Properties

Label 6422.2.a.bm.1.13
Level $6422$
Weight $2$
Character 6422.1
Self dual yes
Analytic conductor $51.280$
Analytic rank $0$
Dimension $14$
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] = [6422,2,Mod(1,6422)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6422, 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("6422.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6422 = 2 \cdot 13^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6422.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: \(51.2799281781\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} - 22 x^{12} + 98 x^{11} + 164 x^{10} - 912 x^{9} - 374 x^{8} + 3996 x^{7} - 817 x^{6} - 8194 x^{5} + 4418 x^{4} + 6430 x^{3} - 4327 x^{2} - 922 x + 358 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 494)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(2.81805\) of defining polynomial
Character \(\chi\) \(=\) 6422.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +2.81805 q^{3} +1.00000 q^{4} +3.87767 q^{5} -2.81805 q^{6} -2.87057 q^{7} -1.00000 q^{8} +4.94139 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +2.81805 q^{3} +1.00000 q^{4} +3.87767 q^{5} -2.81805 q^{6} -2.87057 q^{7} -1.00000 q^{8} +4.94139 q^{9} -3.87767 q^{10} -3.63541 q^{11} +2.81805 q^{12} +2.87057 q^{14} +10.9275 q^{15} +1.00000 q^{16} +6.24396 q^{17} -4.94139 q^{18} -1.00000 q^{19} +3.87767 q^{20} -8.08939 q^{21} +3.63541 q^{22} -8.25037 q^{23} -2.81805 q^{24} +10.0364 q^{25} +5.47093 q^{27} -2.87057 q^{28} +4.57590 q^{29} -10.9275 q^{30} +3.50294 q^{31} -1.00000 q^{32} -10.2448 q^{33} -6.24396 q^{34} -11.1311 q^{35} +4.94139 q^{36} -0.0117786 q^{37} +1.00000 q^{38} -3.87767 q^{40} +7.96875 q^{41} +8.08939 q^{42} +11.3957 q^{43} -3.63541 q^{44} +19.1611 q^{45} +8.25037 q^{46} +1.96020 q^{47} +2.81805 q^{48} +1.24015 q^{49} -10.0364 q^{50} +17.5958 q^{51} +8.92897 q^{53} -5.47093 q^{54} -14.0969 q^{55} +2.87057 q^{56} -2.81805 q^{57} -4.57590 q^{58} +5.83570 q^{59} +10.9275 q^{60} -0.214830 q^{61} -3.50294 q^{62} -14.1846 q^{63} +1.00000 q^{64} +10.2448 q^{66} -8.68818 q^{67} +6.24396 q^{68} -23.2499 q^{69} +11.1311 q^{70} +6.62648 q^{71} -4.94139 q^{72} +13.4954 q^{73} +0.0117786 q^{74} +28.2829 q^{75} -1.00000 q^{76} +10.4357 q^{77} +6.33212 q^{79} +3.87767 q^{80} +0.593170 q^{81} -7.96875 q^{82} -7.39527 q^{83} -8.08939 q^{84} +24.2120 q^{85} -11.3957 q^{86} +12.8951 q^{87} +3.63541 q^{88} -7.06857 q^{89} -19.1611 q^{90} -8.25037 q^{92} +9.87145 q^{93} -1.96020 q^{94} -3.87767 q^{95} -2.81805 q^{96} -6.09538 q^{97} -1.24015 q^{98} -17.9640 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} + 4 q^{3} + 14 q^{4} + 2 q^{5} - 4 q^{6} - 2 q^{7} - 14 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} + 4 q^{3} + 14 q^{4} + 2 q^{5} - 4 q^{6} - 2 q^{7} - 14 q^{8} + 18 q^{9} - 2 q^{10} - 10 q^{11} + 4 q^{12} + 2 q^{14} + 4 q^{15} + 14 q^{16} + 2 q^{17} - 18 q^{18} - 14 q^{19} + 2 q^{20} + 18 q^{21} + 10 q^{22} + 12 q^{23} - 4 q^{24} + 44 q^{25} + 10 q^{27} - 2 q^{28} + 4 q^{29} - 4 q^{30} - 4 q^{31} - 14 q^{32} - 12 q^{33} - 2 q^{34} + 14 q^{35} + 18 q^{36} - 18 q^{37} + 14 q^{38} - 2 q^{40} - 6 q^{41} - 18 q^{42} + 28 q^{43} - 10 q^{44} - 8 q^{45} - 12 q^{46} + 20 q^{47} + 4 q^{48} + 28 q^{49} - 44 q^{50} + 10 q^{51} + 12 q^{53} - 10 q^{54} - 2 q^{55} + 2 q^{56} - 4 q^{57} - 4 q^{58} - 16 q^{59} + 4 q^{60} + 30 q^{61} + 4 q^{62} - 28 q^{63} + 14 q^{64} + 12 q^{66} - 2 q^{67} + 2 q^{68} - 42 q^{69} - 14 q^{70} - 44 q^{71} - 18 q^{72} + 36 q^{73} + 18 q^{74} + 46 q^{75} - 14 q^{76} + 68 q^{77} + 34 q^{79} + 2 q^{80} - 6 q^{81} + 6 q^{82} - 2 q^{83} + 18 q^{84} + 30 q^{85} - 28 q^{86} + 52 q^{87} + 10 q^{88} - 42 q^{89} + 8 q^{90} + 12 q^{92} + 12 q^{93} - 20 q^{94} - 2 q^{95} - 4 q^{96} - 40 q^{97} - 28 q^{98} - 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 2.81805 1.62700 0.813500 0.581565i \(-0.197559\pi\)
0.813500 + 0.581565i \(0.197559\pi\)
\(4\) 1.00000 0.500000
\(5\) 3.87767 1.73415 0.867074 0.498179i \(-0.165998\pi\)
0.867074 + 0.498179i \(0.165998\pi\)
\(6\) −2.81805 −1.15046
\(7\) −2.87057 −1.08497 −0.542486 0.840065i \(-0.682517\pi\)
−0.542486 + 0.840065i \(0.682517\pi\)
\(8\) −1.00000 −0.353553
\(9\) 4.94139 1.64713
\(10\) −3.87767 −1.22623
\(11\) −3.63541 −1.09612 −0.548059 0.836440i \(-0.684633\pi\)
−0.548059 + 0.836440i \(0.684633\pi\)
\(12\) 2.81805 0.813500
\(13\) 0 0
\(14\) 2.87057 0.767191
\(15\) 10.9275 2.82146
\(16\) 1.00000 0.250000
\(17\) 6.24396 1.51438 0.757191 0.653193i \(-0.226571\pi\)
0.757191 + 0.653193i \(0.226571\pi\)
\(18\) −4.94139 −1.16470
\(19\) −1.00000 −0.229416
\(20\) 3.87767 0.867074
\(21\) −8.08939 −1.76525
\(22\) 3.63541 0.775073
\(23\) −8.25037 −1.72032 −0.860161 0.510023i \(-0.829637\pi\)
−0.860161 + 0.510023i \(0.829637\pi\)
\(24\) −2.81805 −0.575232
\(25\) 10.0364 2.00727
\(26\) 0 0
\(27\) 5.47093 1.05288
\(28\) −2.87057 −0.542486
\(29\) 4.57590 0.849724 0.424862 0.905258i \(-0.360323\pi\)
0.424862 + 0.905258i \(0.360323\pi\)
\(30\) −10.9275 −1.99507
\(31\) 3.50294 0.629146 0.314573 0.949233i \(-0.398139\pi\)
0.314573 + 0.949233i \(0.398139\pi\)
\(32\) −1.00000 −0.176777
\(33\) −10.2448 −1.78338
\(34\) −6.24396 −1.07083
