Properties

Label 429.2.a.h.1.1
Level $429$
Weight $2$
Character 429.1
Self dual yes
Analytic conductor $3.426$
Analytic rank $0$
Dimension $4$
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] = [429,2,Mod(1,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.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: \(3.42558224671\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.8468.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 3x + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.31743\) of defining polynomial
Character \(\chi\) \(=\) 429.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.58181 q^{2} -1.00000 q^{3} +4.66573 q^{4} -2.71878 q^{5} +2.58181 q^{6} -4.30059 q^{7} -6.88240 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.58181 q^{2} -1.00000 q^{3} +4.66573 q^{4} -2.71878 q^{5} +2.58181 q^{6} -4.30059 q^{7} -6.88240 q^{8} +1.00000 q^{9} +7.01937 q^{10} -1.00000 q^{11} -4.66573 q^{12} +1.00000 q^{13} +11.1033 q^{14} +2.71878 q^{15} +8.43756 q^{16} -6.11056 q^{17} -2.58181 q^{18} -0.300590 q^{19} -12.6851 q^{20} +4.30059 q^{21} +2.58181 q^{22} +1.49789 q^{23} +6.88240 q^{24} +2.39178 q^{25} -2.58181 q^{26} -1.00000 q^{27} -20.0654 q^{28} -0.312085 q^{29} -7.01937 q^{30} +4.55094 q^{31} -8.01937 q^{32} +1.00000 q^{33} +15.7763 q^{34} +11.6924 q^{35} +4.66573 q^{36} -7.96632 q^{37} +0.776064 q^{38} -1.00000 q^{39} +18.7117 q^{40} +1.03087 q^{41} -11.1033 q^{42} +12.4906 q^{43} -4.66573 q^{44} -2.71878 q^{45} -3.86725 q^{46} +13.1299 q^{47} -8.43756 q^{48} +11.4951 q^{49} -6.17511 q^{50} +6.11056 q^{51} +4.66573 q^{52} +8.49507 q^{53} +2.58181 q^{54} +2.71878 q^{55} +29.5984 q^{56} +0.300590 q^{57} +0.805743 q^{58} +9.22535 q^{59} +12.6851 q^{60} -9.86021 q^{61} -11.7496 q^{62} -4.30059 q^{63} +3.82934 q^{64} -2.71878 q^{65} -2.58181 q^{66} -8.05024 q^{67} -28.5102 q^{68} -1.49789 q^{69} -30.1874 q^{70} -0.696596 q^{71} -6.88240 q^{72} +4.57454 q^{73} +20.5675 q^{74} -2.39178 q^{75} -1.40247 q^{76} +4.30059 q^{77} +2.58181 q^{78} -17.0918 q^{79} -22.9399 q^{80} +1.00000 q^{81} -2.66150 q^{82} +8.76902 q^{83} +20.0654 q^{84} +16.6133 q^{85} -32.2484 q^{86} +0.312085 q^{87} +6.88240 q^{88} -3.24754 q^{89} +7.01937 q^{90} -4.30059 q^{91} +6.98873 q^{92} -4.55094 q^{93} -33.8990 q^{94} +0.817238 q^{95} +8.01937 q^{96} +7.63063 q^{97} -29.6781 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 4 q^{3} + 8 q^{4} + 2 q^{6} + 2 q^{7} + 4 q^{9} - 2 q^{10} - 4 q^{11} - 8 q^{12} + 4 q^{13} + 12 q^{14} + 12 q^{16} - 8 q^{17} - 2 q^{18} + 18 q^{19} - 10 q^{20} - 2 q^{21} + 2 q^{22} + 4 q^{25} - 2 q^{26} - 4 q^{27} - 6 q^{28} - 10 q^{29} + 2 q^{30} + 12 q^{31} - 2 q^{32} + 4 q^{33} + 36 q^{34} + 22 q^{35} + 8 q^{36} - 2 q^{37} + 4 q^{38} - 4 q^{39} + 20 q^{40} + 2 q^{41} - 12 q^{42} + 28 q^{43} - 8 q^{44} - 30 q^{46} + 6 q^{47} - 12 q^{48} + 8 q^{49} - 36 q^{50} + 8 q^{51} + 8 q^{52} - 4 q^{53} + 2 q^{54} + 48 q^{56} - 18 q^{57} - 6 q^{58} + 16 q^{59} + 10 q^{60} - 10 q^{61} - 34 q^{62} + 2 q^{63} - 12 q^{64} - 2 q^{66} - 34 q^{68} - 58 q^{70} + 10 q^{71} - 6 q^{73} + 14 q^{74} - 4 q^{75} + 26 q^{76} - 2 q^{77} + 2 q^{78} - 8 q^{79} - 48 q^{80} + 4 q^{81} + 12 q^{82} - 8 q^{83} + 6 q^{84} - 18 q^{85} - 8 q^{86} + 10 q^{87} + 6 q^{89} - 2 q^{90} + 2 q^{91} - 28 q^{92} - 12 q^{93} - 46 q^{94} + 22 q^{95} + 2 q^{96} + 10 q^{97} - 34 q^{98} - 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.58181 −1.82561 −0.912807 0.408392i \(-0.866090\pi\)
−0.912807 + 0.408392i \(0.866090\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.66573 2.33286
\(5\) −2.71878 −1.21588 −0.607938 0.793984i \(-0.708003\pi\)
−0.607938 + 0.793984i \(0.708003\pi\)
\(6\) 2.58181 1.05402
\(7\) −4.30059 −1.62547 −0.812735 0.582633i \(-0.802022\pi\)
−0.812735 + 0.582633i \(0.802022\pi\)
\(8\) −6.88240 −2.43329
\(9\) 1.00000 0.333333
\(10\) 7.01937 2.21972
\(11\) −1.00000 −0.301511
\(12\) −4.66573 −1.34688
\(13\) 1.00000 0.277350
\(14\) 11.1033 2.96748
\(15\) 2.71878 0.701987
\(16\) 8.43756 2.10939
\(17\) −6.11056 −1.48203 −0.741014 0.671489i \(-0.765655\pi\)
−0.741014 + 0.671489i \(0.765655\pi\)
\(18\) −2.58181 −0.608538
\(19\) −0.300590 −0.0689600 −0.0344800 0.999405i \(-0.510977\pi\)
−0.0344800 + 0.999405i \(0.510977\pi\)
\(20\) −12.6851 −2.83647
\(21\) 4.30059 0.938466
\(22\) 2.58181 0.550443
\(23\) 1.49789 0.312331 0.156165 0.987731i \(-0.450087\pi\)
0.156165 + 0.987731i \(0.450087\pi\)
\(24\) 6.88240 1.40486
\(25\) 2.39178 0.478356
\(26\) −2.58181 −0.506334
\(27\) −1.00000 −0.192450
\(28\) −20.0654 −3.79200
\(29\) −0.312085 −0.0579527 −0.0289763 0.999580i \(-0.509225\pi\)
−0.0289763 + 0.999580i \(0.509225\pi\)
\(30\) −7.01937 −1.28156
\(31\) 4.55094 0.817373 0.408686 0.912675i \(-0.365987\pi\)
0.408686 + 0.912675i \(0.365987\pi\)
\(32\) −8.01937 −1.41764
\(33\) 1.00000 0.174078
\(34\) 15.7763 2.70561
\(35\) 11.6924 1.97637
\(36\) 4.66573 0.777621
\(37\) −7.96632 −1.30965 −0.654827 0.755779i \(-0.727259\pi\)
−0.654827 + 0.755779i \(0.727259\pi\)
\(38\) 0.776064 0.125894
\(39\) −1.00000 −0.160128
\(40\) 18.7117 2.95859
\(41\) 1.03087 0.160994 0.0804972 0.996755i \(-0.474349\pi\)
0.0804972 + 0.996755i \(0.474349\pi\)
\(42\) −11.1033 −1.71328
\(43\) 12.4906 1.90480 0.952401 0.304849i \(-0.0986059\pi\)
