Properties

Label 6025.2.a.q.1.20
Level $6025$
Weight $2$
Character 6025.1
Self dual yes
Analytic conductor $48.110$
Analytic rank $0$
Dimension $66$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6025,2,Mod(1,6025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6025, 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("6025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6025 = 5^{2} \cdot 241 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6025.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: \(48.1098672178\)
Analytic rank: \(0\)
Dimension: \(66\)
Twist minimal: no (minimal twist has level 1205)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.20
Character \(\chi\) \(=\) 6025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.38861 q^{2} -0.0366779 q^{3} -0.0717484 q^{4} +0.0509315 q^{6} +3.69082 q^{7} +2.87686 q^{8} -2.99865 q^{9} +O(q^{10})\) \(q-1.38861 q^{2} -0.0366779 q^{3} -0.0717484 q^{4} +0.0509315 q^{6} +3.69082 q^{7} +2.87686 q^{8} -2.99865 q^{9} +2.61704 q^{11} +0.00263158 q^{12} +5.39014 q^{13} -5.12512 q^{14} -3.85136 q^{16} +5.52357 q^{17} +4.16398 q^{18} -8.11267 q^{19} -0.135371 q^{21} -3.63406 q^{22} -7.71557 q^{23} -0.105517 q^{24} -7.48483 q^{26} +0.220018 q^{27} -0.264810 q^{28} +7.36101 q^{29} +5.97766 q^{31} -0.405672 q^{32} -0.0959874 q^{33} -7.67012 q^{34} +0.215149 q^{36} -2.83895 q^{37} +11.2654 q^{38} -0.197699 q^{39} +1.56023 q^{41} +0.187979 q^{42} +9.68988 q^{43} -0.187768 q^{44} +10.7140 q^{46} -0.834859 q^{47} +0.141260 q^{48} +6.62212 q^{49} -0.202593 q^{51} -0.386734 q^{52} +8.02986 q^{53} -0.305520 q^{54} +10.6180 q^{56} +0.297556 q^{57} -10.2216 q^{58} +6.01249 q^{59} +4.78566 q^{61} -8.30066 q^{62} -11.0675 q^{63} +8.26603 q^{64} +0.133290 q^{66} +5.31555 q^{67} -0.396308 q^{68} +0.282991 q^{69} -4.61386 q^{71} -8.62671 q^{72} -5.51666 q^{73} +3.94221 q^{74} +0.582071 q^{76} +9.65901 q^{77} +0.274528 q^{78} +5.27223 q^{79} +8.98789 q^{81} -2.16656 q^{82} -2.54694 q^{83} +0.00971267 q^{84} -13.4555 q^{86} -0.269986 q^{87} +7.52886 q^{88} +12.7951 q^{89} +19.8940 q^{91} +0.553580 q^{92} -0.219248 q^{93} +1.15930 q^{94} +0.0148792 q^{96} -10.9674 q^{97} -9.19557 q^{98} -7.84759 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 66 q + 78 q^{4} + 16 q^{6} + 90 q^{9} + 48 q^{11} + 30 q^{14} + 98 q^{16} + 12 q^{19} + 18 q^{21} + 42 q^{24} + 48 q^{26} + 56 q^{29} + 48 q^{31} + 8 q^{34} + 158 q^{36} + 84 q^{39} + 56 q^{41} + 144 q^{44} + 36 q^{46} + 98 q^{49} + 44 q^{51} + 86 q^{54} + 104 q^{56} + 108 q^{59} + 22 q^{61} + 136 q^{64} + 74 q^{66} + 20 q^{69} + 212 q^{71} + 84 q^{74} + 6 q^{76} + 66 q^{79} + 162 q^{81} - 52 q^{84} + 100 q^{86} + 54 q^{89} + 72 q^{91} - 96 q^{94} + 122 q^{96} + 112 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.38861 −0.981899 −0.490950 0.871188i \(-0.663350\pi\)
−0.490950 + 0.871188i \(0.663350\pi\)
\(3\) −0.0366779 −0.0211760 −0.0105880 0.999944i \(-0.503370\pi\)
−0.0105880 + 0.999944i \(0.503370\pi\)
\(4\) −0.0717484 −0.0358742
\(5\) 0 0
\(6\) 0.0509315 0.0207927
\(7\) 3.69082 1.39500 0.697499 0.716586i \(-0.254296\pi\)
0.697499 + 0.716586i \(0.254296\pi\)
\(8\) 2.87686 1.01712
\(9\) −2.99865 −0.999552
\(10\) 0 0
\(11\) 2.61704 0.789067 0.394533 0.918882i \(-0.370906\pi\)
0.394533 + 0.918882i \(0.370906\pi\)
\(12\) 0.00263158 0.000759672 0
\(13\) 5.39014 1.49496 0.747478 0.664286i \(-0.231264\pi\)
0.747478 + 0.664286i \(0.231264\pi\)
\(14\) −5.12512 −1.36975
\(15\) 0 0
\(16\) −3.85136 −0.962839
\(17\) 5.52357 1.33966 0.669832 0.742513i \(-0.266366\pi\)
0.669832 + 0.742513i \(0.266366\pi\)
\(18\) 4.16398 0.981459
\(19\) −8.11267 −1.86117 −0.930587 0.366071i \(-0.880703\pi\)
−0.930587 + 0.366071i \(0.880703\pi\)
\(20\) 0 0
\(21\) −0.135371 −0.0295404
\(22\) −3.63406 −0.774784
\(23\) −7.71557 −1.60881 −0.804404 0.594082i \(-0.797515\pi\)
−0.804404 + 0.594082i \(0.797515\pi\)
\(24\) −0.105517 −0.0215386
\(25\) 0 0
\(26\) −7.48483 −1.46790
\(27\) 0.220018 0.0423425
\(28\) −0.264810 −0.0500444
\(29\) 7.36101 1.36690 0.683452 0.729995i \(-0.260478\pi\)
0.683452 + 0.729995i \(0.260478\pi\)
\(30\) 0 0
\(31\) 5.97766 1.07362 0.536809 0.843704i \(-0.319629\pi\)
0.536809 + 0.843704i \(0.319629\pi\)
\(32\) −0.405672 −0.0717133
\(33\) −0.0959874 −0.0167093
\(34\) −7.67012 −1.31541
\(35\) 0 0
\(36\) 0.215149 0.0358581
\(37\) −2.83895 −0.466720 −0.233360 0.972390i \(-0.574972\pi\)
−0.233360 + 0.972390i \(0.574972\pi\)
\(38\) 11.2654 1.82748
\(39\) −0.197699 −0.0316572
\(40\) 0 0
\(41\) 1.56023 0.243667 0.121833 0.992551i \(-0.461123\pi\)
0.121833 + 0.992551i \(0.461123\pi\)
\(42\) 0.187979 0.0290057
\(43\) 9.68988 1.47769 0.738846 0.673874i \(-0.235371\pi\)
0.738846 + 0.673874i \(0.235371\pi\)
\(44\) −0.187768 −0.0283071
\(45\) 0 0
\(46\) 10.7140 1.57969
\(47\) −0.834859 −0.121777 −0.0608884 0.998145i \(-0.519393\pi\)
−0.0608884 + 0.998145i \(0.519393\pi\)
\(48\) 0.141260 0.0203891
\(49\) 6.62212 0.946017
