Properties

Label 1183.2.a.q.1.9
Level $1183$
Weight $2$
Character 1183.1
Self dual yes
Analytic conductor $9.446$
Analytic rank $0$
Dimension $12$
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] = [1183,2,Mod(1,1183)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1183.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.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: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 15 x^{10} + 46 x^{9} + 80 x^{8} - 246 x^{7} - 199 x^{6} + 562 x^{5} + 262 x^{4} - 542 x^{3} - 157 x^{2} + 183 x + 29 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-1.10989\) of defining polynomial
Character \(\chi\) \(=\) 1183.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.10989 q^{2} +0.955760 q^{3} -0.768150 q^{4} +3.55862 q^{5} +1.06079 q^{6} -1.00000 q^{7} -3.07233 q^{8} -2.08652 q^{9} +O(q^{10})\) \(q+1.10989 q^{2} +0.955760 q^{3} -0.768150 q^{4} +3.55862 q^{5} +1.06079 q^{6} -1.00000 q^{7} -3.07233 q^{8} -2.08652 q^{9} +3.94966 q^{10} +4.00483 q^{11} -0.734167 q^{12} -1.10989 q^{14} +3.40118 q^{15} -1.87364 q^{16} +1.86471 q^{17} -2.31581 q^{18} +6.34660 q^{19} -2.73355 q^{20} -0.955760 q^{21} +4.44491 q^{22} +4.50206 q^{23} -2.93641 q^{24} +7.66376 q^{25} -4.86149 q^{27} +0.768150 q^{28} +8.63110 q^{29} +3.77493 q^{30} +3.22435 q^{31} +4.06514 q^{32} +3.82766 q^{33} +2.06962 q^{34} -3.55862 q^{35} +1.60276 q^{36} -1.83406 q^{37} +7.04401 q^{38} -10.9333 q^{40} -10.3590 q^{41} -1.06079 q^{42} -10.3318 q^{43} -3.07631 q^{44} -7.42514 q^{45} +4.99678 q^{46} +0.832988 q^{47} -1.79075 q^{48} +1.00000 q^{49} +8.50590 q^{50} +1.78221 q^{51} +6.53512 q^{53} -5.39571 q^{54} +14.2517 q^{55} +3.07233 q^{56} +6.06582 q^{57} +9.57955 q^{58} -5.20780 q^{59} -2.61262 q^{60} -1.83794 q^{61} +3.57866 q^{62} +2.08652 q^{63} +8.25913 q^{64} +4.24827 q^{66} -3.05008 q^{67} -1.43238 q^{68} +4.30289 q^{69} -3.94966 q^{70} -14.4391 q^{71} +6.41050 q^{72} -12.9055 q^{73} -2.03559 q^{74} +7.32471 q^{75} -4.87514 q^{76} -4.00483 q^{77} +0.0931919 q^{79} -6.66758 q^{80} +1.61315 q^{81} -11.4973 q^{82} -3.17863 q^{83} +0.734167 q^{84} +6.63578 q^{85} -11.4671 q^{86} +8.24926 q^{87} -12.3042 q^{88} -12.2290 q^{89} -8.24106 q^{90} -3.45826 q^{92} +3.08170 q^{93} +0.924522 q^{94} +22.5851 q^{95} +3.88529 q^{96} +13.1916 q^{97} +1.10989 q^{98} -8.35617 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{2} + 8 q^{3} + 15 q^{4} + 4 q^{5} - 2 q^{6} - 12 q^{7} - 12 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{2} + 8 q^{3} + 15 q^{4} + 4 q^{5} - 2 q^{6} - 12 q^{7} - 12 q^{8} + 26 q^{9} - 6 q^{10} - 12 q^{11} + 13 q^{12} + 3 q^{14} - 11 q^{15} + 13 q^{16} + 31 q^{17} + 29 q^{18} + 3 q^{19} + 18 q^{20} - 8 q^{21} - 4 q^{22} + 18 q^{23} + 6 q^{24} + 32 q^{25} + 32 q^{27} - 15 q^{28} + 15 q^{29} - 10 q^{30} + 21 q^{31} + 3 q^{32} + 29 q^{33} - 3 q^{34} - 4 q^{35} + 49 q^{36} - 5 q^{37} + 45 q^{38} - 20 q^{40} + 16 q^{41} + 2 q^{42} - 22 q^{43} - 35 q^{44} - 5 q^{45} - 2 q^{46} + 4 q^{47} + 11 q^{48} + 12 q^{49} - 13 q^{50} + 18 q^{51} + 53 q^{53} + 5 q^{54} - 26 q^{55} + 12 q^{56} + 8 q^{57} - 32 q^{58} + 26 q^{59} - 38 q^{60} + 22 q^{61} + 19 q^{62} - 26 q^{63} + 2 q^{64} - 34 q^{66} - 12 q^{67} + 34 q^{68} + 3 q^{69} + 6 q^{70} - 21 q^{71} + 4 q^{72} + 15 q^{73} - 40 q^{74} + 15 q^{75} - 43 q^{76} + 12 q^{77} + 2 q^{79} - 13 q^{80} + 36 q^{81} - 32 q^{82} + 9 q^{83} - 13 q^{84} + 39 q^{85} - 44 q^{86} + 27 q^{87} - 48 q^{88} + 22 q^{89} - 26 q^{90} + 52 q^{92} + 53 q^{93} - 44 q^{94} + 29 q^{95} + 114 q^{96} - 9 q^{97} - 3 q^{98} - 37 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.10989 0.784809 0.392404 0.919793i \(-0.371643\pi\)
0.392404 + 0.919793i \(0.371643\pi\)
\(3\) 0.955760 0.551808 0.275904 0.961185i \(-0.411023\pi\)
0.275904 + 0.961185i \(0.411023\pi\)
\(4\) −0.768150 −0.384075
\(5\) 3.55862 1.59146 0.795731 0.605650i \(-0.207087\pi\)
0.795731 + 0.605650i \(0.207087\pi\)
\(6\) 1.06079 0.433064
\(7\) −1.00000 −0.377964
\(8\) −3.07233 −1.08623
\(9\) −2.08652 −0.695508
\(10\) 3.94966 1.24899
\(11\) 4.00483 1.20750 0.603751 0.797173i \(-0.293672\pi\)
0.603751 + 0.797173i \(0.293672\pi\)
\(12\) −0.734167 −0.211936
\(13\) 0 0
\(14\) −1.10989 −0.296630
\(15\) 3.40118 0.878182
\(16\) −1.87364 −0.468411
\(17\) 1.86471 0.452258 0.226129 0.974097i \(-0.427393\pi\)
0.226129 + 0.974097i \(0.427393\pi\)
\(18\) −2.31581 −0.545841
\(19\) 6.34660 1.45601 0.728005 0.685572i \(-0.240448\pi\)
0.728005 + 0.685572i \(0.240448\pi\)
\(20\) −2.73355 −0.611241
\(21\) −0.955760 −0.208564
\(22\) 4.44491 0.947658
\(23\) 4.50206 0.938744 0.469372 0.883001i \(-0.344480\pi\)
0.469372 + 0.883001i \(0.344480\pi\)
\(24\) −2.93641 −0.599393
\(25\) 7.66376 1.53275
\(26\) 0 0
\(27\) −4.86149 −0.935595
\(28\) 0.768150 0.145167
\(29\) 8.63110 1.60275 0.801377 0.598159i \(-0.204101\pi\)
0.801377 + 0.598159i \(0.204101\pi\)
\(30\) 3.77493 0.689205
\(31\) 3.22435 0.579110 0.289555 0.957161i \(-0.406493\pi\)
0.289555 + 0.957161i \(0.406493\pi\)
\(32\) 4.06514 0.718621
\(33\) 3.82766 0.666309
\(34\) 2.06962 0.354936
\(35\) −3.55862 −0.601516
\(36\) 1.60276 0.267127
\(37\) −1.83406 −0.301517 −0.150758 0.988571i \(-0.548172\pi\)