0.952401 + 0.304849i \(0.0986059\pi\)
\(44\) −4.66573 −0.703385
\(45\) −2.71878 −0.405292
\(46\) −3.86725 −0.570195
\(47\) 13.1299 1.91520 0.957599 0.288105i \(-0.0930253\pi\)
0.957599 + 0.288105i \(0.0930253\pi\)
\(48\) −8.43756 −1.21786
\(49\) 11.4951 1.64215
\(50\) −6.17511 −0.873292
\(51\) 6.11056 0.855650
\(52\) 4.66573 0.647020
\(53\) 8.49507 1.16689 0.583444 0.812153i \(-0.301705\pi\)
0.583444 + 0.812153i \(0.301705\pi\)
\(54\) 2.58181 0.351339
\(55\) 2.71878 0.366601
\(56\) 29.5984 3.95525
\(57\) 0.300590 0.0398141
\(58\) 0.805743 0.105799
\(59\) 9.22535 1.20104 0.600519 0.799610i \(-0.294961\pi\)
0.600519 + 0.799610i \(0.294961\pi\)
\(60\) 12.6851 1.63764
\(61\) −9.86021 −1.26247 −0.631235 0.775591i \(-0.717452\pi\)
−0.631235 + 0.775591i \(0.717452\pi\)
\(62\) −11.7496 −1.49221
\(63\) −4.30059 −0.541823
\(64\) 3.82934 0.478668
\(65\) −2.71878 −0.337223
\(66\) −2.58181 −0.317798
\(67\) −8.05024 −0.983493 −0.491747 0.870738i \(-0.663641\pi\)
−0.491747 + 0.870738i \(0.663641\pi\)
\(68\) −28.5102 −3.45737
\(69\) −1.49789 −0.180324
\(70\) −30.1874 −3.60809
\(71\) −0.696596 −0.0826707 −0.0413353 0.999145i \(-0.513161\pi\)
−0.0413353 + 0.999145i \(0.513161\pi\)
\(72\) −6.88240 −0.811098
\(73\) 4.57454 0.535409 0.267705 0.963501i \(-0.413735\pi\)
0.267705 + 0.963501i \(0.413735\pi\)
\(74\) 20.5675 2.39092
\(75\) −2.39178 −0.276179
\(76\) −1.40247 −0.160874
\(77\) 4.30059 0.490098
\(78\) 2.58181 0.292332
\(79\) −17.0918 −1.92298 −0.961489 0.274844i \(-0.911374\pi\)
−0.961489 + 0.274844i \(0.911374\pi\)
\(80\) −22.9399 −2.56476
\(81\) 1.00000 0.111111
\(82\) −2.66150 −0.293914
\(83\) 8.76902 0.962525 0.481263 0.876576i \(-0.340178\pi\)
0.481263 + 0.876576i \(0.340178\pi\)
\(84\) 20.0654 2.18931
\(85\) 16.6133 1.80196
\(86\) −32.2484 −3.47743
\(87\) 0.312085 0.0334590
\(88\) 6.88240 0.733666
\(89\) −3.24754 −0.344238 −0.172119 0.985076i \(-0.555061\pi\)
−0.172119 + 0.985076i \(0.555061\pi\)
\(90\) 7.01937 0.739907
\(91\) −4.30059 −0.450824
\(92\) 6.98873 0.728625
\(93\) −4.55094 −0.471910
\(94\) −33.8990 −3.49641
\(95\) 0.817238 0.0838468
\(96\) 8.01937 0.818474
\(97\) 7.63063 0.774773 0.387387 0.921917i \(-0.373378\pi\)
0.387387 + 0.921917i \(0.373378\pi\)
\(98\) −29.6781 −2.99794
\(99\) −1.00000 −0.100504
\(100\) 11.1594 1.11594
\(101\) 3.83661 0.381757 0.190878 0.981614i \(-0.438866\pi\)
0.190878 + 0.981614i \(0.438866\pi\)
\(102\) −15.7763 −1.56209
\(103\) 11.3315 1.11652 0.558261 0.829666i \(-0.311469\pi\)
0.558261 + 0.829666i \(0.311469\pi\)
\(104\) −6.88240 −0.674875
\(105\) −11.6924 −1.14106
\(106\) −21.9326 −2.13029
\(107\) −15.5526 −1.50352 −0.751762 0.659434i \(-0.770796\pi\)
−0.751762 + 0.659434i \(0.770796\pi\)
\(108\) −4.66573 −0.448960
\(109\) −0.0266392 −0.00255157 −0.00127578 0.999999i \(-0.500406\pi\)
−0.00127578 + 0.999999i \(0.500406\pi\)
\(110\) −7.01937 −0.669271
\(111\) 7.96632 0.756129
\(112\) −36.2865 −3.42875
\(113\) −18.4333 −1.73406 −0.867031 0.498254i \(-0.833975\pi\)
−0.867031 + 0.498254i \(0.833975\pi\)
\(114\) −0.776064 −0.0726851
\(115\) −4.07243 −0.379756
\(116\) −1.45610 −0.135196
\(117\) 1.00000 0.0924500
\(118\) −23.8181 −2.19263
\(119\) 26.2790 2.40899
\(120\) −18.7117 −1.70814
\(121\) 1.00000 0.0909091
\(122\) 25.4572 2.30478
\(123\) −1.03087 −0.0929502
\(124\) 21.2335 1.90682
\(125\) 7.09119 0.634255
\(126\) 11.1033 0.989160
\(127\) −8.01514 −0.711229 −0.355615 0.934633i \(-0.615728\pi\)
−0.355615 + 0.934633i \(0.615728\pi\)
\(128\) 6.15212 0.543776
\(129\) −12.4906 −1.09974
\(130\) 7.01937 0.615640
\(131\) 11.5437 1.00858 0.504288 0.863536i \(-0.331755\pi\)
0.504288 + 0.863536i \(0.331755\pi\)
\(132\) 4.66573 0.406100
\(133\) 1.29271 0.112092
\(134\) 20.7842 1.79548
\(135\) 2.71878 0.233996
\(136\) 42.0553 3.60621
\(137\) −5.97781 −0.510719 −0.255360 0.966846i \(-0.582194\pi\)
−0.255360 + 0.966846i \(0.582194\pi\)
\(138\) 3.86725 0.329202
\(139\) 9.85576 0.835954 0.417977 0.908458i \(-0.362739\pi\)
0.417977 + 0.908458i \(0.362739\pi\)
\(140\) 54.5534 4.61060
\(141\) −13.1299 −1.10574
\(142\) 1.79848 0.150925
\(143\) −1.00000 −0.0836242
\(144\) 8.43756 0.703130
\(145\) 0.848491 0.0704633
\(146\) −11.8106 −0.977451
\(147\) −11.4951 −0.948097
\(148\) −37.1687 −3.05525
\(149\) −11.5703 −0.947877 −0.473938 0.880558i \(-0.657168\pi\)
−0.473938 + 0.880558i \(0.657168\pi\)
\(150\) 6.17511 0.504196
\(151\) −3.24308 −0.263918 −0.131959 0.991255i \(-0.542127\pi\)
−0.131959 + 0.991255i \(0.542127\pi\)
\(152\) 2.06878 0.167800
\(153\) −6.11056 −0.494010
\(154\) −11.1033 −0.894729
\(155\) −12.3730 −0.993825
\(156\) −4.66573 −0.373557
\(157\) 8.21363 0.655519 0.327759 0.944761i \(-0.393706\pi\)
0.327759 + 0.944761i \(0.393706\pi\)
\(158\) 44.1277 3.51061
\(159\) −8.49507 −0.673703
\(160\) 21.8029 1.72367
\(161\) −6.44179 −0.507684
\(162\) −2.58181 −0.202846
\(163\) −2.80574 −0.219763 −0.109881 0.993945i \(-0.535047\pi\)
−0.109881 + 0.993945i \(0.535047\pi\)
\(164\) 4.80975 0.375578
\(165\) −2.71878 −0.211657
\(166\) −22.6399 −1.75720
\(167\) −1.93827 −0.149987 −0.0749937 0.997184i \(-0.523894\pi\)
−0.0749937 + 0.997184i \(0.523894\pi\)
\(168\) −29.5984 −2.28356
\(169\) 1.00000 0.0769231
\(170\) −42.8923 −3.28969
\(171\) −0.300590 −0.0229867
\(172\) 58.2778 4.44364
\(173\) −8.55235 −0.650223 −0.325112 0.945676i \(-0.605402\pi\)
