Properties

Label 8041.2.a.c.1.3
Level $8041$
Weight $2$
Character 8041.1
Self dual yes
Analytic conductor $64.208$
Analytic rank $1$
Dimension $60$
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] = [8041,2,Mod(1,8041)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, 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("8041.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8041 = 11 \cdot 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8041.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(64.2077082653\)
Analytic rank: \(1\)
Dimension: \(60\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Character \(\chi\) \(=\) 8041.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.54628 q^{2} +1.17861 q^{3} +4.48352 q^{4} +1.63315 q^{5} -3.00107 q^{6} -0.708864 q^{7} -6.32373 q^{8} -1.61088 q^{9} +O(q^{10})\) \(q-2.54628 q^{2} +1.17861 q^{3} +4.48352 q^{4} +1.63315 q^{5} -3.00107 q^{6} -0.708864 q^{7} -6.32373 q^{8} -1.61088 q^{9} -4.15845 q^{10} +1.00000 q^{11} +5.28433 q^{12} -0.231092 q^{13} +1.80496 q^{14} +1.92485 q^{15} +7.13491 q^{16} +1.00000 q^{17} +4.10173 q^{18} +2.82304 q^{19} +7.32226 q^{20} -0.835475 q^{21} -2.54628 q^{22} -4.91189 q^{23} -7.45322 q^{24} -2.33282 q^{25} +0.588424 q^{26} -5.43443 q^{27} -3.17820 q^{28} +5.74697 q^{29} -4.90120 q^{30} +10.6463 q^{31} -5.52000 q^{32} +1.17861 q^{33} -2.54628 q^{34} -1.15768 q^{35} -7.22239 q^{36} -5.71901 q^{37} -7.18823 q^{38} -0.272368 q^{39} -10.3276 q^{40} -3.74440 q^{41} +2.12735 q^{42} -1.00000 q^{43} +4.48352 q^{44} -2.63080 q^{45} +12.5070 q^{46} -8.08760 q^{47} +8.40929 q^{48} -6.49751 q^{49} +5.94001 q^{50} +1.17861 q^{51} -1.03611 q^{52} +13.1046 q^{53} +13.8376 q^{54} +1.63315 q^{55} +4.48266 q^{56} +3.32726 q^{57} -14.6334 q^{58} -10.6936 q^{59} +8.63010 q^{60} -7.14368 q^{61} -27.1084 q^{62} +1.14189 q^{63} -0.214376 q^{64} -0.377408 q^{65} -3.00107 q^{66} +1.17446 q^{67} +4.48352 q^{68} -5.78921 q^{69} +2.94777 q^{70} -16.1972 q^{71} +10.1867 q^{72} +4.11659 q^{73} +14.5622 q^{74} -2.74949 q^{75} +12.6571 q^{76} -0.708864 q^{77} +0.693523 q^{78} -6.14710 q^{79} +11.6524 q^{80} -1.57246 q^{81} +9.53427 q^{82} +7.46155 q^{83} -3.74587 q^{84} +1.63315 q^{85} +2.54628 q^{86} +6.77345 q^{87} -6.32373 q^{88} -9.20969 q^{89} +6.69874 q^{90} +0.163813 q^{91} -22.0226 q^{92} +12.5478 q^{93} +20.5933 q^{94} +4.61044 q^{95} -6.50594 q^{96} +0.962798 q^{97} +16.5445 q^{98} -1.61088 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 60 q - 9 q^{2} - 6 q^{3} + 43 q^{4} - 15 q^{5} - 4 q^{6} - 17 q^{7} - 21 q^{8} + 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 60 q - 9 q^{2} - 6 q^{3} + 43 q^{4} - 15 q^{5} - 4 q^{6} - 17 q^{7} - 21 q^{8} + 40 q^{9} + q^{10} + 60 q^{11} - 13 q^{12} - 2 q^{13} - 15 q^{14} - 32 q^{15} + 21 q^{16} + 60 q^{17} - 41 q^{18} - 24 q^{19} - 33 q^{20} - 12 q^{21} - 9 q^{22} - 20 q^{23} + 9 q^{24} + 25 q^{25} - 40 q^{26} - 18 q^{27} - 3 q^{28} - 54 q^{29} - 18 q^{30} - 50 q^{31} - 51 q^{32} - 6 q^{33} - 9 q^{34} - 30 q^{35} + 27 q^{36} - 63 q^{37} - 38 q^{38} - 55 q^{39} - 5 q^{40} - 53 q^{41} - 47 q^{42} - 60 q^{43} + 43 q^{44} - 36 q^{45} - 27 q^{46} - 47 q^{47} - 38 q^{48} + 5 q^{49} - 4 q^{50} - 6 q^{51} - 13 q^{52} - 24 q^{53} - 58 q^{54} - 15 q^{55} - 45 q^{56} + 23 q^{57} + 16 q^{58} - 57 q^{59} - 4 q^{60} - 8 q^{61} - 3 q^{62} - 57 q^{63} - 5 q^{64} - 27 q^{65} - 4 q^{66} - 49 q^{67} + 43 q^{68} - 57 q^{69} - 7 q^{70} - 151 q^{71} - 29 q^{72} - 12 q^{73} + 18 q^{74} - 23 q^{75} - 38 q^{76} - 17 q^{77} + 37 q^{78} - 11 q^{79} - 21 q^{80} + 28 q^{81} + 44 q^{82} - 42 q^{83} + 16 q^{84} - 15 q^{85} + 9 q^{86} - 56 q^{87} - 21 q^{88} - 88 q^{89} + 47 q^{90} - 60 q^{91} + 26 q^{92} - 16 q^{93} + 37 q^{94} - 57 q^{95} + 108 q^{96} - 22 q^{97} + 8 q^{98} + 40 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.54628 −1.80049 −0.900244 0.435385i \(-0.856612\pi\)
−0.900244 + 0.435385i \(0.856612\pi\)
\(3\) 1.17861 0.680472 0.340236 0.940340i \(-0.389493\pi\)
0.340236 + 0.940340i \(0.389493\pi\)
\(4\) 4.48352 2.24176
\(5\) 1.63315 0.730367 0.365183 0.930936i \(-0.381006\pi\)
0.365183 + 0.930936i \(0.381006\pi\)
\(6\) −3.00107 −1.22518
\(7\) −0.708864 −0.267925 −0.133963 0.990986i \(-0.542770\pi\)
−0.133963 + 0.990986i \(0.542770\pi\)
\(8\) −6.32373 −2.23578
\(9\) −1.61088 −0.536958
\(10\) −4.15845 −1.31502
\(11\) 1.00000 0.301511
\(12\) 5.28433 1.52545
\(13\) −0.231092 −0.0640934 −0.0320467 0.999486i \(-0.510203\pi\)
−0.0320467 + 0.999486i \(0.510203\pi\)
\(14\) 1.80496 0.482396
\(15\) 1.92485 0.496994
\(16\) 7.13491 1.78373
\(17\) 1.00000 0.242536
\(18\) 4.10173 0.966788
\(19\) 2.82304 0.647649 0.323824 0.946117i \(-0.395031\pi\)
0.323824 + 0.946117i \(0.395031\pi\)
\(20\) 7.32226 1.63731
\(21\) −0.835475 −0.182316
\(22\) −2.54628 −0.542868
\(23\) −4.91189 −1.02420 −0.512100 0.858926i \(-0.671132\pi\)
−0.512100 + 0.858926i \(0.671132\pi\)
\(24\) −7.45322 −1.52138
\(25\) −2.33282 −0.466564
\(26\) 0.588424 0.115399
\(27\) −5.43443 −1.04586
\(28\) −3.17820 −0.600624
\(29\) 5.74697 1.06719 0.533593 0.845741i \(-0.320841\pi\)
0.533593 + 0.845741i \(0.320841\pi\)
\(30\) −4.90120 −0.894832
\(31\) 10.6463 1.91213 0.956066 0.293152i \(-0.0947041\pi\)
0.956066 + 0.293152i \(0.0947041\pi\)
\(32\) −5.52000 −0.975808
\(33\) 1.17861 0.205170
\(34\) −2.54628 −0.436683