−0.150758 + 0.988571i \(0.548172\pi\)
\(38\) 7.04401 1.14269
\(39\) 0 0
\(40\) −10.9333 −1.72870
\(41\) −10.3590 −1.61780 −0.808899 0.587947i \(-0.799936\pi\)
−0.808899 + 0.587947i \(0.799936\pi\)
\(42\) −1.06079 −0.163683
\(43\) −10.3318 −1.57559 −0.787793 0.615941i \(-0.788776\pi\)
−0.787793 + 0.615941i \(0.788776\pi\)
\(44\) −3.07631 −0.463771
\(45\) −7.42514 −1.10687
\(46\) 4.99678 0.736735
\(47\) 0.832988 0.121504 0.0607519 0.998153i \(-0.480650\pi\)
0.0607519 + 0.998153i \(0.480650\pi\)
\(48\) −1.79075 −0.258473
\(49\) 1.00000 0.142857
\(50\) 8.50590 1.20292
\(51\) 1.78221 0.249560
\(52\) 0 0
\(53\) 6.53512 0.897667 0.448834 0.893615i \(-0.351840\pi\)
0.448834 + 0.893615i \(0.351840\pi\)
\(54\) −5.39571 −0.734263
\(55\) 14.2517 1.92169
\(56\) 3.07233 0.410558
\(57\) 6.06582 0.803438
\(58\) 9.57955 1.25786
\(59\) −5.20780 −0.677997 −0.338999 0.940787i \(-0.610088\pi\)
−0.338999 + 0.940787i \(0.610088\pi\)
\(60\) −2.61262 −0.337288
\(61\) −1.83794 −0.235324 −0.117662 0.993054i \(-0.537540\pi\)
−0.117662 + 0.993054i \(0.537540\pi\)
\(62\) 3.57866 0.454491
\(63\) 2.08652 0.262877
\(64\) 8.25913 1.03239
\(65\) 0 0
\(66\) 4.24827 0.522925
\(67\) −3.05008 −0.372626 −0.186313 0.982490i \(-0.559654\pi\)
−0.186313 + 0.982490i \(0.559654\pi\)
\(68\) −1.43238 −0.173701
\(69\) 4.30289 0.518007
\(70\) −3.94966 −0.472075
\(71\) −14.4391 −1.71361 −0.856805 0.515640i \(-0.827554\pi\)
−0.856805 + 0.515640i \(0.827554\pi\)
\(72\) 6.41050 0.755484
\(73\) −12.9055 −1.51047 −0.755235 0.655454i \(-0.772477\pi\)
−0.755235 + 0.655454i \(0.772477\pi\)
\(74\) −2.03559 −0.236633
\(75\) 7.32471 0.845785
\(76\) −4.87514 −0.559217
\(77\) −4.00483 −0.456393
\(78\) 0 0
\(79\) 0.0931919 0.0104849 0.00524245 0.999986i \(-0.498331\pi\)
0.00524245 + 0.999986i \(0.498331\pi\)
\(80\) −6.66758 −0.745458
\(81\) 1.61315 0.179239
\(82\) −11.4973 −1.26966
\(83\) −3.17863 −0.348900 −0.174450 0.984666i \(-0.555815\pi\)
−0.174450 + 0.984666i \(0.555815\pi\)
\(84\) 0.734167 0.0801042
\(85\) 6.63578 0.719752
\(86\) −11.4671 −1.23653
\(87\) 8.24926 0.884413
\(88\) −12.3042 −1.31163
\(89\) −12.2290 −1.29627 −0.648137 0.761524i \(-0.724451\pi\)
−0.648137 + 0.761524i \(0.724451\pi\)
\(90\) −8.24106 −0.868684
\(91\) 0 0
\(92\) −3.45826 −0.360548
\(93\) 3.08170 0.319558
\(94\) 0.924522 0.0953572
\(95\) 22.5851 2.31718
\(96\) 3.88529 0.396541
\(97\) 13.1916 1.33941 0.669704 0.742628i \(-0.266421\pi\)
0.669704 + 0.742628i \(0.266421\pi\)
\(98\) 1.10989 0.112116
\(99\) −8.35617 −0.839827
\(100\) −5.88692 −0.588692
\(101\) −7.73025 −0.769188 −0.384594 0.923086i \(-0.625659\pi\)
−0.384594 + 0.923086i \(0.625659\pi\)
\(102\) 1.97806 0.195857
\(103\) −0.359154 −0.0353885 −0.0176943 0.999843i \(-0.505633\pi\)
−0.0176943 + 0.999843i \(0.505633\pi\)
\(104\) 0 0
\(105\) −3.40118 −0.331922
\(106\) 7.25324 0.704497
\(107\) −3.99044 −0.385770 −0.192885 0.981221i \(-0.561784\pi\)
−0.192885 + 0.981221i \(0.561784\pi\)
\(108\) 3.73436 0.359339
\(109\) 9.65395 0.924681 0.462341 0.886702i \(-0.347010\pi\)
0.462341 + 0.886702i \(0.347010\pi\)
\(110\) 15.8177 1.50816
\(111\) −1.75292 −0.166379
\(112\) 1.87364 0.177043
\(113\) 2.08248 0.195904 0.0979518 0.995191i \(-0.468771\pi\)
0.0979518 + 0.995191i \(0.468771\pi\)
\(114\) 6.73238 0.630545
\(115\) 16.0211 1.49398
\(116\) −6.62998 −0.615578
\(117\) 0 0
\(118\) −5.78007 −0.532098
\(119\) −1.86471 −0.170937
\(120\) −10.4496 −0.953911
\(121\) 5.03866 0.458060
\(122\) −2.03991 −0.184685
\(123\) −9.90068 −0.892714
\(124\) −2.47678 −0.222422
\(125\) 9.47929 0.847853
\(126\) 2.31581 0.206308
\(127\) 7.18287 0.637376 0.318688 0.947860i \(-0.396758\pi\)
0.318688 + 0.947860i \(0.396758\pi\)
\(128\) 1.03643 0.0916084
\(129\) −9.87472 −0.869421
\(130\) 0 0
\(131\) −1.48085 −0.129383 −0.0646913 0.997905i \(-0.520606\pi\)
−0.0646913 + 0.997905i \(0.520606\pi\)
\(132\) −2.94022 −0.255913
\(133\) −6.34660 −0.550320
\(134\) −3.38524 −0.292440
\(135\) −17.3002 −1.48896
\(136\) −5.72901 −0.491258
\(137\) 4.43864 0.379219 0.189609 0.981860i \(-0.439278\pi\)
0.189609 + 0.981860i \(0.439278\pi\)
\(138\) 4.77572 0.406536
\(139\) 17.3529 1.47186 0.735929 0.677059i \(-0.236746\pi\)
0.735929 + 0.677059i \(0.236746\pi\)
\(140\) 2.73355 0.231027
\(141\) 0.796136 0.0670468
\(142\) −16.0258 −1.34486
\(143\) 0 0
\(144\) 3.90940 0.325783
\(145\) 30.7148 2.55072
\(146\) −14.3236 −1.18543
\(147\) 0.955760 0.0788297
\(148\) 1.40883 0.115805
\(149\) −23.1381 −1.89554 −0.947771 0.318950i \(-0.896670\pi\)
−0.947771 + 0.318950i \(0.896670\pi\)
\(150\) 8.12960 0.663779
\(151\) −10.8846 −0.885778 −0.442889 0.896576i \(-0.646047\pi\)
−0.442889 + 0.896576i \(0.646047\pi\)
\(152\) −19.4989 −1.58157
\(153\) −3.89076 −0.314549
\(154\) −4.44491 −0.358181
\(155\) 11.4742 0.921632
\(156\) 0 0
\(157\) 14.2555 1.13771 0.568855 0.822438i \(-0.307387\pi\)
0.568855 + 0.822438i \(0.307387\pi\)
\(158\) 0.103432 0.00822865
\(159\) 6.24600 0.495340
\(160\) 14.4663 1.14366
\(161\) −4.50206 −0.354812
\(162\) 1.79041 0.140668
\(163\) 2.01430 0.157772 0.0788861 0.996884i \(-0.474864\pi\)
0.0788861 + 0.996884i \(0.474864\pi\)
\(164\) 7.95724 0.621356
\(165\) 13.6212 1.06041
\(166\) −3.52792 −0.273820