\(35\) −11.1311 −1.88150
\(36\) 4.94139 0.823565
\(37\) −0.0117786 −0.00193638 −0.000968192 1.00000i \(-0.500308\pi\)
−0.000968192 1.00000i \(0.500308\pi\)
\(38\) 1.00000 0.162221
\(39\) 0 0
\(40\) −3.87767 −0.613114
\(41\) 7.96875 1.24451 0.622255 0.782815i \(-0.286217\pi\)
0.622255 + 0.782815i \(0.286217\pi\)
\(42\) 8.08939 1.24822
\(43\) 11.3957 1.73783 0.868915 0.494962i \(-0.164818\pi\)
0.868915 + 0.494962i \(0.164818\pi\)
\(44\) −3.63541 −0.548059
\(45\) 19.1611 2.85637
\(46\) 8.25037 1.21645
\(47\) 1.96020 0.285924 0.142962 0.989728i \(-0.454337\pi\)
0.142962 + 0.989728i \(0.454337\pi\)
\(48\) 2.81805 0.406750
\(49\) 1.24015 0.177164
\(50\) −10.0364 −1.41936
\(51\) 17.5958 2.46390
\(52\) 0 0
\(53\) 8.92897 1.22649 0.613244 0.789893i \(-0.289864\pi\)
0.613244 + 0.789893i \(0.289864\pi\)
\(54\) −5.47093 −0.744499
\(55\) −14.0969 −1.90083
\(56\) 2.87057 0.383596
\(57\) −2.81805 −0.373259
\(58\) −4.57590 −0.600845
\(59\) 5.83570 0.759743 0.379872 0.925039i \(-0.375968\pi\)
0.379872 + 0.925039i \(0.375968\pi\)
\(60\) 10.9275 1.41073
\(61\) −0.214830 −0.0275062 −0.0137531 0.999905i \(-0.504378\pi\)
−0.0137531 + 0.999905i \(0.504378\pi\)
\(62\) −3.50294 −0.444874
\(63\) −14.1846 −1.78709
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 10.2448 1.26104
\(67\) −8.68818 −1.06143 −0.530715 0.847550i \(-0.678077\pi\)
−0.530715 + 0.847550i \(0.678077\pi\)
\(68\) 6.24396 0.757191
\(69\) −23.2499 −2.79896
\(70\) 11.1311 1.33042
\(71\) 6.62648 0.786419 0.393209 0.919449i \(-0.371365\pi\)
0.393209 + 0.919449i \(0.371365\pi\)
\(72\) −4.94139 −0.582348
\(73\) 13.4954 1.57951 0.789756 0.613421i \(-0.210207\pi\)
0.789756 + 0.613421i \(0.210207\pi\)
\(74\) 0.0117786 0.00136923
\(75\) 28.2829 3.26583
\(76\) −1.00000 −0.114708
\(77\) 10.4357 1.18926
\(78\) 0 0
\(79\) 6.33212 0.712419 0.356209 0.934406i \(-0.384069\pi\)
0.356209 + 0.934406i \(0.384069\pi\)
\(80\) 3.87767 0.433537
\(81\) 0.593170 0.0659078
\(82\) −7.96875 −0.880002
\(83\) −7.39527 −0.811736 −0.405868 0.913932i \(-0.633031\pi\)
−0.405868 + 0.913932i \(0.633031\pi\)
\(84\) −8.08939 −0.882625
\(85\) 24.2120 2.62616
\(86\) −11.3957 −1.22883
\(87\) 12.8951 1.38250
\(88\) 3.63541 0.387536
\(89\) −7.06857 −0.749267 −0.374633 0.927173i \(-0.622231\pi\)
−0.374633 + 0.927173i \(0.622231\pi\)
\(90\) −19.1611 −2.01976
\(91\) 0 0
\(92\) −8.25037 −0.860161
\(93\) 9.87145 1.02362
\(94\) −1.96020 −0.202179
\(95\) −3.87767 −0.397841
\(96\) −2.81805 −0.287616
\(97\) −6.09538 −0.618892 −0.309446 0.950917i \(-0.600144\pi\)
−0.309446 + 0.950917i \(0.600144\pi\)
\(98\) −1.24015 −0.125274
\(99\) −17.9640 −1.80545
\(100\) 10.0364 1.00364
\(101\) −0.590697 −0.0587765 −0.0293883 0.999568i \(-0.509356\pi\)
−0.0293883 + 0.999568i \(0.509356\pi\)
\(102\) −17.5958 −1.74224
\(103\) 0.525578 0.0517868 0.0258934 0.999665i \(-0.491757\pi\)
0.0258934 + 0.999665i \(0.491757\pi\)
\(104\) 0 0
\(105\) −31.3680 −3.06121
\(106\) −8.92897 −0.867259
\(107\) −8.94396 −0.864645 −0.432322 0.901719i \(-0.642306\pi\)
−0.432322 + 0.901719i \(0.642306\pi\)
\(108\) 5.47093 0.526441
\(109\) 0.271028 0.0259598 0.0129799 0.999916i \(-0.495868\pi\)
0.0129799 + 0.999916i \(0.495868\pi\)
\(110\) 14.0969 1.34409
\(111\) −0.0331926 −0.00315050
\(112\) −2.87057 −0.271243
\(113\) 0.112771 0.0106086 0.00530430 0.999986i \(-0.498312\pi\)
0.00530430 + 0.999986i \(0.498312\pi\)
\(114\) 2.81805 0.263934
\(115\) −31.9923 −2.98329
\(116\) 4.57590 0.424862
\(117\) 0 0
\(118\) −5.83570 −0.537219
\(119\) −17.9237 −1.64306
\(120\) −10.9275 −0.997537
\(121\) 2.21623 0.201475
\(122\) 0.214830 0.0194498
\(123\) 22.4563 2.02482
\(124\) 3.50294 0.314573
\(125\) 19.5294 1.74676
\(126\) 14.1846 1.26366
\(127\) 13.4369 1.19233 0.596165 0.802862i \(-0.296691\pi\)
0.596165 + 0.802862i \(0.296691\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 32.1137 2.82745
\(130\) 0 0
\(131\) 7.52443 0.657412 0.328706 0.944432i \(-0.393387\pi\)
0.328706 + 0.944432i \(0.393387\pi\)
\(132\) −10.2448 −0.891692
\(133\) 2.87057 0.248910
\(134\) 8.68818 0.750545
\(135\) 21.2145 1.82585
\(136\) −6.24396 −0.535415
\(137\) 0.458171 0.0391442 0.0195721 0.999808i \(-0.493770\pi\)
0.0195721 + 0.999808i \(0.493770\pi\)
\(138\) 23.2499 1.97917
\(139\) 11.9096 1.01016 0.505078 0.863074i \(-0.331464\pi\)
0.505078 + 0.863074i \(0.331464\pi\)
\(140\) −11.1311 −0.940751
\(141\) 5.52393 0.465199
\(142\) −6.62648 −0.556082
\(143\) 0 0
\(144\) 4.94139 0.411783
\(145\) 17.7439 1.47355
\(146\) −13.4954 −1.11688
\(147\) 3.49480 0.288246
\(148\) −0.0117786 −0.000968192 0
\(149\) −6.67440 −0.546788 −0.273394 0.961902i \(-0.588146\pi\)
−0.273394 + 0.961902i \(0.588146\pi\)
\(150\) −28.2829 −2.30929
\(151\) −5.22614 −0.425298 −0.212649 0.977129i \(-0.568209\pi\)
−0.212649 + 0.977129i \(0.568209\pi\)
\(152\) 1.00000 0.0811107
\(153\) 30.8538 2.49439
\(154\) −10.4357 −0.840932
\(155\) 13.5833 1.09103
\(156\) 0 0
\(157\) −8.58330 −0.685022 −0.342511 0.939514i \(-0.611277\pi\)
−0.342511 + 0.939514i \(0.611277\pi\)
\(158\) −6.33212 −0.503756
\(159\) 25.1623 1.99550
\(160\) −3.87767 −0.306557
\(161\) 23.6832 1.86650
\(162\) −0.593170 −0.0466038
\(163\) −22.1972 −1.73862 −0.869308 0.494271i \(-0.835435\pi\)