\(50\) 0 0
\(51\) −0.202593 −0.0283687
\(52\) −0.386734 −0.0536304
\(53\) 8.02986 1.10299 0.551493 0.834179i \(-0.314058\pi\)
0.551493 + 0.834179i \(0.314058\pi\)
\(54\) −0.305520 −0.0415760
\(55\) 0 0
\(56\) 10.6180 1.41888
\(57\) 0.297556 0.0394122
\(58\) −10.2216 −1.34216
\(59\) 6.01249 0.782759 0.391380 0.920229i \(-0.371998\pi\)
0.391380 + 0.920229i \(0.371998\pi\)
\(60\) 0 0
\(61\) 4.78566 0.612741 0.306371 0.951912i \(-0.400885\pi\)
0.306371 + 0.951912i \(0.400885\pi\)
\(62\) −8.30066 −1.05419
\(63\) −11.0675 −1.39437
\(64\) 8.26603 1.03325
\(65\) 0 0
\(66\) 0.133290 0.0164068
\(67\) 5.31555 0.649398 0.324699 0.945817i \(-0.394737\pi\)
0.324699 + 0.945817i \(0.394737\pi\)
\(68\) −0.396308 −0.0480594
\(69\) 0.282991 0.0340681
\(70\) 0 0
\(71\) −4.61386 −0.547565 −0.273782 0.961792i \(-0.588275\pi\)
−0.273782 + 0.961792i \(0.588275\pi\)
\(72\) −8.62671 −1.01667
\(73\) −5.51666 −0.645676 −0.322838 0.946454i \(-0.604637\pi\)
−0.322838 + 0.946454i \(0.604637\pi\)
\(74\) 3.94221 0.458272
\(75\) 0 0
\(76\) 0.582071 0.0667681
\(77\) 9.65901 1.10075
\(78\) 0.274528 0.0310842
\(79\) 5.27223 0.593172 0.296586 0.955006i \(-0.404152\pi\)
0.296586 + 0.955006i \(0.404152\pi\)
\(80\) 0 0
\(81\) 8.98789 0.998655
\(82\) −2.16656 −0.239256
\(83\) −2.54694 −0.279563 −0.139782 0.990182i \(-0.544640\pi\)
−0.139782 + 0.990182i \(0.544640\pi\)
\(84\) 0.00971267 0.00105974
\(85\) 0 0
\(86\) −13.4555 −1.45094
\(87\) −0.269986 −0.0289456
\(88\) 7.52886 0.802579
\(89\) 12.7951 1.35627 0.678137 0.734936i \(-0.262788\pi\)
0.678137 + 0.734936i \(0.262788\pi\)
\(90\) 0 0
\(91\) 19.8940 2.08546
\(92\) 0.553580 0.0577147
\(93\) −0.219248 −0.0227349
\(94\) 1.15930 0.119572
\(95\) 0 0
\(96\) 0.0148792 0.00151860
\(97\) −10.9674 −1.11357 −0.556787 0.830656i \(-0.687966\pi\)
−0.556787 + 0.830656i \(0.687966\pi\)
\(98\) −9.19557 −0.928893
\(99\) −7.84759 −0.788713
\(100\) 0 0
\(101\) −16.4616 −1.63800 −0.818998 0.573797i \(-0.805470\pi\)
−0.818998 + 0.573797i \(0.805470\pi\)
\(102\) 0.281324 0.0278552
\(103\) 14.6984 1.44827 0.724136 0.689657i \(-0.242239\pi\)
0.724136 + 0.689657i \(0.242239\pi\)
\(104\) 15.5067 1.52056
\(105\) 0 0
\(106\) −11.1504 −1.08302
\(107\) −0.783974 −0.0757896 −0.0378948 0.999282i \(-0.512065\pi\)
−0.0378948 + 0.999282i \(0.512065\pi\)
\(108\) −0.0157859 −0.00151900
\(109\) −12.3667 −1.18451 −0.592256 0.805750i \(-0.701763\pi\)
−0.592256 + 0.805750i \(0.701763\pi\)
\(110\) 0 0
\(111\) 0.104127 0.00988327
\(112\) −14.2146 −1.34316
\(113\) −20.2824 −1.90801 −0.954003 0.299796i \(-0.903081\pi\)
−0.954003 + 0.299796i \(0.903081\pi\)
\(114\) −0.413190 −0.0386988
\(115\) 0 0
\(116\) −0.528141 −0.0490366
\(117\) −16.1632 −1.49429
\(118\) −8.34903 −0.768590
\(119\) 20.3865 1.86883
\(120\) 0 0
\(121\) −4.15111 −0.377374
\(122\) −6.64544 −0.601650
\(123\) −0.0572259 −0.00515988
\(124\) −0.428887 −0.0385152
\(125\) 0 0
\(126\) 15.3685 1.36913
\(127\) −5.27769 −0.468319 −0.234160 0.972198i \(-0.575234\pi\)
−0.234160 + 0.972198i \(0.575234\pi\)
\(128\) −10.6670 −0.942838
\(129\) −0.355404 −0.0312916
\(130\) 0 0
\(131\) −1.74621 −0.152567 −0.0762835 0.997086i \(-0.524305\pi\)
−0.0762835 + 0.997086i \(0.524305\pi\)
\(132\) 0.00688695 0.000599432 0
\(133\) −29.9424 −2.59633
\(134\) −7.38126 −0.637643
\(135\) 0 0
\(136\) 15.8906 1.36260
\(137\) −2.84655 −0.243197 −0.121599 0.992579i \(-0.538802\pi\)
−0.121599 + 0.992579i \(0.538802\pi\)
\(138\) −0.392965 −0.0334514
\(139\) 3.73442 0.316749 0.158375 0.987379i \(-0.449375\pi\)
0.158375 + 0.987379i \(0.449375\pi\)
\(140\) 0 0
\(141\) 0.0306209 0.00257874
\(142\) 6.40688 0.537653
\(143\) 14.1062 1.17962
\(144\) 11.5489 0.962407
\(145\) 0 0
\(146\) 7.66051 0.633988
\(147\) −0.242885 −0.0200328
\(148\) 0.203690 0.0167432
\(149\) −10.3087 −0.844520 −0.422260 0.906475i \(-0.638763\pi\)
−0.422260 + 0.906475i \(0.638763\pi\)
\(150\) 0 0
\(151\) −2.96022 −0.240899 −0.120450 0.992719i \(-0.538434\pi\)
−0.120450 + 0.992719i \(0.538434\pi\)
\(152\) −23.3390 −1.89304
\(153\) −16.5633 −1.33906
\(154\) −13.4126 −1.08082
\(155\) 0 0
\(156\) 0.0141846 0.00113568
\(157\) 4.61712 0.368486 0.184243 0.982881i \(-0.441017\pi\)
0.184243 + 0.982881i \(0.441017\pi\)
\(158\) −7.32109 −0.582435
\(159\) −0.294518 −0.0233568
\(160\) 0 0
\(161\) −28.4768 −2.24428
\(162\) −12.4807 −0.980578
\(163\) −4.19155 −0.328307 −0.164154 0.986435i \(-0.552489\pi\)
−0.164154 + 0.986435i \(0.552489\pi\)
\(164\) −0.111944 −0.00874135
\(165\) 0 0
\(166\) 3.53672 0.274503
\(167\) 15.1804 1.17469 0.587346 0.809336i \(-0.300173\pi\)
0.587346 + 0.809336i \(0.300173\pi\)
\(168\) −0.389444 −0.0300463
\(169\) 16.0536 1.23490
\(170\) 0 0
\(171\) 24.3271 1.86034
\(172\) −0.695233 −0.0530110
\(173\) 11.5652 0.879284 0.439642 0.898173i \(-0.355105\pi\)
0.439642 + 0.898173i \(0.355105\pi\)
\(174\) 0.374907 0.0284216
\(175\) 0 0
\(176\) −10.0791 −0.759744
\(177\) −0.220525 −0.0165757
\(178\) −17.7674 −1.33172