−0.325112 + 0.945676i \(0.605402\pi\)
\(174\) −0.805743 −0.0610832
\(175\) −10.2861 −0.777553
\(176\) −8.43756 −0.636005
\(177\) −9.22535 −0.693420
\(178\) 8.38451 0.628446
\(179\) 17.9312 1.34024 0.670121 0.742252i \(-0.266242\pi\)
0.670121 + 0.742252i \(0.266242\pi\)
\(180\) −12.6851 −0.945492
\(181\) 0.334272 0.0248462 0.0124231 0.999923i \(-0.496046\pi\)
0.0124231 + 0.999923i \(0.496046\pi\)
\(182\) 11.1033 0.823031
\(183\) 9.86021 0.728888
\(184\) −10.3090 −0.759993
\(185\) 21.6587 1.59238
\(186\) 11.7496 0.861526
\(187\) 6.11056 0.446848
\(188\) 61.2607 4.46790
\(189\) 4.30059 0.312822
\(190\) −2.10995 −0.153072
\(191\) −20.1538 −1.45827 −0.729137 0.684367i \(-0.760078\pi\)
−0.729137 + 0.684367i \(0.760078\pi\)
\(192\) −3.82934 −0.276359
\(193\) 22.5689 1.62455 0.812273 0.583278i \(-0.198230\pi\)
0.812273 + 0.583278i \(0.198230\pi\)
\(194\) −19.7008 −1.41444
\(195\) 2.71878 0.194696
\(196\) 53.6329 3.83092
\(197\) −26.6133 −1.89612 −0.948059 0.318095i \(-0.896957\pi\)
−0.948059 + 0.318095i \(0.896957\pi\)
\(198\) 2.58181 0.183481
\(199\) 4.81340 0.341213 0.170606 0.985339i \(-0.445427\pi\)
0.170606 + 0.985339i \(0.445427\pi\)
\(200\) −16.4612 −1.16398
\(201\) 8.05024 0.567820
\(202\) −9.90539 −0.696941
\(203\) 1.34215 0.0942004
\(204\) 28.5102 1.99611
\(205\) −2.80270 −0.195749
\(206\) −29.2556 −2.03834
\(207\) 1.49789 0.104110
\(208\) 8.43756 0.585040
\(209\) 0.300590 0.0207922
\(210\) 30.1874 2.08313
\(211\) 25.2596 1.73895 0.869473 0.493981i \(-0.164459\pi\)
0.869473 + 0.493981i \(0.164459\pi\)
\(212\) 39.6357 2.72219
\(213\) 0.696596 0.0477299
\(214\) 40.1538 2.74485
\(215\) −33.9593 −2.31600
\(216\) 6.88240 0.468288
\(217\) −19.5717 −1.32862
\(218\) 0.0687772 0.00465818
\(219\) −4.57454 −0.309119
\(220\) 12.6851 0.855229
\(221\) −6.11056 −0.411041
\(222\) −20.5675 −1.38040
\(223\) −11.8291 −0.792136 −0.396068 0.918221i \(-0.629626\pi\)
−0.396068 + 0.918221i \(0.629626\pi\)
\(224\) 34.4880 2.30433
\(225\) 2.39178 0.159452
\(226\) 47.5913 3.16573
\(227\) 21.1370 1.40291 0.701455 0.712714i \(-0.252534\pi\)
0.701455 + 0.712714i \(0.252534\pi\)
\(228\) 1.40247 0.0928808
\(229\) −11.6778 −0.771693 −0.385846 0.922563i \(-0.626091\pi\)
−0.385846 + 0.922563i \(0.626091\pi\)
\(230\) 10.5142 0.693287
\(231\) −4.30059 −0.282958
\(232\) 2.14789 0.141016
\(233\) 10.1156 0.662696 0.331348 0.943509i \(-0.392497\pi\)
0.331348 + 0.943509i \(0.392497\pi\)
\(234\) −2.58181 −0.168778
\(235\) −35.6974 −2.32864
\(236\) 43.0430 2.80186
\(237\) 17.0918 1.11023
\(238\) −67.8473 −4.39789
\(239\) 2.65081 0.171467 0.0857333 0.996318i \(-0.472677\pi\)
0.0857333 + 0.996318i \(0.472677\pi\)
\(240\) 22.9399 1.48076
\(241\) 4.15939 0.267930 0.133965 0.990986i \(-0.457229\pi\)
0.133965 + 0.990986i \(0.457229\pi\)
\(242\) −2.58181 −0.165965
\(243\) −1.00000 −0.0641500
\(244\) −46.0051 −2.94517
\(245\) −31.2526 −1.99666
\(246\) 2.66150 0.169691
\(247\) −0.300590 −0.0191261
\(248\) −31.3214 −1.98891
\(249\) −8.76902 −0.555714
\(250\) −18.3081 −1.15790
\(251\) 15.7115 0.991702 0.495851 0.868408i \(-0.334856\pi\)
0.495851 + 0.868408i \(0.334856\pi\)
\(252\) −20.0654 −1.26400
\(253\) −1.49789 −0.0941713
\(254\) 20.6936 1.29843
\(255\) −16.6133 −1.04036
\(256\) −23.5423 −1.47139
\(257\) 4.53944 0.283163 0.141581 0.989927i \(-0.454781\pi\)
0.141581 + 0.989927i \(0.454781\pi\)
\(258\) 32.2484 2.00770
\(259\) 34.2599 2.12880
\(260\) −12.6851 −0.786697
\(261\) −0.312085 −0.0193176
\(262\) −29.8035 −1.84127
\(263\) 8.38006 0.516737 0.258368 0.966046i \(-0.416815\pi\)
0.258368 + 0.966046i \(0.416815\pi\)
\(264\) −6.88240 −0.423582
\(265\) −23.0963 −1.41879
\(266\) −3.33753 −0.204637
\(267\) 3.24754 0.198746
\(268\) −37.5602 −2.29436
\(269\) −4.95140 −0.301892 −0.150946 0.988542i \(-0.548232\pi\)
−0.150946 + 0.988542i \(0.548232\pi\)
\(270\) −7.01937 −0.427185
\(271\) −24.5750 −1.49282 −0.746412 0.665484i \(-0.768225\pi\)
−0.746412 + 0.665484i \(0.768225\pi\)
\(272\) −51.5582 −3.12618
\(273\) 4.30059 0.260284
\(274\) 15.4336 0.932376
\(275\) −2.39178 −0.144230
\(276\) −6.98873 −0.420672
\(277\) 27.5913 1.65780 0.828901 0.559395i \(-0.188967\pi\)
0.828901 + 0.559395i \(0.188967\pi\)
\(278\) −25.4457 −1.52613
\(279\) 4.55094 0.272458
\(280\) −80.4715 −4.80909
\(281\) 10.3478 0.617297 0.308649 0.951176i \(-0.400123\pi\)
0.308649 + 0.951176i \(0.400123\pi\)
\(282\) 33.8990 2.01865
\(283\) −1.97640 −0.117485 −0.0587424 0.998273i \(-0.518709\pi\)
−0.0587424 + 0.998273i \(0.518709\pi\)
\(284\) −3.25013 −0.192859
\(285\) −0.817238 −0.0484090
\(286\) 2.58181 0.152665
\(287\) −4.43334 −0.261692
\(288\) −8.01937 −0.472546
\(289\) 20.3389 1.19641
\(290\) −2.19064 −0.128639
\(291\) −7.63063 −0.447316
\(292\) 21.3436 1.24904
\(293\) 13.5877 0.793800 0.396900 0.917862i \(-0.370086\pi\)
0.396900 + 0.917862i \(0.370086\pi\)
\(294\) 29.6781 1.73086
\(295\) −25.0817 −1.46031
\(296\) 54.8274 3.18678
\(297\) 1.00000 0.0580259
\(298\) 29.8723 1.73046
\(299\) 1.49789 0.0866250
\(300\) −11.1594 −0.644287
\(301\) −53.7170 −3.09620
\(302\) 8.37302 0.481813
\(303\) −3.83661 −0.220407
\(304\) −2.53624 −0.145464
\(305\) 26.8078 1.53501
\(306\) 15.7763 0.901870
\(307\) 31.8302 1.81664 0.908322 0.418271i \(-0.137364\pi\)
0.908322 + 0.418271i \(0.137364\pi\)
