693199
\(426\) 15.3168 0.742103
\(427\) 1.83794i 0.0889442i
\(428\) −3.06526 −0.148165
\(429\) 0 0
\(430\) −40.8071 −1.96790
\(431\) 37.9100i 1.82606i 0.407889 + 0.913031i \(0.366265\pi\)
−0.407889 + 0.913031i \(0.633735\pi\)
\(432\) 9.10871 0.438243
\(433\) −8.51058 −0.408993 −0.204496 0.978867i \(-0.565556\pi\)
−0.204496 + 0.978867i \(0.565556\pi\)
\(434\) 3.57866i 0.171781i
\(435\) − 29.3559i − 1.40751i
\(436\) − 7.41569i − 0.355147i
\(437\) 28.5728i 1.36682i
\(438\) −13.6899 −0.654130
\(439\) −18.6101 −0.888213 −0.444107 0.895974i \(-0.646479\pi\)
−0.444107 + 0.895974i \(0.646479\pi\)
\(440\) − 43.7859i − 2.08741i
\(441\) 2.08652 0.0993582
\(442\) 0 0
\(443\) 26.8582 1.27607 0.638035 0.770007i \(-0.279747\pi\)
0.638035 + 0.770007i \(0.279747\pi\)
\(444\) − 1.34650i − 0.0639022i
\(445\) −43.5184 −2.06297
\(446\) 17.3851 0.823208
\(447\) 22.1144i 1.04598i
\(448\) 8.25913i 0.390207i
\(449\) − 15.5278i − 0.732804i −0.930457 0.366402i \(-0.880589\pi\)
0.930457 0.366402i \(-0.119411\pi\)
\(450\) − 17.7478i − 0.836638i
\(451\) −41.4859 −1.95349
\(452\) 1.59966 0.0752418
\(453\) − 10.4031i − 0.488780i
\(454\) −8.93986 −0.419568
\(455\) 0 0
\(456\) −18.6362 −0.872722
\(457\) 33.5563i 1.56970i 0.619688 + 0.784848i \(0.287259\pi\)
−0.619688 + 0.784848i \(0.712741\pi\)
\(458\) 22.1612 1.03552
\(459\) 9.06527 0.423130
\(460\) 12.3066i 0.573799i
\(461\) − 4.34899i − 0.202553i −0.994858 0.101276i \(-0.967707\pi\)
0.994858 0.101276i \(-0.0322926\pi\)
\(462\) − 4.24827i − 0.197647i
\(463\) − 23.2754i − 1.08170i −0.841119 0.540850i \(-0.818103\pi\)
0.841119 0.540850i \(-0.181897\pi\)
\(464\) −16.1716 −0.750748
\(465\) −10.9666 −0.508564
\(466\) 13.5173i 0.626176i
\(467\) −12.2510 −0.566909 −0.283454 0.958986i \(-0.591480\pi\)
−0.283454 + 0.958986i \(0.591480\pi\)
\(468\) 0 0
\(469\) 3.05008 0.140840
\(470\) − 3.29002i − 0.151757i
\(471\) 13.6248 0.627797
\(472\) −16.0001 −0.736464
\(473\) 41.3771i 1.90252i
\(474\) − 0.0988566i − 0.00454063i
\(475\) 48.6388i 2.23170i
\(476\) 1.43238i 0.0656529i
\(477\) −13.6357 −0.624334
\(478\) 17.2814 0.790434
\(479\) 0.683706i 0.0312393i 0.999878 + 0.0156197i \(0.00497209\pi\)
−0.999878 + 0.0156197i \(0.995028\pi\)
\(480\) −13.8263 −0.631080
\(481\) 0 0
\(482\) −1.44930 −0.0660140
\(483\) 4.30289i 0.195788i
\(484\) −3.87045 −0.175930
\(485\) −46.9440 −2.13162
\(486\) − 17.8983i − 0.811885i
\(487\) − 2.18545i − 0.0990323i −0.998773 0.0495162i \(-0.984232\pi\)
0.998773 0.0495162i \(-0.0157679\pi\)
\(488\) 5.64677i 0.255617i
\(489\) 1.92519i 0.0870601i
\(490\) 3.94966 0.178428
\(491\) −8.81707 −0.397909 −0.198954 0.980009i \(-0.563755\pi\)
−0.198954 + 0.980009i \(0.563755\pi\)
\(492\) 7.60521i 0.342870i
\(493\) −16.0945 −0.724859
\(494\) 0 0
\(495\) −29.7364 −1.33655