\(35\) −1.15768 −0.195684
\(36\) −7.22239 −1.20373
\(37\) −5.71901 −0.940200 −0.470100 0.882613i \(-0.655782\pi\)
−0.470100 + 0.882613i \(0.655782\pi\)
\(38\) −7.18823 −1.16608
\(39\) −0.272368 −0.0436137
\(40\) −10.3276 −1.63294
\(41\) −3.74440 −0.584777 −0.292389 0.956300i \(-0.594450\pi\)
−0.292389 + 0.956300i \(0.594450\pi\)
\(42\) 2.12735 0.328257
\(43\) −1.00000 −0.152499
\(44\) 4.48352 0.675916
\(45\) −2.63080 −0.392177
\(46\) 12.5070 1.84406
\(47\) −8.08760 −1.17970 −0.589849 0.807513i \(-0.700813\pi\)
−0.589849 + 0.807513i \(0.700813\pi\)
\(48\) 8.40929 1.21378
\(49\) −6.49751 −0.928216
\(50\) 5.94001 0.840044
\(51\) 1.17861 0.165039
\(52\) −1.03611 −0.143682
\(53\) 13.1046 1.80005 0.900024 0.435840i \(-0.143549\pi\)
0.900024 + 0.435840i \(0.143549\pi\)
\(54\) 13.8376 1.88305
\(55\) 1.63315 0.220214
\(56\) 4.48266 0.599021
\(57\) 3.32726 0.440707
\(58\) −14.6334 −1.92146
\(59\) −10.6936 −1.39219 −0.696097 0.717948i \(-0.745082\pi\)
−0.696097 + 0.717948i \(0.745082\pi\)
\(60\) 8.63010 1.11414
\(61\) −7.14368 −0.914654 −0.457327 0.889299i \(-0.651193\pi\)
−0.457327 + 0.889299i \(0.651193\pi\)
\(62\) −27.1084 −3.44277
\(63\) 1.14189 0.143865
\(64\) −0.214376 −0.0267970
\(65\) −0.377408 −0.0468117
\(66\) −3.00107 −0.369406
\(67\) 1.17446 0.143483 0.0717415 0.997423i \(-0.477144\pi\)
0.0717415 + 0.997423i \(0.477144\pi\)
\(68\) 4.48352 0.543707
\(69\) −5.78921 −0.696939
\(70\) 2.94777 0.352326
\(71\) −16.1972 −1.92226 −0.961130 0.276097i \(-0.910959\pi\)
−0.961130 + 0.276097i \(0.910959\pi\)
\(72\) 10.1867 1.20052
\(73\) 4.11659 0.481811 0.240905 0.970549i \(-0.422556\pi\)
0.240905 + 0.970549i \(0.422556\pi\)
\(74\) 14.5622 1.69282
\(75\) −2.74949 −0.317484
\(76\) 12.6571 1.45187
\(77\) −0.708864 −0.0807825
\(78\) 0.693523 0.0785260
\(79\) −6.14710 −0.691603 −0.345801 0.938308i \(-0.612393\pi\)
−0.345801 + 0.938308i \(0.612393\pi\)
\(80\) 11.6524 1.30278
\(81\) −1.57246 −0.174717
\(82\) 9.53427 1.05288
\(83\) 7.46155 0.819012 0.409506 0.912307i \(-0.365701\pi\)
0.409506 + 0.912307i \(0.365701\pi\)
\(84\) −3.74587 −0.408708
\(85\) 1.63315 0.177140
\(86\) 2.54628 0.274572
\(87\) 6.77345 0.726190
\(88\) −6.32373 −0.674112
\(89\) −9.20969 −0.976225 −0.488112 0.872781i \(-0.662314\pi\)
−0.488112 + 0.872781i \(0.662314\pi\)
\(90\) 6.69874 0.706110
\(91\) 0.163813 0.0171722
\(92\) −22.0226 −2.29601
\(93\) 12.5478 1.30115
\(94\) 20.5933 2.12403
\(95\) 4.61044 0.473021
\(96\) −6.50594 −0.664010
\(97\) 0.962798 0.0977573 0.0488786 0.998805i \(-0.484435\pi\)
0.0488786 + 0.998805i \(0.484435\pi\)
\(98\) 16.5445 1.67124
\(99\) −1.61088 −0.161899
\(100\) −10.4593 −1.04593
\(101\) −11.2770 −1.12210 −0.561052 0.827780i \(-0.689603\pi\)
−0.561052 + 0.827780i \(0.689603\pi\)
\(102\) −3.00107 −0.297150
\(103\) 2.45797 0.242191 0.121096 0.992641i \(-0.461359\pi\)
0.121096 + 0.992641i \(0.461359\pi\)
\(104\) 1.46136 0.143298
\(105\) −1.36446 −0.133157
\(106\) −33.3678 −3.24097
\(107\) 11.7185 1.13287 0.566435 0.824107i \(-0.308322\pi\)
0.566435 + 0.824107i \(0.308322\pi\)
\(108\) −24.3654 −2.34456
\(109\) 13.6569 1.30809 0.654044 0.756456i \(-0.273071\pi\)
0.654044 + 0.756456i \(0.273071\pi\)
\(110\) −4.15845 −0.396493
\(111\) −6.74049 −0.639779
\(112\) −5.05768 −0.477906
\(113\) 6.22921 0.585994 0.292997 0.956113i \(-0.405347\pi\)
0.292997 + 0.956113i \(0.405347\pi\)
\(114\) −8.47213 −0.793487
\(115\) −8.02185 −0.748041
\(116\) 25.7667 2.39237
\(117\) 0.372260 0.0344155
\(118\) 27.2290 2.50663
\(119\) −0.708864 −0.0649814
\(120\) −12.1722 −1.11117
\(121\) 1.00000 0.0909091
\(122\) 18.1898 1.64682
\(123\) −4.41319 −0.397924
\(124\) 47.7329 4.28654
\(125\) −11.9756 −1.07113
\(126\) −2.90757 −0.259027
\(127\) 4.24032 0.376268 0.188134 0.982143i \(-0.439756\pi\)
0.188134 + 0.982143i \(0.439756\pi\)
\(128\) 11.5859 1.02406
\(129\) −1.17861 −0.103771
\(130\) 0.960984 0.0842839
\(131\) 2.94489 0.257296 0.128648 0.991690i \(-0.458936\pi\)
0.128648 + 0.991690i \(0.458936\pi\)
\(132\) 5.28433 0.459942
\(133\) −2.00115 −0.173521
\(134\) −2.99050 −0.258339
\(135\) −8.87524 −0.763859
\(136\) −6.32373 −0.542255
\(137\) 17.1564 1.46577 0.732885 0.680352i \(-0.238173\pi\)
0.732885 + 0.680352i \(0.238173\pi\)
\(138\) 14.7409 1.25483
\(139\) −4.16976 −0.353674 −0.176837 0.984240i \(-0.556587\pi\)
−0.176837 + 0.984240i \(0.556587\pi\)
\(140\) −5.19048 −0.438676
\(141\) −9.53214 −0.802751
\(142\) 41.2427 3.46101
\(143\) −0.231092 −0.0193249
\(144\) −11.4935 −0.957788
\(145\) 9.38567 0.779437
\(146\) −10.4820 −0.867495
\(147\) −7.65804 −0.631625
\(148\) −25.6413 −2.10770
\(149\) −7.36718 −0.603543 −0.301772 0.953380i \(-0.597578\pi\)
−0.301772 + 0.953380i \(0.597578\pi\)
\(150\) 7.00096 0.571626
\(151\) 6.73449 0.548045 0.274023 0.961723i \(-0.411646\pi\)
0.274023 + 0.961723i \(0.411646\pi\)
\(152\) −17.8521 −1.44800
\(153\) −1.61088 −0.130232
\(154\) 1.80496 0.145448
\(155\) 17.3870 1.39656
\(156\) −1.22117 −0.0977715
\(157\) 1.06032 0.0846225 0.0423113 0.999104i \(-0.486528\pi\)
0.0423113 + 0.999104i \(0.486528\pi\)
\(158\) 15.6522 1.24522
\(159\) 15.4452 1.22488
\(160\) −9.01499 −0.712698
\(161\) 3.48186 0.274409
\(162\) 4.00390 0.314576
\(163\) −2.77089 −0.217033 −0.108517 0.994095i \(-0.534610\pi\)
−0.108517 + 0.994095i \(0.534610\pi\)
\(164\) −16.7881 −1.31093
\(165\) 1.92485 0.149849