\(308\) 20.0654 1.14333
\(309\) −11.3315 −0.644624
\(310\) 31.9447 1.81434
\(311\) 6.64697 0.376915 0.188457 0.982081i \(-0.439651\pi\)
0.188457 + 0.982081i \(0.439651\pi\)
\(312\) 6.88240 0.389639
\(313\) −22.6171 −1.27840 −0.639198 0.769042i \(-0.720733\pi\)
−0.639198 + 0.769042i \(0.720733\pi\)
\(314\) −21.2060 −1.19672
\(315\) 11.6924 0.658790
\(316\) −79.7457 −4.48605
\(317\) 26.3971 1.48261 0.741303 0.671170i \(-0.234208\pi\)
0.741303 + 0.671170i \(0.234208\pi\)
\(318\) 21.9326 1.22992
\(319\) 0.312085 0.0174734
\(320\) −10.4112 −0.582001
\(321\) 15.5526 0.868060
\(322\) 16.6315 0.926835
\(323\) 1.83677 0.102201
\(324\) 4.66573 0.259207
\(325\) 2.39178 0.132672
\(326\) 7.24389 0.401202
\(327\) 0.0266392 0.00147315
\(328\) −7.09484 −0.391747
\(329\) −56.4665 −3.11310
\(330\) 7.01937 0.386404
\(331\) 20.1227 1.10604 0.553021 0.833167i \(-0.313475\pi\)
0.553021 + 0.833167i \(0.313475\pi\)
\(332\) 40.9139 2.24544
\(333\) −7.96632 −0.436552
\(334\) 5.00423 0.273819
\(335\) 21.8868 1.19581
\(336\) 36.2865 1.97959
\(337\) 13.5526 0.738256 0.369128 0.929379i \(-0.379656\pi\)
0.369128 + 0.929379i \(0.379656\pi\)
\(338\) −2.58181 −0.140432
\(339\) 18.4333 1.00116
\(340\) 77.5131 4.20374
\(341\) −4.55094 −0.246447
\(342\) 0.776064 0.0419648
\(343\) −19.3315 −1.04380
\(344\) −85.9654 −4.63494
\(345\) 4.07243 0.219252
\(346\) 22.0805 1.18706
\(347\) −2.24590 −0.120566 −0.0602831 0.998181i \(-0.519200\pi\)
−0.0602831 + 0.998181i \(0.519200\pi\)
\(348\) 1.45610 0.0780553
\(349\) 28.7634 1.53967 0.769835 0.638244i \(-0.220339\pi\)
0.769835 + 0.638244i \(0.220339\pi\)
\(350\) 26.5566 1.41951
\(351\) −1.00000 −0.0533761
\(352\) 8.01937 0.427434
\(353\) −24.4920 −1.30358 −0.651790 0.758400i \(-0.725982\pi\)
−0.651790 + 0.758400i \(0.725982\pi\)
\(354\) 23.8181 1.26592
\(355\) 1.89389 0.100517
\(356\) −15.1521 −0.803061
\(357\) −26.2790 −1.39083
\(358\) −46.2950 −2.44676
\(359\) 7.12852 0.376229 0.188114 0.982147i \(-0.439762\pi\)
0.188114 + 0.982147i \(0.439762\pi\)
\(360\) 18.7117 0.986195
\(361\) −18.9096 −0.995245
\(362\) −0.863025 −0.0453596
\(363\) −1.00000 −0.0524864
\(364\) −20.0654 −1.05171
\(365\) −12.4372 −0.650992
\(366\) −25.4572 −1.33067
\(367\) 6.56244 0.342556 0.171278 0.985223i \(-0.445210\pi\)
0.171278 + 0.985223i \(0.445210\pi\)
\(368\) 12.6385 0.658828
\(369\) 1.03087 0.0536648
\(370\) −55.9186 −2.90707
\(371\) −36.5338 −1.89674
\(372\) −21.2335 −1.10090
\(373\) 19.0345 0.985570 0.492785 0.870151i \(-0.335979\pi\)
0.492785 + 0.870151i \(0.335979\pi\)
\(374\) −15.7763 −0.815772
\(375\) −7.09119 −0.366187
\(376\) −90.3654 −4.66024
\(377\) −0.312085 −0.0160732
\(378\) −11.1033 −0.571092
\(379\) 9.46866 0.486372 0.243186 0.969980i \(-0.421807\pi\)
0.243186 + 0.969980i \(0.421807\pi\)
\(380\) 3.81301 0.195603
\(381\) 8.01514 0.410628
\(382\) 52.0331 2.66225
\(383\) 1.89389 0.0967734 0.0483867 0.998829i \(-0.484592\pi\)
0.0483867 + 0.998829i \(0.484592\pi\)
\(384\) −6.15212 −0.313949
\(385\) −11.6924 −0.595898
\(386\) −58.2686 −2.96579
\(387\) 12.4906 0.634934
\(388\) 35.6025 1.80744
\(389\) −6.31269 −0.320066 −0.160033 0.987112i \(-0.551160\pi\)
−0.160033 + 0.987112i \(0.551160\pi\)
\(390\) −7.01937 −0.355440
\(391\) −9.15292 −0.462883
\(392\) −79.1136 −3.99584
\(393\) −11.5437 −0.582301
\(394\) 68.7104 3.46158
\(395\) 46.4689 2.33810
\(396\) −4.66573 −0.234462
\(397\) −15.2949 −0.767631 −0.383816 0.923410i \(-0.625390\pi\)
−0.383816 + 0.923410i \(0.625390\pi\)
\(398\) −12.4273 −0.622922
\(399\) −1.29271 −0.0647166
\(400\) 20.1808 1.00904
\(401\) 2.45329 0.122511 0.0612557 0.998122i \(-0.480489\pi\)
0.0612557 + 0.998122i \(0.480489\pi\)
\(402\) −20.7842 −1.03662
\(403\) 4.55094 0.226698
\(404\) 17.9006 0.890587
\(405\) −2.71878 −0.135097
\(406\) −3.46517 −0.171973
\(407\) 7.96632 0.394876
\(408\) −42.0553 −2.08205
\(409\) −3.76114 −0.185977 −0.0929883 0.995667i \(-0.529642\pi\)
−0.0929883 + 0.995667i \(0.529642\pi\)
\(410\) 7.23604 0.357363
\(411\) 5.97781 0.294864
\(412\) 52.8695 2.60469
\(413\) −39.6744 −1.95225
\(414\) −3.86725 −0.190065
\(415\) −23.8411 −1.17031
\(416\) −8.01937 −0.393182
\(417\) −9.85576 −0.482639
\(418\) −0.776064 −0.0379586
\(419\) 32.2080 1.57346 0.786732 0.617295i \(-0.211771\pi\)
0.786732 + 0.617295i \(0.211771\pi\)
\(420\) −54.5534 −2.66193
\(421\) −26.3216 −1.28284 −0.641418 0.767191i \(-0.721654\pi\)
−0.641418 + 0.767191i \(0.721654\pi\)
\(422\) −65.2155 −3.17464
\(423\) 13.1299 0.638399
\(424\) −58.4665 −2.83938
\(425\) −14.6151 −0.708937
\(426\) −1.79848 −0.0871364
\(427\) 42.4047 2.05211
\(428\) −72.5641 −3.50752
\(429\) 1.00000 0.0482805
\(430\) 87.6763 4.22813
\(431\) 2.71619 0.130834 0.0654172 0.997858i \(-0.479162\pi\)
0.0654172 + 0.997858i \(0.479162\pi\)
\(432\) −8.43756 −0.405953
\(433\) 6.54790 0.314672 0.157336 0.987545i \(-0.449709\pi\)
0.157336 + 0.987545i \(0.449709\pi\)
\(434\) 50.5304 2.42554
\(435\) −0.848491 −0.0406820
\(436\) −0.124291 −0.00595246
\(437\) −0.450249 −0.0215383
\(438\) 11.8106 0.564331
\(439\) 18.9413 0.904020 0.452010 0.892013i \(-0.350707\pi\)
0.452010 + 0.892013i \(0.350707\pi\)
\(440\) −18.7117 −0.892047
\(441\) 11.4951 0.547384
\(442\) 15.7763 0.750402
\(443\) −33.8568 −1.60859 −0.804293 0.594233i \(-0.797456\pi\)
−0.804293 + 0.594233i \(0.797456\pi\)