−0.869308 + 0.494271i \(0.835435\pi\)
\(164\) 7.96875 0.622255
\(165\) −39.7259 −3.09265
\(166\) 7.39527 0.573984
\(167\) 22.0652 1.70746 0.853730 0.520716i \(-0.174335\pi\)
0.853730 + 0.520716i \(0.174335\pi\)
\(168\) 8.08939 0.624110
\(169\) 0 0
\(170\) −24.2120 −1.85698
\(171\) −4.94139 −0.377878
\(172\) 11.3957 0.868915
\(173\) 21.2498 1.61559 0.807796 0.589462i \(-0.200660\pi\)
0.807796 + 0.589462i \(0.200660\pi\)
\(174\) −12.8951 −0.977576
\(175\) −28.8100 −2.17783
\(176\) −3.63541 −0.274030
\(177\) 16.4453 1.23610
\(178\) 7.06857 0.529812
\(179\) −22.6071 −1.68974 −0.844868 0.534975i \(-0.820321\pi\)
−0.844868 + 0.534975i \(0.820321\pi\)
\(180\) 19.1611 1.42818
\(181\) −4.83570 −0.359435 −0.179717 0.983718i \(-0.557518\pi\)
−0.179717 + 0.983718i \(0.557518\pi\)
\(182\) 0 0
\(183\) −0.605401 −0.0447525
\(184\) 8.25037 0.608225
\(185\) −0.0456734 −0.00335798
\(186\) −9.87145 −0.723810
\(187\) −22.6994 −1.65994
\(188\) 1.96020 0.142962
\(189\) −15.7047 −1.14235
\(190\) 3.87767 0.281316
\(191\) 18.0400 1.30533 0.652666 0.757646i \(-0.273651\pi\)
0.652666 + 0.757646i \(0.273651\pi\)
\(192\) 2.81805 0.203375
\(193\) 6.52151 0.469428 0.234714 0.972064i \(-0.424585\pi\)
0.234714 + 0.972064i \(0.424585\pi\)
\(194\) 6.09538 0.437623
\(195\) 0 0
\(196\) 1.24015 0.0885820
\(197\) 16.0992 1.14702 0.573510 0.819199i \(-0.305581\pi\)
0.573510 + 0.819199i \(0.305581\pi\)
\(198\) 17.9640 1.27665
\(199\) −6.01255 −0.426218 −0.213109 0.977028i \(-0.568359\pi\)
−0.213109 + 0.977028i \(0.568359\pi\)
\(200\) −10.0364 −0.709678
\(201\) −24.4837 −1.72695
\(202\) 0.590697 0.0415613
\(203\) −13.1354 −0.921926
\(204\) 17.5958 1.23195
\(205\) 30.9002 2.15817
\(206\) −0.525578 −0.0366188
\(207\) −40.7683 −2.83359
\(208\) 0 0
\(209\) 3.63541 0.251467
\(210\) 31.3680 2.16460
\(211\) −4.97594 −0.342558 −0.171279 0.985223i \(-0.554790\pi\)
−0.171279 + 0.985223i \(0.554790\pi\)
\(212\) 8.92897 0.613244
\(213\) 18.6737 1.27950
\(214\) 8.94396 0.611396
\(215\) 44.1889 3.01366
\(216\) −5.47093 −0.372250
\(217\) −10.0554 −0.682606
\(218\) −0.271028 −0.0183563
\(219\) 38.0306 2.56987
\(220\) −14.0969 −0.950416
\(221\) 0 0
\(222\) 0.0331926 0.00222774
\(223\) 18.1338 1.21433 0.607163 0.794577i \(-0.292307\pi\)
0.607163 + 0.794577i \(0.292307\pi\)
\(224\) 2.87057 0.191798
\(225\) 49.5936 3.30624
\(226\) −0.112771 −0.00750141
\(227\) 12.3853 0.822040 0.411020 0.911626i \(-0.365173\pi\)
0.411020 + 0.911626i \(0.365173\pi\)
\(228\) −2.81805 −0.186630
\(229\) 2.20870 0.145955 0.0729774 0.997334i \(-0.476750\pi\)
0.0729774 + 0.997334i \(0.476750\pi\)
\(230\) 31.9923 2.10951
\(231\) 29.4083 1.93492
\(232\) −4.57590 −0.300423
\(233\) 3.73725 0.244835 0.122418 0.992479i \(-0.460935\pi\)
0.122418 + 0.992479i \(0.460935\pi\)
\(234\) 0 0
\(235\) 7.60101 0.495835
\(236\) 5.83570 0.379872
\(237\) 17.8442 1.15911
\(238\) 17.9237 1.16182
\(239\) 2.20869 0.142868 0.0714342 0.997445i \(-0.477242\pi\)
0.0714342 + 0.997445i \(0.477242\pi\)
\(240\) 10.9275 0.705365
\(241\) −22.9135 −1.47599 −0.737995 0.674806i \(-0.764227\pi\)
−0.737995 + 0.674806i \(0.764227\pi\)
\(242\) −2.21623 −0.142464
\(243\) −14.7412 −0.945649
\(244\) −0.214830 −0.0137531
\(245\) 4.80889 0.307229
\(246\) −22.4563 −1.43176
\(247\) 0 0
\(248\) −3.50294 −0.222437
\(249\) −20.8402 −1.32070
\(250\) −19.5294 −1.23515
\(251\) −1.22889 −0.0775666 −0.0387833 0.999248i \(-0.512348\pi\)
−0.0387833 + 0.999248i \(0.512348\pi\)
\(252\) −14.1846 −0.893545
\(253\) 29.9935 1.88568
\(254\) −13.4369 −0.843104
\(255\) 68.2307 4.27277
\(256\) 1.00000 0.0625000
\(257\) −12.3311 −0.769190 −0.384595 0.923085i \(-0.625659\pi\)
−0.384595 + 0.923085i \(0.625659\pi\)
\(258\) −32.1137 −1.99931
\(259\) 0.0338111 0.00210092
\(260\) 0 0
\(261\) 22.6113 1.39961
\(262\) −7.52443 −0.464861
\(263\) −7.51111 −0.463155 −0.231577 0.972816i \(-0.574389\pi\)
−0.231577 + 0.972816i \(0.574389\pi\)
\(264\) 10.2448 0.630522
\(265\) 34.6237 2.12691
\(266\) −2.87057 −0.176006
\(267\) −19.9196 −1.21906
\(268\) −8.68818 −0.530715
\(269\) −16.8519 −1.02748 −0.513739 0.857946i \(-0.671740\pi\)
−0.513739 + 0.857946i \(0.671740\pi\)
\(270\) −21.2145 −1.29107
\(271\) −9.80221 −0.595442 −0.297721 0.954653i \(-0.596227\pi\)
−0.297721 + 0.954653i \(0.596227\pi\)
\(272\) 6.24396 0.378596
\(273\) 0 0
\(274\) −0.458171 −0.0276791
\(275\) −36.4863 −2.20021
\(276\) −23.2499 −1.39948
\(277\) −21.3755 −1.28433 −0.642165 0.766566i \(-0.721964\pi\)
−0.642165 + 0.766566i \(0.721964\pi\)
\(278\) −11.9096 −0.714289
\(279\) 17.3094 1.03629
\(280\) 11.1311 0.665212
\(281\) −3.05650 −0.182336 −0.0911678 0.995836i \(-0.529060\pi\)
−0.0911678 + 0.995836i \(0.529060\pi\)
\(282\) −5.52393 −0.328945
\(283\) −2.99153 −0.177828 −0.0889141 0.996039i \(-0.528340\pi\)
−0.0889141 + 0.996039i \(0.528340\pi\)
\(284\) 6.62648 0.393209
\(285\) −10.9275 −0.647288
\(286\) 0 0
\(287\) −22.8748 −1.35026
\(288\) −4.94139 −0.291174
\(289\) 21.9870 1.29335
\(290\) −17.7439 −1.04196
\(291\) −17.1771 −1.00694
\(292\) 13.4954 0.789756
\(293\) −10.1240 −0.591449 −0.295724 0.955273i \(-0.595561\pi\)
−0.295724 + 0.955273i \(0.595561\pi\)
\(294\) −3.49480 −0.203821
\(295\) 22.6289 1.31751
\(296\) 0.0117786 0.000684615 0
\(297\) −19.8891 −1.15408