\(166\) −18.9992 −1.47462
\(167\) −22.1394 −1.71320 −0.856598 0.515985i \(-0.827426\pi\)
−0.856598 + 0.515985i \(0.827426\pi\)
\(168\) 5.28331 0.407617
\(169\) −12.9466 −0.995892
\(170\) −4.15845 −0.318939
\(171\) −4.54756 −0.347761
\(172\) −4.48352 −0.341865
\(173\) −5.09869 −0.387646 −0.193823 0.981036i \(-0.562089\pi\)
−0.193823 + 0.981036i \(0.562089\pi\)
\(174\) −17.2471 −1.30750
\(175\) 1.65365 0.125004
\(176\) 7.13491 0.537814
\(177\) −12.6037 −0.947349
\(178\) 23.4504 1.75768
\(179\) −14.5637 −1.08854 −0.544270 0.838910i \(-0.683193\pi\)
−0.544270 + 0.838910i \(0.683193\pi\)
\(180\) −11.7953 −0.879166
\(181\) 3.07250 0.228377 0.114189 0.993459i \(-0.463573\pi\)
0.114189 + 0.993459i \(0.463573\pi\)
\(182\) −0.417112 −0.0309184
\(183\) −8.41962 −0.622396
\(184\) 31.0614 2.28988
\(185\) −9.34001 −0.686691
\(186\) −31.9503 −2.34271
\(187\) 1.00000 0.0731272
\(188\) −36.2609 −2.64460
\(189\) 3.85227 0.280211
\(190\) −11.7395 −0.851670
\(191\) −16.2062 −1.17264 −0.586320 0.810079i \(-0.699424\pi\)
−0.586320 + 0.810079i \(0.699424\pi\)
\(192\) −0.252666 −0.0182346
\(193\) −2.05024 −0.147580 −0.0737899 0.997274i \(-0.523509\pi\)
−0.0737899 + 0.997274i \(0.523509\pi\)
\(194\) −2.45155 −0.176011
\(195\) −0.444817 −0.0318540
\(196\) −29.1317 −2.08084
\(197\) −0.0852862 −0.00607639 −0.00303819 0.999995i \(-0.500967\pi\)
−0.00303819 + 0.999995i \(0.500967\pi\)
\(198\) 4.10173 0.291497
\(199\) −13.7575 −0.975240 −0.487620 0.873056i \(-0.662135\pi\)
−0.487620 + 0.873056i \(0.662135\pi\)
\(200\) 14.7521 1.04313
\(201\) 1.38423 0.0976361
\(202\) 28.7144 2.02034
\(203\) −4.07382 −0.285926
\(204\) 5.28433 0.369977
\(205\) −6.11517 −0.427102
\(206\) −6.25867 −0.436062
\(207\) 7.91244 0.549953
\(208\) −1.64882 −0.114325
\(209\) 2.82304 0.195273
\(210\) 3.47428 0.239748
\(211\) −13.1591 −0.905910 −0.452955 0.891533i \(-0.649630\pi\)
−0.452955 + 0.891533i \(0.649630\pi\)
\(212\) 58.7545 4.03528
\(213\) −19.0903 −1.30804
\(214\) −29.8385 −2.03972
\(215\) −1.63315 −0.111380
\(216\) 34.3659 2.33830
\(217\) −7.54677 −0.512308
\(218\) −34.7741 −2.35520
\(219\) 4.85186 0.327859
\(220\) 7.32226 0.493667
\(221\) −0.231092 −0.0155449
\(222\) 17.1632 1.15192
\(223\) 11.6518 0.780260 0.390130 0.920760i \(-0.372430\pi\)
0.390130 + 0.920760i \(0.372430\pi\)
\(224\) 3.91293 0.261444
\(225\) 3.75788 0.250526
\(226\) −15.8613 −1.05508
\(227\) 9.90372 0.657333 0.328666 0.944446i \(-0.393401\pi\)
0.328666 + 0.944446i \(0.393401\pi\)
\(228\) 14.9178 0.987959
\(229\) −28.0539 −1.85385 −0.926926 0.375243i \(-0.877559\pi\)
−0.926926 + 0.375243i \(0.877559\pi\)
\(230\) 20.4258 1.34684
\(231\) −0.835475 −0.0549702
\(232\) −36.3423 −2.38599
\(233\) −12.5694 −0.823448 −0.411724 0.911309i \(-0.635073\pi\)
−0.411724 + 0.911309i \(0.635073\pi\)
\(234\) −0.947877 −0.0619647
\(235\) −13.2083 −0.861613
\(236\) −47.9452 −3.12097
\(237\) −7.24504 −0.470616
\(238\) 1.80496 0.116998
\(239\) 23.7323 1.53511 0.767556 0.640982i \(-0.221473\pi\)
0.767556 + 0.640982i \(0.221473\pi\)
\(240\) 13.7336 0.886502
\(241\) 23.0797 1.48670 0.743348 0.668905i \(-0.233237\pi\)
0.743348 + 0.668905i \(0.233237\pi\)
\(242\) −2.54628 −0.163681
\(243\) 14.4500 0.926966
\(244\) −32.0288 −2.05044
\(245\) −10.6114 −0.677938
\(246\) 11.2372 0.716458
\(247\) −0.652381 −0.0415100
\(248\) −67.3243 −4.27510
\(249\) 8.79427 0.557314
\(250\) 30.4932 1.92856
\(251\) 1.14001 0.0719571 0.0359785 0.999353i \(-0.488545\pi\)
0.0359785 + 0.999353i \(0.488545\pi\)
\(252\) 5.11969 0.322510
\(253\) −4.91189 −0.308808
\(254\) −10.7970 −0.677466
\(255\) 1.92485 0.120539
\(256\) −29.0721 −1.81700
\(257\) −5.65618 −0.352823 −0.176412 0.984316i \(-0.556449\pi\)
−0.176412 + 0.984316i \(0.556449\pi\)
\(258\) 3.00107 0.186838
\(259\) 4.05400 0.251903
\(260\) −1.69212 −0.104941
\(261\) −9.25765 −0.573034
\(262\) −7.49850 −0.463259
\(263\) −20.9695 −1.29304 −0.646518 0.762898i \(-0.723776\pi\)
−0.646518 + 0.762898i \(0.723776\pi\)
\(264\) −7.45322 −0.458714
\(265\) 21.4017 1.31470
\(266\) 5.09547 0.312424
\(267\) −10.8546 −0.664293
\(268\) 5.26571 0.321654
\(269\) −7.89417 −0.481316 −0.240658 0.970610i \(-0.577363\pi\)
−0.240658 + 0.970610i \(0.577363\pi\)
\(270\) 22.5988 1.37532
\(271\) −21.2402 −1.29025 −0.645124 0.764078i \(-0.723194\pi\)
−0.645124 + 0.764078i \(0.723194\pi\)
\(272\) 7.13491 0.432618
\(273\) 0.193071 0.0116852
\(274\) −43.6849 −2.63910
\(275\) −2.33282 −0.140674
\(276\) −25.9560 −1.56237
\(277\) 9.98539 0.599964 0.299982 0.953945i \(-0.403019\pi\)
0.299982 + 0.953945i \(0.403019\pi\)
\(278\) 10.6174 0.636787
\(279\) −17.1499 −1.02674
\(280\) 7.32086 0.437505
\(281\) −28.8502 −1.72106 −0.860528 0.509403i \(-0.829866\pi\)
−0.860528 + 0.509403i \(0.829866\pi\)
\(282\) 24.2715 1.44534
\(283\) 2.25801 0.134225 0.0671125 0.997745i \(-0.478621\pi\)
0.0671125 + 0.997745i \(0.478621\pi\)
\(284\) −72.6207 −4.30924
\(285\) 5.43392 0.321878
\(286\) 0.588424 0.0347942
\(287\) 2.65427 0.156677
\(288\) 8.89204 0.523968
\(289\) 1.00000 0.0588235
\(290\) −23.8985 −1.40337
\(291\) 1.13476 0.0665211
\(292\) 18.4568 1.08010
\(293\) −6.53576 −0.381823 −0.190912 0.981607i \(-0.561144\pi\)
−0.190912 + 0.981607i \(0.561144\pi\)
\(294\) 19.4995 1.13723
\(295\) −17.4643 −1.01681
\(296\) 36.1655 2.10208
\(297\) −5.43443 −0.315338
\(298\) 18.7589 1.08667