\(444\) 37.1687 1.76395
\(445\) 8.82934 0.418551
\(446\) 30.5405 1.44613
\(447\) 11.5703 0.547257
\(448\) −16.4684 −0.778060
\(449\) −2.62721 −0.123986 −0.0619928 0.998077i \(-0.519746\pi\)
−0.0619928 + 0.998077i \(0.519746\pi\)
\(450\) −6.17511 −0.291097
\(451\) −1.03087 −0.0485416
\(452\) −86.0049 −4.04533
\(453\) 3.24308 0.152373
\(454\) −54.5716 −2.56117
\(455\) 11.6924 0.548147
\(456\) −2.06878 −0.0968794
\(457\) 19.3230 0.903892 0.451946 0.892045i \(-0.350730\pi\)
0.451946 + 0.892045i \(0.350730\pi\)
\(458\) 30.1499 1.40881
\(459\) 6.11056 0.285217
\(460\) −19.0008 −0.885918
\(461\) −6.17995 −0.287829 −0.143914 0.989590i \(-0.545969\pi\)
−0.143914 + 0.989590i \(0.545969\pi\)
\(462\) 11.1033 0.516572
\(463\) −3.36773 −0.156512 −0.0782558 0.996933i \(-0.524935\pi\)
−0.0782558 + 0.996933i \(0.524935\pi\)
\(464\) −2.63324 −0.122245
\(465\) 12.3730 0.573785
\(466\) −26.1166 −1.20983
\(467\) 37.3632 1.72896 0.864480 0.502667i \(-0.167648\pi\)
0.864480 + 0.502667i \(0.167648\pi\)
\(468\) 4.66573 0.215673
\(469\) 34.6208 1.59864
\(470\) 92.1639 4.25120
\(471\) −8.21363 −0.378464
\(472\) −63.4925 −2.92248
\(473\) −12.4906 −0.574319
\(474\) −44.1277 −2.02685
\(475\) −0.718944 −0.0329874
\(476\) 122.611 5.61985
\(477\) 8.49507 0.388963
\(478\) −6.84388 −0.313032
\(479\) −20.6133 −0.941845 −0.470922 0.882175i \(-0.656079\pi\)
−0.470922 + 0.882175i \(0.656079\pi\)
\(480\) −21.8029 −0.995163
\(481\) −7.96632 −0.363233
\(482\) −10.7387 −0.489136
\(483\) 6.44179 0.293112
\(484\) 4.66573 0.212079
\(485\) −20.7460 −0.942029
\(486\) 2.58181 0.117113
\(487\) 7.47711 0.338820 0.169410 0.985546i \(-0.445814\pi\)
0.169410 + 0.985546i \(0.445814\pi\)
\(488\) 67.8619 3.07196
\(489\) 2.80574 0.126880
\(490\) 80.6882 3.64512
\(491\) −11.6250 −0.524629 −0.262315 0.964982i \(-0.584486\pi\)
−0.262315 + 0.964982i \(0.584486\pi\)
\(492\) −4.80975 −0.216840
\(493\) 1.90701 0.0858875
\(494\) 0.776064 0.0349168
\(495\) 2.71878 0.122200
\(496\) 38.3989 1.72416
\(497\) 2.99577 0.134379
\(498\) 22.6399 1.01452
\(499\) 24.2714 1.08654 0.543268 0.839560i \(-0.317187\pi\)
0.543268 + 0.839560i \(0.317187\pi\)
\(500\) 33.0856 1.47963
\(501\) 1.93827 0.0865953
\(502\) −40.5641 −1.81046
\(503\) −31.7648 −1.41632 −0.708161 0.706051i \(-0.750475\pi\)
−0.708161 + 0.706051i \(0.750475\pi\)
\(504\) 29.5984 1.31842
\(505\) −10.4309 −0.464169
\(506\) 3.86725 0.171920
\(507\) −1.00000 −0.0444116
\(508\) −37.3965 −1.65920
\(509\) −8.06984 −0.357689 −0.178845 0.983877i \(-0.557236\pi\)
−0.178845 + 0.983877i \(0.557236\pi\)
\(510\) 42.8923 1.89930
\(511\) −19.6732 −0.870292
\(512\) 48.4774 2.14242
\(513\) 0.300590 0.0132714
\(514\) −11.7200 −0.516946
\(515\) −30.8078 −1.35755
\(516\) −58.2778 −2.56554
\(517\) −13.1299 −0.577454
\(518\) −88.4524 −3.88637
\(519\) 8.55235 0.375407
\(520\) 18.7117 0.820564
\(521\) 16.9986 0.744722 0.372361 0.928088i \(-0.378548\pi\)
0.372361 + 0.928088i \(0.378548\pi\)
\(522\) 0.805743 0.0352664
\(523\) 7.38591 0.322963 0.161482 0.986876i \(-0.448373\pi\)
0.161482 + 0.986876i \(0.448373\pi\)
\(524\) 53.8596 2.35287
\(525\) 10.2861 0.448920
\(526\) −21.6357 −0.943361
\(527\) −27.8088 −1.21137
\(528\) 8.43756 0.367198
\(529\) −20.7563 −0.902449
\(530\) 59.6301 2.59016
\(531\) 9.22535 0.400346
\(532\) 6.03145 0.261496
\(533\) 1.03087 0.0446518
\(534\) −8.38451 −0.362833
\(535\) 42.2841 1.82810
\(536\) 55.4049 2.39313
\(537\) −17.9312 −0.773789
\(538\) 12.7836 0.551138
\(539\) −11.4951 −0.495128
\(540\) 12.6851 0.545880
\(541\) 34.9901 1.50434 0.752172 0.658967i \(-0.229006\pi\)
0.752172 + 0.658967i \(0.229006\pi\)
\(542\) 63.4479 2.72532
\(543\) −0.334272 −0.0143450
\(544\) 49.0029 2.10098
\(545\) 0.0724261 0.00310239
\(546\) −11.1033 −0.475177
\(547\) 9.93203 0.424663 0.212331 0.977198i \(-0.431894\pi\)
0.212331 + 0.977198i \(0.431894\pi\)
\(548\) −27.8909 −1.19144
\(549\) −9.86021 −0.420824
\(550\) 6.17511 0.263308
\(551\) 0.0938094 0.00399642
\(552\) 10.3090 0.438782
\(553\) 73.5048 3.12574
\(554\) −71.2355 −3.02651
\(555\) −21.6587 −0.919360
\(556\) 45.9843 1.95017
\(557\) 4.53624 0.192207 0.0961034 0.995371i \(-0.469362\pi\)
0.0961034 + 0.995371i \(0.469362\pi\)
\(558\) −11.7496 −0.497402
\(559\) 12.4906 0.528297
\(560\) 98.6551 4.16894
\(561\) −6.11056 −0.257988
\(562\) −26.7160 −1.12695
\(563\) −24.0916 −1.01534 −0.507669 0.861552i \(-0.669493\pi\)
−0.507669 + 0.861552i \(0.669493\pi\)
\(564\) −61.2607 −2.57954
\(565\) 50.1162 2.10841
\(566\) 5.10268 0.214482
\(567\) −4.30059 −0.180608
\(568\) 4.79425 0.201162
\(569\) 2.72583 0.114273 0.0571363 0.998366i \(-0.481803\pi\)
0.0571363 + 0.998366i \(0.481803\pi\)
\(570\) 2.10995 0.0883761
\(571\) 37.1221 1.55351 0.776755 0.629802i \(-0.216864\pi\)
0.776755 + 0.629802i \(0.216864\pi\)
\(572\) −4.66573 −0.195084
\(573\) 20.1538 0.841935
\(574\) 11.4460 0.477748
\(575\) 3.58261 0.149405
\(576\) 3.82934 0.159556
\(577\) 8.06173 0.335614 0.167807 0.985820i \(-0.446331\pi\)
0.167807 + 0.985820i \(0.446331\pi\)
\(578\) −52.5112 −2.18418
\(579\) −22.5689 −0.937932
\(580\) 3.95883 0.164381
\(581\) −37.7120 −1.56456
\(582\) 19.7008 0.816625
\(583\) −8.49507 −0.351830
\(584\) −31.4838 −1.30281
\(585\) −2.71878 −0.112408
\(586\) −35.0808 −1.44917
\(587\) 1.16924 0.0482599 0.0241299 0.999709i \(-0.492318\pi\)