\(496\) 6.04128i 0.271262i
\(497\) 14.4391 0.647684
\(498\) 3.37185 0.151096
\(499\) 16.3801i 0.733275i 0.930364 + 0.366637i \(0.119491\pi\)
−0.930364 + 0.366637i \(0.880509\pi\)
\(500\) 7.28152i 0.325639i
\(501\) 18.0618i 0.806940i
\(502\) − 22.5143i − 1.00486i
\(503\) −11.5765 −0.516168 −0.258084 0.966122i \(-0.583091\pi\)
−0.258084 + 0.966122i \(0.583091\pi\)
\(504\) 6.41050 0.285546
\(505\) − 27.5090i − 1.22413i
\(506\) −20.0112 −0.889608
\(507\) 0 0
\(508\) −5.51752 −0.244800
\(509\) 13.3984i 0.593874i 0.954897 + 0.296937i \(0.0959651\pi\)
−0.954897 + 0.296937i \(0.904035\pi\)
\(510\) 7.03914 0.311698
\(511\) −12.9055 −0.570904
\(512\) 19.1295i 0.845414i
\(513\) 30.8540i 1.36224i
\(514\) 19.2510i 0.849127i
\(515\) − 1.27809i − 0.0563195i
\(516\) 7.58527 0.333923
\(517\) −3.33597 −0.146716
\(518\) 2.03559i 0.0894389i
\(519\) 11.8654 0.520834
\(520\) 0 0
\(521\) 38.0891 1.66871 0.834356 0.551226i \(-0.185840\pi\)
0.834356 + 0.551226i \(0.185840\pi\)
\(522\) 19.9879i 0.874848i
\(523\) 1.91936 0.0839276 0.0419638 0.999119i \(-0.486639\pi\)
0.0419638 + 0.999119i \(0.486639\pi\)
\(524\) −1.13752 −0.0496927
\(525\) 7.32471i 0.319677i
\(526\) 12.0856i 0.526959i
\(527\) 6.01247i 0.261907i
\(528\) − 7.17166i − 0.312107i
\(529\) −2.73147 −0.118760
\(530\) −25.8115 −1.12118
\(531\) 10.8662i 0.471552i
\(532\) −4.87514 −0.211364
\(533\) 0 0
\(534\) −12.9724 −0.561369
\(535\) 14.2004i 0.613938i
\(536\) 9.37086 0.404760
\(537\) 8.95923 0.386619
\(538\) − 9.98248i − 0.430375i
\(539\) − 4.00483i − 0.172500i
\(540\) 13.2892i 0.571874i
\(541\) 7.49222i 0.322116i 0.986945 + 0.161058i \(0.0514905\pi\)
−0.986945 + 0.161058i \(0.948509\pi\)
\(542\) −5.48597 −0.235642
\(543\) −5.57731 −0.239345
\(544\) 7.58029i 0.325002i
\(545\) −34.3547 −1.47159
\(546\) 0 0
\(547\) 14.7156 0.629192 0.314596 0.949226i \(-0.398131\pi\)
0.314596 + 0.949226i \(0.398131\pi\)
\(548\) 3.40955i 0.145649i
\(549\) 3.83491 0.163670
\(550\) −34.0647 −1.45252
\(551\) − 54.7781i − 2.33363i
\(552\) 13.2199i 0.562677i
\(553\) − 0.0931919i − 0.00396292i
\(554\) 21.5995i 0.917677i
\(555\) −6.23796 −0.264787
\(556\) 13.3297 0.565304
\(557\) 5.45353i 0.231073i 0.993303 + 0.115537i \(0.0368588\pi\)
−0.993303 + 0.115537i \(0.963141\pi\)
\(558\) 7.46696 0.316102
\(559\) 0 0
\(560\) 6.66758 0.281757
\(561\) − 7.13746i − 0.301344i
\(562\) 22.3606 0.943226
\(563\) −36.8364 −1.55247 −0.776235 0.630444i \(-0.782873\pi\)
−0.776235 + 0.630444i \(0.782873\pi\)
\(564\) 0.611552i 0.0257510i
\(565\) − 7.41077i − 0.311773i
\(566\) − 25.3367i − 1.06498i
\(567\) − 1.61315i − 0.0677458i
\(568\) 44.3619 1.86138
\(569\) −15.0251 −0.629885 −0.314942 0.949111i \(-0.601985\pi\)
−0.314942 + 0.949111i \(0.601985\pi\)
\(570\) 23.9580i 1.00349i
\(571\) −33.2518 −1.39155 −0.695773 0.718262i \(-0.744938\pi\)