\(299\) 1.13510 0.0656444
\(300\) −12.3274 −0.711722
\(301\) 0.708864 0.0408582
\(302\) −17.1479 −0.986749
\(303\) −13.2912 −0.763560
\(304\) 20.1421 1.15523
\(305\) −11.6667 −0.668033
\(306\) 4.10173 0.234480
\(307\) 14.8500 0.847533 0.423767 0.905771i \(-0.360708\pi\)
0.423767 + 0.905771i \(0.360708\pi\)
\(308\) −3.17820 −0.181095
\(309\) 2.89699 0.164804
\(310\) −44.2721 −2.51449
\(311\) −0.386557 −0.0219196 −0.0109598 0.999940i \(-0.503489\pi\)
−0.0109598 + 0.999940i \(0.503489\pi\)
\(312\) 1.72238 0.0975105
\(313\) −6.02934 −0.340799 −0.170399 0.985375i \(-0.554506\pi\)
−0.170399 + 0.985375i \(0.554506\pi\)
\(314\) −2.69986 −0.152362
\(315\) 1.86488 0.105074
\(316\) −27.5606 −1.55041
\(317\) −15.2913 −0.858846 −0.429423 0.903104i \(-0.641283\pi\)
−0.429423 + 0.903104i \(0.641283\pi\)
\(318\) −39.3277 −2.20539
\(319\) 5.74697 0.321769
\(320\) −0.350108 −0.0195717
\(321\) 13.8115 0.770885
\(322\) −8.86577 −0.494070
\(323\) 2.82304 0.157078
\(324\) −7.05013 −0.391674
\(325\) 0.539096 0.0299037
\(326\) 7.05546 0.390766
\(327\) 16.0961 0.890117
\(328\) 23.6786 1.30743
\(329\) 5.73301 0.316071
\(330\) −4.90120 −0.269802
\(331\) 25.0003 1.37414 0.687070 0.726591i \(-0.258896\pi\)
0.687070 + 0.726591i \(0.258896\pi\)
\(332\) 33.4540 1.83603
\(333\) 9.21262 0.504848
\(334\) 56.3729 3.08459
\(335\) 1.91807 0.104795
\(336\) −5.96104 −0.325201
\(337\) −9.13469 −0.497598 −0.248799 0.968555i \(-0.580036\pi\)
−0.248799 + 0.968555i \(0.580036\pi\)
\(338\) 32.9656 1.79309
\(339\) 7.34181 0.398753
\(340\) 7.32226 0.397105
\(341\) 10.6463 0.576529
\(342\) 11.5793 0.626139
\(343\) 9.56790 0.516618
\(344\) 6.32373 0.340953
\(345\) −9.45464 −0.509021
\(346\) 12.9827 0.697953
\(347\) −13.3251 −0.715331 −0.357666 0.933850i \(-0.616427\pi\)
−0.357666 + 0.933850i \(0.616427\pi\)
\(348\) 30.3689 1.62794
\(349\) 7.86651 0.421085 0.210542 0.977585i \(-0.432477\pi\)
0.210542 + 0.977585i \(0.432477\pi\)
\(350\) −4.21065 −0.225069
\(351\) 1.25585 0.0670325
\(352\) −5.52000 −0.294217
\(353\) −18.9598 −1.00913 −0.504564 0.863374i \(-0.668347\pi\)
−0.504564 + 0.863374i \(0.668347\pi\)
\(354\) 32.0924 1.70569
\(355\) −26.4525 −1.40395
\(356\) −41.2918 −2.18846
\(357\) −0.835475 −0.0442180
\(358\) 37.0831 1.95990
\(359\) −6.01951 −0.317697 −0.158849 0.987303i \(-0.550778\pi\)
−0.158849 + 0.987303i \(0.550778\pi\)
\(360\) 16.6365 0.876819
\(361\) −11.0305 −0.580551
\(362\) −7.82344 −0.411191
\(363\) 1.17861 0.0618611
\(364\) 0.734457 0.0384960
\(365\) 6.72302 0.351899
\(366\) 21.4387 1.12062
\(367\) 11.3785 0.593950 0.296975 0.954885i \(-0.404022\pi\)
0.296975 + 0.954885i \(0.404022\pi\)
\(368\) −35.0459 −1.82689
\(369\) 6.03176 0.314001
\(370\) 23.7822 1.23638
\(371\) −9.28934 −0.482278
\(372\) 56.2585 2.91687
\(373\) 16.1380 0.835596 0.417798 0.908540i \(-0.362802\pi\)
0.417798 + 0.908540i \(0.362802\pi\)
\(374\) −2.54628 −0.131665
\(375\) −14.1146 −0.728873
\(376\) 51.1438 2.63754
\(377\) −1.32808 −0.0683995
\(378\) −9.80894 −0.504517
\(379\) −11.0914 −0.569729 −0.284864 0.958568i \(-0.591949\pi\)
−0.284864 + 0.958568i \(0.591949\pi\)
\(380\) 20.6710 1.06040
\(381\) 4.99769 0.256040
\(382\) 41.2655 2.11133
\(383\) 36.6621 1.87335 0.936674 0.350202i \(-0.113887\pi\)
0.936674 + 0.350202i \(0.113887\pi\)
\(384\) 13.6552 0.696841
\(385\) −1.15768 −0.0590009
\(386\) 5.22049 0.265716
\(387\) 1.61088 0.0818854
\(388\) 4.31672 0.219148
\(389\) 10.8004 0.547600 0.273800 0.961787i \(-0.411719\pi\)
0.273800 + 0.961787i \(0.411719\pi\)
\(390\) 1.13263 0.0573528
\(391\) −4.91189 −0.248405
\(392\) 41.0885 2.07528
\(393\) 3.47088 0.175083
\(394\) 0.217162 0.0109405
\(395\) −10.0391 −0.505124
\(396\) −7.22239 −0.362939
\(397\) −0.957048 −0.0480329 −0.0240164 0.999712i \(-0.507645\pi\)
−0.0240164 + 0.999712i \(0.507645\pi\)
\(398\) 35.0303 1.75591
\(399\) −2.35858 −0.118076
\(400\) −16.6445 −0.832224
\(401\) −21.9974 −1.09850 −0.549249 0.835659i \(-0.685086\pi\)
−0.549249 + 0.835659i \(0.685086\pi\)
\(402\) −3.52463 −0.175793
\(403\) −2.46027 −0.122555
\(404\) −50.5607 −2.51549
\(405\) −2.56806 −0.127608
\(406\) 10.3731 0.514807
\(407\) −5.71901 −0.283481
\(408\) −7.45322 −0.368989
\(409\) 1.59262 0.0787498 0.0393749 0.999225i \(-0.487463\pi\)
0.0393749 + 0.999225i \(0.487463\pi\)
\(410\) 15.5709 0.768992
\(411\) 20.2207 0.997415
\(412\) 11.0204 0.542934
\(413\) 7.58034 0.373004
\(414\) −20.1473 −0.990183
\(415\) 12.1858 0.598179
\(416\) 1.27563 0.0625428
\(417\) −4.91453 −0.240665
\(418\) −7.18823 −0.351588
\(419\) −9.17654 −0.448303 −0.224152 0.974554i \(-0.571961\pi\)
−0.224152 + 0.974554i \(0.571961\pi\)
\(420\) −6.11756 −0.298507
\(421\) 19.1887 0.935202 0.467601 0.883940i \(-0.345118\pi\)
0.467601 + 0.883940i \(0.345118\pi\)
\(422\) 33.5067 1.63108
\(423\) 13.0281 0.633449
\(424\) −82.8696 −4.02450
\(425\) −2.33282 −0.113158
\(426\) 48.6091 2.35512
\(427\) 5.06389 0.245059
\(428\) 52.5401 2.53962
\(429\) −0.272368 −0.0131500
\(430\) 4.15845 0.200538
\(431\) 30.5385 1.47099 0.735493 0.677532i \(-0.236951\pi\)
0.735493 + 0.677532i \(0.236951\pi\)
\(432\) −38.7742 −1.86552
\(433\) 40.2091 1.93233 0.966163 0.257933i \(-0.0830413\pi\)
0.966163 + 0.257933i \(0.0830413\pi\)
\(434\) 19.2162 0.922406
\(435\) 11.0621 0.530385
\(436\) 61.2308 2.93242
\(437\) −13.8664 −0.663322
\(438\) −12.3542 −0.590306