\(179\) −10.9935 −0.821694 −0.410847 0.911704i \(-0.634767\pi\)
−0.410847 + 0.911704i \(0.634767\pi\)
\(180\) 0 0
\(181\) −0.352647 −0.0262120 −0.0131060 0.999914i \(-0.504172\pi\)
−0.0131060 + 0.999914i \(0.504172\pi\)
\(182\) −27.6251 −2.04771
\(183\) −0.175528 −0.0129754
\(184\) −22.1966 −1.63636
\(185\) 0 0
\(186\) 0.304451 0.0223234
\(187\) 14.4554 1.05708
\(188\) 0.0598998 0.00436864
\(189\) 0.812046 0.0590676
\(190\) 0 0
\(191\) 15.1867 1.09887 0.549437 0.835535i \(-0.314842\pi\)
0.549437 + 0.835535i \(0.314842\pi\)
\(192\) −0.303181 −0.0218802
\(193\) 0.151159 0.0108807 0.00544033 0.999985i \(-0.498268\pi\)
0.00544033 + 0.999985i \(0.498268\pi\)
\(194\) 15.2295 1.09342
\(195\) 0 0
\(196\) −0.475126 −0.0339376
\(197\) 26.5412 1.89098 0.945490 0.325653i \(-0.105584\pi\)
0.945490 + 0.325653i \(0.105584\pi\)
\(198\) 10.8973 0.774437
\(199\) 17.6403 1.25048 0.625242 0.780431i \(-0.285000\pi\)
0.625242 + 0.780431i \(0.285000\pi\)
\(200\) 0 0
\(201\) −0.194963 −0.0137516
\(202\) 22.8589 1.60835
\(203\) 27.1681 1.90683
\(204\) 0.0145357 0.00101770
\(205\) 0 0
\(206\) −20.4104 −1.42206
\(207\) 23.1363 1.60809
\(208\) −20.7594 −1.43940
\(209\) −21.2312 −1.46859
\(210\) 0 0
\(211\) −3.78020 −0.260240 −0.130120 0.991498i \(-0.541536\pi\)
−0.130120 + 0.991498i \(0.541536\pi\)
\(212\) −0.576130 −0.0395688
\(213\) 0.169227 0.0115952
\(214\) 1.08864 0.0744177
\(215\) 0 0
\(216\) 0.632961 0.0430676
\(217\) 22.0624 1.49769
\(218\) 17.1725 1.16307
\(219\) 0.202339 0.0136728
\(220\) 0 0
\(221\) 29.7729 2.00274
\(222\) −0.144592 −0.00970437
\(223\) −16.3491 −1.09482 −0.547410 0.836865i \(-0.684386\pi\)
−0.547410 + 0.836865i \(0.684386\pi\)
\(224\) −1.49726 −0.100040
\(225\) 0 0
\(226\) 28.1644 1.87347
\(227\) 20.9681 1.39170 0.695851 0.718186i \(-0.255027\pi\)
0.695851 + 0.718186i \(0.255027\pi\)
\(228\) −0.0213491 −0.00141388
\(229\) −15.2467 −1.00753 −0.503764 0.863841i \(-0.668052\pi\)
−0.503764 + 0.863841i \(0.668052\pi\)
\(230\) 0 0
\(231\) −0.354272 −0.0233094
\(232\) 21.1766 1.39031
\(233\) −17.5288 −1.14835 −0.574175 0.818732i \(-0.694677\pi\)
−0.574175 + 0.818732i \(0.694677\pi\)
\(234\) 22.4444 1.46724
\(235\) 0 0
\(236\) −0.431386 −0.0280809
\(237\) −0.193374 −0.0125610
\(238\) −28.3090 −1.83500
\(239\) 0.809356 0.0523529 0.0261764 0.999657i \(-0.491667\pi\)
0.0261764 + 0.999657i \(0.491667\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157
\(242\) 5.76429 0.370543
\(243\) −0.989711 −0.0634900
\(244\) −0.343364 −0.0219816
\(245\) 0 0
\(246\) 0.0794647 0.00506648
\(247\) −43.7284 −2.78237
\(248\) 17.1969 1.09200
\(249\) 0.0934164 0.00592003
\(250\) 0 0
\(251\) −19.0352 −1.20149 −0.600745 0.799441i \(-0.705129\pi\)
−0.600745 + 0.799441i \(0.705129\pi\)
\(252\) 0.794074 0.0500220
\(253\) −20.1920 −1.26946
\(254\) 7.32868 0.459842
\(255\) 0 0
\(256\) −1.71972 −0.107482
\(257\) 19.3841 1.20915 0.604573 0.796550i \(-0.293344\pi\)
0.604573 + 0.796550i \(0.293344\pi\)
\(258\) 0.493520 0.0307252
\(259\) −10.4780 −0.651074
\(260\) 0 0
\(261\) −22.0731 −1.36629
\(262\) 2.42481 0.149805
\(263\) 8.35304 0.515071 0.257535 0.966269i \(-0.417090\pi\)
0.257535 + 0.966269i \(0.417090\pi\)
\(264\) −0.276143 −0.0169954
\(265\) 0 0
\(266\) 41.5784 2.54934
\(267\) −0.469296 −0.0287204
\(268\) −0.381382 −0.0232966
\(269\) 7.93093 0.483557 0.241779 0.970331i \(-0.422269\pi\)
0.241779 + 0.970331i \(0.422269\pi\)
\(270\) 0 0
\(271\) −9.54415 −0.579766 −0.289883 0.957062i \(-0.593616\pi\)
−0.289883 + 0.957062i \(0.593616\pi\)
\(272\) −21.2732 −1.28988
\(273\) −0.729671 −0.0441617
\(274\) 3.95277 0.238795
\(275\) 0 0
\(276\) −0.0203041 −0.00122217
\(277\) −27.9456 −1.67909 −0.839544 0.543291i \(-0.817178\pi\)
−0.839544 + 0.543291i \(0.817178\pi\)
\(278\) −5.18567 −0.311016
\(279\) −17.9249 −1.07314
\(280\) 0 0
\(281\) 28.6875 1.71135 0.855676 0.517512i \(-0.173142\pi\)
0.855676 + 0.517512i \(0.173142\pi\)
\(282\) −0.0425206 −0.00253207
\(283\) −14.4245 −0.857449 −0.428725 0.903435i \(-0.641037\pi\)
−0.428725 + 0.903435i \(0.641037\pi\)
\(284\) 0.331037 0.0196435
\(285\) 0 0
\(286\) −19.5881 −1.15827
\(287\) 5.75851 0.339914
\(288\) 1.21647 0.0716812
\(289\) 13.5099 0.794698
\(290\) 0 0
\(291\) 0.402262 0.0235810
\(292\) 0.395811 0.0231631
\(293\) 28.2391 1.64975 0.824874 0.565316i \(-0.191246\pi\)
0.824874 + 0.565316i \(0.191246\pi\)
\(294\) 0.337274 0.0196702
\(295\) 0 0
\(296\) −8.16726 −0.474712
\(297\) 0.575796 0.0334110
\(298\) 14.3148 0.829234
\(299\) −41.5880 −2.40510
\(300\) 0 0
\(301\) 35.7635 2.06138
\(302\) 4.11060 0.236539
\(303\) 0.603778 0.0346862
\(304\) 31.2448 1.79201
\(305\) 0 0
\(306\) 23.0000 1.31482
\(307\) 29.8363 1.70285 0.851423 0.524479i \(-0.175740\pi\)
0.851423 + 0.524479i \(0.175740\pi\)
\(308\) −0.693018 −0.0394884
\(309\) −0.539105 −0.0306686
\(310\) 0 0
\(311\) 20.0603 1.13752 0.568759 0.822504i \(-0.307424\pi\)
0.568759 + 0.822504i \(0.307424\pi\)
\(312\) −0.568753 −0.0321993