−0.695773 + 0.718262i \(0.744938\pi\)
\(572\) 0 0
\(573\) −15.5838 −0.651023
\(574\) − 11.4973i − 0.479887i
\(575\) 34.5027 1.43886
\(576\) 17.2329 0.718036
\(577\) 34.7983i 1.44867i 0.689448 + 0.724335i \(0.257853\pi\)
−0.689448 + 0.724335i \(0.742147\pi\)
\(578\) 15.0089i 0.624286i
\(579\) − 20.2861i − 0.843063i
\(580\) − 23.5936i − 0.979670i
\(581\) 3.17863 0.131872
\(582\) −13.9935 −0.580049
\(583\) 26.1720i 1.08393i
\(584\) −39.6499 −1.64072
\(585\) 0 0
\(586\) −29.4915 −1.21828
\(587\) − 19.7267i − 0.814210i −0.913381 0.407105i \(-0.866538\pi\)
0.913381 0.407105i \(-0.133462\pi\)
\(588\) −0.734167 −0.0302766
\(589\) −20.4636 −0.843190
\(590\) 20.5690i 0.846814i
\(591\) − 15.3684i − 0.632171i
\(592\) 3.43637i 0.141234i
\(593\) − 7.56122i − 0.310502i −0.987875 0.155251i \(-0.950381\pi\)
0.987875 0.155251i \(-0.0496186\pi\)
\(594\) −21.6089 −0.886624
\(595\) 6.63578 0.272041
\(596\) 17.7735i 0.728031i
\(597\) 14.7554 0.603896
\(598\) 0 0
\(599\) 3.48332 0.142325 0.0711623 0.997465i \(-0.477329\pi\)
0.0711623 + 0.997465i \(0.477329\pi\)
\(600\) 22.5040i 0.918720i
\(601\) −43.3614 −1.76875 −0.884374 0.466778i \(-0.845415\pi\)
−0.884374 + 0.466778i \(0.845415\pi\)
\(602\) −11.4671 −0.467366
\(603\) − 6.36406i − 0.259165i
\(604\) − 8.36103i − 0.340206i
\(605\) 17.9307i 0.728985i
\(606\) − 8.20013i − 0.333108i
\(607\) −24.5108 −0.994864 −0.497432 0.867503i \(-0.665724\pi\)
−0.497432 + 0.867503i \(0.665724\pi\)
\(608\) −25.7998 −1.04632
\(609\) − 8.24926i − 0.334277i
\(610\) 7.25925 0.293918
\(611\) 0 0
\(612\) 2.98869 0.120810
\(613\) − 0.842003i − 0.0340082i −0.999855 0.0170041i \(-0.994587\pi\)
0.999855 0.0170041i \(-0.00541283\pi\)
\(614\) 7.95604 0.321080
\(615\) 35.2327 1.42072
\(616\) − 12.3042i − 0.495749i
\(617\) 11.6951i 0.470828i 0.971895 + 0.235414i \(0.0756446\pi\)
−0.971895 + 0.235414i \(0.924355\pi\)
\(618\) − 0.380986i − 0.0153255i
\(619\) − 30.4915i − 1.22556i −0.790254 0.612779i \(-0.790052\pi\)
0.790254 0.612779i \(-0.209948\pi\)
\(620\) −8.81393 −0.353976
\(621\) 21.8867 0.878284
\(622\) − 15.1655i − 0.608083i
\(623\) −12.2290 −0.489945
\(624\) 0 0
\(625\) −4.58563 −0.183425
\(626\) − 11.7450i − 0.469423i
\(627\) 24.2926 0.970153
\(628\) 10.9503 0.436966
\(629\) 3.41998i 0.136363i
\(630\) − 8.24106i − 0.328332i
\(631\) 19.0797i 0.759552i 0.925078 + 0.379776i \(0.123999\pi\)
−0.925078 + 0.379776i \(0.876001\pi\)
\(632\) − 0.286317i − 0.0113891i
\(633\) 3.30097 0.131202
\(634\) −4.05154 −0.160907
\(635\) 25.5611i 1.01436i
\(636\) 4.79787 0.190248
\(637\) 0 0
\(638\) 38.3644 1.51886
\(639\) − 30.1276i − 1.19183i
\(640\) 3.68826 0.145791
\(641\) −26.9166 −1.06314 −0.531571 0.847014i \(-0.678398\pi\)
−0.531571 + 0.847014i \(0.678398\pi\)
\(642\) 4.23300i 0.167063i