\(439\) 21.5005 1.02616 0.513081 0.858340i \(-0.328504\pi\)
0.513081 + 0.858340i \(0.328504\pi\)
\(440\) −10.3276 −0.492349
\(441\) 10.4667 0.498413
\(442\) 0.588424 0.0279885
\(443\) −30.2540 −1.43741 −0.718707 0.695313i \(-0.755266\pi\)
−0.718707 + 0.695313i \(0.755266\pi\)
\(444\) −30.2211 −1.43423
\(445\) −15.0408 −0.713002
\(446\) −29.6686 −1.40485
\(447\) −8.68304 −0.410694
\(448\) 0.151963 0.00717960
\(449\) −10.8353 −0.511348 −0.255674 0.966763i \(-0.582297\pi\)
−0.255674 + 0.966763i \(0.582297\pi\)
\(450\) −9.56861 −0.451068
\(451\) −3.74440 −0.176317
\(452\) 27.9288 1.31366
\(453\) 7.93735 0.372929
\(454\) −25.2176 −1.18352
\(455\) 0.267531 0.0125420
\(456\) −21.0407 −0.985321
\(457\) −27.6430 −1.29309 −0.646543 0.762877i \(-0.723786\pi\)
−0.646543 + 0.762877i \(0.723786\pi\)
\(458\) 71.4329 3.33784
\(459\) −5.43443 −0.253657
\(460\) −35.9661 −1.67693
\(461\) 4.65593 0.216848 0.108424 0.994105i \(-0.465420\pi\)
0.108424 + 0.994105i \(0.465420\pi\)
\(462\) 2.12735 0.0989732
\(463\) 2.87153 0.133451 0.0667257 0.997771i \(-0.478745\pi\)
0.0667257 + 0.997771i \(0.478745\pi\)
\(464\) 41.0041 1.90357
\(465\) 20.4925 0.950318
\(466\) 32.0051 1.48261
\(467\) −18.4667 −0.854537 −0.427269 0.904125i \(-0.640524\pi\)
−0.427269 + 0.904125i \(0.640524\pi\)
\(468\) 1.66904 0.0771512
\(469\) −0.832531 −0.0384427
\(470\) 33.6319 1.55132
\(471\) 1.24970 0.0575832
\(472\) 67.6237 3.11263
\(473\) −1.00000 −0.0459800
\(474\) 18.4479 0.847339
\(475\) −6.58564 −0.302170
\(476\) −3.17820 −0.145673
\(477\) −21.1098 −0.966551
\(478\) −60.4289 −2.76395
\(479\) −8.36694 −0.382295 −0.191148 0.981561i \(-0.561221\pi\)
−0.191148 + 0.981561i \(0.561221\pi\)
\(480\) −10.6252 −0.484971
\(481\) 1.32162 0.0602606
\(482\) −58.7674 −2.67678
\(483\) 4.10376 0.186727
\(484\) 4.48352 0.203796
\(485\) 1.57239 0.0713987
\(486\) −36.7936 −1.66899
\(487\) −6.74347 −0.305576 −0.152788 0.988259i \(-0.548825\pi\)
−0.152788 + 0.988259i \(0.548825\pi\)
\(488\) 45.1747 2.04496
\(489\) −3.26580 −0.147685
\(490\) 27.0196 1.22062
\(491\) −10.1207 −0.456740 −0.228370 0.973574i \(-0.573340\pi\)
−0.228370 + 0.973574i \(0.573340\pi\)
\(492\) −19.7866 −0.892050
\(493\) 5.74697 0.258831
\(494\) 1.66114 0.0747383
\(495\) −2.63080 −0.118246
\(496\) 75.9604 3.41072
\(497\) 11.4816 0.515022
\(498\) −22.3926 −1.00344
\(499\) 12.1933 0.545845 0.272923 0.962036i \(-0.412010\pi\)
0.272923 + 0.962036i \(0.412010\pi\)
\(500\) −53.6928 −2.40122
\(501\) −26.0937 −1.16578
\(502\) −2.90279 −0.129558
\(503\) −23.1526 −1.03232 −0.516161 0.856491i \(-0.672639\pi\)
−0.516161 + 0.856491i \(0.672639\pi\)
\(504\) −7.22101 −0.321649
\(505\) −18.4171 −0.819548
\(506\) 12.5070 0.556005
\(507\) −15.2590 −0.677676
\(508\) 19.0116 0.843502
\(509\) −21.1603 −0.937916 −0.468958 0.883221i \(-0.655370\pi\)
−0.468958 + 0.883221i \(0.655370\pi\)
\(510\) −4.90120 −0.217029
\(511\) −2.91810 −0.129089
\(512\) 50.8538 2.24744
\(513\) −15.3416 −0.677348
\(514\) 14.4022 0.635254
\(515\) 4.01424 0.176888
\(516\) −5.28433 −0.232630
\(517\) −8.08760 −0.355692
\(518\) −10.3226 −0.453549
\(519\) −6.00937 −0.263782
\(520\) 2.38662 0.104660
\(521\) −16.4659 −0.721384 −0.360692 0.932685i \(-0.617459\pi\)
−0.360692 + 0.932685i \(0.617459\pi\)
\(522\) 23.5725 1.03174
\(523\) −30.3934 −1.32901 −0.664505 0.747284i \(-0.731357\pi\)
−0.664505 + 0.747284i \(0.731357\pi\)
\(524\) 13.2035 0.576796
\(525\) 1.94901 0.0850619
\(526\) 53.3942 2.32810
\(527\) 10.6463 0.463760
\(528\) 8.40929 0.365967
\(529\) 1.12665 0.0489847
\(530\) −54.4946 −2.36710
\(531\) 17.2261 0.747551
\(532\) −8.97219 −0.388994
\(533\) 0.865301 0.0374803
\(534\) 27.6389 1.19605
\(535\) 19.1381 0.827410
\(536\) −7.42696 −0.320796
\(537\) −17.1649 −0.740720
\(538\) 20.1007 0.866604
\(539\) −6.49751 −0.279868
\(540\) −39.7923 −1.71239
\(541\) −10.3622 −0.445507 −0.222753 0.974875i \(-0.571504\pi\)
−0.222753 + 0.974875i \(0.571504\pi\)
\(542\) 54.0833 2.32308
\(543\) 3.62129 0.155404
\(544\) −5.52000 −0.236668
\(545\) 22.3037 0.955385
\(546\) −0.491613 −0.0210391
\(547\) −26.6802 −1.14076 −0.570382 0.821379i \(-0.693205\pi\)
−0.570382 + 0.821379i \(0.693205\pi\)
\(548\) 76.9211 3.28591
\(549\) 11.5076 0.491131
\(550\) 5.94001 0.253283
\(551\) 16.2239 0.691162
\(552\) 36.6094 1.55820
\(553\) 4.35746 0.185298
\(554\) −25.4256 −1.08023
\(555\) −11.0082 −0.467274
\(556\) −18.6952 −0.792853
\(557\) 18.5243 0.784901 0.392451 0.919773i \(-0.371627\pi\)
0.392451 + 0.919773i \(0.371627\pi\)
\(558\) 43.6683 1.84863
\(559\) 0.231092 0.00977415
\(560\) −8.25995 −0.349047
\(561\) 1.17861 0.0497610
\(562\) 73.4605 3.09874
\(563\) 31.0547 1.30880 0.654399 0.756149i \(-0.272921\pi\)
0.654399 + 0.756149i \(0.272921\pi\)
\(564\) −42.7376 −1.79958
\(565\) 10.1732 0.427991
\(566\) −5.74952 −0.241670
\(567\) 1.11466 0.0468112
\(568\) 102.427 4.29774
\(569\) −17.2048 −0.721262 −0.360631 0.932709i \(-0.617439\pi\)
−0.360631 + 0.932709i \(0.617439\pi\)
\(570\) −13.8363 −0.579537
\(571\) −24.1044 −1.00874 −0.504369 0.863488i \(-0.668275\pi\)
−0.504369 + 0.863488i \(0.668275\pi\)
\(572\) −1.03611 −0.0433217
\(573\) −19.1008 −0.797948
\(574\) −6.75850 −0.282094
\(575\) 11.4586 0.477855
\(576\) 0.345333 0.0143889
\(577\) 15.9565 0.664280 0.332140 0.943230i \(-0.392229\pi\)
0.332140 + 0.943230i \(0.392229\pi\)