\(313\) −11.9865 −0.677516 −0.338758 0.940874i \(-0.610007\pi\)
−0.338758 + 0.940874i \(0.610007\pi\)
\(314\) −6.41140 −0.361816
\(315\) 0 0
\(316\) −0.378274 −0.0212796
\(317\) −27.2280 −1.52928 −0.764639 0.644459i \(-0.777083\pi\)
−0.764639 + 0.644459i \(0.777083\pi\)
\(318\) 0.408973 0.0229341
\(319\) 19.2640 1.07858
\(320\) 0 0
\(321\) 0.0287545 0.00160492
\(322\) 39.5432 2.20366
\(323\) −44.8109 −2.49335
\(324\) −0.644867 −0.0358259
\(325\) 0 0
\(326\) 5.82044 0.322365
\(327\) 0.453583 0.0250832
\(328\) 4.48856 0.247839
\(329\) −3.08131 −0.169878
\(330\) 0 0
\(331\) 17.1507 0.942688 0.471344 0.881950i \(-0.343769\pi\)
0.471344 + 0.881950i \(0.343769\pi\)
\(332\) 0.182739 0.0100291
\(333\) 8.51303 0.466511
\(334\) −21.0797 −1.15343
\(335\) 0 0
\(336\) 0.521363 0.0284427
\(337\) 4.40209 0.239797 0.119899 0.992786i \(-0.461743\pi\)
0.119899 + 0.992786i \(0.461743\pi\)
\(338\) −22.2923 −1.21254
\(339\) 0.743915 0.0404039
\(340\) 0 0
\(341\) 15.6438 0.847157
\(342\) −33.7810 −1.82667
\(343\) −1.39470 −0.0753065
\(344\) 27.8764 1.50300
\(345\) 0 0
\(346\) −16.0596 −0.863368
\(347\) 4.06906 0.218438 0.109219 0.994018i \(-0.465165\pi\)
0.109219 + 0.994018i \(0.465165\pi\)
\(348\) 0.0193711 0.00103840
\(349\) 34.5033 1.84692 0.923459 0.383697i \(-0.125349\pi\)
0.923459 + 0.383697i \(0.125349\pi\)
\(350\) 0 0
\(351\) 1.18593 0.0633002
\(352\) −1.06166 −0.0565866
\(353\) −8.89963 −0.473679 −0.236840 0.971549i \(-0.576112\pi\)
−0.236840 + 0.971549i \(0.576112\pi\)
\(354\) 0.306225 0.0162757
\(355\) 0 0
\(356\) −0.918025 −0.0486552
\(357\) −0.747733 −0.0395743
\(358\) 15.2658 0.806820
\(359\) 10.9527 0.578060 0.289030 0.957320i \(-0.406667\pi\)
0.289030 + 0.957320i \(0.406667\pi\)
\(360\) 0 0
\(361\) 46.8154 2.46397
\(362\) 0.489691 0.0257376
\(363\) 0.152254 0.00799126
\(364\) −1.42736 −0.0748142
\(365\) 0 0
\(366\) 0.243741 0.0127405
\(367\) −9.11480 −0.475789 −0.237894 0.971291i \(-0.576457\pi\)
−0.237894 + 0.971291i \(0.576457\pi\)
\(368\) 29.7154 1.54902
\(369\) −4.67859 −0.243557
\(370\) 0 0
\(371\) 29.6367 1.53866
\(372\) 0.0157307 0.000815598 0
\(373\) 6.33820 0.328180 0.164090 0.986445i \(-0.447531\pi\)
0.164090 + 0.986445i \(0.447531\pi\)
\(374\) −20.0730 −1.03795
\(375\) 0 0
\(376\) −2.40177 −0.123862
\(377\) 39.6769 2.04346
\(378\) −1.12762 −0.0579985
\(379\) 20.0791 1.03139 0.515696 0.856772i \(-0.327533\pi\)
0.515696 + 0.856772i \(0.327533\pi\)
\(380\) 0 0
\(381\) 0.193575 0.00991712
\(382\) −21.0885 −1.07898
\(383\) −13.2925 −0.679214 −0.339607 0.940567i \(-0.610294\pi\)
−0.339607 + 0.940567i \(0.610294\pi\)
\(384\) 0.391243 0.0199655
\(385\) 0 0
\(386\) −0.209901 −0.0106837
\(387\) −29.0566 −1.47703
\(388\) 0.786895 0.0399486
\(389\) −3.03200 −0.153729 −0.0768644 0.997042i \(-0.524491\pi\)
−0.0768644 + 0.997042i \(0.524491\pi\)
\(390\) 0 0
\(391\) −42.6175 −2.15526
\(392\) 19.0509 0.962216
\(393\) 0.0640473 0.00323076
\(394\) −36.8554 −1.85675
\(395\) 0 0
\(396\) 0.563052 0.0282944
\(397\) −2.31275 −0.116073 −0.0580367 0.998314i \(-0.518484\pi\)
−0.0580367 + 0.998314i \(0.518484\pi\)
\(398\) −24.4955 −1.22785
\(399\) 1.09822 0.0549799
\(400\) 0 0
\(401\) 18.0279 0.900269 0.450134 0.892961i \(-0.351376\pi\)
0.450134 + 0.892961i \(0.351376\pi\)
\(402\) 0.270729 0.0135027
\(403\) 32.2204 1.60501
\(404\) 1.18110 0.0587618
\(405\) 0 0
\(406\) −37.7261 −1.87231
\(407\) −7.42964 −0.368274
\(408\) −0.582832 −0.0288545
\(409\) −14.6411 −0.723956 −0.361978 0.932187i \(-0.617899\pi\)
−0.361978 + 0.932187i \(0.617899\pi\)
\(410\) 0 0
\(411\) 0.104406 0.00514995
\(412\) −1.05458 −0.0519556
\(413\) 22.1910 1.09195
\(414\) −32.1275 −1.57898
\(415\) 0 0
\(416\) −2.18663 −0.107208
\(417\) −0.136971 −0.00670748
\(418\) 29.4819 1.44201
\(419\) 11.8773 0.580242 0.290121 0.956990i \(-0.406304\pi\)
0.290121 + 0.956990i \(0.406304\pi\)
\(420\) 0 0
\(421\) −13.8898 −0.676945 −0.338473 0.940976i \(-0.609910\pi\)
−0.338473 + 0.940976i \(0.609910\pi\)
\(422\) 5.24924 0.255529
\(423\) 2.50346 0.121722
\(424\) 23.1008 1.12187
\(425\) 0 0
\(426\) −0.234991 −0.0113853
\(427\) 17.6630 0.854772
\(428\) 0.0562489 0.00271889
\(429\) −0.517386 −0.0249796
\(430\) 0 0
\(431\) 14.5438 0.700551 0.350276 0.936647i \(-0.386088\pi\)
0.350276 + 0.936647i \(0.386088\pi\)
\(432\) −0.847367 −0.0407690
\(433\) −34.3657 −1.65151 −0.825754 0.564030i \(-0.809250\pi\)
−0.825754 + 0.564030i \(0.809250\pi\)
\(434\) −30.6362 −1.47059
\(435\) 0 0
\(436\) 0.887288 0.0424934
\(437\) 62.5939 2.99427
\(438\) −0.280971 −0.0134253
\(439\) −9.72345 −0.464075 −0.232037 0.972707i \(-0.574539\pi\)
−0.232037 + 0.972707i \(0.574539\pi\)
\(440\) 0 0
\(441\) −19.8574 −0.945593
\(442\) −41.3430 −1.96649
\(443\) 23.9451 1.13767 0.568833 0.822453i \(-0.307395\pi\)
0.568833 + 0.822453i \(0.307395\pi\)
\(444\) −0.00747092 −0.000354554 0
\(445\) 0 0
\(446\) 22.7027 1.07500
\(447\) 0.378101 0.0178836
\(448\) 30.5084 1.44139
\(449\) 14.7162 0.694499 0.347250 0.937773i \(-0.387116\pi\)