\(167\) 18.8978 1.46236 0.731178 0.682187i \(-0.238971\pi\)
0.731178 + 0.682187i \(0.238971\pi\)
\(168\) 2.93641 0.226549
\(169\) 0 0
\(170\) 7.36497 0.564867
\(171\) −13.2423 −1.01267
\(172\) 7.93638 0.605143
\(173\) −12.4146 −0.943868 −0.471934 0.881634i \(-0.656444\pi\)
−0.471934 + 0.881634i \(0.656444\pi\)
\(174\) 9.15575 0.694095
\(175\) −7.66376 −0.579325
\(176\) −7.50363 −0.565607
\(177\) −4.97740 −0.374125
\(178\) −13.5728 −1.01733
\(179\) −9.37393 −0.700641 −0.350320 0.936630i \(-0.613927\pi\)
−0.350320 + 0.936630i \(0.613927\pi\)
\(180\) 5.70362 0.425123
\(181\) 5.83547 0.433747 0.216873 0.976200i \(-0.430414\pi\)
0.216873 + 0.976200i \(0.430414\pi\)
\(182\) 0 0
\(183\) −1.75663 −0.129854
\(184\) −13.8318 −1.01970
\(185\) −6.52670 −0.479852
\(186\) 3.42034 0.250792
\(187\) 7.46784 0.546102
\(188\) −0.639860 −0.0466666
\(189\) 4.86149 0.353622
\(190\) 25.0669 1.81855
\(191\) −16.3051 −1.17980 −0.589899 0.807477i \(-0.700833\pi\)
−0.589899 + 0.807477i \(0.700833\pi\)
\(192\) 7.89375 0.569682
\(193\) −21.2251 −1.52782 −0.763910 0.645323i \(-0.776723\pi\)
−0.763910 + 0.645323i \(0.776723\pi\)
\(194\) 14.6412 1.05118
\(195\) 0 0
\(196\) −0.768150 −0.0548679
\(197\) 16.0798 1.14564 0.572818 0.819683i \(-0.305850\pi\)
0.572818 + 0.819683i \(0.305850\pi\)
\(198\) −9.27441 −0.659103
\(199\) −15.4383 −1.09440 −0.547198 0.837003i \(-0.684305\pi\)
−0.547198 + 0.837003i \(0.684305\pi\)
\(200\) −23.5456 −1.66493
\(201\) −2.91514 −0.205618
\(202\) −8.57970 −0.603666
\(203\) −8.63110 −0.605784
\(204\) −1.36901 −0.0958497
\(205\) −36.8636 −2.57466
\(206\) −0.398621 −0.0277732
\(207\) −9.39365 −0.652904
\(208\) 0 0
\(209\) 25.4170 1.75813
\(210\) −3.77493 −0.260495
\(211\) 3.45377 0.237767 0.118884 0.992908i \(-0.462068\pi\)
0.118884 + 0.992908i \(0.462068\pi\)
\(212\) −5.01995 −0.344772
\(213\) −13.8004 −0.945584
\(214\) −4.42893 −0.302756
\(215\) −36.7669 −2.50748
\(216\) 14.9361 1.01628
\(217\) −3.22435 −0.218883
\(218\) 10.7148 0.725698
\(219\) −12.3345 −0.833490
\(220\) −10.9474 −0.738075
\(221\) 0 0
\(222\) −1.94554 −0.130576
\(223\) −15.6638 −1.04893 −0.524464 0.851433i \(-0.675734\pi\)
−0.524464 + 0.851433i \(0.675734\pi\)
\(224\) −4.06514 −0.271613
\(225\) −15.9906 −1.06604
\(226\) 2.31132 0.153747
\(227\) 8.05474 0.534612 0.267306 0.963612i \(-0.413867\pi\)
0.267306 + 0.963612i \(0.413867\pi\)
\(228\) −4.65947 −0.308581
\(229\) 19.9670 1.31946 0.659730 0.751503i \(-0.270671\pi\)
0.659730 + 0.751503i \(0.270671\pi\)
\(230\) 17.7816 1.17248
\(231\) −3.82766 −0.251841
\(232\) −26.5176 −1.74097
\(233\) 12.1790 0.797871 0.398935 0.916979i \(-0.369380\pi\)
0.398935 + 0.916979i \(0.369380\pi\)
\(234\) 0 0
\(235\) 2.96428 0.193369
\(236\) 4.00037 0.260402
\(237\) 0.0890691 0.00578566
\(238\) −2.06962 −0.134153
\(239\) −15.5704 −1.00717 −0.503584 0.863946i \(-0.667986\pi\)
−0.503584 + 0.863946i \(0.667986\pi\)
\(240\) −6.37261 −0.411350
\(241\) −1.30581 −0.0841148 −0.0420574 0.999115i \(-0.513391\pi\)
−0.0420574 + 0.999115i \(0.513391\pi\)
\(242\) 5.59235 0.359490
\(243\) 16.1263 1.03450
\(244\) 1.41182 0.0903822
\(245\) 3.55862 0.227352
\(246\) −10.9886 −0.700610
\(247\) 0 0
\(248\) −9.90628 −0.629049
\(249\) −3.03801 −0.192526
\(250\) 10.5209 0.665402
\(251\) −20.2852 −1.28039 −0.640196 0.768212i \(-0.721147\pi\)
−0.640196 + 0.768212i \(0.721147\pi\)
\(252\) −1.60276 −0.100965
\(253\) 18.0300 1.13353
\(254\) 7.97217 0.500219
\(255\) 6.34221 0.397165
\(256\) −15.3679 −0.960496
\(257\) 17.3450 1.08195 0.540977 0.841038i \(-0.318055\pi\)
0.540977 + 0.841038i \(0.318055\pi\)
\(258\) −10.9598 −0.682329
\(259\) 1.83406 0.113963
\(260\) 0 0
\(261\) −18.0090 −1.11473
\(262\) −1.64358 −0.101541
\(263\) −10.8891 −0.671448 −0.335724 0.941960i \(-0.608981\pi\)
−0.335724 + 0.941960i \(0.608981\pi\)
\(264\) −11.7598 −0.723768
\(265\) 23.2560 1.42860
\(266\) −7.04401 −0.431896
\(267\) −11.6880 −0.715294
\(268\) 2.34292 0.143117
\(269\) 8.99414 0.548382 0.274191 0.961675i \(-0.411590\pi\)
0.274191 + 0.961675i \(0.411590\pi\)
\(270\) −19.2013 −1.16855
\(271\) −4.94282 −0.300255 −0.150127 0.988667i \(-0.547968\pi\)
−0.150127 + 0.988667i \(0.547968\pi\)
\(272\) −3.49380 −0.211843
\(273\) 0 0
\(274\) 4.92639 0.297614
\(275\) 30.6920 1.85080
\(276\) −3.30526 −0.198954
\(277\) 19.4610 1.16930 0.584650 0.811286i \(-0.301232\pi\)
0.584650 + 0.811286i \(0.301232\pi\)
\(278\) 19.2598 1.15513
\(279\) −6.72768 −0.402776
\(280\) 10.9333 0.653387
\(281\) 20.1468 1.20185 0.600927 0.799304i \(-0.294798\pi\)
0.600927 + 0.799304i \(0.294798\pi\)
\(282\) 0.883621 0.0526189
\(283\) −22.8282 −1.35699 −0.678496 0.734604i \(-0.737368\pi\)
−0.678496 + 0.734604i \(0.737368\pi\)
\(284\) 11.0914 0.658155
\(285\) 21.5859 1.27864
\(286\) 0 0
\(287\) 10.3590 0.611470
\(288\) −8.48200 −0.499807
\(289\) −13.5229 −0.795463
\(290\) 34.0899 2.00183
\(291\) 12.6080 0.739096
\(292\) 9.91334 0.580134
\(293\) −26.5717 −1.55233 −0.776166 0.630528i \(-0.782838\pi\)
−0.776166 + 0.630528i \(0.782838\pi\)
\(294\) 1.06079 0.0618663
\(295\) −18.5326 −1.07901
\(296\) 5.63483 0.327518
\(297\) −19.4695 −1.12973
\(298\) −25.6806 −1.48764
\(299\) 0 0
\(300\) −5.62648 −0.324845
\(301\) 10.3318 0.595515
\(302\) −12.0807 −0.695167