\(578\) −2.54628 −0.105911
\(579\) −2.41644 −0.100424
\(580\) 42.0808 1.74731
\(581\) −5.28922 −0.219434
\(582\) −2.88942 −0.119770
\(583\) 13.1046 0.542735
\(584\) −26.0322 −1.07722
\(585\) 0.607957 0.0251359
\(586\) 16.6418 0.687468
\(587\) −30.1529 −1.24455 −0.622273 0.782801i \(-0.713790\pi\)
−0.622273 + 0.782801i \(0.713790\pi\)
\(588\) −34.3350 −1.41595
\(589\) 30.0549 1.23839
\(590\) 44.4690 1.83076
\(591\) −0.100519 −0.00413481
\(592\) −40.8047 −1.67706
\(593\) −23.1415 −0.950307 −0.475154 0.879903i \(-0.657608\pi\)
−0.475154 + 0.879903i \(0.657608\pi\)
\(594\) 13.8376 0.567762
\(595\) −1.15768 −0.0474603
\(596\) −33.0309 −1.35300
\(597\) −16.2147 −0.663623
\(598\) −2.89027 −0.118192
\(599\) 37.6241 1.53728 0.768640 0.639681i \(-0.220934\pi\)
0.768640 + 0.639681i \(0.220934\pi\)
\(600\) 17.3870 0.709822
\(601\) 35.6946 1.45601 0.728007 0.685570i \(-0.240447\pi\)
0.728007 + 0.685570i \(0.240447\pi\)
\(602\) −1.80496 −0.0735648
\(603\) −1.89191 −0.0770444
\(604\) 30.1942 1.22859
\(605\) 1.63315 0.0663970
\(606\) 33.8431 1.37478
\(607\) −16.2869 −0.661063 −0.330532 0.943795i \(-0.607228\pi\)
−0.330532 + 0.943795i \(0.607228\pi\)
\(608\) −15.5832 −0.631981
\(609\) −4.80145 −0.194565
\(610\) 29.7066 1.20279
\(611\) 1.86898 0.0756108
\(612\) −7.22239 −0.291948
\(613\) 44.0803 1.78039 0.890193 0.455584i \(-0.150569\pi\)
0.890193 + 0.455584i \(0.150569\pi\)
\(614\) −37.8122 −1.52597
\(615\) −7.20741 −0.290631
\(616\) 4.48266 0.180612
\(617\) 13.5115 0.543952 0.271976 0.962304i \(-0.412323\pi\)
0.271976 + 0.962304i \(0.412323\pi\)
\(618\) −7.37654 −0.296728
\(619\) 16.6828 0.670536 0.335268 0.942123i \(-0.391173\pi\)
0.335268 + 0.942123i \(0.391173\pi\)
\(620\) 77.9550 3.13075
\(621\) 26.6933 1.07117
\(622\) 0.984281 0.0394661
\(623\) 6.52841 0.261555
\(624\) −1.94332 −0.0777950
\(625\) −7.89384 −0.315754
\(626\) 15.3524 0.613604
\(627\) 3.32726 0.132878
\(628\) 4.75395 0.189703
\(629\) −5.71901 −0.228032
\(630\) −4.74850 −0.189185
\(631\) −27.8685 −1.10943 −0.554714 0.832041i \(-0.687172\pi\)
−0.554714 + 0.832041i \(0.687172\pi\)
\(632\) 38.8726 1.54627
\(633\) −15.5095 −0.616446
\(634\) 38.9359 1.54634
\(635\) 6.92509 0.274814
\(636\) 69.2487 2.74589
\(637\) 1.50152 0.0594925
\(638\) −14.6334 −0.579341
\(639\) 26.0917 1.03217
\(640\) 18.9215 0.747936
\(641\) −17.8423 −0.704728 −0.352364 0.935863i \(-0.614622\pi\)
−0.352364 + 0.935863i \(0.614622\pi\)
\(642\) −35.1680 −1.38797
\(643\) 19.8841 0.784151 0.392076 0.919933i \(-0.371757\pi\)
0.392076 + 0.919933i \(0.371757\pi\)
\(644\) 15.6110 0.615159
\(645\) −1.92485 −0.0757909
\(646\) −7.18823 −0.282817
\(647\) 9.45606 0.371756 0.185878 0.982573i \(-0.440487\pi\)
0.185878 + 0.982573i \(0.440487\pi\)
\(648\) 9.94378 0.390629
\(649\) −10.6936 −0.419762
\(650\) −1.37269 −0.0538412
\(651\) −8.89471 −0.348611
\(652\) −12.4234 −0.486536
\(653\) −28.9421 −1.13259 −0.566296 0.824202i \(-0.691624\pi\)
−0.566296 + 0.824202i \(0.691624\pi\)
\(654\) −40.9852 −1.60265
\(655\) 4.80945 0.187921
\(656\) −26.7160 −1.04308
\(657\) −6.63132 −0.258712
\(658\) −14.5978 −0.569082
\(659\) −33.0650 −1.28803 −0.644015 0.765013i \(-0.722733\pi\)
−0.644015 + 0.765013i \(0.722733\pi\)
\(660\) 8.63010 0.335926
\(661\) 6.93782 0.269850 0.134925 0.990856i \(-0.456921\pi\)
0.134925 + 0.990856i \(0.456921\pi\)
\(662\) −63.6576 −2.47412
\(663\) −0.272368 −0.0105779
\(664\) −47.1848 −1.83113
\(665\) −3.26817 −0.126734
\(666\) −23.4579 −0.908974
\(667\) −28.2285 −1.09301
\(668\) −99.2623 −3.84057
\(669\) 13.7329 0.530945
\(670\) −4.88393 −0.188683
\(671\) −7.14368 −0.275779
\(672\) 4.61182 0.177905
\(673\) −17.0941 −0.658930 −0.329465 0.944168i \(-0.606868\pi\)
−0.329465 + 0.944168i \(0.606868\pi\)
\(674\) 23.2594 0.895920
\(675\) 12.6776 0.487959
\(676\) −58.0463 −2.23255
\(677\) 15.7718 0.606158 0.303079 0.952965i \(-0.401985\pi\)
0.303079 + 0.952965i \(0.401985\pi\)
\(678\) −18.6943 −0.717950
\(679\) −0.682492 −0.0261916
\(680\) −10.3276 −0.396045
\(681\) 11.6726 0.447296
\(682\) −27.1084 −1.03803
\(683\) −41.0020 −1.56890 −0.784449 0.620193i \(-0.787054\pi\)
−0.784449 + 0.620193i \(0.787054\pi\)
\(684\) −20.3891 −0.779596
\(685\) 28.0190 1.07055
\(686\) −24.3625 −0.930165
\(687\) −33.0646 −1.26149
\(688\) −7.13491 −0.272016
\(689\) −3.02836 −0.115371
\(690\) 24.0741 0.916487
\(691\) −0.482298 −0.0183475 −0.00917375 0.999958i \(-0.502920\pi\)
−0.00917375 + 0.999958i \(0.502920\pi\)
\(692\) −22.8601 −0.869010
\(693\) 1.14189 0.0433768
\(694\) 33.9295 1.28795
\(695\) −6.80984 −0.258312
\(696\) −42.8334 −1.62360
\(697\) −3.74440 −0.141829
\(698\) −20.0303 −0.758158
\(699\) −14.8144 −0.560333
\(700\) 7.41418 0.280230
\(701\) −0.593044 −0.0223990 −0.0111995 0.999937i \(-0.503565\pi\)
−0.0111995 + 0.999937i \(0.503565\pi\)
\(702\) −3.19775 −0.120691
\(703\) −16.1450 −0.608920
\(704\) −0.214376 −0.00807960
\(705\) −15.5674 −0.586303
\(706\) 48.2769 1.81692
\(707\) 7.99386 0.300640
\(708\) −56.5087 −2.12373
\(709\) −16.9986 −0.638397 −0.319199 0.947688i \(-0.603414\pi\)
−0.319199 + 0.947688i \(0.603414\pi\)
\(710\) 67.3554 2.52780
\(711\) 9.90221 0.371362
\(712\) 58.2396 2.18262
\(713\) −52.2934 −1.95840
\(714\) 2.12735 0.0796140
\(715\) −0.377408 −0.0141143
\(716\) −65.2965 −2.44024
\(717\) 27.9711 1.04460
\(718\) 15.3273 0.572011