0.347250 + 0.937773i \(0.387116\pi\)
\(450\) 0 0
\(451\) 4.08318 0.192269
\(452\) 1.45523 0.0684482
\(453\) 0.108575 0.00510128
\(454\) −29.1166 −1.36651
\(455\) 0 0
\(456\) 0.856026 0.0400871
\(457\) −24.2946 −1.13646 −0.568228 0.822871i \(-0.692371\pi\)
−0.568228 + 0.822871i \(0.692371\pi\)
\(458\) 21.1717 0.989291
\(459\) 1.21529 0.0567247
\(460\) 0 0
\(461\) 11.5144 0.536281 0.268140 0.963380i \(-0.413591\pi\)
0.268140 + 0.963380i \(0.413591\pi\)
\(462\) 0.491947 0.0228875
\(463\) 5.82777 0.270840 0.135420 0.990788i \(-0.456762\pi\)
0.135420 + 0.990788i \(0.456762\pi\)
\(464\) −28.3499 −1.31611
\(465\) 0 0
\(466\) 24.3408 1.12756
\(467\) −10.0402 −0.464605 −0.232302 0.972644i \(-0.574626\pi\)
−0.232302 + 0.972644i \(0.574626\pi\)
\(468\) 1.15968 0.0536063
\(469\) 19.6187 0.905909
\(470\) 0 0
\(471\) −0.169346 −0.00780306
\(472\) 17.2971 0.796163
\(473\) 25.3588 1.16600
\(474\) 0.268522 0.0123336
\(475\) 0 0
\(476\) −1.46270 −0.0670427
\(477\) −24.0788 −1.10249
\(478\) −1.12388 −0.0514053
\(479\) 7.83576 0.358025 0.179012 0.983847i \(-0.442710\pi\)
0.179012 + 0.983847i \(0.442710\pi\)
\(480\) 0 0
\(481\) −15.3023 −0.697727
\(482\) 1.38861 0.0632497
\(483\) 1.04447 0.0475249
\(484\) 0.297835 0.0135380
\(485\) 0 0
\(486\) 1.37433 0.0623408
\(487\) −42.5597 −1.92856 −0.964281 0.264880i \(-0.914668\pi\)
−0.964281 + 0.264880i \(0.914668\pi\)
\(488\) 13.7677 0.623234
\(489\) 0.153737 0.00695223
\(490\) 0 0
\(491\) −12.8949 −0.581938 −0.290969 0.956732i \(-0.593978\pi\)
−0.290969 + 0.956732i \(0.593978\pi\)
\(492\) 0.00410587 0.000185107 0
\(493\) 40.6591 1.83119
\(494\) 60.7220 2.73201
\(495\) 0 0
\(496\) −23.0221 −1.03372
\(497\) −17.0289 −0.763851
\(498\) −0.129719 −0.00581287
\(499\) −3.36405 −0.150595 −0.0752977 0.997161i \(-0.523991\pi\)
−0.0752977 + 0.997161i \(0.523991\pi\)
\(500\) 0 0
\(501\) −0.556784 −0.0248753
\(502\) 26.4325 1.17974
\(503\) −34.4241 −1.53489 −0.767447 0.641113i \(-0.778473\pi\)
−0.767447 + 0.641113i \(0.778473\pi\)
\(504\) −31.8396 −1.41825
\(505\) 0 0
\(506\) 28.0388 1.24648
\(507\) −0.588813 −0.0261501
\(508\) 0.378666 0.0168006
\(509\) 37.5965 1.66643 0.833217 0.552947i \(-0.186497\pi\)
0.833217 + 0.552947i \(0.186497\pi\)
\(510\) 0 0
\(511\) −20.3610 −0.900716
\(512\) 23.7220 1.04837
\(513\) −1.78493 −0.0788067
\(514\) −26.9170 −1.18726
\(515\) 0 0
\(516\) 0.0254997 0.00112256
\(517\) −2.18486 −0.0960900
\(518\) 14.5500 0.639289
\(519\) −0.424186 −0.0186197
\(520\) 0 0
\(521\) −38.9298 −1.70555 −0.852774 0.522281i \(-0.825081\pi\)
−0.852774 + 0.522281i \(0.825081\pi\)
\(522\) 30.6511 1.34156
\(523\) −0.192415 −0.00841374 −0.00420687 0.999991i \(-0.501339\pi\)
−0.00420687 + 0.999991i \(0.501339\pi\)
\(524\) 0.125288 0.00547322
\(525\) 0 0
\(526\) −11.5992 −0.505747
\(527\) 33.0180 1.43829
\(528\) 0.369682 0.0160883
\(529\) 36.5301 1.58826
\(530\) 0 0
\(531\) −18.0294 −0.782408
\(532\) 2.14832 0.0931413
\(533\) 8.40985 0.364271
\(534\) 0.651671 0.0282006
\(535\) 0 0
\(536\) 15.2921 0.660518
\(537\) 0.403219 0.0174002
\(538\) −11.0130 −0.474805
\(539\) 17.3303 0.746470
\(540\) 0 0
\(541\) −22.9233 −0.985548 −0.492774 0.870157i \(-0.664017\pi\)
−0.492774 + 0.870157i \(0.664017\pi\)
\(542\) 13.2532 0.569272
\(543\) 0.0129343 0.000555066 0
\(544\) −2.24076 −0.0960718
\(545\) 0 0
\(546\) 1.01323 0.0433623
\(547\) 18.0310 0.770950 0.385475 0.922718i \(-0.374038\pi\)
0.385475 + 0.922718i \(0.374038\pi\)
\(548\) 0.204236 0.00872452
\(549\) −14.3506 −0.612467
\(550\) 0 0
\(551\) −59.7174 −2.54405
\(552\) 0.814126 0.0346515
\(553\) 19.4588 0.827473
\(554\) 38.8057 1.64870
\(555\) 0 0
\(556\) −0.267939 −0.0113631
\(557\) 23.7519 1.00640 0.503201 0.864170i \(-0.332156\pi\)
0.503201 + 0.864170i \(0.332156\pi\)
\(558\) 24.8908 1.05371
\(559\) 52.2298 2.20909
\(560\) 0 0
\(561\) −0.530194 −0.0223848
\(562\) −39.8359 −1.68038
\(563\) 6.47734 0.272987 0.136494 0.990641i \(-0.456417\pi\)
0.136494 + 0.990641i \(0.456417\pi\)
\(564\) −0.00219700 −9.25104e−5 0
\(565\) 0 0
\(566\) 20.0301 0.841929
\(567\) 33.1727 1.39312
\(568\) −13.2734 −0.556941
\(569\) 16.7745 0.703223 0.351611 0.936146i \(-0.385634\pi\)
0.351611 + 0.936146i \(0.385634\pi\)
\(570\) 0 0
\(571\) 28.8775 1.20849 0.604243 0.796800i \(-0.293476\pi\)
0.604243 + 0.796800i \(0.293476\pi\)
\(572\) −1.01210 −0.0423179
\(573\) −0.557017 −0.0232697
\(574\) −7.99636 −0.333762
\(575\) 0 0
\(576\) −24.7870 −1.03279
\(577\) −15.4881 −0.644779 −0.322390 0.946607i \(-0.604486\pi\)
−0.322390 + 0.946607i \(0.604486\pi\)
\(578\) −18.7600 −0.780313
\(579\) −0.00554419 −0.000230409 0
\(580\) 0 0
\(581\) −9.40029 −0.389990
\(582\) −0.558587 −0.0231542
\(583\) 21.0145 0.870330
\(584\) −15.8707 −0.656732
\(585\) 0 0
\(586\) −39.2133 −1.61989
\(587\) −10.7554 −0.443922 −0.221961 0.975056i \(-0.571246\pi\)
−0.221961 + 0.975056i \(0.571246\pi\)
\(588\) 0.0174266 0.000718662 0
\(589\) −48.4947 −1.99819
\(590\) 0 0