\(298\) 6.67440 0.386638
\(299\) 0 0
\(300\) 28.2829 1.63292
\(301\) −32.7121 −1.88550
\(302\) 5.22614 0.300731
\(303\) −1.66461 −0.0956295
\(304\) −1.00000 −0.0573539
\(305\) −0.833041 −0.0476998
\(306\) −30.8538 −1.76380
\(307\) −19.5892 −1.11802 −0.559008 0.829162i \(-0.688818\pi\)
−0.559008 + 0.829162i \(0.688818\pi\)
\(308\) 10.4357 0.594629
\(309\) 1.48110 0.0842571
\(310\) −13.5833 −0.771477
\(311\) −19.4946 −1.10544 −0.552719 0.833368i \(-0.686410\pi\)
−0.552719 + 0.833368i \(0.686410\pi\)
\(312\) 0 0
\(313\) −20.3035 −1.14762 −0.573811 0.818987i \(-0.694536\pi\)
−0.573811 + 0.818987i \(0.694536\pi\)
\(314\) 8.58330 0.484383
\(315\) −55.0032 −3.09908
\(316\) 6.33212 0.356209
\(317\) 11.2881 0.634001 0.317000 0.948425i \(-0.397324\pi\)
0.317000 + 0.948425i \(0.397324\pi\)
\(318\) −25.1623 −1.41103
\(319\) −16.6353 −0.931397
\(320\) 3.87767 0.216769
\(321\) −25.2045 −1.40678
\(322\) −23.6832 −1.31982
\(323\) −6.24396 −0.347423
\(324\) 0.593170 0.0329539
\(325\) 0 0
\(326\) 22.1972 1.22939
\(327\) 0.763769 0.0422365
\(328\) −7.96875 −0.440001
\(329\) −5.62688 −0.310220
\(330\) 39.7259 2.18684
\(331\) 0.417820 0.0229654 0.0114827 0.999934i \(-0.496345\pi\)
0.0114827 + 0.999934i \(0.496345\pi\)
\(332\) −7.39527 −0.405868
\(333\) −0.0582025 −0.00318948
\(334\) −22.0652 −1.20736
\(335\) −33.6900 −1.84068
\(336\) −8.08939 −0.441312
\(337\) 17.1029 0.931652 0.465826 0.884876i \(-0.345757\pi\)
0.465826 + 0.884876i \(0.345757\pi\)
\(338\) 0 0
\(339\) 0.317794 0.0172602
\(340\) 24.2120 1.31308
\(341\) −12.7346 −0.689619
\(342\) 4.94139 0.267200
\(343\) 16.5340 0.892754
\(344\) −11.3957 −0.614416
\(345\) −90.1557 −4.85382
\(346\) −21.2498 −1.14240
\(347\) −20.8002 −1.11661 −0.558305 0.829636i \(-0.688548\pi\)
−0.558305 + 0.829636i \(0.688548\pi\)
\(348\) 12.8951 0.691250
\(349\) −3.29845 −0.176562 −0.0882810 0.996096i \(-0.528137\pi\)
−0.0882810 + 0.996096i \(0.528137\pi\)
\(350\) 28.8100 1.53996
\(351\) 0 0
\(352\) 3.63541 0.193768
\(353\) −9.00845 −0.479472 −0.239736 0.970838i \(-0.577061\pi\)
−0.239736 + 0.970838i \(0.577061\pi\)
\(354\) −16.4453 −0.874056
\(355\) 25.6953 1.36377
\(356\) −7.06857 −0.374633
\(357\) −50.5098 −2.67326
\(358\) 22.6071 1.19482
\(359\) −17.6429 −0.931154 −0.465577 0.885007i \(-0.654153\pi\)
−0.465577 + 0.885007i \(0.654153\pi\)
\(360\) −19.1611 −1.00988
\(361\) 1.00000 0.0526316
\(362\) 4.83570 0.254159
\(363\) 6.24543 0.327800
\(364\) 0 0
\(365\) 52.3306 2.73911
\(366\) 0.605401 0.0316448
\(367\) −17.9215 −0.935493 −0.467746 0.883863i \(-0.654934\pi\)
−0.467746 + 0.883863i \(0.654934\pi\)
\(368\) −8.25037 −0.430080
\(369\) 39.3767 2.04987
\(370\) 0.0456734 0.00237445
\(371\) −25.6312 −1.33071
\(372\) 9.87145 0.511811
\(373\) 7.67742 0.397522 0.198761 0.980048i \(-0.436308\pi\)
0.198761 + 0.980048i \(0.436308\pi\)
\(374\) 22.6994 1.17376
\(375\) 55.0347 2.84198
\(376\) −1.96020 −0.101090
\(377\) 0 0
\(378\) 15.7047 0.807761
\(379\) 14.0050 0.719390 0.359695 0.933070i \(-0.382881\pi\)
0.359695 + 0.933070i \(0.382881\pi\)
\(380\) −3.87767 −0.198921
\(381\) 37.8657 1.93992
\(382\) −18.0400 −0.923009
\(383\) −7.25286 −0.370604 −0.185302 0.982682i \(-0.559326\pi\)
−0.185302 + 0.982682i \(0.559326\pi\)
\(384\) −2.81805 −0.143808
\(385\) 40.4662 2.06235
\(386\) −6.52151 −0.331936
\(387\) 56.3107 2.86243
\(388\) −6.09538 −0.309446
\(389\) 9.44818 0.479042 0.239521 0.970891i \(-0.423010\pi\)
0.239521 + 0.970891i \(0.423010\pi\)
\(390\) 0 0
\(391\) −51.5150 −2.60522
\(392\) −1.24015 −0.0626370
\(393\) 21.2042 1.06961
\(394\) −16.0992 −0.811066
\(395\) 24.5539 1.23544
\(396\) −17.9640 −0.902725
\(397\) 1.04634 0.0525144 0.0262572 0.999655i \(-0.491641\pi\)
0.0262572 + 0.999655i \(0.491641\pi\)
\(398\) 6.01255 0.301382
\(399\) 8.08939 0.404976
\(400\) 10.0364 0.501818
\(401\) −23.9797 −1.19749 −0.598744 0.800940i \(-0.704333\pi\)
−0.598744 + 0.800940i \(0.704333\pi\)
\(402\) 24.4837 1.22114
\(403\) 0 0
\(404\) −0.590697 −0.0293883
\(405\) 2.30012 0.114294
\(406\) 13.1354 0.651900
\(407\) 0.0428199 0.00212251
\(408\) −17.5958 −0.871121
\(409\) −38.8529 −1.92115 −0.960575 0.278021i \(-0.910322\pi\)
−0.960575 + 0.278021i \(0.910322\pi\)
\(410\) −30.9002 −1.52605
\(411\) 1.29115 0.0636876
\(412\) 0.525578 0.0258934
\(413\) −16.7518 −0.824300
\(414\) 40.7683 2.00365
\(415\) −28.6765 −1.40767
\(416\) 0 0
\(417\) 33.5617 1.64352
\(418\) −3.63541 −0.177814
\(419\) 29.2905 1.43094 0.715468 0.698645i \(-0.246213\pi\)
0.715468 + 0.698645i \(0.246213\pi\)
\(420\) −31.3680 −1.53060
\(421\) 20.6974 1.00873 0.504365 0.863490i \(-0.331727\pi\)
0.504365 + 0.863490i \(0.331727\pi\)
\(422\) 4.97594 0.242225
\(423\) 9.68611 0.470955
\(424\) −8.92897 −0.433629
\(425\) 62.6666 3.03978
\(426\) −18.6737 −0.904746
\(427\) 0.616684 0.0298434
\(428\) −8.94396 −0.432322
\(429\) 0 0
\(430\) −44.1889 −2.13098
\(431\) −23.9545 −1.15385 −0.576923 0.816799i \(-0.695747\pi\)
−0.576923 + 0.816799i \(0.695747\pi\)
\(432\) 5.47093 0.263220
\(433\) −32.1161 −1.54340 −0.771701 0.635986i \(-0.780594\pi\)
−0.771701 + 0.635986i \(0.780594\pi\)
\(434\) 10.0554 0.482675
\(435\) 50.0030 2.39746
\(436\) 0.271028 0.0129799
\(437\) 8.25037 0.394669
\(438\) −38.0306 −1.81717