0.0241299 + 0.999709i \(0.492318\pi\)
\(588\) −53.6329 −2.21178
\(589\) −1.36797 −0.0563660
\(590\) 64.7562 2.66597
\(591\) 26.6133 1.09472
\(592\) −67.2163 −2.76257
\(593\) 13.6550 0.560745 0.280373 0.959891i \(-0.409542\pi\)
0.280373 + 0.959891i \(0.409542\pi\)
\(594\) −2.58181 −0.105933
\(595\) −71.4469 −2.92904
\(596\) −53.9839 −2.21127
\(597\) −4.81340 −0.196999
\(598\) −3.86725 −0.158144
\(599\) −23.0289 −0.940935 −0.470467 0.882417i \(-0.655915\pi\)
−0.470467 + 0.882417i \(0.655915\pi\)
\(600\) 16.4612 0.672024
\(601\) 1.70261 0.0694509 0.0347255 0.999397i \(-0.488944\pi\)
0.0347255 + 0.999397i \(0.488944\pi\)
\(602\) 138.687 5.65246
\(603\) −8.05024 −0.327831
\(604\) −15.1313 −0.615686
\(605\) −2.71878 −0.110534
\(606\) 9.90539 0.402379
\(607\) 8.51584 0.345647 0.172824 0.984953i \(-0.444711\pi\)
0.172824 + 0.984953i \(0.444711\pi\)
\(608\) 2.41054 0.0977603
\(609\) −1.34215 −0.0543866
\(610\) −69.2125 −2.80233
\(611\) 13.1299 0.531180
\(612\) −28.5102 −1.15246
\(613\) −0.185577 −0.00749538 −0.00374769 0.999993i \(-0.501193\pi\)
−0.00374769 + 0.999993i \(0.501193\pi\)
\(614\) −82.1794 −3.31649
\(615\) 2.80270 0.113016
\(616\) −29.5984 −1.19255
\(617\) −11.7230 −0.471951 −0.235975 0.971759i \(-0.575828\pi\)
−0.235975 + 0.971759i \(0.575828\pi\)
\(618\) 29.2556 1.17683
\(619\) −44.5649 −1.79121 −0.895607 0.444845i \(-0.853259\pi\)
−0.895607 + 0.444845i \(0.853259\pi\)
\(620\) −57.7291 −2.31846
\(621\) −1.49789 −0.0601081
\(622\) −17.1612 −0.688101
\(623\) 13.9663 0.559549
\(624\) −8.43756 −0.337773
\(625\) −31.2383 −1.24953
\(626\) 58.3931 2.33386
\(627\) −0.300590 −0.0120044
\(628\) 38.3226 1.52924
\(629\) 48.6787 1.94095
\(630\) −30.1874 −1.20270
\(631\) −5.29191 −0.210668 −0.105334 0.994437i \(-0.533591\pi\)
−0.105334 + 0.994437i \(0.533591\pi\)
\(632\) 117.633 4.67917
\(633\) −25.2596 −1.00398
\(634\) −68.1521 −2.70667
\(635\) 21.7914 0.864767
\(636\) −39.6357 −1.57166
\(637\) 11.4951 0.455451
\(638\) −0.805743 −0.0318997
\(639\) −0.696596 −0.0275569
\(640\) −16.7263 −0.661164
\(641\) 13.1406 0.519023 0.259512 0.965740i \(-0.416438\pi\)
0.259512 + 0.965740i \(0.416438\pi\)
\(642\) −40.1538 −1.58474
\(643\) 23.8594 0.940923 0.470462 0.882420i \(-0.344087\pi\)
0.470462 + 0.882420i \(0.344087\pi\)
\(644\) −30.0557 −1.18436
\(645\) 33.9593 1.33715
\(646\) −4.74219 −0.186579
\(647\) −12.0472 −0.473624 −0.236812 0.971555i \(-0.576103\pi\)
−0.236812 + 0.971555i \(0.576103\pi\)
\(648\) −6.88240 −0.270366
\(649\) −9.22535 −0.362127
\(650\) −6.17511 −0.242208
\(651\) 19.5717 0.767076
\(652\) −13.0908 −0.512677
\(653\) 2.21222 0.0865707 0.0432854 0.999063i \(-0.486218\pi\)
0.0432854 + 0.999063i \(0.486218\pi\)
\(654\) −0.0687772 −0.00268940
\(655\) −31.3847 −1.22630
\(656\) 8.69801 0.339600
\(657\) 4.57454 0.178470
\(658\) 145.785 5.68331
\(659\) 28.6764 1.11708 0.558538 0.829479i \(-0.311363\pi\)
0.558538 + 0.829479i \(0.311363\pi\)
\(660\) −12.6851 −0.493767
\(661\) −27.9500 −1.08713 −0.543564 0.839367i \(-0.682926\pi\)
−0.543564 + 0.839367i \(0.682926\pi\)
\(662\) −51.9528 −2.01920
\(663\) 6.11056 0.237315
\(664\) −60.3519 −2.34211
\(665\) −3.51460 −0.136291
\(666\) 20.5675 0.796974
\(667\) −0.467467 −0.0181004
\(668\) −9.04342 −0.349900
\(669\) 11.8291 0.457340
\(670\) −56.5076 −2.18308
\(671\) 9.86021 0.380649
\(672\) −34.4880 −1.33040
\(673\) −28.5534 −1.10065 −0.550327 0.834949i \(-0.685497\pi\)
−0.550327 + 0.834949i \(0.685497\pi\)
\(674\) −34.9901 −1.34777
\(675\) −2.39178 −0.0920596
\(676\) 4.66573 0.179451
\(677\) 7.52290 0.289129 0.144564 0.989495i \(-0.453822\pi\)
0.144564 + 0.989495i \(0.453822\pi\)
\(678\) −47.5913 −1.82773
\(679\) −32.8162 −1.25937
\(680\) −114.339 −4.38471
\(681\) −21.1370 −0.809971
\(682\) 11.7496 0.449917
\(683\) −1.33324 −0.0510152 −0.0255076 0.999675i \(-0.508120\pi\)
−0.0255076 + 0.999675i \(0.508120\pi\)
\(684\) −1.40247 −0.0536248
\(685\) 16.2524 0.620971
\(686\) 49.9101 1.90558
\(687\) 11.6778 0.445537
\(688\) 105.390 4.01797
\(689\) 8.49507 0.323636
\(690\) −10.5142 −0.400269
\(691\) 8.96796 0.341157 0.170579 0.985344i \(-0.445436\pi\)
0.170579 + 0.985344i \(0.445436\pi\)
\(692\) −39.9030 −1.51688
\(693\) 4.30059 0.163366
\(694\) 5.79848 0.220107
\(695\) −26.7957 −1.01642
\(696\) −2.14789 −0.0814156
\(697\) −6.29918 −0.238598
\(698\) −74.2615 −2.81084
\(699\) −10.1156 −0.382608
\(700\) −47.9919 −1.81392
\(701\) 7.66269 0.289416 0.144708 0.989474i \(-0.453776\pi\)
0.144708 + 0.989474i \(0.453776\pi\)
\(702\) 2.58181 0.0974440
\(703\) 2.39459 0.0903138
\(704\) −3.82934 −0.144324
\(705\) 35.6974 1.34444
\(706\) 63.2337 2.37983
\(707\) −16.4997 −0.620535
\(708\) −43.0430 −1.61765
\(709\) 17.4253 0.654420 0.327210 0.944952i \(-0.393892\pi\)
0.327210 + 0.944952i \(0.393892\pi\)
\(710\) −4.88966 −0.183506
\(711\) −17.0918 −0.640993
\(712\) 22.3508 0.837633
\(713\) 6.81679 0.255291
\(714\) 67.8473 2.53912
\(715\) 2.71878 0.101677
\(716\) 83.6622 3.12660
\(717\) −2.65081 −0.0989963
\(718\) −18.4045 −0.686848
\(719\) 6.79764 0.253509 0.126755 0.991934i \(-0.459544\pi\)
0.126755 + 0.991934i \(0.459544\pi\)
\(720\) −22.9399 −0.854920
\(721\) −48.7319 −1.81487
\(722\) 48.8211 1.81693
\(723\) −4.15939 −0.154689
\(724\) 1.55962 0.0579628
\(725\) −0.746437 −0.0277220