\(591\) −0.973474 −0.0400434
\(592\) 10.9338 0.449377
\(593\) 11.9957 0.492604 0.246302 0.969193i \(-0.420785\pi\)
0.246302 + 0.969193i \(0.420785\pi\)
\(594\) −0.799558 −0.0328063
\(595\) 0 0
\(596\) 0.739632 0.0302965
\(597\) −0.647007 −0.0264803
\(598\) 57.7498 2.36156
\(599\) 26.0997 1.06641 0.533203 0.845987i \(-0.320988\pi\)
0.533203 + 0.845987i \(0.320988\pi\)
\(600\) 0 0
\(601\) 24.1996 0.987124 0.493562 0.869711i \(-0.335695\pi\)
0.493562 + 0.869711i \(0.335695\pi\)
\(602\) −49.6618 −2.02406
\(603\) −15.9395 −0.649107
\(604\) 0.212391 0.00864206
\(605\) 0 0
\(606\) −0.838416 −0.0340583
\(607\) 38.7637 1.57337 0.786685 0.617355i \(-0.211796\pi\)
0.786685 + 0.617355i \(0.211796\pi\)
\(608\) 3.29108 0.133471
\(609\) −0.996469 −0.0403790
\(610\) 0 0
\(611\) −4.50001 −0.182051
\(612\) 1.18839 0.0480378
\(613\) −0.0930290 −0.00375741 −0.00187870 0.999998i \(-0.500598\pi\)
−0.00187870 + 0.999998i \(0.500598\pi\)
\(614\) −41.4311 −1.67202
\(615\) 0 0
\(616\) 27.7876 1.11959
\(617\) −21.3227 −0.858419 −0.429209 0.903205i \(-0.641208\pi\)
−0.429209 + 0.903205i \(0.641208\pi\)
\(618\) 0.748609 0.0301135
\(619\) 25.9696 1.04381 0.521904 0.853004i \(-0.325222\pi\)
0.521904 + 0.853004i \(0.325222\pi\)
\(620\) 0 0
\(621\) −1.69756 −0.0681209
\(622\) −27.8561 −1.11693
\(623\) 47.2242 1.89200
\(624\) 0.761409 0.0304808
\(625\) 0 0
\(626\) 16.6446 0.665252
\(627\) 0.778714 0.0310989
\(628\) −0.331271 −0.0132192
\(629\) −15.6811 −0.625248
\(630\) 0 0
\(631\) −18.6418 −0.742120 −0.371060 0.928609i \(-0.621006\pi\)
−0.371060 + 0.928609i \(0.621006\pi\)
\(632\) 15.1675 0.603329
\(633\) 0.138650 0.00551083
\(634\) 37.8092 1.50160
\(635\) 0 0
\(636\) 0.0211312 0.000837908 0
\(637\) 35.6942 1.41425
\(638\) −26.7503 −1.05906
\(639\) 13.8354 0.547319
\(640\) 0 0
\(641\) −22.4174 −0.885432 −0.442716 0.896662i \(-0.645985\pi\)
−0.442716 + 0.896662i \(0.645985\pi\)
\(642\) −0.0399289 −0.00157587
\(643\) 35.0341 1.38161 0.690805 0.723041i \(-0.257256\pi\)
0.690805 + 0.723041i \(0.257256\pi\)
\(644\) 2.04316 0.0805118
\(645\) 0 0
\(646\) 62.2251 2.44821
\(647\) −13.4824 −0.530047 −0.265023 0.964242i \(-0.585380\pi\)
−0.265023 + 0.964242i \(0.585380\pi\)
\(648\) 25.8569 1.01576
\(649\) 15.7349 0.617649
\(650\) 0 0
\(651\) −0.809203 −0.0317152
\(652\) 0.300737 0.0117778
\(653\) 38.6066 1.51079 0.755397 0.655268i \(-0.227444\pi\)
0.755397 + 0.655268i \(0.227444\pi\)
\(654\) −0.629852 −0.0246292
\(655\) 0 0
\(656\) −6.00899 −0.234612
\(657\) 16.5425 0.645386
\(658\) 4.27876 0.166803
\(659\) −10.6663 −0.415501 −0.207750 0.978182i \(-0.566614\pi\)
−0.207750 + 0.978182i \(0.566614\pi\)
\(660\) 0 0
\(661\) −4.68886 −0.182376 −0.0911878 0.995834i \(-0.529066\pi\)
−0.0911878 + 0.995834i \(0.529066\pi\)
\(662\) −23.8157 −0.925624
\(663\) −1.09201 −0.0424100
\(664\) −7.32720 −0.284350
\(665\) 0 0
\(666\) −11.8213 −0.458067
\(667\) −56.7944 −2.19909
\(668\) −1.08917 −0.0421412
\(669\) 0.599652 0.0231839
\(670\) 0 0
\(671\) 12.5243 0.483494
\(672\) 0.0549163 0.00211844
\(673\) −19.9709 −0.769820 −0.384910 0.922954i \(-0.625768\pi\)
−0.384910 + 0.922954i \(0.625768\pi\)
\(674\) −6.11281 −0.235457
\(675\) 0 0
\(676\) −1.15182 −0.0443009
\(677\) −10.1509 −0.390132 −0.195066 0.980790i \(-0.562492\pi\)
−0.195066 + 0.980790i \(0.562492\pi\)
\(678\) −1.03301 −0.0396726
\(679\) −40.4787 −1.55343
\(680\) 0 0
\(681\) −0.769066 −0.0294707
\(682\) −21.7232 −0.831823
\(683\) 34.4938 1.31987 0.659934 0.751324i \(-0.270584\pi\)
0.659934 + 0.751324i \(0.270584\pi\)
\(684\) −1.74543 −0.0667382
\(685\) 0 0
\(686\) 1.93669 0.0739433
\(687\) 0.559216 0.0213354
\(688\) −37.3192 −1.42278
\(689\) 43.2821 1.64892
\(690\) 0 0
\(691\) −26.0773 −0.992027 −0.496013 0.868315i \(-0.665203\pi\)
−0.496013 + 0.868315i \(0.665203\pi\)
\(692\) −0.829782 −0.0315436
\(693\) −28.9640 −1.10025
\(694\) −5.65035 −0.214484
\(695\) 0 0
\(696\) −0.776713 −0.0294412
\(697\) 8.61804 0.326431
\(698\) −47.9118 −1.81349
\(699\) 0.642920 0.0243175
\(700\) 0 0
\(701\) 28.3525 1.07086 0.535430 0.844579i \(-0.320149\pi\)
0.535430 + 0.844579i \(0.320149\pi\)
\(702\) −1.64680 −0.0621544
\(703\) 23.0315 0.868648
\(704\) 21.6325 0.815307
\(705\) 0 0
\(706\) 12.3582 0.465105
\(707\) −60.7569 −2.28500
\(708\) 0.0158223 0.000594640 0
\(709\) −4.52985 −0.170122 −0.0850610 0.996376i \(-0.527109\pi\)
−0.0850610 + 0.996376i \(0.527109\pi\)
\(710\) 0 0
\(711\) −15.8096 −0.592906
\(712\) 36.8096 1.37950
\(713\) −46.1210 −1.72725
\(714\) 1.03831 0.0388579
\(715\) 0 0
\(716\) 0.788767 0.0294776
\(717\) −0.0296855 −0.00110862
\(718\) −15.2091 −0.567597
\(719\) −32.1437 −1.19876 −0.599379 0.800466i \(-0.704586\pi\)
−0.599379 + 0.800466i \(0.704586\pi\)
\(720\) 0 0
\(721\) 54.2489 2.02033
\(722\) −65.0085 −2.41937
\(723\) 0.0366779 0.00136407
\(724\) 0.0253018 0.000940336 0
\(725\) 0 0
\(726\) −0.211422 −0.00784661
\(727\) 1.21358 0.0450090 0.0225045 0.999747i \(-0.492836\pi\)