\(439\) −22.7654 −1.08653 −0.543266 0.839561i \(-0.682813\pi\)
−0.543266 + 0.839561i \(0.682813\pi\)
\(440\) 14.0969 0.672046
\(441\) 6.12806 0.291812
\(442\) 0 0
\(443\) 41.0644 1.95103 0.975515 0.219934i \(-0.0705842\pi\)
0.975515 + 0.219934i \(0.0705842\pi\)
\(444\) −0.0331926 −0.00157525
\(445\) −27.4096 −1.29934
\(446\) −18.1338 −0.858659
\(447\) −18.8088 −0.889625
\(448\) −2.87057 −0.135621
\(449\) 19.7108 0.930209 0.465104 0.885256i \(-0.346017\pi\)
0.465104 + 0.885256i \(0.346017\pi\)
\(450\) −49.5936 −2.33786
\(451\) −28.9697 −1.36413
\(452\) 0.112771 0.00530430
\(453\) −14.7275 −0.691959
\(454\) −12.3853 −0.581270
\(455\) 0 0
\(456\) 2.81805 0.131967
\(457\) 13.3513 0.624548 0.312274 0.949992i \(-0.398909\pi\)
0.312274 + 0.949992i \(0.398909\pi\)
\(458\) −2.20870 −0.103206
\(459\) 34.1603 1.59446
\(460\) −31.9923 −1.49165
\(461\) 11.0636 0.515285 0.257642 0.966240i \(-0.417054\pi\)
0.257642 + 0.966240i \(0.417054\pi\)
\(462\) −29.4083 −1.36820
\(463\) 12.7149 0.590913 0.295456 0.955356i \(-0.404528\pi\)
0.295456 + 0.955356i \(0.404528\pi\)
\(464\) 4.57590 0.212431
\(465\) 38.2783 1.77511
\(466\) −3.73725 −0.173125
\(467\) −15.3190 −0.708877 −0.354438 0.935079i \(-0.615328\pi\)
−0.354438 + 0.935079i \(0.615328\pi\)
\(468\) 0 0
\(469\) 24.9400 1.15162
\(470\) −7.60101 −0.350609
\(471\) −24.1881 −1.11453
\(472\) −5.83570 −0.268610
\(473\) −41.4281 −1.90487
\(474\) −17.8442 −0.819611
\(475\) −10.0364 −0.460500
\(476\) −17.9237 −0.821531
\(477\) 44.1216 2.02019
\(478\) −2.20869 −0.101023
\(479\) 15.5689 0.711363 0.355682 0.934607i \(-0.384249\pi\)
0.355682 + 0.934607i \(0.384249\pi\)
\(480\) −10.9275 −0.498769
\(481\) 0 0
\(482\) 22.9135 1.04368
\(483\) 66.7405 3.03680
\(484\) 2.21623 0.100738
\(485\) −23.6359 −1.07325
\(486\) 14.7412 0.668675
\(487\) −30.8664 −1.39869 −0.699346 0.714784i \(-0.746525\pi\)
−0.699346 + 0.714784i \(0.746525\pi\)
\(488\) 0.214830 0.00972489
\(489\) −62.5526 −2.82873
\(490\) −4.80889 −0.217244
\(491\) −29.6957 −1.34015 −0.670074 0.742294i \(-0.733738\pi\)
−0.670074 + 0.742294i \(0.733738\pi\)
\(492\) 22.4563 1.01241
\(493\) 28.5717 1.28681
\(494\) 0 0
\(495\) −69.6585 −3.13092
\(496\) 3.50294 0.157287
\(497\) −19.0218 −0.853242
\(498\) 20.8402 0.933873
\(499\) 16.4665 0.737141 0.368571 0.929600i \(-0.379847\pi\)
0.368571 + 0.929600i \(0.379847\pi\)
\(500\) 19.5294 0.873380
\(501\) 62.1809 2.77804
\(502\) 1.22889 0.0548479
\(503\) −26.3001 −1.17267 −0.586333 0.810070i \(-0.699429\pi\)
−0.586333 + 0.810070i \(0.699429\pi\)
\(504\) 14.1846 0.631832
\(505\) −2.29053 −0.101927
\(506\) −29.9935 −1.33337
\(507\) 0 0
\(508\) 13.4369 0.596165
\(509\) 32.6972 1.44928 0.724639 0.689128i \(-0.242006\pi\)
0.724639 + 0.689128i \(0.242006\pi\)
\(510\) −68.2307 −3.02131
\(511\) −38.7393 −1.71373
\(512\) −1.00000 −0.0441942
\(513\) −5.47093 −0.241547
\(514\) 12.3311 0.543900
\(515\) 2.03802 0.0898060
\(516\) 32.1137 1.41372
\(517\) −7.12613 −0.313407
\(518\) −0.0338111 −0.00148558
\(519\) 59.8829 2.62857
\(520\) 0 0
\(521\) −42.1295 −1.84573 −0.922863 0.385129i \(-0.874157\pi\)
−0.922863 + 0.385129i \(0.874157\pi\)
\(522\) −22.6113 −0.989670
\(523\) −14.2840 −0.624597 −0.312298 0.949984i \(-0.601099\pi\)
−0.312298 + 0.949984i \(0.601099\pi\)
\(524\) 7.52443 0.328706
\(525\) −81.1881 −3.54334
\(526\) 7.51111 0.327500
\(527\) 21.8722 0.952768
\(528\) −10.2448 −0.445846
\(529\) 45.0686 1.95951
\(530\) −34.6237 −1.50396
\(531\) 28.8365 1.25140
\(532\) 2.87057 0.124455
\(533\) 0 0
\(534\) 19.9196 0.862004
\(535\) −34.6818 −1.49942
\(536\) 8.68818 0.375272
\(537\) −63.7080 −2.74920
\(538\) 16.8519 0.726537
\(539\) −4.50845 −0.194193
\(540\) 21.2145 0.912926
\(541\) 17.2406 0.741232 0.370616 0.928786i \(-0.379147\pi\)
0.370616 + 0.928786i \(0.379147\pi\)
\(542\) 9.80221 0.421041
\(543\) −13.6272 −0.584800
\(544\) −6.24396 −0.267708
\(545\) 1.05096 0.0450181
\(546\) 0 0
\(547\) −16.5996 −0.709748 −0.354874 0.934914i \(-0.615476\pi\)
−0.354874 + 0.934914i \(0.615476\pi\)
\(548\) 0.458171 0.0195721
\(549\) −1.06156 −0.0453062
\(550\) 36.4863 1.55578
\(551\) −4.57590 −0.194940
\(552\) 23.2499 0.989583
\(553\) −18.1768 −0.772954
\(554\) 21.3755 0.908159
\(555\) −0.128710 −0.00546343
\(556\) 11.9096 0.505078
\(557\) 21.0519 0.891997 0.445998 0.895034i \(-0.352849\pi\)
0.445998 + 0.895034i \(0.352849\pi\)
\(558\) −17.3094 −0.732765
\(559\) 0 0
\(560\) −11.1311 −0.470376
\(561\) −63.9679 −2.70073
\(562\) 3.05650 0.128931
\(563\) −15.0002 −0.632185 −0.316092 0.948728i \(-0.602371\pi\)
−0.316092 + 0.948728i \(0.602371\pi\)
\(564\) 5.52393 0.232599
\(565\) 0.437289 0.0183969
\(566\) 2.99153 0.125744
\(567\) −1.70273 −0.0715081
\(568\) −6.62648 −0.278041
\(569\) 0.572295 0.0239919 0.0119959 0.999928i \(-0.496181\pi\)
0.0119959 + 0.999928i \(0.496181\pi\)
\(570\) 10.9275 0.457701
\(571\) −38.8818 −1.62715 −0.813577 0.581457i \(-0.802483\pi\)
−0.813577 + 0.581457i \(0.802483\pi\)
\(572\) 0 0
\(573\) 50.8377 2.12378
\(574\) 22.8748 0.954777
\(575\) −82.8037 −3.45315
\(576\) 4.94139 0.205891
\(577\) −3.71329 −0.154586 −0.0772931 0.997008i \(-0.524628\pi\)
−0.0772931 + 0.997008i \(0.524628\pi\)
\(578\) −21.9870 −0.914540
\(579\) 18.3779 0.763760
\(580\) 17.7439 0.736774