\(643\) − 50.6462i − 1.99729i −0.0520237 0.998646i \(-0.516567\pi\)
0.0520237 0.998646i \(-0.483433\pi\)
\(644\) 3.45826i 0.136274i
\(645\) − 35.1404i − 1.38365i
\(646\) 13.1350 0.516790
\(647\) 38.2321 1.50306 0.751530 0.659699i \(-0.229316\pi\)
0.751530 + 0.659699i \(0.229316\pi\)
\(648\) − 4.95613i − 0.194695i
\(649\) 20.8563 0.818683
\(650\) 0 0
\(651\) −3.08170 −0.120781
\(652\) 1.54729i 0.0605964i
\(653\) −10.3427 −0.404742 −0.202371 0.979309i \(-0.564865\pi\)
−0.202371 + 0.979309i \(0.564865\pi\)
\(654\) −10.2408 −0.400446
\(655\) 5.26979i 0.205908i
\(656\) − 19.4090i − 0.757795i
\(657\) 26.9275i 1.05054i
\(658\) − 0.924522i − 0.0360416i
\(659\) 12.8844 0.501904 0.250952 0.968000i \(-0.419256\pi\)
0.250952 + 0.968000i \(0.419256\pi\)
\(660\) 10.4631 0.407276
\(661\) 19.1895i 0.746385i 0.927754 + 0.373192i \(0.121737\pi\)
−0.927754 + 0.373192i \(0.878263\pi\)
\(662\) 24.1484 0.938555
\(663\) 0 0
\(664\) 9.76582 0.378987
\(665\) 22.5851i 0.875813i
\(666\) 4.24731 0.164580
\(667\) −38.8577 −1.50458
\(668\) 14.5163i 0.561654i
\(669\) 14.9709i 0.578807i
\(670\) − 12.0468i − 0.465408i
\(671\) − 7.36064i − 0.284154i
\(672\) −3.88529 −0.149878
\(673\) −11.1856 −0.431172 −0.215586 0.976485i \(-0.569166\pi\)
−0.215586 + 0.976485i \(0.569166\pi\)
\(674\) − 19.8533i − 0.764721i
\(675\) 37.2573 1.43403
\(676\) 0 0
\(677\) −16.4527 −0.632330 −0.316165 0.948704i \(-0.602395\pi\)
−0.316165 + 0.948704i \(0.602395\pi\)
\(678\) − 2.20907i − 0.0848388i
\(679\) −13.1916 −0.506248
\(680\) 20.3873 0.781819
\(681\) − 7.69840i − 0.295003i
\(682\) − 14.3319i − 0.548798i
\(683\) − 13.5271i − 0.517600i −0.965931 0.258800i \(-0.916673\pi\)
0.965931 0.258800i \(-0.0833271\pi\)
\(684\) 10.1721i 0.388940i
\(685\) 15.7954 0.603512
\(686\) 1.10989 0.0423757
\(687\) 19.0837i 0.728089i
\(688\) −19.3581 −0.738021
\(689\) 0 0
\(690\) 16.9950 0.646987
\(691\) 20.5171i 0.780507i 0.920707 + 0.390254i \(0.127613\pi\)
−0.920707 + 0.390254i \(0.872387\pi\)
\(692\) 9.53632 0.362516
\(693\) −8.35617 −0.317425
\(694\) − 12.4738i − 0.473500i
\(695\) − 61.7525i − 2.34241i
\(696\) − 25.3445i − 0.960680i
\(697\) − 19.3164i − 0.731662i
\(698\) 27.1016 1.02581
\(699\) −11.6402 −0.440272
\(700\) 5.88692i 0.222505i
\(701\) −46.6380 −1.76149 −0.880747 0.473587i \(-0.842959\pi\)
−0.880747 + 0.473587i \(0.842959\pi\)
\(702\) 0 0
\(703\) −11.6400 −0.439011
\(704\) − 33.0764i − 1.24661i
\(705\) 2.83314 0.106702
\(706\) −17.6552 −0.664462
\(707\) − 7.73025i − 0.290726i
\(708\) − 3.82339i − 0.143692i
\(709\) − 42.2314i − 1.58603i −0.609199 0.793017i \(-0.708509\pi\)
0.609199 0.793017i \(-0.291491\pi\)
\(710\) − 57.0297i − 2.14029i
\(711\) −0.194447 −0.00729233
\(712\) −37.5716 −1.40806
\(713\) 14.5162i 0.543636i
\(714\) 1.97806 0.0740269
\(715\) 0 0