0.0225045 + 0.999747i \(0.492836\pi\)
\(728\) 57.2323 2.12117
\(729\) −26.9274 −0.997310
\(730\) 0 0
\(731\) 53.5227 1.97961
\(732\) 0.0125939 0.000465482 0
\(733\) −15.9118 −0.587715 −0.293858 0.955849i \(-0.594939\pi\)
−0.293858 + 0.955849i \(0.594939\pi\)
\(734\) 12.6570 0.467177
\(735\) 0 0
\(736\) 3.12999 0.115373
\(737\) 13.9110 0.512419
\(738\) 6.49675 0.239149
\(739\) 38.0619 1.40013 0.700066 0.714079i \(-0.253154\pi\)
0.700066 + 0.714079i \(0.253154\pi\)
\(740\) 0 0
\(741\) 1.60387 0.0589195
\(742\) −41.1540 −1.51081
\(743\) 19.2213 0.705161 0.352581 0.935781i \(-0.385304\pi\)
0.352581 + 0.935781i \(0.385304\pi\)
\(744\) −0.630745 −0.0231242
\(745\) 0 0
\(746\) −8.80132 −0.322239
\(747\) 7.63740 0.279438
\(748\) −1.03715 −0.0379220
\(749\) −2.89350 −0.105726
\(750\) 0 0
\(751\) −5.13459 −0.187364 −0.0936820 0.995602i \(-0.529864\pi\)
−0.0936820 + 0.995602i \(0.529864\pi\)
\(752\) 3.21534 0.117251
\(753\) 0.698170 0.0254427
\(754\) −55.0959 −2.00647
\(755\) 0 0
\(756\) −0.0582630 −0.00211900
\(757\) −9.04957 −0.328912 −0.164456 0.986384i \(-0.552587\pi\)
−0.164456 + 0.986384i \(0.552587\pi\)
\(758\) −27.8821 −1.01272
\(759\) 0.740598 0.0268820
\(760\) 0 0
\(761\) 24.1984 0.877193 0.438596 0.898684i \(-0.355476\pi\)
0.438596 + 0.898684i \(0.355476\pi\)
\(762\) −0.268800 −0.00973761
\(763\) −45.6431 −1.65239
\(764\) −1.08962 −0.0394212
\(765\) 0 0
\(766\) 18.4581 0.666919
\(767\) 32.4082 1.17019
\(768\) 0.0630757 0.00227605
\(769\) −18.9691 −0.684042 −0.342021 0.939692i \(-0.611111\pi\)
−0.342021 + 0.939692i \(0.611111\pi\)
\(770\) 0 0
\(771\) −0.710967 −0.0256048
\(772\) −0.0108454 −0.000390335 0
\(773\) −7.02815 −0.252785 −0.126393 0.991980i \(-0.540340\pi\)
−0.126393 + 0.991980i \(0.540340\pi\)
\(774\) 40.3484 1.45029
\(775\) 0 0
\(776\) −31.5518 −1.13264
\(777\) 0.384312 0.0137871
\(778\) 4.21029 0.150946
\(779\) −12.6576 −0.453506
\(780\) 0 0
\(781\) −12.0747 −0.432065
\(782\) 59.1794 2.11625
\(783\) 1.61955 0.0578781
\(784\) −25.5041 −0.910862
\(785\) 0 0
\(786\) −0.0889370 −0.00317228
\(787\) 36.9565 1.31736 0.658678 0.752425i \(-0.271116\pi\)
0.658678 + 0.752425i \(0.271116\pi\)
\(788\) −1.90429 −0.0678374
\(789\) −0.306372 −0.0109071
\(790\) 0 0
\(791\) −74.8585 −2.66166
\(792\) −22.5764 −0.802219
\(793\) 25.7954 0.916022
\(794\) 3.21152 0.113972
\(795\) 0 0
\(796\) −1.26566 −0.0448601
\(797\) 32.9379 1.16672 0.583359 0.812214i \(-0.301738\pi\)
0.583359 + 0.812214i \(0.301738\pi\)
\(798\) −1.52501 −0.0539847
\(799\) −4.61141 −0.163140
\(800\) 0 0
\(801\) −38.3680 −1.35567
\(802\) −25.0338 −0.883973
\(803\) −14.4373 −0.509481
\(804\) 0.0139883 0.000493329 0
\(805\) 0 0
\(806\) −44.7418 −1.57596
\(807\) −0.290890 −0.0102398
\(808\) −47.3579 −1.66604
\(809\) 34.4463 1.21107 0.605534 0.795819i \(-0.292959\pi\)
0.605534 + 0.795819i \(0.292959\pi\)
\(810\) 0 0
\(811\) 49.2181 1.72828 0.864140 0.503251i \(-0.167863\pi\)
0.864140 + 0.503251i \(0.167863\pi\)
\(812\) −1.94927 −0.0684059
\(813\) 0.350059 0.0122771
\(814\) 10.3169 0.361607
\(815\) 0 0
\(816\) 0.780258 0.0273145
\(817\) −78.6107 −2.75024
\(818\) 20.3309 0.710852
\(819\) −59.6553 −2.08452
\(820\) 0 0
\(821\) −12.0723 −0.421328 −0.210664 0.977559i \(-0.567563\pi\)
−0.210664 + 0.977559i \(0.567563\pi\)
\(822\) −0.144979 −0.00505673
\(823\) −27.1896 −0.947770 −0.473885 0.880587i \(-0.657149\pi\)
−0.473885 + 0.880587i \(0.657149\pi\)
\(824\) 42.2851 1.47307
\(825\) 0 0
\(826\) −30.8147 −1.07218
\(827\) −25.6452 −0.891772 −0.445886 0.895090i \(-0.647111\pi\)
−0.445886 + 0.895090i \(0.647111\pi\)
\(828\) −1.66000 −0.0576888
\(829\) −21.3080 −0.740057 −0.370029 0.929020i \(-0.620652\pi\)
−0.370029 + 0.929020i \(0.620652\pi\)
\(830\) 0 0
\(831\) 1.02499 0.0355564
\(832\) 44.5551 1.54467
\(833\) 36.5778 1.26734
\(834\) 0.190199 0.00658607
\(835\) 0 0
\(836\) 1.52330 0.0526845
\(837\) 1.31519 0.0454597
\(838\) −16.4929 −0.569739
\(839\) 16.2665 0.561582 0.280791 0.959769i \(-0.409403\pi\)
0.280791 + 0.959769i \(0.409403\pi\)
\(840\) 0 0
\(841\) 25.1844 0.868429
\(842\) 19.2875 0.664692
\(843\) −1.05220 −0.0362396
\(844\) 0.271223 0.00933589
\(845\) 0 0
\(846\) −3.47634 −0.119519
\(847\) −15.3210 −0.526435
\(848\) −30.9259 −1.06200
\(849\) 0.529061 0.0181573
\(850\) 0 0
\(851\) 21.9041 0.750864
\(852\) −0.0121418 −0.000415970 0
\(853\) −30.7507 −1.05288 −0.526442 0.850211i \(-0.676474\pi\)
−0.526442 + 0.850211i \(0.676474\pi\)
\(854\) −24.5271 −0.839300
\(855\) 0 0
\(856\) −2.25538 −0.0770874
\(857\) 7.02517 0.239975 0.119988 0.992775i \(-0.461715\pi\)
0.119988 + 0.992775i \(0.461715\pi\)
\(858\) 0.718450 0.0245275
\(859\) −2.64135 −0.0901215 −0.0450608 0.998984i \(-0.514348\pi\)
−0.0450608 + 0.998984i \(0.514348\pi\)
\(860\) 0 0
\(861\) −0.211210 −0.00719802
\(862\) −20.1958 −0.687871
\(863\) −0.483719 −0.0164660 −0.00823299 0.999966i \(-0.502621\pi\)
−0.00823299 + 0.999966i \(0.502621\pi\)
\(864\) −0.0892551 −0.00303652
\(865\) 0 0