\(303\) −7.38826 −0.424444
\(304\) −11.8913 −0.682011
\(305\) −6.54053 −0.374510
\(306\) −4.31830 −0.246861
\(307\) 7.16833 0.409118 0.204559 0.978854i \(-0.434424\pi\)
0.204559 + 0.978854i \(0.434424\pi\)
\(308\) 3.07631 0.175289
\(309\) −0.343265 −0.0195277
\(310\) 12.7351 0.723305
\(311\) −13.6640 −0.774817 −0.387408 0.921908i \(-0.626630\pi\)
−0.387408 + 0.921908i \(0.626630\pi\)
\(312\) 0 0
\(313\) 10.5821 0.598137 0.299068 0.954232i \(-0.403324\pi\)
0.299068 + 0.954232i \(0.403324\pi\)
\(314\) 15.8219 0.892884
\(315\) 7.42514 0.418359
\(316\) −0.0715854 −0.00402699
\(317\) 3.65041 0.205027 0.102514 0.994732i \(-0.467311\pi\)
0.102514 + 0.994732i \(0.467311\pi\)
\(318\) 6.93236 0.388747
\(319\) 34.5661 1.93533
\(320\) 29.3911 1.64301
\(321\) −3.81390 −0.212871
\(322\) −4.99678 −0.278459
\(323\) 11.8346 0.658492
\(324\) −1.23914 −0.0688411
\(325\) 0 0
\(326\) 2.23565 0.123821
\(327\) 9.22686 0.510247
\(328\) 31.8262 1.75731
\(329\) −0.832988 −0.0459241
\(330\) 15.1180 0.832216
\(331\) −21.7576 −1.19590 −0.597952 0.801532i \(-0.704019\pi\)
−0.597952 + 0.801532i \(0.704019\pi\)
\(332\) 2.44167 0.134004
\(333\) 3.82680 0.209707
\(334\) 20.9744 1.14767
\(335\) −10.8541 −0.593021
\(336\) 1.79075 0.0976936
\(337\) −17.8877 −0.974405 −0.487202 0.873289i \(-0.661983\pi\)
−0.487202 + 0.873289i \(0.661983\pi\)
\(338\) 0 0
\(339\) 1.99036 0.108101
\(340\) −5.09728 −0.276439
\(341\) 12.9130 0.699276
\(342\) −14.6975 −0.794749
\(343\) −1.00000 −0.0539949
\(344\) 31.7428 1.71145
\(345\) 15.3123 0.824388
\(346\) −13.7789 −0.740756
\(347\) 11.2388 0.603331 0.301666 0.953414i \(-0.402457\pi\)
0.301666 + 0.953414i \(0.402457\pi\)
\(348\) −6.33667 −0.339681
\(349\) 24.4183 1.30708 0.653542 0.756891i \(-0.273282\pi\)
0.653542 + 0.756891i \(0.273282\pi\)
\(350\) −8.50590 −0.454660
\(351\) 0 0
\(352\) 16.2802 0.867736
\(353\) 15.9072 0.846655 0.423327 0.905977i \(-0.360862\pi\)
0.423327 + 0.905977i \(0.360862\pi\)
\(354\) −5.52436 −0.293616
\(355\) −51.3834 −2.72715
\(356\) 9.39372 0.497866
\(357\) −1.78221 −0.0943247
\(358\) −10.4040 −0.549869
\(359\) −12.4456 −0.656853 −0.328427 0.944529i \(-0.606518\pi\)
−0.328427 + 0.944529i \(0.606518\pi\)
\(360\) 22.8125 1.20232
\(361\) 21.2793 1.11996
\(362\) 6.47671 0.340408
\(363\) 4.81575 0.252761
\(364\) 0 0
\(365\) −45.9256 −2.40386
\(366\) −1.94966 −0.101910
\(367\) −18.2038 −0.950231 −0.475115 0.879923i \(-0.657594\pi\)
−0.475115 + 0.879923i \(0.657594\pi\)
\(368\) −8.43525 −0.439718
\(369\) 21.6142 1.12519
\(370\) −7.24390 −0.376592
\(371\) −6.53512 −0.339286
\(372\) −2.36721 −0.122734
\(373\) 32.6828 1.69225 0.846126 0.532983i \(-0.178929\pi\)
0.846126 + 0.532983i \(0.178929\pi\)
\(374\) 8.28846 0.428586
\(375\) 9.05992 0.467852
\(376\) −2.55922 −0.131982
\(377\) 0 0
\(378\) 5.39571 0.277525
\(379\) −8.81792 −0.452946 −0.226473 0.974017i \(-0.572720\pi\)
−0.226473 + 0.974017i \(0.572720\pi\)
\(380\) −17.3488 −0.889973
\(381\) 6.86510 0.351710
\(382\) −18.0969 −0.925916
\(383\) 2.54364 0.129974 0.0649870 0.997886i \(-0.479299\pi\)
0.0649870 + 0.997886i \(0.479299\pi\)
\(384\) 0.990579 0.0505502
\(385\) −14.2517 −0.726332
\(386\) −23.5575 −1.19905
\(387\) 21.5575 1.09583
\(388\) −10.1332 −0.514433
\(389\) −0.516707 −0.0261981 −0.0130990 0.999914i \(-0.504170\pi\)
−0.0130990 + 0.999914i \(0.504170\pi\)
\(390\) 0 0
\(391\) 8.39502 0.424555
\(392\) −3.07233 −0.155176
\(393\) −1.41534 −0.0713944
\(394\) 17.8467 0.899104
\(395\) 0.331634 0.0166863
\(396\) 6.41880 0.322557
\(397\) 25.8956 1.29967 0.649833 0.760077i \(-0.274839\pi\)
0.649833 + 0.760077i \(0.274839\pi\)
\(398\) −17.1348 −0.858891
\(399\) −6.06582 −0.303671
\(400\) −14.3591 −0.717957
\(401\) 10.2493 0.511828 0.255914 0.966700i \(-0.417624\pi\)
0.255914 + 0.966700i \(0.417624\pi\)
\(402\) −3.23548 −0.161371
\(403\) 0 0
\(404\) 5.93799 0.295426
\(405\) 5.74057 0.285251
\(406\) −9.57955 −0.475425
\(407\) −7.34508 −0.364082
\(408\) −5.47556 −0.271080
\(409\) −20.5100 −1.01416 −0.507078 0.861900i \(-0.669274\pi\)
−0.507078 + 0.861900i \(0.669274\pi\)
\(410\) −40.9144 −2.02062
\(411\) 4.24228 0.209256
\(412\) 0.275884 0.0135918
\(413\) 5.20780 0.256259
\(414\) −10.4259 −0.512405
\(415\) −11.3115 −0.555261
\(416\) 0 0
\(417\) 16.5852 0.812183
\(418\) 28.2101 1.37980
\(419\) 15.2380 0.744426 0.372213 0.928147i \(-0.378599\pi\)
0.372213 + 0.928147i \(0.378599\pi\)
\(420\) 2.61262 0.127483
\(421\) 26.3846 1.28590 0.642952 0.765906i \(-0.277709\pi\)
0.642952 + 0.765906i \(0.277709\pi\)
\(422\) 3.83329 0.186602
\(423\) −1.73805 −0.0845068
\(424\) −20.0781 −0.975077
\(425\) 14.2907 0.693199
\(426\) −15.3168 −0.742103
\(427\) 1.83794 0.0889442
\(428\) 3.06526 0.148165
\(429\) 0 0
\(430\) −40.8071 −1.96790
\(431\) −37.9100 −1.82606 −0.913031 0.407889i \(-0.866265\pi\)
−0.913031 + 0.407889i \(0.866265\pi\)
\(432\) 9.10871 0.438243
\(433\) 8.51058 0.408993 0.204496 0.978867i \(-0.434444\pi\)
0.204496 + 0.978867i \(0.434444\pi\)
\(434\) −3.57866 −0.171781
\(435\) 29.3559 1.40751
\(436\) −7.41569 −0.355147
\(437\) 28.5728 1.36682
\(438\) −13.6899 −0.654130
\(439\) 18.6101 0.888213 0.444107 0.895974i \(-0.353521\pi\)
0.444107 + 0.895974i \(0.353521\pi\)