\(719\) 24.5830 0.916790 0.458395 0.888748i \(-0.348424\pi\)
0.458395 + 0.888748i \(0.348424\pi\)
\(720\) −18.7705 −0.699537
\(721\) −1.74237 −0.0648891
\(722\) 28.0866 1.04528
\(723\) 27.2020 1.01165
\(724\) 13.7756 0.511967
\(725\) −13.4067 −0.497911
\(726\) −3.00107 −0.111380
\(727\) 21.0699 0.781440 0.390720 0.920510i \(-0.372226\pi\)
0.390720 + 0.920510i \(0.372226\pi\)
\(728\) −1.03591 −0.0383933
\(729\) 21.7483 0.805492
\(730\) −17.1187 −0.633590
\(731\) −1.00000 −0.0369863
\(732\) −37.7495 −1.39526
\(733\) 5.60797 0.207135 0.103567 0.994622i \(-0.466974\pi\)
0.103567 + 0.994622i \(0.466974\pi\)
\(734\) −28.9727 −1.06940
\(735\) −12.5067 −0.461318
\(736\) 27.1136 0.999422
\(737\) 1.17446 0.0432617
\(738\) −15.3585 −0.565355
\(739\) 0.450844 0.0165846 0.00829229 0.999966i \(-0.497360\pi\)
0.00829229 + 0.999966i \(0.497360\pi\)
\(740\) −41.8761 −1.53940
\(741\) −0.768903 −0.0282464
\(742\) 23.6532 0.868337
\(743\) −17.6038 −0.645819 −0.322910 0.946430i \(-0.604661\pi\)
−0.322910 + 0.946430i \(0.604661\pi\)
\(744\) −79.3492 −2.90908
\(745\) −12.0317 −0.440808
\(746\) −41.0919 −1.50448
\(747\) −12.0196 −0.439775
\(748\) 4.48352 0.163934
\(749\) −8.30681 −0.303524
\(750\) 35.9396 1.31233
\(751\) −16.4118 −0.598874 −0.299437 0.954116i \(-0.596799\pi\)
−0.299437 + 0.954116i \(0.596799\pi\)
\(752\) −57.7044 −2.10426
\(753\) 1.34363 0.0489648
\(754\) 3.38165 0.123153
\(755\) 10.9984 0.400274
\(756\) 17.2717 0.628167
\(757\) −33.4257 −1.21488 −0.607438 0.794367i \(-0.707803\pi\)
−0.607438 + 0.794367i \(0.707803\pi\)
\(758\) 28.2419 1.02579
\(759\) −5.78921 −0.210135
\(760\) −29.1552 −1.05757
\(761\) −18.6102 −0.674620 −0.337310 0.941394i \(-0.609517\pi\)
−0.337310 + 0.941394i \(0.609517\pi\)
\(762\) −12.7255 −0.460997
\(763\) −9.68084 −0.350470
\(764\) −72.6609 −2.62878
\(765\) −2.63080 −0.0951168
\(766\) −93.3519 −3.37294
\(767\) 2.47122 0.0892304
\(768\) −34.2647 −1.23642
\(769\) −30.3092 −1.09298 −0.546489 0.837467i \(-0.684036\pi\)
−0.546489 + 0.837467i \(0.684036\pi\)
\(770\) 2.94777 0.106230
\(771\) −6.66644 −0.240086
\(772\) −9.19231 −0.330839
\(773\) 40.8955 1.47091 0.735455 0.677574i \(-0.236969\pi\)
0.735455 + 0.677574i \(0.236969\pi\)
\(774\) −4.10173 −0.147434
\(775\) −24.8359 −0.892132
\(776\) −6.08847 −0.218563
\(777\) 4.77809 0.171413
\(778\) −27.5007 −0.985949
\(779\) −10.5706 −0.378730
\(780\) −1.99435 −0.0714091
\(781\) −16.1972 −0.579583
\(782\) 12.5070 0.447250
\(783\) −31.2315 −1.11612
\(784\) −46.3592 −1.65569
\(785\) 1.73166 0.0618055
\(786\) −8.83782 −0.315235
\(787\) 23.7482 0.846532 0.423266 0.906005i \(-0.360883\pi\)
0.423266 + 0.906005i \(0.360883\pi\)
\(788\) −0.382382 −0.0136218
\(789\) −24.7149 −0.879875
\(790\) 25.5624 0.909470
\(791\) −4.41566 −0.157003
\(792\) 10.1867 0.361970
\(793\) 1.65085 0.0586233
\(794\) 2.43691 0.0864827
\(795\) 25.2243 0.894613
\(796\) −61.6818 −2.18625
\(797\) −30.4224 −1.07762 −0.538808 0.842428i \(-0.681125\pi\)
−0.538808 + 0.842428i \(0.681125\pi\)
\(798\) 6.00558 0.212595
\(799\) −8.08760 −0.286119
\(800\) 12.8772 0.455277
\(801\) 14.8357 0.524192
\(802\) 56.0115 1.97783
\(803\) 4.11659 0.145271
\(804\) 6.20622 0.218877
\(805\) 5.68640 0.200419
\(806\) 6.26454 0.220659
\(807\) −9.30416 −0.327522
\(808\) 71.3128 2.50877
\(809\) 4.74344 0.166770 0.0833852 0.996517i \(-0.473427\pi\)
0.0833852 + 0.996517i \(0.473427\pi\)
\(810\) 6.53898 0.229756
\(811\) 30.4007 1.06751 0.533756 0.845638i \(-0.320780\pi\)
0.533756 + 0.845638i \(0.320780\pi\)
\(812\) −18.2651 −0.640978
\(813\) −25.0339 −0.877977
\(814\) 14.5622 0.510404
\(815\) −4.52528 −0.158514
\(816\) 8.40929 0.294384
\(817\) −2.82304 −0.0987655
\(818\) −4.05524 −0.141788
\(819\) −0.263882 −0.00922077
\(820\) −27.4175 −0.957460
\(821\) −9.40907 −0.328379 −0.164189 0.986429i \(-0.552501\pi\)
−0.164189 + 0.986429i \(0.552501\pi\)
\(822\) −51.4876 −1.79583
\(823\) 42.3813 1.47732 0.738660 0.674078i \(-0.235459\pi\)
0.738660 + 0.674078i \(0.235459\pi\)
\(824\) −15.5435 −0.541485
\(825\) −2.74949 −0.0957249
\(826\) −19.3016 −0.671590
\(827\) −46.3265 −1.61093 −0.805465 0.592643i \(-0.798085\pi\)
−0.805465 + 0.592643i \(0.798085\pi\)
\(828\) 35.4756 1.23286
\(829\) −34.6287 −1.20271 −0.601353 0.798983i \(-0.705371\pi\)
−0.601353 + 0.798983i \(0.705371\pi\)
\(830\) −31.0285 −1.07701
\(831\) 11.7689 0.408258
\(832\) 0.0495406 0.00171751
\(833\) −6.49751 −0.225125
\(834\) 12.5137 0.433315
\(835\) −36.1569 −1.25126
\(836\) 12.6571 0.437756
\(837\) −57.8566 −1.99982
\(838\) 23.3660 0.807165
\(839\) −8.22244 −0.283870 −0.141935 0.989876i \(-0.545332\pi\)
−0.141935 + 0.989876i \(0.545332\pi\)
\(840\) 8.62845 0.297710
\(841\) 4.02768 0.138885
\(842\) −48.8598 −1.68382
\(843\) −34.0031 −1.17113
\(844\) −58.9991 −2.03083
\(845\) −21.1437 −0.727367
\(846\) −33.1732 −1.14052
\(847\) −0.708864 −0.0243568
\(848\) 93.4998 3.21080
\(849\) 2.66132 0.0913362
\(850\) 5.94001 0.203741
\(851\) 28.0912 0.962952
\(852\) −85.5916 −2.93232
\(853\) 45.5009 1.55792 0.778960 0.627074i \(-0.215747\pi\)
0.778960 + 0.627074i \(0.215747\pi\)
\(854\) −12.8941 −0.441226
\(855\) −7.42685 −0.253993
\(856\) −74.1046 −2.53284
\(857\) 2.98251 0.101881 0.0509403 0.998702i \(-0.483778\pi\)
0.0509403 + 0.998702i \(0.483778\pi\)
\(858\) 0.693523 0.0236765
\(859\) −23.4095 −0.798722 −0.399361 0.916794i \(-0.630768\pi\)