\(866\) 47.7207 1.62162
\(867\) −0.495513 −0.0168285
\(868\) −1.58294 −0.0537286
\(869\) 13.7976 0.468052
\(870\) 0 0
\(871\) 28.6516 0.970822
\(872\) −35.5772 −1.20480
\(873\) 32.8875 1.11307
\(874\) −86.9188 −2.94007
\(875\) 0 0
\(876\) −0.0145175 −0.000490502 0
\(877\) −12.5713 −0.424501 −0.212251 0.977215i \(-0.568079\pi\)
−0.212251 + 0.977215i \(0.568079\pi\)
\(878\) 13.5021 0.455675
\(879\) −1.03575 −0.0349351
\(880\) 0 0
\(881\) −18.5085 −0.623567 −0.311784 0.950153i \(-0.600926\pi\)
−0.311784 + 0.950153i \(0.600926\pi\)
\(882\) 27.5743 0.928476
\(883\) 4.07979 0.137296 0.0686479 0.997641i \(-0.478132\pi\)
0.0686479 + 0.997641i \(0.478132\pi\)
\(884\) −2.13615 −0.0718466
\(885\) 0 0
\(886\) −33.2505 −1.11707
\(887\) −6.49890 −0.218212 −0.109106 0.994030i \(-0.534799\pi\)
−0.109106 + 0.994030i \(0.534799\pi\)
\(888\) 0.299558 0.0100525
\(889\) −19.4790 −0.653304
\(890\) 0 0
\(891\) 23.5217 0.788005
\(892\) 1.17302 0.0392758
\(893\) 6.77294 0.226648
\(894\) −0.525036 −0.0175598
\(895\) 0 0
\(896\) −39.3699 −1.31526
\(897\) 1.52536 0.0509303
\(898\) −20.4351 −0.681928
\(899\) 44.0016 1.46753
\(900\) 0 0
\(901\) 44.3535 1.47763
\(902\) −5.66996 −0.188789
\(903\) −1.31173 −0.0436517
\(904\) −58.3496 −1.94068
\(905\) 0 0
\(906\) −0.150768 −0.00500894
\(907\) 4.17469 0.138618 0.0693091 0.997595i \(-0.477921\pi\)
0.0693091 + 0.997595i \(0.477921\pi\)
\(908\) −1.50443 −0.0499262
\(909\) 49.3628 1.63726
\(910\) 0 0
\(911\) 55.0766 1.82477 0.912385 0.409333i \(-0.134239\pi\)
0.912385 + 0.409333i \(0.134239\pi\)
\(912\) −1.14599 −0.0379476
\(913\) −6.66544 −0.220594
\(914\) 33.7359 1.11588
\(915\) 0 0
\(916\) 1.09392 0.0361443
\(917\) −6.44494 −0.212831
\(918\) −1.68756 −0.0556979
\(919\) −29.2769 −0.965755 −0.482877 0.875688i \(-0.660408\pi\)
−0.482877 + 0.875688i \(0.660408\pi\)
\(920\) 0 0
\(921\) −1.09433 −0.0360595
\(922\) −15.9891 −0.526574
\(923\) −24.8694 −0.818586
\(924\) 0.0254184 0.000836205 0
\(925\) 0 0
\(926\) −8.09253 −0.265937
\(927\) −44.0753 −1.44762
\(928\) −2.98615 −0.0980253
\(929\) 24.0911 0.790403 0.395201 0.918595i \(-0.370675\pi\)
0.395201 + 0.918595i \(0.370675\pi\)
\(930\) 0 0
\(931\) −53.7230 −1.76070
\(932\) 1.25766 0.0411962
\(933\) −0.735771 −0.0240881
\(934\) 13.9420 0.456195
\(935\) 0 0
\(936\) −46.4992 −1.51987
\(937\) 44.1001 1.44069 0.720344 0.693617i \(-0.243984\pi\)
0.720344 + 0.693617i \(0.243984\pi\)
\(938\) −27.2429 −0.889511
\(939\) 0.439639 0.0143471
\(940\) 0 0
\(941\) −33.9838 −1.10784 −0.553921 0.832569i \(-0.686869\pi\)
−0.553921 + 0.832569i \(0.686869\pi\)
\(942\) 0.235157 0.00766182
\(943\) −12.0381 −0.392013
\(944\) −23.1562 −0.753671
\(945\) 0 0
\(946\) −35.2136 −1.14489
\(947\) 6.98838 0.227092 0.113546 0.993533i \(-0.463779\pi\)
0.113546 + 0.993533i \(0.463779\pi\)
\(948\) 0.0138743 0.000450616 0
\(949\) −29.7356 −0.965257
\(950\) 0 0
\(951\) 0.998666 0.0323840
\(952\) 58.6491 1.90083
\(953\) 51.8095 1.67828 0.839138 0.543919i \(-0.183060\pi\)
0.839138 + 0.543919i \(0.183060\pi\)
\(954\) 33.4362 1.08254
\(955\) 0 0
\(956\) −0.0580700 −0.00187812
\(957\) −0.706564 −0.0228400
\(958\) −10.8809 −0.351544
\(959\) −10.5061 −0.339260
\(960\) 0 0
\(961\) 4.73237 0.152657
\(962\) 21.2491 0.685097
\(963\) 2.35087 0.0757556
\(964\) 0.0717484 0.00231086
\(965\) 0 0
\(966\) −1.45036 −0.0466647
\(967\) 19.3056 0.620826 0.310413 0.950602i \(-0.399533\pi\)
0.310413 + 0.950602i \(0.399533\pi\)
\(968\) −11.9422 −0.383836
\(969\) 1.64357 0.0527991
\(970\) 0 0
\(971\) 48.7263 1.56370 0.781851 0.623465i \(-0.214276\pi\)
0.781851 + 0.623465i \(0.214276\pi\)
\(972\) 0.0710102 0.00227765
\(973\) 13.7831 0.441864
\(974\) 59.0990 1.89365
\(975\) 0 0
\(976\) −18.4313 −0.589971
\(977\) 13.0658 0.418012 0.209006 0.977914i \(-0.432977\pi\)
0.209006 + 0.977914i \(0.432977\pi\)
\(978\) −0.213482 −0.00682639
\(979\) 33.4852 1.07019
\(980\) 0 0
\(981\) 37.0834 1.18398
\(982\) 17.9060 0.571405
\(983\) 8.81862 0.281270 0.140635 0.990061i \(-0.455086\pi\)
0.140635 + 0.990061i \(0.455086\pi\)
\(984\) −0.164631 −0.00524824
\(985\) 0 0
\(986\) −56.4598 −1.79805
\(987\) 0.113016 0.00359734
\(988\) 3.13745 0.0998154
\(989\) −74.7630 −2.37732
\(990\) 0 0
\(991\) 45.8928 1.45783 0.728916 0.684603i \(-0.240024\pi\)
0.728916 + 0.684603i \(0.240024\pi\)
\(992\) −2.42497 −0.0769928
\(993\) −0.629051 −0.0199623
\(994\) 23.6466 0.750025
\(995\) 0 0
\(996\) −0.00670248 −0.000212376 0
\(997\) −42.5992 −1.34913 −0.674565 0.738215i \(-0.735669\pi\)
−0.674565 + 0.738215i \(0.735669\pi\)
\(998\) 4.67136 0.147869
\(999\) −0.624620 −0.0197621
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 6025.2.a.q.1.20 66
5.2 odd 4 1205.2.b.d.724.20 66
5.3 odd 4 1205.2.b.d.724.47 yes 66
5.4 even 2 inner 6025.2.a.q.1.47 66
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1205.2.b.d.724.20 66 5.2 odd 4
1205.2.b.d.724.47 yes 66 5.3 odd 4
6025.2.a.q.1.20 66 1.1 even 1 trivial
6025.2.a.q.1.47 66 5.4 even 2 inner