\(581\) 21.2286 0.880711
\(582\) 17.1771 0.712012
\(583\) −32.4605 −1.34438
\(584\) −13.4954 −0.558442
\(585\) 0 0
\(586\) 10.1240 0.418217
\(587\) −5.88784 −0.243017 −0.121509 0.992590i \(-0.538773\pi\)
−0.121509 + 0.992590i \(0.538773\pi\)
\(588\) 3.49480 0.144123
\(589\) −3.50294 −0.144336
\(590\) −22.6289 −0.931619
\(591\) 45.3683 1.86620
\(592\) −0.0117786 −0.000484096 0
\(593\) −12.5482 −0.515293 −0.257647 0.966239i \(-0.582947\pi\)
−0.257647 + 0.966239i \(0.582947\pi\)
\(594\) 19.8891 0.816059
\(595\) −69.5023 −2.84932
\(596\) −6.67440 −0.273394
\(597\) −16.9436 −0.693457
\(598\) 0 0
\(599\) 11.9454 0.488075 0.244038 0.969766i \(-0.421528\pi\)
0.244038 + 0.969766i \(0.421528\pi\)
\(600\) −28.2829 −1.15465
\(601\) 35.0150 1.42829 0.714147 0.699996i \(-0.246815\pi\)
0.714147 + 0.699996i \(0.246815\pi\)
\(602\) 32.7121 1.33325
\(603\) −42.9317 −1.74831
\(604\) −5.22614 −0.212649
\(605\) 8.59380 0.349388
\(606\) 1.66461 0.0676202
\(607\) 7.18015 0.291433 0.145717 0.989326i \(-0.453451\pi\)
0.145717 + 0.989326i \(0.453451\pi\)
\(608\) 1.00000 0.0405554
\(609\) −37.0163 −1.49997
\(610\) 0.833041 0.0337288
\(611\) 0 0
\(612\) 30.8538 1.24719
\(613\) −25.1763 −1.01686 −0.508430 0.861103i \(-0.669774\pi\)
−0.508430 + 0.861103i \(0.669774\pi\)
\(614\) 19.5892 0.790556
\(615\) 87.0783 3.51134
\(616\) −10.4357 −0.420466
\(617\) 6.50472 0.261870 0.130935 0.991391i \(-0.458202\pi\)
0.130935 + 0.991391i \(0.458202\pi\)
\(618\) −1.48110 −0.0595788
\(619\) 19.3211 0.776580 0.388290 0.921537i \(-0.373066\pi\)
0.388290 + 0.921537i \(0.373066\pi\)
\(620\) 13.5833 0.545517
\(621\) −45.1372 −1.81129
\(622\) 19.4946 0.781662
\(623\) 20.2908 0.812933
\(624\) 0 0
\(625\) 25.5468 1.02187
\(626\) 20.3035 0.811492
\(627\) 10.2448 0.409137
\(628\) −8.58330 −0.342511
\(629\) −0.0735449 −0.00293243
\(630\) 55.0032 2.19138
\(631\) −20.6678 −0.822772 −0.411386 0.911461i \(-0.634955\pi\)
−0.411386 + 0.911461i \(0.634955\pi\)
\(632\) −6.33212 −0.251878
\(633\) −14.0224 −0.557341
\(634\) −11.2881 −0.448306
\(635\) 52.1038 2.06768
\(636\) 25.1623 0.997749
\(637\) 0 0
\(638\) 16.6353 0.658597
\(639\) 32.7440 1.29533
\(640\) −3.87767 −0.153279
\(641\) −23.0621 −0.910897 −0.455448 0.890262i \(-0.650521\pi\)
−0.455448 + 0.890262i \(0.650521\pi\)
\(642\) 25.2045 0.994742
\(643\) −28.3674 −1.11870 −0.559351 0.828931i \(-0.688949\pi\)
−0.559351 + 0.828931i \(0.688949\pi\)
\(644\) 23.6832 0.933250
\(645\) 124.526 4.90322
\(646\) 6.24396 0.245665
\(647\) 0.127715 0.00502100 0.00251050 0.999997i \(-0.499201\pi\)
0.00251050 + 0.999997i \(0.499201\pi\)
\(648\) −0.593170 −0.0233019
\(649\) −21.2152 −0.832768
\(650\) 0 0
\(651\) −28.3366 −1.11060
\(652\) −22.1972 −0.869308
\(653\) −34.3776 −1.34530 −0.672649 0.739961i \(-0.734844\pi\)
−0.672649 + 0.739961i \(0.734844\pi\)
\(654\) −0.763769 −0.0298657
\(655\) 29.1773 1.14005
\(656\) 7.96875 0.311128
\(657\) 66.6858 2.60166
\(658\) 5.62688 0.219359
\(659\) −0.0515489 −0.00200806 −0.00100403 0.999999i \(-0.500320\pi\)
−0.00100403 + 0.999999i \(0.500320\pi\)
\(660\) −39.7259 −1.54633
\(661\) −4.51042 −0.175435 −0.0877174 0.996145i \(-0.527957\pi\)
−0.0877174 + 0.996145i \(0.527957\pi\)
\(662\) −0.417820 −0.0162390
\(663\) 0 0
\(664\) 7.39527 0.286992
\(665\) 11.1311 0.431646
\(666\) 0.0582025 0.00225530
\(667\) −37.7529 −1.46180
\(668\) 22.0652 0.853730
\(669\) 51.1018 1.97571
\(670\) 33.6900 1.30156
\(671\) 0.780995 0.0301500
\(672\) 8.08939 0.312055
\(673\) 8.33824 0.321416 0.160708 0.987002i \(-0.448622\pi\)
0.160708 + 0.987002i \(0.448622\pi\)
\(674\) −17.1029 −0.658778
\(675\) 54.9082 2.11342
\(676\) 0 0
\(677\) −31.4629 −1.20922 −0.604609 0.796522i \(-0.706671\pi\)
−0.604609 + 0.796522i \(0.706671\pi\)
\(678\) −0.317794 −0.0122048
\(679\) 17.4972 0.671480
\(680\) −24.2120 −0.928489
\(681\) 34.9023 1.33746
\(682\) 12.7346 0.487634
\(683\) 11.9211 0.456150 0.228075 0.973644i \(-0.426757\pi\)
0.228075 + 0.973644i \(0.426757\pi\)
\(684\) −4.94139 −0.188939
\(685\) 1.77664 0.0678818
\(686\) −16.5340 −0.631272
\(687\) 6.22422 0.237469
\(688\) 11.3957 0.434457
\(689\) 0 0
\(690\) 90.1557 3.43217
\(691\) 21.6103 0.822096 0.411048 0.911614i \(-0.365163\pi\)
0.411048 + 0.911614i \(0.365163\pi\)
\(692\) 21.2498 0.807796
\(693\) 51.5668 1.95886
\(694\) 20.8002 0.789563
\(695\) 46.1814 1.75176
\(696\) −12.8951 −0.488788
\(697\) 49.7566 1.88466
\(698\) 3.29845 0.124848
\(699\) 10.5317 0.398347
\(700\) −28.8100 −1.08892
\(701\) −52.0824 −1.96713 −0.983563 0.180563i \(-0.942208\pi\)
−0.983563 + 0.180563i \(0.942208\pi\)
\(702\) 0 0
\(703\) 0.0117786 0.000444237 0
\(704\) −3.63541 −0.137015
\(705\) 21.4200 0.806724
\(706\) 9.00845 0.339038
\(707\) 1.69563 0.0637709
\(708\) 16.4453 0.618051
\(709\) −13.1704 −0.494625 −0.247312 0.968936i \(-0.579547\pi\)
−0.247312 + 0.968936i \(0.579547\pi\)
\(710\) −25.6953 −0.964329
\(711\) 31.2895 1.17345
\(712\) 7.06857 0.264906
\(713\) −28.9005 −1.08233
\(714\) 50.5098 1.89028
\(715\) 0 0
\(716\) −22.6071 −0.844868
\(717\) 6.22420 0.232447
\(718\) 17.6429 0.658425
\(719\) −2.90129 −0.108200 −0.0540999 0.998536i \(-0.517229\pi\)
−0.0540999 + 0.998536i \(0.517229\pi\)
\(720\) 19.1611 0.714092
\(721\) −1.50871 −0.0561872