\(440\) −43.7859 −2.08741
\(441\) −2.08652 −0.0993582
\(442\) 0 0
\(443\) 26.8582 1.27607 0.638035 0.770007i \(-0.279747\pi\)
0.638035 + 0.770007i \(0.279747\pi\)
\(444\) 1.34650 0.0639022
\(445\) −43.5184 −2.06297
\(446\) −17.3851 −0.823208
\(447\) −22.1144 −1.04598
\(448\) −8.25913 −0.390207
\(449\) −15.5278 −0.732804 −0.366402 0.930457i \(-0.619411\pi\)
−0.366402 + 0.930457i \(0.619411\pi\)
\(450\) −17.7478 −0.836638
\(451\) −41.4859 −1.95349
\(452\) −1.59966 −0.0752418
\(453\) −10.4031 −0.488780
\(454\) 8.93986 0.419568
\(455\) 0 0
\(456\) −18.6362 −0.872722
\(457\) −33.5563 −1.56970 −0.784848 0.619688i \(-0.787259\pi\)
−0.784848 + 0.619688i \(0.787259\pi\)
\(458\) 22.1612 1.03552
\(459\) −9.06527 −0.423130
\(460\) −12.3066 −0.573799
\(461\) 4.34899 0.202553 0.101276 0.994858i \(-0.467707\pi\)
0.101276 + 0.994858i \(0.467707\pi\)
\(462\) −4.24827 −0.197647
\(463\) −23.2754 −1.08170 −0.540850 0.841119i \(-0.681897\pi\)
−0.540850 + 0.841119i \(0.681897\pi\)
\(464\) −16.1716 −0.750748
\(465\) 10.9666 0.508564
\(466\) 13.5173 0.626176
\(467\) 12.2510 0.566909 0.283454 0.958986i \(-0.408520\pi\)
0.283454 + 0.958986i \(0.408520\pi\)
\(468\) 0 0
\(469\) 3.05008 0.140840
\(470\) 3.29002 0.151757
\(471\) 13.6248 0.627797
\(472\) 16.0001 0.736464
\(473\) −41.3771 −1.90252
\(474\) 0.0988566 0.00454063
\(475\) 48.6388 2.23170
\(476\) 1.43238 0.0656529
\(477\) −13.6357 −0.624334
\(478\) −17.2814 −0.790434
\(479\) 0.683706 0.0312393 0.0156197 0.999878i \(-0.495028\pi\)
0.0156197 + 0.999878i \(0.495028\pi\)
\(480\) 13.8263 0.631080
\(481\) 0 0
\(482\) −1.44930 −0.0660140
\(483\) −4.30289 −0.195788
\(484\) −3.87045 −0.175930
\(485\) 46.9440 2.13162
\(486\) 17.8983 0.811885
\(487\) 2.18545 0.0990323 0.0495162 0.998773i \(-0.484232\pi\)
0.0495162 + 0.998773i \(0.484232\pi\)
\(488\) 5.64677 0.255617
\(489\) 1.92519 0.0870601
\(490\) 3.94966 0.178428
\(491\) 8.81707 0.397909 0.198954 0.980009i \(-0.436245\pi\)
0.198954 + 0.980009i \(0.436245\pi\)
\(492\) 7.60521 0.342870
\(493\) 16.0945 0.724859
\(494\) 0 0
\(495\) −29.7364 −1.33655
\(496\) −6.04128 −0.271262
\(497\) 14.4391 0.647684
\(498\) −3.37185 −0.151096
\(499\) −16.3801 −0.733275 −0.366637 0.930364i \(-0.619491\pi\)
−0.366637 + 0.930364i \(0.619491\pi\)
\(500\) −7.28152 −0.325639
\(501\) 18.0618 0.806940
\(502\) −22.5143 −1.00486
\(503\) −11.5765 −0.516168 −0.258084 0.966122i \(-0.583091\pi\)
−0.258084 + 0.966122i \(0.583091\pi\)
\(504\) −6.41050 −0.285546
\(505\) −27.5090 −1.22413
\(506\) 20.0112 0.889608
\(507\) 0 0
\(508\) −5.51752 −0.244800
\(509\) −13.3984 −0.593874 −0.296937 0.954897i \(-0.595965\pi\)
−0.296937 + 0.954897i \(0.595965\pi\)
\(510\) 7.03914 0.311698
\(511\) 12.9055 0.570904
\(512\) −19.1295 −0.845414
\(513\) −30.8540 −1.36224
\(514\) 19.2510 0.849127
\(515\) −1.27809 −0.0563195
\(516\) 7.58527 0.333923
\(517\) 3.33597 0.146716
\(518\) 2.03559 0.0894389
\(519\) −11.8654 −0.520834
\(520\) 0 0
\(521\) 38.0891 1.66871 0.834356 0.551226i \(-0.185840\pi\)
0.834356 + 0.551226i \(0.185840\pi\)
\(522\) −19.9879 −0.874848
\(523\) 1.91936 0.0839276 0.0419638 0.999119i \(-0.486639\pi\)
0.0419638 + 0.999119i \(0.486639\pi\)
\(524\) 1.13752 0.0496927
\(525\) −7.32471 −0.319677
\(526\) −12.0856 −0.526959
\(527\) 6.01247 0.261907
\(528\) −7.17166 −0.312107
\(529\) −2.73147 −0.118760
\(530\) 25.8115 1.12118
\(531\) 10.8662 0.471552
\(532\) 4.87514 0.211364
\(533\) 0 0
\(534\) −12.9724 −0.561369
\(535\) −14.2004 −0.613938
\(536\) 9.37086 0.404760
\(537\) −8.95923 −0.386619
\(538\) 9.98248 0.430375
\(539\) 4.00483 0.172500
\(540\) 13.2892 0.571874
\(541\) 7.49222 0.322116 0.161058 0.986945i \(-0.448509\pi\)
0.161058 + 0.986945i \(0.448509\pi\)
\(542\) −5.48597 −0.235642
\(543\) 5.57731 0.239345
\(544\) 7.58029 0.325002
\(545\) 34.3547 1.47159
\(546\) 0 0
\(547\) 14.7156 0.629192 0.314596 0.949226i \(-0.398131\pi\)
0.314596 + 0.949226i \(0.398131\pi\)
\(548\) −3.40955 −0.145649
\(549\) 3.83491 0.163670
\(550\) 34.0647 1.45252
\(551\) 54.7781 2.33363
\(552\) −13.2199 −0.562677
\(553\) −0.0931919 −0.00396292
\(554\) 21.5995 0.917677
\(555\) −6.23796 −0.264787
\(556\) −13.3297 −0.565304
\(557\) 5.45353 0.231073 0.115537 0.993303i \(-0.463141\pi\)
0.115537 + 0.993303i \(0.463141\pi\)
\(558\) −7.46696 −0.316102
\(559\) 0 0
\(560\) 6.66758 0.281757
\(561\) 7.13746 0.301344
\(562\) 22.3606 0.943226
\(563\) 36.8364 1.55247 0.776235 0.630444i \(-0.217127\pi\)
0.776235 + 0.630444i \(0.217127\pi\)
\(564\) −0.611552 −0.0257510
\(565\) 7.41077 0.311773
\(566\) −25.3367 −1.06498
\(567\) −1.61315 −0.0677458
\(568\) 44.3619 1.86138
\(569\) 15.0251 0.629885 0.314942 0.949111i \(-0.398015\pi\)
0.314942 + 0.949111i \(0.398015\pi\)
\(570\) 23.9580 1.00349
\(571\) 33.2518 1.39155 0.695773 0.718262i \(-0.255062\pi\)
0.695773 + 0.718262i \(0.255062\pi\)
\(572\) 0 0
\(573\) −15.5838 −0.651023
\(574\) 11.4973 0.479887
\(575\) 34.5027 1.43886
\(576\) −17.2329 −0.718036
\(577\) −34.7983 −1.44867 −0.724335 0.689448i \(-0.757853\pi\)
−0.724335 + 0.689448i \(0.757853\pi\)
\(578\) −15.0089 −0.624286
\(579\) −20.2861 −0.843063
\(580\) −23.5936 −0.979670
\(581\) 3.17863 0.131872
\(582\) 13.9935 0.580049
\(583\) 26.1720 1.08393