\(716\) 7.20059 0.269099
\(717\) 14.8816i 0.555764i
\(718\) −13.8132 −0.515504
\(719\) −37.9404 −1.41494 −0.707469 0.706744i \(-0.750163\pi\)
−0.707469 + 0.706744i \(0.750163\pi\)
\(720\) − 13.9121i − 0.518472i
\(721\) − 0.359154i − 0.0133756i
\(722\) 23.6176i 0.878957i
\(723\) − 1.24804i − 0.0464152i
\(724\) −4.48252 −0.166591
\(725\) −66.1466 −2.45662
\(726\) 5.34494i 0.198369i
\(727\) 16.5215 0.612748 0.306374 0.951911i \(-0.400884\pi\)
0.306374 + 0.951911i \(0.400884\pi\)
\(728\) 0 0
\(729\) 10.5734 0.391607
\(730\) 50.9722i 1.88657i
\(731\) −19.2658 −0.712571
\(732\) −1.34936 −0.0498737
\(733\) 17.6511i 0.651957i 0.945377 + 0.325978i \(0.105694\pi\)
−0.945377 + 0.325978i \(0.894306\pi\)
\(734\) 20.2042i 0.745750i
\(735\) 3.40118i 0.125455i
\(736\) 18.3015i 0.674601i
\(737\) −12.2150 −0.449947
\(738\) −23.9893 −0.883060
\(739\) 0.370373i 0.0136244i 0.999977 + 0.00681220i \(0.00216841\pi\)
−0.999977 + 0.00681220i \(0.997832\pi\)
\(740\) −5.01349 −0.184299
\(741\) 0 0
\(742\) −7.25324 −0.266275
\(743\) 17.5579i 0.644137i 0.946716 + 0.322068i \(0.104378\pi\)
−0.946716 + 0.322068i \(0.895622\pi\)
\(744\) −9.46802 −0.347115
\(745\) 82.3395 3.01668
\(746\) − 36.2742i − 1.32809i
\(747\) − 6.63229i − 0.242663i
\(748\) − 5.73642i − 0.209744i
\(749\) 3.99044i 0.145807i
\(750\) 10.0555 0.367175
\(751\) 15.0706 0.549933 0.274967 0.961454i \(-0.411333\pi\)
0.274967 + 0.961454i \(0.411333\pi\)
\(752\) − 1.56072i − 0.0569137i
\(753\) 19.3878 0.706530
\(754\) 0 0
\(755\) −38.7342 −1.40968
\(756\) 3.73436i 0.135817i
\(757\) 39.8030 1.44666 0.723332 0.690500i \(-0.242609\pi\)
0.723332 + 0.690500i \(0.242609\pi\)
\(758\) 9.78689 0.355476
\(759\) − 17.2323i − 0.625494i
\(760\) 69.3890i 2.51700i
\(761\) − 20.7066i − 0.750613i −0.926901 0.375306i \(-0.877538\pi\)
0.926901 0.375306i \(-0.122462\pi\)
\(762\) 7.61948i 0.276025i
\(763\) −9.65395 −0.349497
\(764\) −12.5248 −0.453131
\(765\) − 13.8457i − 0.500593i
\(766\) −2.82315 −0.102005
\(767\) 0 0
\(768\) −14.6881 −0.530010
\(769\) − 13.2638i − 0.478306i −0.970982 0.239153i \(-0.923130\pi\)
0.970982 0.239153i \(-0.0768698\pi\)
\(770\) −15.8177 −0.570031
\(771\) −16.5777 −0.597031
\(772\) − 16.3041i − 0.586798i
\(773\) − 11.6627i − 0.419478i −0.977757 0.209739i \(-0.932739\pi\)
0.977757 0.209739i \(-0.0672614\pi\)
\(774\) 23.9264i 0.860018i
\(775\) 24.7106i 0.887632i
\(776\) −40.5291 −1.45491
\(777\) −1.75292 −0.0628855
\(778\) − 0.573486i − 0.0205605i
\(779\) 65.7442 2.35553
\(780\) 0 0
\(781\) −57.8263 −2.06919
\(782\) − 9.31753i − 0.333194i
\(783\) −41.9600 −1.49953
\(784\) 1.87364 0.0669159
\(785\) − 50.7297i − 1.81062i
\(786\) 1.57087i 0.0560310i
\(787\) 5.00642i 0.178460i 0.996011 + 0.0892298i \(0.0284405\pi\)
−0.996011 + 0.0892298i \(0.971559\pi\)