\(722\) −1.00000 −0.0372161
\(723\) −64.5714 −2.40144
\(724\) −4.83570 −0.179717
\(725\) 45.9254 1.70563
\(726\) −6.24543 −0.231790
\(727\) −32.5084 −1.20567 −0.602835 0.797866i \(-0.705962\pi\)
−0.602835 + 0.797866i \(0.705962\pi\)
\(728\) 0 0
\(729\) −43.3209 −1.60448
\(730\) −52.3306 −1.93684
\(731\) 71.1544 2.63174
\(732\) −0.605401 −0.0223763
\(733\) 35.7248 1.31952 0.659762 0.751475i \(-0.270657\pi\)
0.659762 + 0.751475i \(0.270657\pi\)
\(734\) 17.9215 0.661493
\(735\) 13.5517 0.499862
\(736\) 8.25037 0.304113
\(737\) 31.5851 1.16345
\(738\) −39.3767 −1.44948
\(739\) 3.02329 0.111213 0.0556067 0.998453i \(-0.482291\pi\)
0.0556067 + 0.998453i \(0.482291\pi\)
\(740\) −0.0456734 −0.00167899
\(741\) 0 0
\(742\) 25.6312 0.940951
\(743\) 5.27755 0.193615 0.0968073 0.995303i \(-0.469137\pi\)
0.0968073 + 0.995303i \(0.469137\pi\)
\(744\) −9.87145 −0.361905
\(745\) −25.8812 −0.948213
\(746\) −7.67742 −0.281090
\(747\) −36.5429 −1.33704
\(748\) −22.6994 −0.829971
\(749\) 25.6742 0.938115
\(750\) −55.0347 −2.00958
\(751\) −18.1364 −0.661808 −0.330904 0.943664i \(-0.607354\pi\)
−0.330904 + 0.943664i \(0.607354\pi\)
\(752\) 1.96020 0.0714811
\(753\) −3.46306 −0.126201
\(754\) 0 0
\(755\) −20.2653 −0.737529
\(756\) −15.7047 −0.571173
\(757\) 46.3593 1.68496 0.842478 0.538731i \(-0.181096\pi\)
0.842478 + 0.538731i \(0.181096\pi\)
\(758\) −14.0050 −0.508685
\(759\) 84.5231 3.06799
\(760\) 3.87767 0.140658
\(761\) 45.5300 1.65046 0.825231 0.564796i \(-0.191045\pi\)
0.825231 + 0.564796i \(0.191045\pi\)
\(762\) −37.8657 −1.37173
\(763\) −0.778003 −0.0281656
\(764\) 18.0400 0.652666
\(765\) 119.641 4.32564
\(766\) 7.25286 0.262057
\(767\) 0 0
\(768\) 2.81805 0.101688
\(769\) 51.1629 1.84498 0.922491 0.386018i \(-0.126150\pi\)
0.922491 + 0.386018i \(0.126150\pi\)
\(770\) −40.4662 −1.45830
\(771\) −34.7495 −1.25147
\(772\) 6.52151 0.234714
\(773\) −23.5929 −0.848578 −0.424289 0.905527i \(-0.639476\pi\)
−0.424289 + 0.905527i \(0.639476\pi\)
\(774\) −56.3107 −2.02404
\(775\) 35.1568 1.26287
\(776\) 6.09538 0.218811
\(777\) 0.0952814 0.00341820
\(778\) −9.44818 −0.338734
\(779\) −7.96875 −0.285510
\(780\) 0 0
\(781\) −24.0900 −0.862008
\(782\) 51.5150 1.84217
\(783\) 25.0344 0.894658
\(784\) 1.24015 0.0442910
\(785\) −33.2832 −1.18793
\(786\) −21.2042 −0.756329
\(787\) 35.9121 1.28013 0.640064 0.768321i \(-0.278908\pi\)
0.640064 + 0.768321i \(0.278908\pi\)
\(788\) 16.0992 0.573510
\(789\) −21.1667 −0.753553
\(790\) −24.5539 −0.873588
\(791\) −0.323716 −0.0115100
\(792\) 17.9640 0.638323
\(793\) 0 0
\(794\) −1.04634 −0.0371333
\(795\) 97.5711 3.46049
\(796\) −6.01255 −0.213109
\(797\) 0.957289 0.0339089 0.0169545 0.999856i \(-0.494603\pi\)
0.0169545 + 0.999856i \(0.494603\pi\)
\(798\) −8.08939 −0.286361
\(799\) 12.2394 0.432999
\(800\) −10.0364 −0.354839
\(801\) −34.9286 −1.23414
\(802\) 23.9797 0.846753
\(803\) −49.0612 −1.73133
\(804\) −24.4837 −0.863474
\(805\) 91.8359 3.23679
\(806\) 0 0
\(807\) −47.4895 −1.67171
\(808\) 0.590697 0.0207806
\(809\) −26.5699 −0.934147 −0.467074 0.884218i \(-0.654692\pi\)
−0.467074 + 0.884218i \(0.654692\pi\)
\(810\) −2.30012 −0.0808180
\(811\) −23.2305 −0.815732 −0.407866 0.913042i \(-0.633727\pi\)
−0.407866 + 0.913042i \(0.633727\pi\)
\(812\) −13.1354 −0.460963
\(813\) −27.6231 −0.968784
\(814\) −0.0428199 −0.00150084
\(815\) −86.0734 −3.01502
\(816\) 17.5958 0.615975
\(817\) −11.3957 −0.398685
\(818\) 38.8529 1.35846
\(819\) 0 0
\(820\) 30.9002 1.07908
\(821\) −39.8660 −1.39133 −0.695666 0.718365i \(-0.744891\pi\)
−0.695666 + 0.718365i \(0.744891\pi\)
\(822\) −1.29115 −0.0450339
\(823\) 24.1750 0.842686 0.421343 0.906901i \(-0.361559\pi\)
0.421343 + 0.906901i \(0.361559\pi\)
\(824\) −0.525578 −0.0183094
\(825\) −102.820 −3.57974
\(826\) 16.7518 0.582868
\(827\) −23.4688 −0.816090 −0.408045 0.912962i \(-0.633789\pi\)
−0.408045 + 0.912962i \(0.633789\pi\)
\(828\) −40.7683 −1.41680
\(829\) 25.2579 0.877243 0.438621 0.898672i \(-0.355467\pi\)
0.438621 + 0.898672i \(0.355467\pi\)
\(830\) 28.6765 0.995374
\(831\) −60.2372 −2.08961
\(832\) 0 0
\(833\) 7.74344 0.268294
\(834\) −33.5617 −1.16215
\(835\) 85.5619 2.96099
\(836\) 3.63541 0.125733
\(837\) 19.1643 0.662416
\(838\) −29.2905 −1.01182
\(839\) 30.6795 1.05917 0.529587 0.848255i \(-0.322347\pi\)
0.529587 + 0.848255i \(0.322347\pi\)
\(840\) 31.3680 1.08230
\(841\) −8.06113 −0.277970
\(842\) −20.6974 −0.713280
\(843\) −8.61337 −0.296660
\(844\) −4.97594 −0.171279
\(845\) 0 0
\(846\) −9.68611 −0.333015
\(847\) −6.36182 −0.218595
\(848\) 8.92897 0.306622
\(849\) −8.43029 −0.289327
\(850\) −62.6666 −2.14945
\(851\) 0.0971775 0.00333120
\(852\) 18.6737 0.639752
\(853\) 13.7824 0.471900 0.235950 0.971765i \(-0.424180\pi\)
0.235950 + 0.971765i \(0.424180\pi\)
\(854\) −0.616684 −0.0211025
\(855\) −19.1611 −0.655296
\(856\) 8.94396 0.305698
\(857\) −23.9708 −0.818827 −0.409414 0.912349i \(-0.634267\pi\)
−0.409414 + 0.912349i \(0.634267\pi\)
\(858\) 0 0
\(859\) −18.3175 −0.624985 −0.312492 0.949920i \(-0.601164\pi\)
−0.312492 + 0.949920i \(0.601164\pi\)
\(860\) 44.1889 1.50683
\(861\) −64.4624 −2.19687
\(862\) 23.9545 0.815892
\(863\) −2.56280 −0.0872389 −0.0436194 0.999048i \(-0.513889\pi\)