\(584\) 39.6499 1.64072
\(585\) 0 0
\(586\) −29.4915 −1.21828
\(587\) 19.7267 0.814210 0.407105 0.913381i \(-0.366538\pi\)
0.407105 + 0.913381i \(0.366538\pi\)
\(588\) −0.734167 −0.0302766
\(589\) 20.4636 0.843190
\(590\) −20.5690 −0.846814
\(591\) 15.3684 0.632171
\(592\) 3.43637 0.141234
\(593\) −7.56122 −0.310502 −0.155251 0.987875i \(-0.549619\pi\)
−0.155251 + 0.987875i \(0.549619\pi\)
\(594\) −21.6089 −0.886624
\(595\) −6.63578 −0.272041
\(596\) 17.7735 0.728031
\(597\) −14.7554 −0.603896
\(598\) 0 0
\(599\) 3.48332 0.142325 0.0711623 0.997465i \(-0.477329\pi\)
0.0711623 + 0.997465i \(0.477329\pi\)
\(600\) −22.5040 −0.918720
\(601\) −43.3614 −1.76875 −0.884374 0.466778i \(-0.845415\pi\)
−0.884374 + 0.466778i \(0.845415\pi\)
\(602\) 11.4671 0.467366
\(603\) 6.36406 0.259165
\(604\) 8.36103 0.340206
\(605\) 17.9307 0.728985
\(606\) −8.20013 −0.333108
\(607\) −24.5108 −0.994864 −0.497432 0.867503i \(-0.665724\pi\)
−0.497432 + 0.867503i \(0.665724\pi\)
\(608\) 25.7998 1.04632
\(609\) −8.24926 −0.334277
\(610\) −7.25925 −0.293918
\(611\) 0 0
\(612\) 2.98869 0.120810
\(613\) 0.842003 0.0340082 0.0170041 0.999855i \(-0.494587\pi\)
0.0170041 + 0.999855i \(0.494587\pi\)
\(614\) 7.95604 0.321080
\(615\) −35.2327 −1.42072
\(616\) 12.3042 0.495749
\(617\) −11.6951 −0.470828 −0.235414 0.971895i \(-0.575645\pi\)
−0.235414 + 0.971895i \(0.575645\pi\)
\(618\) −0.380986 −0.0153255
\(619\) −30.4915 −1.22556 −0.612779 0.790254i \(-0.709948\pi\)
−0.612779 + 0.790254i \(0.709948\pi\)
\(620\) −8.81393 −0.353976
\(621\) −21.8867 −0.878284
\(622\) −15.1655 −0.608083
\(623\) 12.2290 0.489945
\(624\) 0 0
\(625\) −4.58563 −0.183425
\(626\) 11.7450 0.469423
\(627\) 24.2926 0.970153
\(628\) −10.9503 −0.436966
\(629\) −3.41998 −0.136363
\(630\) 8.24106 0.328332
\(631\) 19.0797 0.759552 0.379776 0.925078i \(-0.376001\pi\)
0.379776 + 0.925078i \(0.376001\pi\)
\(632\) −0.286317 −0.0113891
\(633\) 3.30097 0.131202
\(634\) 4.05154 0.160907
\(635\) 25.5611 1.01436
\(636\) −4.79787 −0.190248
\(637\) 0 0
\(638\) 38.3644 1.51886
\(639\) 30.1276 1.19183
\(640\) 3.68826 0.145791
\(641\) 26.9166 1.06314 0.531571 0.847014i \(-0.321602\pi\)
0.531571 + 0.847014i \(0.321602\pi\)
\(642\) −4.23300 −0.167063
\(643\) 50.6462 1.99729 0.998646 0.0520237i \(-0.0165671\pi\)
0.998646 + 0.0520237i \(0.0165671\pi\)
\(644\) 3.45826 0.136274
\(645\) −35.1404 −1.38365
\(646\) 13.1350 0.516790
\(647\) −38.2321 −1.50306 −0.751530 0.659699i \(-0.770684\pi\)
−0.751530 + 0.659699i \(0.770684\pi\)
\(648\) −4.95613 −0.194695
\(649\) −20.8563 −0.818683
\(650\) 0 0
\(651\) −3.08170 −0.120781
\(652\) −1.54729 −0.0605964
\(653\) −10.3427 −0.404742 −0.202371 0.979309i \(-0.564865\pi\)
−0.202371 + 0.979309i \(0.564865\pi\)
\(654\) 10.2408 0.400446
\(655\) −5.26979 −0.205908
\(656\) 19.4090 0.757795
\(657\) 26.9275 1.05054
\(658\) −0.924522 −0.0360416
\(659\) 12.8844 0.501904 0.250952 0.968000i \(-0.419256\pi\)
0.250952 + 0.968000i \(0.419256\pi\)
\(660\) −10.4631 −0.407276
\(661\) 19.1895 0.746385 0.373192 0.927754i \(-0.378263\pi\)
0.373192 + 0.927754i \(0.378263\pi\)
\(662\) −24.1484 −0.938555
\(663\) 0 0
\(664\) 9.76582 0.378987
\(665\) −22.5851 −0.875813
\(666\) 4.24731 0.164580
\(667\) 38.8577 1.50458
\(668\) −14.5163 −0.561654
\(669\) −14.9709 −0.578807
\(670\) −12.0468 −0.465408
\(671\) −7.36064 −0.284154
\(672\) −3.88529 −0.149878
\(673\) 11.1856 0.431172 0.215586 0.976485i \(-0.430834\pi\)
0.215586 + 0.976485i \(0.430834\pi\)
\(674\) −19.8533 −0.764721
\(675\) −37.2573 −1.43403
\(676\) 0 0
\(677\) −16.4527 −0.632330 −0.316165 0.948704i \(-0.602395\pi\)
−0.316165 + 0.948704i \(0.602395\pi\)
\(678\) 2.20907 0.0848388
\(679\) −13.1916 −0.506248
\(680\) −20.3873 −0.781819
\(681\) 7.69840 0.295003
\(682\) 14.3319 0.548798
\(683\) −13.5271 −0.517600 −0.258800 0.965931i \(-0.583327\pi\)
−0.258800 + 0.965931i \(0.583327\pi\)
\(684\) 10.1721 0.388940
\(685\) 15.7954 0.603512
\(686\) −1.10989 −0.0423757
\(687\) 19.0837 0.728089
\(688\) 19.3581 0.738021
\(689\) 0 0
\(690\) 16.9950 0.646987
\(691\) −20.5171 −0.780507 −0.390254 0.920707i \(-0.627613\pi\)
−0.390254 + 0.920707i \(0.627613\pi\)
\(692\) 9.53632 0.362516
\(693\) 8.35617 0.317425
\(694\) 12.4738 0.473500
\(695\) 61.7525 2.34241
\(696\) −25.3445 −0.960680
\(697\) −19.3164 −0.731662
\(698\) 27.1016 1.02581
\(699\) 11.6402 0.440272
\(700\) 5.88692 0.222505
\(701\) 46.6380 1.76149 0.880747 0.473587i \(-0.157041\pi\)
0.880747 + 0.473587i \(0.157041\pi\)
\(702\) 0 0
\(703\) −11.6400 −0.439011
\(704\) 33.0764 1.24661
\(705\) 2.83314 0.106702
\(706\) 17.6552 0.664462
\(707\) 7.73025 0.290726
\(708\) 3.82339 0.143692
\(709\) −42.2314 −1.58603 −0.793017 0.609199i \(-0.791491\pi\)
−0.793017 + 0.609199i \(0.791491\pi\)
\(710\) −57.0297 −2.14029
\(711\) −0.194447 −0.00729233
\(712\) 37.5716 1.40806
\(713\) 14.5162 0.543636
\(714\) −1.97806 −0.0740269
\(715\) 0 0
\(716\) 7.20059 0.269099
\(717\) −14.8816 −0.555764
\(718\) −13.8132 −0.515504
\(719\) 37.9404 1.41494 0.707469 0.706744i \(-0.249837\pi\)
0.707469 + 0.706744i \(0.249837\pi\)
\(720\) 13.9121 0.518472
\(721\) 0.359154 0.0133756
\(722\) 23.6176 0.878957
\(723\) −1.24804 −0.0464152
\(724\) −4.48252 −0.166591