\(788\) − 12.3517i − 0.440010i
\(789\) −10.4073 −0.370511
\(790\) −0.368077 −0.0130956
\(791\) − 2.08248i − 0.0740446i
\(792\) −25.6730 −0.912249
\(793\) 0 0
\(794\) 28.7412 1.01999
\(795\) − 22.2271i − 0.788315i
\(796\) 11.8590 0.420330
\(797\) −9.95893 −0.352763 −0.176382 0.984322i \(-0.556439\pi\)
−0.176382 + 0.984322i \(0.556439\pi\)
\(798\) 6.73238i 0.238324i
\(799\) − 1.55328i − 0.0549511i
\(800\) 31.1542i 1.10147i
\(801\) 25.5161i 0.901568i
\(802\) 11.3756 0.401687
\(803\) 51.6842 1.82389
\(804\) 2.23927i 0.0789729i
\(805\) 16.0211 0.564670
\(806\) 0 0
\(807\) 8.59623 0.302602
\(808\) − 23.7499i − 0.835519i
\(809\) 12.0200 0.422601 0.211300 0.977421i \(-0.432230\pi\)
0.211300 + 0.977421i \(0.432230\pi\)
\(810\) −6.37139 −0.223868
\(811\) 2.91856i 0.102484i 0.998686 + 0.0512422i \(0.0163180\pi\)
−0.998686 + 0.0512422i \(0.983682\pi\)
\(812\) − 6.62998i − 0.232667i
\(813\) − 4.72414i − 0.165683i
\(814\) − 8.15221i − 0.285735i
\(815\) 7.16813 0.251089
\(816\) 3.33923 0.116897
\(817\) − 65.5718i − 2.29407i
\(818\) 22.7638 0.795919
\(819\) 0 0
\(820\) 28.3168 0.988865
\(821\) 12.9996i 0.453689i 0.973931 + 0.226844i \(0.0728409\pi\)
−0.973931 + 0.226844i \(0.927159\pi\)
\(822\) 4.70845 0.164226
\(823\) 39.6283 1.38136 0.690678 0.723163i \(-0.257312\pi\)
0.690678 + 0.723163i \(0.257312\pi\)
\(824\) − 1.10344i − 0.0384402i
\(825\) − 29.3342i − 1.02129i
\(826\) 5.78007i 0.201114i
\(827\) 31.5319i 1.09647i 0.836323 + 0.548237i \(0.184701\pi\)
−0.836323 + 0.548237i \(0.815299\pi\)
\(828\) 7.21574 0.250764
\(829\) −36.2352 −1.25850 −0.629251 0.777202i \(-0.716638\pi\)
−0.629251 + 0.777202i \(0.716638\pi\)
\(830\) − 12.5545i − 0.435774i
\(831\) −18.6001 −0.645229
\(832\) 0 0
\(833\) 1.86471 0.0646083
\(834\) − 18.4078i − 0.637408i
\(835\) 67.2500 2.32728
\(836\) 19.5241 0.675256
\(837\) 15.6752i 0.541813i
\(838\) − 16.9125i − 0.584232i
\(839\) 12.0623i 0.416438i 0.978082 + 0.208219i \(0.0667667\pi\)
−0.978082 + 0.208219i \(0.933233\pi\)
\(840\) 10.4496i 0.360545i
\(841\) 45.4959 1.56882
\(842\) −29.2839 −1.00919
\(843\) 19.2555i 0.663193i
\(844\) 2.65301 0.0913204
\(845\) 0 0
\(846\) −1.92904 −0.0663217
\(847\) 5.03866i 0.173130i
\(848\) −12.2445 −0.420477
\(849\) 21.8182 0.748800
\(850\) − 15.8610i − 0.544029i
\(851\) 8.25702i 0.283047i
\(852\) 10.6007i 0.363176i
\(853\) 11.7255i 0.401475i 0.979645 + 0.200738i \(0.0643339\pi\)
−0.979645 + 0.200738i \(0.935666\pi\)
\(854\) 2.03991 0.0698042
\(855\) 47.1244 1.61162
\(856\) 12.2600i 0.419037i
\(857\) −28.0128 −0.956900 −0.478450 0.878115i \(-0.658801\pi\)
−0.478450 + 0.878115i \(0.658801\pi\)
\(858\) 0 0
\(859\) 5.70001 0.194482 0.0972408 0.995261i \(-0.468998\pi\)
0.0972408 + 0.995261i \(0.468998\pi\)
\(860\) − 28.2425i − 0.963063i
\(861\) 9.90068 0.337414