−0.0436194 + 0.999048i \(0.513889\pi\)
\(864\) −5.47093 −0.186125
\(865\) 82.3998 2.80168
\(866\) 32.1161 1.09135
\(867\) 61.9605 2.10429
\(868\) −10.0554 −0.341303
\(869\) −23.0199 −0.780895
\(870\) −50.0030 −1.69526
\(871\) 0 0
\(872\) −0.271028 −0.00917816
\(873\) −30.1196 −1.01940
\(874\) −8.25037 −0.279073
\(875\) −56.0604 −1.89519
\(876\) 38.0306 1.28493
\(877\) 5.80341 0.195967 0.0979837 0.995188i \(-0.468761\pi\)
0.0979837 + 0.995188i \(0.468761\pi\)
\(878\) 22.7654 0.768294
\(879\) −28.5298 −0.962287
\(880\) −14.0969 −0.475208
\(881\) 14.3781 0.484411 0.242205 0.970225i \(-0.422129\pi\)
0.242205 + 0.970225i \(0.422129\pi\)
\(882\) −6.12806 −0.206342
\(883\) 38.7080 1.30263 0.651313 0.758809i \(-0.274218\pi\)
0.651313 + 0.758809i \(0.274218\pi\)
\(884\) 0 0
\(885\) 63.7694 2.14359
\(886\) −41.0644 −1.37959
\(887\) 22.0494 0.740347 0.370174 0.928963i \(-0.379298\pi\)
0.370174 + 0.928963i \(0.379298\pi\)
\(888\) 0.0331926 0.00111387
\(889\) −38.5714 −1.29364
\(890\) 27.4096 0.918772
\(891\) −2.15642 −0.0722427
\(892\) 18.1338 0.607163
\(893\) −1.96020 −0.0655955
\(894\) 18.8088 0.629060
\(895\) −87.6631 −2.93025
\(896\) 2.87057 0.0958989
\(897\) 0 0
\(898\) −19.7108 −0.657757
\(899\) 16.0291 0.534600
\(900\) 49.5936 1.65312
\(901\) 55.7521 1.85737
\(902\) 28.9697 0.964586
\(903\) −92.1844 −3.06770
\(904\) −0.112771 −0.00375071
\(905\) −18.7513 −0.623313
\(906\) 14.7275 0.489289
\(907\) 29.5199 0.980192 0.490096 0.871668i \(-0.336962\pi\)
0.490096 + 0.871668i \(0.336962\pi\)
\(908\) 12.3853 0.411020
\(909\) −2.91886 −0.0968126
\(910\) 0 0
\(911\) −14.8931 −0.493429 −0.246715 0.969088i \(-0.579351\pi\)
−0.246715 + 0.969088i \(0.579351\pi\)
\(912\) −2.81805 −0.0933149
\(913\) 26.8849 0.889759
\(914\) −13.3513 −0.441622
\(915\) −2.34755 −0.0776075
\(916\) 2.20870 0.0729774
\(917\) −21.5994 −0.713274
\(918\) −34.1603 −1.12746
\(919\) 17.9143 0.590940 0.295470 0.955352i \(-0.404524\pi\)
0.295470 + 0.955352i \(0.404524\pi\)
\(920\) 31.9923 1.05475
\(921\) −55.2033 −1.81901
\(922\) −11.0636 −0.364361
\(923\) 0 0
\(924\) 29.4083 0.967461
\(925\) −0.118214 −0.00388685
\(926\) −12.7149 −0.417838
\(927\) 2.59709 0.0852996
\(928\) −4.57590 −0.150211
\(929\) −50.0404 −1.64177 −0.820886 0.571092i \(-0.806520\pi\)
−0.820886 + 0.571092i \(0.806520\pi\)
\(930\) −38.2783 −1.25519
\(931\) −1.24015 −0.0406442
\(932\) 3.73725 0.122418
\(933\) −54.9367 −1.79855
\(934\) 15.3190 0.501251
\(935\) −88.0208 −2.87859
\(936\) 0 0
\(937\) 46.3212 1.51325 0.756624 0.653850i \(-0.226847\pi\)
0.756624 + 0.653850i \(0.226847\pi\)
\(938\) −24.9400 −0.814320
\(939\) −57.2163 −1.86718
\(940\) 7.60101 0.247918
\(941\) −0.505450 −0.0164772 −0.00823859 0.999966i \(-0.502622\pi\)
−0.00823859 + 0.999966i \(0.502622\pi\)
\(942\) 24.1881 0.788092
\(943\) −65.7452 −2.14096
\(944\) 5.83570 0.189936
\(945\) −60.8976 −1.98100
\(946\) 41.4281 1.34694
\(947\) 33.7211 1.09579 0.547894 0.836548i \(-0.315430\pi\)
0.547894 + 0.836548i \(0.315430\pi\)
\(948\) 17.8442 0.579553
\(949\) 0 0
\(950\) 10.0364 0.325623
\(951\) 31.8103 1.03152
\(952\) 17.9237 0.580910
\(953\) −8.77691 −0.284312 −0.142156 0.989844i \(-0.545403\pi\)
−0.142156 + 0.989844i \(0.545403\pi\)
\(954\) −44.1216 −1.42849
\(955\) 69.9534 2.26364
\(956\) 2.20869 0.0714342
\(957\) −46.8790 −1.51538
\(958\) −15.5689 −0.503010
\(959\) −1.31521 −0.0424703
\(960\) 10.9275 0.352683
\(961\) −18.7294 −0.604175
\(962\) 0 0
\(963\) −44.1956 −1.42418
\(964\) −22.9135 −0.737995
\(965\) 25.2883 0.814058
\(966\) −66.7405 −2.14734
\(967\) 17.1197 0.550533 0.275267 0.961368i \(-0.411234\pi\)
0.275267 + 0.961368i \(0.411234\pi\)
\(968\) −2.21623 −0.0712322
\(969\) −17.5958 −0.565258
\(970\) 23.6359 0.758903
\(971\) 27.5662 0.884641 0.442320 0.896857i \(-0.354155\pi\)
0.442320 + 0.896857i \(0.354155\pi\)
\(972\) −14.7412 −0.472825
\(973\) −34.1872 −1.09599
\(974\) 30.8664 0.989024
\(975\) 0 0
\(976\) −0.214830 −0.00687654
\(977\) −39.0758 −1.25015 −0.625074 0.780566i \(-0.714931\pi\)
−0.625074 + 0.780566i \(0.714931\pi\)
\(978\) 62.5526 2.00021
\(979\) 25.6972 0.821285
\(980\) 4.80889 0.153614
\(981\) 1.33925 0.0427591
\(982\) 29.6957 0.947628
\(983\) −5.61172 −0.178986 −0.0894931 0.995987i \(-0.528525\pi\)
−0.0894931 + 0.995987i \(0.528525\pi\)
\(984\) −22.4563 −0.715882
\(985\) 62.4274 1.98910
\(986\) −28.5717 −0.909910
\(987\) −15.8568 −0.504728
\(988\) 0 0
\(989\) −94.0189 −2.98963
\(990\) 69.6585 2.21389
\(991\) 17.7487 0.563805 0.281902 0.959443i \(-0.409035\pi\)
0.281902 + 0.959443i \(0.409035\pi\)
\(992\) −3.50294 −0.111218
\(993\) 1.17744 0.0373648
\(994\) 19.0218 0.603333
\(995\) −23.3147 −0.739126
\(996\) −20.8402 −0.660348
\(997\) −40.6808 −1.28837 −0.644186 0.764869i \(-0.722804\pi\)
−0.644186 + 0.764869i \(0.722804\pi\)
\(998\) −16.4665 −0.521238
\(999\) −0.0644397 −0.00203878
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 6422.2.a.bm.1.13 14
13.2 odd 12 494.2.m.b.381.1 yes 28
13.7 odd 12 494.2.m.b.153.1 28
13.12 even 2 6422.2.a.bn.1.13 14
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
494.2.m.b.153.1 28 13.7 odd 12
494.2.m.b.381.1 yes 28 13.2 odd 12
6422.2.a.bm.1.13 14 1.1 even 1 trivial
6422.2.a.bn.1.13 14 13.12 even 2