\(725\) 66.1466 2.45662
\(726\) 5.34494 0.198369
\(727\) −16.5215 −0.612748 −0.306374 0.951911i \(-0.599116\pi\)
−0.306374 + 0.951911i \(0.599116\pi\)
\(728\) 0 0
\(729\) 10.5734 0.391607
\(730\) −50.9722 −1.88657
\(731\) −19.2658 −0.712571
\(732\) 1.34936 0.0498737
\(733\) −17.6511 −0.651957 −0.325978 0.945377i \(-0.605694\pi\)
−0.325978 + 0.945377i \(0.605694\pi\)
\(734\) −20.2042 −0.745750
\(735\) 3.40118 0.125455
\(736\) 18.3015 0.674601
\(737\) −12.2150 −0.449947
\(738\) 23.9893 0.883060
\(739\) 0.370373 0.0136244 0.00681220 0.999977i \(-0.497832\pi\)
0.00681220 + 0.999977i \(0.497832\pi\)
\(740\) 5.01349 0.184299
\(741\) 0 0
\(742\) −7.25324 −0.266275
\(743\) −17.5579 −0.644137 −0.322068 0.946716i \(-0.604378\pi\)
−0.322068 + 0.946716i \(0.604378\pi\)
\(744\) −9.46802 −0.347115
\(745\) −82.3395 −3.01668
\(746\) 36.2742 1.32809
\(747\) 6.63229 0.242663
\(748\) −5.73642 −0.209744
\(749\) 3.99044 0.145807
\(750\) 10.0555 0.367175
\(751\) −15.0706 −0.549933 −0.274967 0.961454i \(-0.588667\pi\)
−0.274967 + 0.961454i \(0.588667\pi\)
\(752\) −1.56072 −0.0569137
\(753\) −19.3878 −0.706530
\(754\) 0 0
\(755\) −38.7342 −1.40968
\(756\) −3.73436 −0.135817
\(757\) 39.8030 1.44666 0.723332 0.690500i \(-0.242609\pi\)
0.723332 + 0.690500i \(0.242609\pi\)
\(758\) −9.78689 −0.355476
\(759\) 17.2323 0.625494
\(760\) −69.3890 −2.51700
\(761\) −20.7066 −0.750613 −0.375306 0.926901i \(-0.622462\pi\)
−0.375306 + 0.926901i \(0.622462\pi\)
\(762\) 7.61948 0.276025
\(763\) −9.65395 −0.349497
\(764\) 12.5248 0.453131
\(765\) −13.8457 −0.500593
\(766\) 2.82315 0.102005
\(767\) 0 0
\(768\) −14.6881 −0.530010
\(769\) 13.2638 0.478306 0.239153 0.970982i \(-0.423130\pi\)
0.239153 + 0.970982i \(0.423130\pi\)
\(770\) −15.8177 −0.570031
\(771\) 16.5777 0.597031
\(772\) 16.3041 0.586798
\(773\) 11.6627 0.419478 0.209739 0.977757i \(-0.432739\pi\)
0.209739 + 0.977757i \(0.432739\pi\)
\(774\) 23.9264 0.860018
\(775\) 24.7106 0.887632
\(776\) −40.5291 −1.45491
\(777\) 1.75292 0.0628855
\(778\) −0.573486 −0.0205605
\(779\) −65.7442 −2.35553
\(780\) 0 0
\(781\) −57.8263 −2.06919
\(782\) 9.31753 0.333194
\(783\) −41.9600 −1.49953
\(784\) −1.87364 −0.0669159
\(785\) 50.7297 1.81062
\(786\) −1.57087 −0.0560310
\(787\) 5.00642 0.178460 0.0892298 0.996011i \(-0.471559\pi\)
0.0892298 + 0.996011i \(0.471559\pi\)
\(788\) −12.3517 −0.440010
\(789\) −10.4073 −0.370511
\(790\) 0.368077 0.0130956
\(791\) −2.08248 −0.0740446
\(792\) 25.6730 0.912249
\(793\) 0 0
\(794\) 28.7412 1.01999
\(795\) 22.2271 0.788315
\(796\) 11.8590 0.420330
\(797\) 9.95893 0.352763 0.176382 0.984322i \(-0.443561\pi\)
0.176382 + 0.984322i \(0.443561\pi\)
\(798\) −6.73238 −0.238324
\(799\) 1.55328 0.0549511
\(800\) 31.1542 1.10147
\(801\) 25.5161 0.901568
\(802\) 11.3756 0.401687
\(803\) −51.6842 −1.82389
\(804\) 2.23927 0.0789729
\(805\) −16.0211 −0.564670
\(806\) 0 0
\(807\) 8.59623 0.302602
\(808\) 23.7499 0.835519
\(809\) 12.0200 0.422601 0.211300 0.977421i \(-0.432230\pi\)
0.211300 + 0.977421i \(0.432230\pi\)
\(810\) 6.37139 0.223868
\(811\) −2.91856 −0.102484 −0.0512422 0.998686i \(-0.516318\pi\)
−0.0512422 + 0.998686i \(0.516318\pi\)
\(812\) 6.62998 0.232667
\(813\) −4.72414 −0.165683
\(814\) −8.15221 −0.285735
\(815\) 7.16813 0.251089
\(816\) −3.33923 −0.116897
\(817\) −65.5718 −2.29407
\(818\) −22.7638 −0.795919
\(819\) 0 0
\(820\) 28.3168 0.988865
\(821\) −12.9996 −0.453689 −0.226844 0.973931i \(-0.572841\pi\)
−0.226844 + 0.973931i \(0.572841\pi\)
\(822\) 4.70845 0.164226
\(823\) −39.6283 −1.38136 −0.690678 0.723163i \(-0.742688\pi\)
−0.690678 + 0.723163i \(0.742688\pi\)
\(824\) 1.10344 0.0384402
\(825\) 29.3342 1.02129
\(826\) 5.78007 0.201114
\(827\) 31.5319 1.09647 0.548237 0.836323i \(-0.315299\pi\)
0.548237 + 0.836323i \(0.315299\pi\)
\(828\) 7.21574 0.250764
\(829\) 36.2352 1.25850 0.629251 0.777202i \(-0.283362\pi\)
0.629251 + 0.777202i \(0.283362\pi\)
\(830\) −12.5545 −0.435774
\(831\) 18.6001 0.645229
\(832\) 0 0
\(833\) 1.86471 0.0646083
\(834\) 18.4078 0.637408
\(835\) 67.2500 2.32728
\(836\) −19.5241 −0.675256
\(837\) −15.6752 −0.541813
\(838\) 16.9125 0.584232
\(839\) 12.0623 0.416438 0.208219 0.978082i \(-0.433233\pi\)
0.208219 + 0.978082i \(0.433233\pi\)
\(840\) 10.4496 0.360545
\(841\) 45.4959 1.56882
\(842\) 29.2839 1.00919
\(843\) 19.2555 0.663193
\(844\) −2.65301 −0.0913204
\(845\) 0 0
\(846\) −1.92904 −0.0663217
\(847\) −5.03866 −0.173130
\(848\) −12.2445 −0.420477
\(849\) −21.8182 −0.748800
\(850\) 15.8610 0.544029
\(851\) −8.25702 −0.283047
\(852\) 10.6007 0.363176
\(853\) 11.7255 0.401475 0.200738 0.979645i \(-0.435666\pi\)
0.200738 + 0.979645i \(0.435666\pi\)
\(854\) 2.03991 0.0698042
\(855\) −47.1244 −1.61162
\(856\) 12.2600 0.419037
\(857\) 28.0128 0.956900 0.478450 0.878115i \(-0.341199\pi\)
0.478450 + 0.878115i \(0.341199\pi\)
\(858\) 0 0
\(859\) 5.70001 0.194482 0.0972408 0.995261i \(-0.468998\pi\)
0.0972408 + 0.995261i \(0.468998\pi\)
\(860\) 28.2425 0.963063
\(861\) 9.90068 0.337414
\(862\) −42.0759 −1.43311
\(863\) −41.0325 −1.39676 −0.698381 0.715726i \(-0.746096\pi\)
−0.698381 + 0.715726i \(0.746096\pi\)
\(864\) −19.7626 −0.672339
\(865\) −44.1790 −1.50213