\(862\) 42.0759 1.43311
\(863\) 41.0325i 1.39676i 0.715726 + 0.698381i \(0.246096\pi\)
−0.715726 + 0.698381i \(0.753904\pi\)
\(864\) 19.7626i 0.672339i
\(865\) − 44.1790i − 1.50213i
\(866\) 9.44579i 0.320981i
\(867\) −12.9246 −0.438943
\(868\) −2.47678 −0.0840676
\(869\) 0.373218i 0.0126605i
\(870\) −32.5818 −1.10463
\(871\) 0 0
\(872\) −29.6602 −1.00442
\(873\) 27.5246i 0.931568i
\(874\) 31.7125 1.07269
\(875\) 9.47929 0.320458
\(876\) − 9.47477i − 0.320123i
\(877\) 10.7401i 0.362668i 0.983422 + 0.181334i \(0.0580415\pi\)
−0.983422 + 0.181334i \(0.941958\pi\)
\(878\) 20.6551i 0.697077i
\(879\) − 25.3961i − 0.856590i
\(880\) −26.7025 −0.900142
\(881\) −37.5517 −1.26515 −0.632574 0.774500i \(-0.718002\pi\)
−0.632574 + 0.774500i \(0.718002\pi\)
\(882\) − 2.31581i − 0.0779772i
\(883\) −47.1134 −1.58549 −0.792745 0.609553i \(-0.791349\pi\)
−0.792745 + 0.609553i \(0.791349\pi\)
\(884\) 0 0
\(885\) −17.7127 −0.595405
\(886\) − 29.8095i − 1.00147i
\(887\) 5.06090 0.169928 0.0849642 0.996384i \(-0.472922\pi\)
0.0849642 + 0.996384i \(0.472922\pi\)
\(888\) −5.38555 −0.180727
\(889\) 7.18287i 0.240906i
\(890\) 48.3005i 1.61904i
\(891\) 6.46038i 0.216431i
\(892\) 12.0322i 0.402867i
\(893\) 5.28664 0.176911
\(894\) 24.5445 0.820891
\(895\) − 33.3582i − 1.11504i
\(896\) 1.03643 0.0346247
\(897\) 0 0
\(898\) −17.2342 −0.575111
\(899\) − 27.8297i − 0.928171i
\(900\) 12.2832 0.409440
\(901\) −12.1861 −0.405977
\(902\) 46.0447i 1.53312i
\(903\) − 9.87472i − 0.328610i
\(904\) − 6.39809i − 0.212797i
\(905\) 20.7662i 0.690292i
\(906\) −11.5463 −0.383599
\(907\) 48.6215 1.61445 0.807226 0.590243i \(-0.200968\pi\)
0.807226 + 0.590243i \(0.200968\pi\)
\(908\) − 6.18726i − 0.205331i
\(909\) −16.1293 −0.534976
\(910\) 0 0
\(911\) 1.76756 0.0585620 0.0292810 0.999571i \(-0.490678\pi\)
0.0292810 + 0.999571i \(0.490678\pi\)
\(912\) 11.3652i 0.376339i
\(913\) −12.7299 −0.421297
\(914\) 37.2437 1.23191
\(915\) 6.25118i 0.206658i
\(916\) 15.3377i 0.506772i
\(917\) 1.48085i 0.0489021i
\(918\) − 10.0614i − 0.332076i
\(919\) 27.4163 0.904380 0.452190 0.891922i \(-0.350643\pi\)
0.452190 + 0.891922i \(0.350643\pi\)
\(920\) 49.2222 1.62281
\(921\) 6.85120i 0.225755i
\(922\) −4.82689 −0.158965
\(923\) 0 0
\(924\) 2.94022 0.0967260
\(925\) 14.0557i 0.462150i
\(926\) −25.8331 −0.848928
\(927\) −0.749383 −0.0246130
\(928\) − 35.0866i − 1.15177i
\(929\) − 31.0105i − 1.01742i −0.860938 0.508710i \(-0.830122\pi\)
0.860938 0.508710i \(-0.169878\pi\)
\(930\) 12.1717i 0.399125i
\(931\) 6.34660i 0.208001i
\(932\) −9.35528 −0.306442
\(933\) 13.0595 0.427550
\(934\) 13.5972i 0.444915i
\(935\) −26.5752 −0.869101
\(936\) 0 0
\(937\) −13.5463 −0.442537 −0.221269 0.975213i \(-0.571020\pi\)
−0.221269 + 0.975213i \(0.571020\pi\)
\(938\) − 3.38524i − 0.110532i