−0.399361 + 0.916794i \(0.630768\pi\)
\(860\) −7.32226 −0.249687
\(861\) 3.12835 0.106614
\(862\) −77.7593 −2.64849
\(863\) −31.6326 −1.07679 −0.538393 0.842694i \(-0.680968\pi\)
−0.538393 + 0.842694i \(0.680968\pi\)
\(864\) 29.9981 1.02056
\(865\) −8.32693 −0.283124
\(866\) −102.383 −3.47913
\(867\) 1.17861 0.0400277
\(868\) −33.8361 −1.14847
\(869\) −6.14710 −0.208526
\(870\) −28.1670 −0.954952
\(871\) −0.271408 −0.00919631
\(872\) −86.3622 −2.92459
\(873\) −1.55095 −0.0524916
\(874\) 35.3078 1.19430
\(875\) 8.48907 0.286983
\(876\) 21.7534 0.734980
\(877\) −6.18409 −0.208822 −0.104411 0.994534i \(-0.533296\pi\)
−0.104411 + 0.994534i \(0.533296\pi\)
\(878\) −54.7462 −1.84759
\(879\) −7.70312 −0.259820
\(880\) 11.6524 0.392802
\(881\) 52.8354 1.78007 0.890034 0.455894i \(-0.150680\pi\)
0.890034 + 0.455894i \(0.150680\pi\)
\(882\) −26.6511 −0.897388
\(883\) −2.83578 −0.0954316 −0.0477158 0.998861i \(-0.515194\pi\)
−0.0477158 + 0.998861i \(0.515194\pi\)
\(884\) −1.03611 −0.0348480
\(885\) −20.5837 −0.691912
\(886\) 77.0352 2.58805
\(887\) 21.8491 0.733619 0.366810 0.930296i \(-0.380450\pi\)
0.366810 + 0.930296i \(0.380450\pi\)
\(888\) 42.6251 1.43040
\(889\) −3.00581 −0.100812
\(890\) 38.2980 1.28375
\(891\) −1.57246 −0.0526792
\(892\) 52.2410 1.74916
\(893\) −22.8316 −0.764030
\(894\) 22.1094 0.739450
\(895\) −23.7847 −0.795033
\(896\) −8.21280 −0.274370
\(897\) 1.33784 0.0446691
\(898\) 27.5896 0.920677
\(899\) 61.1840 2.04060
\(900\) 16.8485 0.561618
\(901\) 13.1046 0.436576
\(902\) 9.53427 0.317457
\(903\) 0.835475 0.0278029
\(904\) −39.3918 −1.31015
\(905\) 5.01786 0.166799
\(906\) −20.2107 −0.671455
\(907\) −13.5528 −0.450012 −0.225006 0.974357i \(-0.572240\pi\)
−0.225006 + 0.974357i \(0.572240\pi\)
\(908\) 44.4035 1.47358
\(909\) 18.1659 0.602523
\(910\) −0.681207 −0.0225818
\(911\) 30.7440 1.01860 0.509298 0.860590i \(-0.329905\pi\)
0.509298 + 0.860590i \(0.329905\pi\)
\(912\) 23.7397 0.786101
\(913\) 7.46155 0.246941
\(914\) 70.3868 2.32819
\(915\) −13.7505 −0.454578
\(916\) −125.780 −4.15589
\(917\) −2.08753 −0.0689362
\(918\) 13.8376 0.456707
\(919\) −27.6083 −0.910713 −0.455356 0.890309i \(-0.650488\pi\)
−0.455356 + 0.890309i \(0.650488\pi\)
\(920\) 50.7280 1.67245
\(921\) 17.5024 0.576722
\(922\) −11.8553 −0.390433
\(923\) 3.74305 0.123204
\(924\) −3.74587 −0.123230
\(925\) 13.3414 0.438664
\(926\) −7.31171 −0.240278
\(927\) −3.95948 −0.130047
\(928\) −31.7233 −1.04137
\(929\) −49.2689 −1.61646 −0.808230 0.588867i \(-0.799574\pi\)
−0.808230 + 0.588867i \(0.799574\pi\)
\(930\) −52.1796 −1.71104
\(931\) −18.3427 −0.601158
\(932\) −56.3551 −1.84597
\(933\) −0.455600 −0.0149157
\(934\) 47.0213 1.53859
\(935\) 1.63315 0.0534097
\(936\) −2.35407 −0.0769453
\(937\) 7.84795 0.256381 0.128191 0.991750i \(-0.459083\pi\)
0.128191 + 0.991750i \(0.459083\pi\)
\(938\) 2.11985 0.0692157
\(939\) −7.10625 −0.231904
\(940\) −59.2195 −1.93153
\(941\) −25.5271 −0.832159 −0.416079 0.909328i \(-0.636596\pi\)
−0.416079 + 0.909328i \(0.636596\pi\)
\(942\) −3.18209 −0.103678
\(943\) 18.3921 0.598928
\(944\) −76.2983 −2.48330
\(945\) 6.29133 0.204657
\(946\) 2.54628 0.0827866
\(947\) −16.2764 −0.528911 −0.264456 0.964398i \(-0.585192\pi\)
−0.264456 + 0.964398i \(0.585192\pi\)
\(948\) −32.4833 −1.05501
\(949\) −0.951312 −0.0308809
\(950\) 16.7688 0.544053
\(951\) −18.0225 −0.584420
\(952\) 4.48266 0.145284
\(953\) −14.8188 −0.480029 −0.240015 0.970769i \(-0.577152\pi\)
−0.240015 + 0.970769i \(0.577152\pi\)
\(954\) 53.7514 1.74026
\(955\) −26.4672 −0.856458
\(956\) 106.404 3.44135
\(957\) 6.77345 0.218954
\(958\) 21.3045 0.688319
\(959\) −12.1616 −0.392717
\(960\) −0.412642 −0.0133180
\(961\) 82.3437 2.65625
\(962\) −3.36520 −0.108499
\(963\) −18.8770 −0.608304
\(964\) 103.478 3.33282
\(965\) −3.34836 −0.107787
\(966\) −10.4493 −0.336201
\(967\) −35.0273 −1.12640 −0.563200 0.826320i \(-0.690430\pi\)
−0.563200 + 0.826320i \(0.690430\pi\)
\(968\) −6.32373 −0.203252
\(969\) 3.32726 0.106887
\(970\) −4.00375 −0.128553
\(971\) 48.1082 1.54387 0.771933 0.635704i \(-0.219290\pi\)
0.771933 + 0.635704i \(0.219290\pi\)
\(972\) 64.7868 2.07804
\(973\) 2.95579 0.0947583
\(974\) 17.1707 0.550186
\(975\) 0.635385 0.0203486
\(976\) −50.9695 −1.63149
\(977\) −3.92085 −0.125439 −0.0627195 0.998031i \(-0.519977\pi\)
−0.0627195 + 0.998031i \(0.519977\pi\)
\(978\) 8.31564 0.265905
\(979\) −9.20969 −0.294343
\(980\) −47.5765 −1.51978
\(981\) −21.9995 −0.702389
\(982\) 25.7700 0.822355
\(983\) −53.9594 −1.72104 −0.860519 0.509419i \(-0.829860\pi\)
−0.860519 + 0.509419i \(0.829860\pi\)
\(984\) 27.9078 0.889669
\(985\) −0.139285 −0.00443799
\(986\) −14.6334 −0.466022
\(987\) 6.75699 0.215077
\(988\) −2.92496 −0.0930555
\(989\) 4.91189 0.156189
\(990\) 6.69874 0.212900
\(991\) 29.8959 0.949673 0.474837 0.880074i \(-0.342507\pi\)
0.474837 + 0.880074i \(0.342507\pi\)
\(992\) −58.7676 −1.86587
\(993\) 29.4656 0.935063
\(994\) −29.2354 −0.927291
\(995\) −22.4680 −0.712283
\(996\) 39.4293 1.24937
\(997\) 3.03654 0.0961682 0.0480841 0.998843i \(-0.484688\pi\)
0.0480841 + 0.998843i \(0.484688\pi\)
\(998\) −31.0474 −0.982788
\(999\) 31.0796 0.983314
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 8041.2.a.c.1.3 60
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8041.2.a.c.1.3 60 1.1 even 1 trivial