\(866\) 9.44579 0.320981
\(867\) −12.9246 −0.438943
\(868\) 2.47678 0.0840676
\(869\) 0.373218 0.0126605
\(870\) 32.5818 1.10463
\(871\) 0 0
\(872\) −29.6602 −1.00442
\(873\) −27.5246 −0.931568
\(874\) 31.7125 1.07269
\(875\) −9.47929 −0.320458
\(876\) 9.47477 0.320123
\(877\) −10.7401 −0.362668 −0.181334 0.983422i \(-0.558042\pi\)
−0.181334 + 0.983422i \(0.558042\pi\)
\(878\) 20.6551 0.697077
\(879\) −25.3961 −0.856590
\(880\) −26.7025 −0.900142
\(881\) 37.5517 1.26515 0.632574 0.774500i \(-0.281998\pi\)
0.632574 + 0.774500i \(0.281998\pi\)
\(882\) −2.31581 −0.0779772
\(883\) 47.1134 1.58549 0.792745 0.609553i \(-0.208651\pi\)
0.792745 + 0.609553i \(0.208651\pi\)
\(884\) 0 0
\(885\) −17.7127 −0.595405
\(886\) 29.8095 1.00147
\(887\) 5.06090 0.169928 0.0849642 0.996384i \(-0.472922\pi\)
0.0849642 + 0.996384i \(0.472922\pi\)
\(888\) 5.38555 0.180727
\(889\) −7.18287 −0.240906
\(890\) −48.3005 −1.61904
\(891\) 6.46038 0.216431
\(892\) 12.0322 0.402867
\(893\) 5.28664 0.176911
\(894\) −24.5445 −0.820891
\(895\) −33.3582 −1.11504
\(896\) −1.03643 −0.0346247
\(897\) 0 0
\(898\) −17.2342 −0.575111
\(899\) 27.8297 0.928171
\(900\) 12.2832 0.409440
\(901\) 12.1861 0.405977
\(902\) −46.0447 −1.53312
\(903\) 9.87472 0.328610
\(904\) −6.39809 −0.212797
\(905\) 20.7662 0.690292
\(906\) −11.5463 −0.383599
\(907\) −48.6215 −1.61445 −0.807226 0.590243i \(-0.799032\pi\)
−0.807226 + 0.590243i \(0.799032\pi\)
\(908\) −6.18726 −0.205331
\(909\) 16.1293 0.534976
\(910\) 0 0
\(911\) 1.76756 0.0585620 0.0292810 0.999571i \(-0.490678\pi\)
0.0292810 + 0.999571i \(0.490678\pi\)
\(912\) −11.3652 −0.376339
\(913\) −12.7299 −0.421297
\(914\) −37.2437 −1.23191
\(915\) −6.25118 −0.206658
\(916\) −15.3377 −0.506772
\(917\) 1.48085 0.0489021
\(918\) −10.0614 −0.332076
\(919\) 27.4163 0.904380 0.452190 0.891922i \(-0.350643\pi\)
0.452190 + 0.891922i \(0.350643\pi\)
\(920\) −49.2222 −1.62281
\(921\) 6.85120 0.225755
\(922\) 4.82689 0.158965
\(923\) 0 0
\(924\) 2.94022 0.0967260
\(925\) −14.0557 −0.462150
\(926\) −25.8331 −0.848928
\(927\) 0.749383 0.0246130
\(928\) 35.0866 1.15177
\(929\) 31.0105 1.01742 0.508710 0.860938i \(-0.330122\pi\)
0.508710 + 0.860938i \(0.330122\pi\)
\(930\) 12.1717 0.399125
\(931\) 6.34660 0.208001
\(932\) −9.35528 −0.306442
\(933\) −13.0595 −0.427550
\(934\) 13.5972 0.444915
\(935\) 26.5752 0.869101
\(936\) 0 0
\(937\) −13.5463 −0.442537 −0.221269 0.975213i \(-0.571020\pi\)
−0.221269 + 0.975213i \(0.571020\pi\)
\(938\) 3.38524 0.110532
\(939\) 10.1140 0.330057
\(940\) −2.27702 −0.0742681
\(941\) 27.1579 0.885321 0.442661 0.896689i \(-0.354035\pi\)
0.442661 + 0.896689i \(0.354035\pi\)
\(942\) 15.1220 0.492701
\(943\) −46.6367 −1.51870
\(944\) 9.75756 0.317581
\(945\) 17.3002 0.562775
\(946\) −45.9239 −1.49312
\(947\) −30.1736 −0.980511 −0.490256 0.871579i \(-0.663097\pi\)
−0.490256 + 0.871579i \(0.663097\pi\)
\(948\) −0.0684184 −0.00222213
\(949\) 0 0
\(950\) 53.9835 1.75146
\(951\) 3.48892 0.113136
\(952\) 5.72901 0.185678
\(953\) 13.0194 0.421738 0.210869 0.977514i \(-0.432371\pi\)
0.210869 + 0.977514i \(0.432371\pi\)
\(954\) −15.1341 −0.489983
\(955\) −58.0237 −1.87760
\(956\) 11.9604 0.386828
\(957\) 33.0369 1.06793
\(958\) 0.758836 0.0245169
\(959\) −4.43864 −0.143331
\(960\) 28.0908 0.906627
\(961\) −20.6036 −0.664632
\(962\) 0 0
\(963\) 8.32614 0.268306
\(964\) 1.00306 0.0323064
\(965\) −75.5322 −2.43147
\(966\) −4.77572 −0.153656
\(967\) −49.5415 −1.59315 −0.796573 0.604542i \(-0.793356\pi\)
−0.796573 + 0.604542i \(0.793356\pi\)
\(968\) −15.4805 −0.497561
\(969\) 11.3110 0.363361
\(970\) 52.1025 1.67291
\(971\) −34.3718 −1.10304 −0.551522 0.834160i \(-0.685953\pi\)
−0.551522 + 0.834160i \(0.685953\pi\)
\(972\) −12.3874 −0.397326
\(973\) −17.3529 −0.556310
\(974\) 2.42561 0.0777214
\(975\) 0 0
\(976\) 3.44365 0.110228
\(977\) −10.8964 −0.348607 −0.174303 0.984692i \(-0.555767\pi\)
−0.174303 + 0.984692i \(0.555767\pi\)
\(978\) 2.13674 0.0683255
\(979\) −48.9751 −1.56525
\(980\) −2.73355 −0.0873202
\(981\) −20.1432 −0.643123
\(982\) 9.78595 0.312282
\(983\) 17.8803 0.570293 0.285146 0.958484i \(-0.407958\pi\)
0.285146 + 0.958484i \(0.407958\pi\)
\(984\) 30.4182 0.969697
\(985\) 57.2217 1.82323
\(986\) 17.8631 0.568876
\(987\) −0.796136 −0.0253413
\(988\) 0 0
\(989\) −46.5144 −1.47907
\(990\) −33.0041 −1.04894
\(991\) −43.2925 −1.37523 −0.687616 0.726074i \(-0.741343\pi\)
−0.687616 + 0.726074i \(0.741343\pi\)
\(992\) 13.1074 0.416161
\(993\) −20.7950 −0.659909
\(994\) 16.0258 0.508308
\(995\) −54.9392 −1.74169
\(996\) 2.33365 0.0739444
\(997\) 25.7004 0.813940 0.406970 0.913442i \(-0.366585\pi\)
0.406970 + 0.913442i \(0.366585\pi\)
\(998\) −18.1801 −0.575480
\(999\) 8.91625 0.282098
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.a.q.1.9 12
7.6 odd 2 8281.2.a.cn.1.9 12
13.5 odd 4 1183.2.c.j.337.8 24
13.8 odd 4 1183.2.c.j.337.17 24
13.12 even 2 1183.2.a.r.1.4 yes 12
91.90 odd 2 8281.2.a.cq.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1183.2.a.q.1.9 12 1.1 even 1 trivial
1183.2.a.r.1.4 yes 12 13.12 even 2
1183.2.c.j.337.8 24 13.5 odd 4
1183.2.c.j.337.17 24 13.8 odd 4
8281.2.a.cn.1.9 12 7.6 odd 2
8281.2.a.cq.1.4 12 91.90 odd 2