\(939\) 10.1140 0.330057
\(940\) 2.27702 0.0742681
\(941\) − 27.1579i − 0.885321i −0.896689 0.442661i \(-0.854035\pi\)
0.896689 0.442661i \(-0.145965\pi\)
\(942\) − 15.1220i − 0.492701i
\(943\) − 46.6367i − 1.51870i
\(944\) 9.75756i 0.317581i
\(945\) 17.3002 0.562775
\(946\) 45.9239 1.49312
\(947\) − 30.1736i − 0.980511i −0.871579 0.490256i \(-0.836903\pi\)
0.871579 0.490256i \(-0.163097\pi\)
\(948\) 0.0684184 0.00222213
\(949\) 0 0
\(950\) 53.9835 1.75146
\(951\) − 3.48892i − 0.113136i
\(952\) 5.72901 0.185678
\(953\) −13.0194 −0.421738 −0.210869 0.977514i \(-0.567629\pi\)
−0.210869 + 0.977514i \(0.567629\pi\)
\(954\) 15.1341i 0.489983i
\(955\) 58.0237i 1.87760i
\(956\) 11.9604i 0.386828i
\(957\) 33.0369i 1.06793i
\(958\) 0.758836 0.0245169
\(959\) 4.43864 0.143331
\(960\) 28.0908i 0.906627i
\(961\) 20.6036 0.664632
\(962\) 0 0
\(963\) 8.32614 0.268306
\(964\) − 1.00306i − 0.0323064i
\(965\) −75.5322 −2.43147
\(966\) 4.77572 0.153656
\(967\) 49.5415i 1.59315i 0.604542 + 0.796573i \(0.293356\pi\)
−0.604542 + 0.796573i \(0.706644\pi\)
\(968\) 15.4805i 0.497561i
\(969\) 11.3110i 0.363361i
\(970\) 52.1025i 1.67291i
\(971\) −34.3718 −1.10304 −0.551522 0.834160i \(-0.685953\pi\)
−0.551522 + 0.834160i \(0.685953\pi\)
\(972\) 12.3874 0.397326
\(973\) − 17.3529i − 0.556310i
\(974\) −2.42561 −0.0777214
\(975\) 0 0
\(976\) 3.44365 0.110228
\(977\) 10.8964i 0.348607i 0.984692 + 0.174303i \(0.0557674\pi\)
−0.984692 + 0.174303i \(0.944233\pi\)
\(978\) 2.13674 0.0683255
\(979\) 48.9751 1.56525
\(980\) 2.73355i 0.0873202i
\(981\) 20.1432i 0.643123i
\(982\) 9.78595i 0.312282i
\(983\) 17.8803i 0.570293i 0.958484 + 0.285146i \(0.0920422\pi\)
−0.958484 + 0.285146i \(0.907958\pi\)
\(984\) 30.4182 0.969697
\(985\) −57.2217 −1.82323
\(986\) 17.8631i 0.568876i
\(987\) 0.796136 0.0253413
\(988\) 0 0
\(989\) −46.5144 −1.47907
\(990\) 33.0041i 1.04894i
\(991\) −43.2925 −1.37523 −0.687616 0.726074i \(-0.741343\pi\)
−0.687616 + 0.726074i \(0.741343\pi\)
\(992\) −13.1074 −0.416161
\(993\) 20.7950i 0.659909i
\(994\) − 16.0258i − 0.508308i
\(995\) − 54.9392i − 1.74169i
\(996\) 2.33365i 0.0739444i
\(997\) 25.7004 0.813940 0.406970 0.913442i \(-0.366585\pi\)
0.406970 + 0.913442i \(0.366585\pi\)
\(998\) 18.1801 0.575480
\(999\) 8.91625i 0.282098i
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 1183.2.c.j.337.8 24
13.5 odd 4 1183.2.a.r.1.4 yes 12
13.8 odd 4 1183.2.a.q.1.9 12
13.12 even 2 inner 1183.2.c.j.337.17 24
91.34 even 4 8281.2.a.cn.1.9 12
91.83 even 4 8281.2.a.cq.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1183.2.a.q.1.9 12 13.8 odd 4
1183.2.a.r.1.4 yes 12 13.5 odd 4
1183.2.c.j.337.8 24 1.1 even 1 trivial
1183.2.c.j.337.17 24 13.12 even 2 inner
8281.2.a.cn.1.9 12 91.34 even 4
8281.2.a.cq.1.4 12 91.83 even 4