\(726\) 2.58181 0.0958199
\(727\) −30.6932 −1.13835 −0.569174 0.822217i \(-0.692737\pi\)
−0.569174 + 0.822217i \(0.692737\pi\)
\(728\) 29.5984 1.09699
\(729\) 1.00000 0.0370370
\(730\) 32.1104 1.18846
\(731\) −76.3247 −2.82297
\(732\) 46.0051 1.70040
\(733\) −16.3482 −0.603836 −0.301918 0.953334i \(-0.597627\pi\)
−0.301918 + 0.953334i \(0.597627\pi\)
\(734\) −16.9429 −0.625375
\(735\) 31.2526 1.15277
\(736\) −12.0121 −0.442772
\(737\) 8.05024 0.296534
\(738\) −2.66150 −0.0979712
\(739\) −19.9677 −0.734525 −0.367262 0.930117i \(-0.619705\pi\)
−0.367262 + 0.930117i \(0.619705\pi\)
\(740\) 101.054 3.71480
\(741\) 0.300590 0.0110424
\(742\) 94.3233 3.46272
\(743\) 18.9756 0.696148 0.348074 0.937467i \(-0.386836\pi\)
0.348074 + 0.937467i \(0.386836\pi\)
\(744\) 31.3214 1.14830
\(745\) 31.4572 1.15250
\(746\) −49.1435 −1.79927
\(747\) 8.76902 0.320842
\(748\) 28.5102 1.04244
\(749\) 66.8853 2.44393
\(750\) 18.3081 0.668517
\(751\) 5.94717 0.217015 0.108508 0.994096i \(-0.465393\pi\)
0.108508 + 0.994096i \(0.465393\pi\)
\(752\) 110.785 4.03990
\(753\) −15.7115 −0.572559
\(754\) 0.805743 0.0293434
\(755\) 8.81724 0.320892
\(756\) 20.0654 0.729771
\(757\) −21.7275 −0.789698 −0.394849 0.918746i \(-0.629203\pi\)
−0.394849 + 0.918746i \(0.629203\pi\)
\(758\) −24.4462 −0.887928
\(759\) 1.49789 0.0543698
\(760\) −5.62456 −0.204024
\(761\) −11.7241 −0.424997 −0.212499 0.977161i \(-0.568160\pi\)
−0.212499 + 0.977161i \(0.568160\pi\)
\(762\) −20.6936 −0.749648
\(763\) 0.114564 0.00414750
\(764\) −94.0320 −3.40196
\(765\) 16.6133 0.600655
\(766\) −4.88966 −0.176671
\(767\) 9.22535 0.333108
\(768\) 23.5423 0.849509
\(769\) 27.0676 0.976084 0.488042 0.872820i \(-0.337711\pi\)
0.488042 + 0.872820i \(0.337711\pi\)
\(770\) 30.1874 1.08788
\(771\) −4.53944 −0.163484
\(772\) 105.300 3.78984
\(773\) 17.6574 0.635094 0.317547 0.948243i \(-0.397141\pi\)
0.317547 + 0.948243i \(0.397141\pi\)
\(774\) −32.2484 −1.15914
\(775\) 10.8848 0.390995
\(776\) −52.5170 −1.88525
\(777\) −34.2599 −1.22907
\(778\) 16.2982 0.584317
\(779\) −0.309868 −0.0111022
\(780\) 12.6851 0.454199
\(781\) 0.696596 0.0249262
\(782\) 23.6311 0.845046
\(783\) 0.312085 0.0111530
\(784\) 96.9904 3.46394
\(785\) −22.3311 −0.797030
\(786\) 29.8035 1.06306
\(787\) −19.9677 −0.711773 −0.355886 0.934529i \(-0.615821\pi\)
−0.355886 + 0.934529i \(0.615821\pi\)
\(788\) −124.170 −4.42339
\(789\) −8.38006 −0.298338
\(790\) −119.974 −4.26847
\(791\) 79.2742 2.81867
\(792\) 6.88240 0.244555
\(793\) −9.86021 −0.350146
\(794\) 39.4886 1.40140
\(795\) 23.0963 0.819140
\(796\) 22.4580 0.796003
\(797\) 15.2343 0.539625 0.269812 0.962913i \(-0.413038\pi\)
0.269812 + 0.962913i \(0.413038\pi\)
\(798\) 3.33753 0.118147
\(799\) −80.2312 −2.83838
\(800\) −19.1806 −0.678135
\(801\) −3.24754 −0.114746
\(802\) −6.33392 −0.223658
\(803\) −4.57454 −0.161432
\(804\) 37.5602 1.32465
\(805\) 17.5138 0.617282
\(806\) −11.7496 −0.413864
\(807\) 4.95140 0.174297
\(808\) −26.4051 −0.928927
\(809\) −34.4551 −1.21138 −0.605689 0.795701i \(-0.707103\pi\)
−0.605689 + 0.795701i \(0.707103\pi\)
\(810\) 7.01937 0.246636
\(811\) 4.75409 0.166939 0.0834693 0.996510i \(-0.473400\pi\)
0.0834693 + 0.996510i \(0.473400\pi\)
\(812\) 6.26210 0.219757
\(813\) 24.5750 0.861882
\(814\) −20.5675 −0.720890
\(815\) 7.62820 0.267204
\(816\) 51.5582 1.80490
\(817\) −3.75455 −0.131355
\(818\) 9.71055 0.339521
\(819\) −4.30059 −0.150275
\(820\) −13.0767 −0.456657
\(821\) 7.36700 0.257110 0.128555 0.991702i \(-0.458966\pi\)
0.128555 + 0.991702i \(0.458966\pi\)
\(822\) −15.4336 −0.538307
\(823\) 35.5385 1.23879 0.619397 0.785078i \(-0.287377\pi\)
0.619397 + 0.785078i \(0.287377\pi\)
\(824\) −77.9876 −2.71683
\(825\) 2.39178 0.0832710
\(826\) 102.432 3.56406
\(827\) −16.1506 −0.561610 −0.280805 0.959765i \(-0.590601\pi\)
−0.280805 + 0.959765i \(0.590601\pi\)
\(828\) 6.98873 0.242875
\(829\) 28.9986 1.00716 0.503581 0.863948i \(-0.332015\pi\)
0.503581 + 0.863948i \(0.332015\pi\)
\(830\) 61.5530 2.13654
\(831\) −27.5913 −0.957132
\(832\) 3.82934 0.132759
\(833\) −70.2413 −2.43372
\(834\) 25.4457 0.881111
\(835\) 5.26972 0.182366
\(836\) 1.40247 0.0485054
\(837\) −4.55094 −0.157303
\(838\) −83.1548 −2.87254
\(839\) −12.7246 −0.439304 −0.219652 0.975578i \(-0.570492\pi\)
−0.219652 + 0.975578i \(0.570492\pi\)
\(840\) 80.4715 2.77653
\(841\) −28.9026 −0.996641
\(842\) 67.9573 2.34196
\(843\) −10.3478 −0.356397
\(844\) 117.855 4.05672
\(845\) −2.71878 −0.0935290
\(846\) −33.8990 −1.16547
\(847\) −4.30059 −0.147770
\(848\) 71.6777 2.46142
\(849\) 1.97640 0.0678299
\(850\) 37.7334 1.29424
\(851\) −11.9326 −0.409045
\(852\) 3.25013 0.111347
\(853\) 31.0434 1.06291 0.531453 0.847088i \(-0.321646\pi\)
0.531453 + 0.847088i \(0.321646\pi\)
\(854\) −109.481 −3.74636
\(855\) 0.817238 0.0279489
\(856\) 107.039 3.65852
\(857\) 36.4658 1.24565 0.622825 0.782361i \(-0.285985\pi\)
0.622825 + 0.782361i \(0.285985\pi\)
\(858\) −2.58181 −0.0881414
\(859\) −44.3200 −1.51218 −0.756089 0.654468i \(-0.772892\pi\)
−0.756089 + 0.654468i \(0.772892\pi\)
\(860\) −158.445 −5.40292
\(861\) 4.43334 0.151088
\(862\) −7.01268 −0.238853
\(863\) −35.1467 −1.19641 −0.598204 0.801344i \(-0.704119\pi\)
−0.598204 + 0.801344i \(0.704119\pi\)
\(864\) 8.01937 0.272825
\(865\) 23.2520 0.790591
\(866\) −16.9054 −0.574469
\(867\) −20.3389 −0.690747
\(868\) −91.3164 −3.09948
\(869\) 17.0918 0.579800
\(870\) 2.19064 0.0742696
\(871\) −8.05024 −0.272772
\(872\) 0.183341 0.00620872
\(873\) 7.63063 0.258258
\(874\) 1.16246 0.0393207
\(875\) −30.4963 −1.03096
\(876\) −21.3436 −0.721132
\(877\) −11.2927 −0.381328 −0.190664 0.981655i \(-0.561064\pi\)
−0.190664 + 0.981655i \(0.561064\pi\)
\(878\) −48.9028 −1.65039
\(879\) −13.5877 −0.458301
\(880\) 22.9399 0.773304
\(881\) 31.7503 1.06969 0.534847 0.844949i \(-0.320369\pi\)
0.534847 + 0.844949i \(0.320369\pi\)
\(882\) −29.6781 −0.999312
\(883\) 42.8824 1.44311 0.721554 0.692358i \(-0.243428\pi\)
0.721554 + 0.692358i \(0.243428\pi\)
\(884\) −28.5102 −0.958902
\(885\) 25.0817 0.843113
\(886\) 87.4118 2.93666
\(887\) 32.8290 1.10229 0.551144 0.834410i \(-0.314191\pi\)
0.551144 + 0.834410i \(0.314191\pi\)
\(888\) −54.8274 −1.83989
\(889\) 34.4698 1.15608
\(890\) −22.7957 −0.764112
\(891\) −1.00000 −0.0335013
\(892\) −55.1915 −1.84795
\(893\) −3.94672 −0.132072
\(894\) −29.8723 −0.999080
\(895\) −48.7511 −1.62957
\(896\) −26.4577 −0.883891
\(897\) −1.49789 −0.0500130
\(898\) 6.78295 0.226350
\(899\) −1.42028 −0.0473690
\(900\) 11.1594 0.371980
\(901\) −51.9096 −1.72936
\(902\) 2.66150 0.0886183
\(903\) 53.7170 1.78759
\(904\) 126.866 4.21949
\(905\) −0.908812 −0.0302099
\(906\) −8.37302 −0.278175
\(907\) 43.1597 1.43309 0.716546 0.697539i \(-0.245722\pi\)
0.716546 + 0.697539i \(0.245722\pi\)
\(908\) 98.6194 3.27280
\(909\) 3.83661 0.127252
\(910\) −30.1874 −1.00070
\(911\) −28.7690 −0.953160 −0.476580 0.879131i \(-0.658124\pi\)
−0.476580 + 0.879131i \(0.658124\pi\)
\(912\) 2.53624 0.0839835
\(913\) −8.76902 −0.290212
\(914\) −49.8883 −1.65016
\(915\) −26.8078 −0.886237
\(916\) −54.4856 −1.80025
\(917\) −49.6446 −1.63941
\(918\) −15.7763 −0.520695
\(919\) 26.5804 0.876807 0.438403 0.898778i \(-0.355544\pi\)
0.438403 + 0.898778i \(0.355544\pi\)
\(920\) 28.0281 0.924058
\(921\) −31.8302 −1.04884
\(922\) 15.9554 0.525464
\(923\) −0.696596 −0.0229287
\(924\) −20.0654 −0.660103
\(925\) −19.0537 −0.626481
\(926\) 8.69483 0.285730
\(927\) 11.3315 0.372174
\(928\) 2.50272 0.0821559
\(929\) 28.1614 0.923946 0.461973 0.886894i \(-0.347142\pi\)
0.461973 + 0.886894i \(0.347142\pi\)
\(930\) −31.9447 −1.04751
\(931\) −3.45530 −0.113243
\(932\) 47.1967 1.54598
\(933\) −6.64697 −0.217612
\(934\) −96.4645 −3.15641
\(935\) −16.6133 −0.543312
\(936\) −6.88240 −0.224958
\(937\) −35.6727 −1.16537 −0.582687 0.812696i \(-0.697999\pi\)
−0.582687 + 0.812696i \(0.697999\pi\)
\(938\) −89.3842 −2.91850
\(939\) 22.6171 0.738082
\(940\) −166.555 −5.43241
\(941\) 18.0210 0.587468 0.293734 0.955887i \(-0.405102\pi\)
0.293734 + 0.955887i \(0.405102\pi\)
\(942\) 21.2060 0.690929
\(943\) 1.54412 0.0502835
\(944\) 77.8395 2.53346
\(945\) −11.6924 −0.380353
\(946\) 32.2484 1.04848
\(947\) −25.0093 −0.812693 −0.406346 0.913719i \(-0.633197\pi\)
−0.406346 + 0.913719i \(0.633197\pi\)
\(948\) 79.7457 2.59002
\(949\) 4.57454 0.148496
\(950\) 1.85617 0.0602222
\(951\) −26.3971 −0.855983
\(952\) −180.863 −5.86179
\(953\) 18.2302 0.590534 0.295267 0.955415i \(-0.404591\pi\)
0.295267 + 0.955415i \(0.404591\pi\)
\(954\) −21.9326 −0.710095
\(955\) 54.7937 1.77308
\(956\) 12.3680 0.400008
\(957\) −0.312085 −0.0100883
\(958\) 53.2195 1.71944
\(959\) 25.7081 0.830159
\(960\) 10.4112 0.336018
\(961\) −10.2889 −0.331901
\(962\) 20.5675 0.663123
\(963\) −15.5526 −0.501175
\(964\) 19.4066 0.625043
\(965\) −61.3600 −1.97525
\(966\) −16.6315 −0.535109
\(967\) 21.5792 0.693941 0.346970 0.937876i \(-0.387210\pi\)
0.346970 + 0.937876i \(0.387210\pi\)
\(968\) −6.88240 −0.221209
\(969\) −1.83677 −0.0590056
\(970\) 53.5623 1.71978
\(971\) 52.4263 1.68244 0.841220 0.540693i \(-0.181838\pi\)
0.841220 + 0.540693i \(0.181838\pi\)
\(972\) −4.66573 −0.149653
\(973\) −42.3856 −1.35882
\(974\) −19.3045 −0.618555
\(975\) −2.39178 −0.0765982
\(976\) −83.1962 −2.66304
\(977\) 50.5189 1.61624 0.808122 0.589015i \(-0.200484\pi\)
0.808122 + 0.589015i \(0.200484\pi\)
\(978\) −7.24389 −0.231634
\(979\) 3.24754 0.103792
\(980\) −145.816 −4.65793
\(981\) −0.0266392 −0.000850523 0
\(982\) 30.0135 0.957770
\(983\) −29.2556 −0.933110 −0.466555 0.884492i \(-0.654505\pi\)
−0.466555 + 0.884492i \(0.654505\pi\)
\(984\) 7.09484 0.226175
\(985\) 72.3557 2.30545
\(986\) −4.92354 −0.156797
\(987\) 56.4665 1.79735
\(988\) −1.40247 −0.0446185
\(989\) 18.7095 0.594928
\(990\) −7.01937 −0.223090
\(991\) −12.1117 −0.384742 −0.192371 0.981322i \(-0.561618\pi\)
−0.192371 + 0.981322i \(0.561618\pi\)
\(992\) −36.4957 −1.15874
\(993\) −20.1227 −0.638573
\(994\) −7.73451 −0.245324
\(995\) −13.0866 −0.414872
\(996\) −40.9139 −1.29641
\(997\) −47.9395 −1.51826 −0.759129 0.650940i \(-0.774375\pi\)
−0.759129 + 0.650940i \(0.774375\pi\)
\(998\) −62.6640 −1.98359
\(999\) 7.96632 0.252043
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 429.2.a.h.1.1 4
3.2 odd 2 1287.2.a.m.1.4 4
4.3 odd 2 6864.2.a.bz.1.1 4
11.10 odd 2 4719.2.a.z.1.4 4
13.12 even 2 5577.2.a.m.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
429.2.a.h.1.1 4 1.1 even 1 trivial
1287.2.a.m.1.4 4 3.2 odd 2
4719.2.a.z.1.4 4 11.10 odd 2
5577.2.a.m.1.4 4 13.12 even 2
6864.2.a.bz.1.1 4 4.3 odd 2