Properties

Label 6042.2.a.z.1.3
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
Analytic rank $0$
Dimension $9$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 4x^{8} - 11x^{7} + 51x^{6} + 25x^{5} - 180x^{4} + 29x^{3} + 119x^{2} - 8x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.71772\) of defining polynomial
Character \(\chi\) \(=\) 6042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.02880 q^{5} +1.00000 q^{6} -1.70084 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.02880 q^{5} +1.00000 q^{6} -1.70084 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.02880 q^{10} -0.223482 q^{11} -1.00000 q^{12} +4.73778 q^{13} +1.70084 q^{14} +2.02880 q^{15} +1.00000 q^{16} -2.06304 q^{17} -1.00000 q^{18} -1.00000 q^{19} -2.02880 q^{20} +1.70084 q^{21} +0.223482 q^{22} +4.59214 q^{23} +1.00000 q^{24} -0.883975 q^{25} -4.73778 q^{26} -1.00000 q^{27} -1.70084 q^{28} +8.61390 q^{29} -2.02880 q^{30} -6.41980 q^{31} -1.00000 q^{32} +0.223482 q^{33} +2.06304 q^{34} +3.45067 q^{35} +1.00000 q^{36} +5.05467 q^{37} +1.00000 q^{38} -4.73778 q^{39} +2.02880 q^{40} -5.36854 q^{41} -1.70084 q^{42} -8.67694 q^{43} -0.223482 q^{44} -2.02880 q^{45} -4.59214 q^{46} +10.7858 q^{47} -1.00000 q^{48} -4.10714 q^{49} +0.883975 q^{50} +2.06304 q^{51} +4.73778 q^{52} +1.00000 q^{53} +1.00000 q^{54} +0.453401 q^{55} +1.70084 q^{56} +1.00000 q^{57} -8.61390 q^{58} +9.34264 q^{59} +2.02880 q^{60} -9.93398 q^{61} +6.41980 q^{62} -1.70084 q^{63} +1.00000 q^{64} -9.61200 q^{65} -0.223482 q^{66} -15.8980 q^{67} -2.06304 q^{68} -4.59214 q^{69} -3.45067 q^{70} +7.18427 q^{71} -1.00000 q^{72} -7.93289 q^{73} -5.05467 q^{74} +0.883975 q^{75} -1.00000 q^{76} +0.380108 q^{77} +4.73778 q^{78} +14.4968 q^{79} -2.02880 q^{80} +1.00000 q^{81} +5.36854 q^{82} +3.88043 q^{83} +1.70084 q^{84} +4.18550 q^{85} +8.67694 q^{86} -8.61390 q^{87} +0.223482 q^{88} -2.21998 q^{89} +2.02880 q^{90} -8.05822 q^{91} +4.59214 q^{92} +6.41980 q^{93} -10.7858 q^{94} +2.02880 q^{95} +1.00000 q^{96} -3.71588 q^{97} +4.10714 q^{98} -0.223482 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} - 9 q^{3} + 9 q^{4} - q^{5} + 9 q^{6} + 4 q^{7} - 9 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{2} - 9 q^{3} + 9 q^{4} - q^{5} + 9 q^{6} + 4 q^{7} - 9 q^{8} + 9 q^{9} + q^{10} + 12 q^{11} - 9 q^{12} - q^{13} - 4 q^{14} + q^{15} + 9 q^{16} + 12 q^{17} - 9 q^{18} - 9 q^{19} - q^{20} - 4 q^{21} - 12 q^{22} + 11 q^{23} + 9 q^{24} + 10 q^{25} + q^{26} - 9 q^{27} + 4 q^{28} + 7 q^{29} - q^{30} - 12 q^{31} - 9 q^{32} - 12 q^{33} - 12 q^{34} + 15 q^{35} + 9 q^{36} - 9 q^{37} + 9 q^{38} + q^{39} + q^{40} - 4 q^{41} + 4 q^{42} + 23 q^{43} + 12 q^{44} - q^{45} - 11 q^{46} + 35 q^{47} - 9 q^{48} + 3 q^{49} - 10 q^{50} - 12 q^{51} - q^{52} + 9 q^{53} + 9 q^{54} + 3 q^{55} - 4 q^{56} + 9 q^{57} - 7 q^{58} + 14 q^{59} + q^{60} + 14 q^{61} + 12 q^{62} + 4 q^{63} + 9 q^{64} + 13 q^{65} + 12 q^{66} - 10 q^{67} + 12 q^{68} - 11 q^{69} - 15 q^{70} + 4 q^{71} - 9 q^{72} - 5 q^{73} + 9 q^{74} - 10 q^{75} - 9 q^{76} + 17 q^{77} - q^{78} - 14 q^{79} - q^{80} + 9 q^{81} + 4 q^{82} + 37 q^{83} - 4 q^{84} - 31 q^{85} - 23 q^{86} - 7 q^{87} - 12 q^{88} - 20 q^{89} + q^{90} - 12 q^{91} + 11 q^{92} + 12 q^{93} - 35 q^{94} + q^{95} + 9 q^{96} - 2 q^{97} - 3 q^{98} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.02880 −0.907306 −0.453653 0.891178i \(-0.649880\pi\)
−0.453653 + 0.891178i \(0.649880\pi\)
\(6\) 1.00000 0.408248
\(7\) −1.70084 −0.642858 −0.321429 0.946934i \(-0.604163\pi\)
−0.321429 + 0.946934i \(0.604163\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.02880 0.641563
\(11\) −0.223482 −0.0673824 −0.0336912 0.999432i \(-0.510726\pi\)
−0.0336912 + 0.999432i \(0.510726\pi\)
\(12\) −1.00000 −0.288675
\(13\) 4.73778 1.31402 0.657012 0.753880i \(-0.271820\pi\)
0.657012 + 0.753880i \(0.271820\pi\)
\(14\) 1.70084 0.454569
\(15\) 2.02880 0.523834
\(16\) 1.00000 0.250000
\(17\) −2.06304 −0.500361 −0.250181 0.968199i \(-0.580490\pi\)
−0.250181 + 0.968199i \(0.580490\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) −2.02880 −0.453653
\(21\) 1.70084 0.371154
\(22\) 0.223482 0.0476466
\(23\) 4.59214 0.957527 0.478763 0.877944i \(-0.341085\pi\)
0.478763 + 0.877944i \(0.341085\pi\)
\(24\) 1.00000 0.204124
\(25\) −0.883975 −0.176795
\(26\) −4.73778 −0.929155
\(27\) −1.00000 −0.192450
\(28\) −1.70084 −0.321429
\(29\) 8.61390 1.59956 0.799781 0.600292i \(-0.204949\pi\)
0.799781 + 0.600292i \(0.204949\pi\)
\(30\) −2.02880 −0.370406
\(31\) −6.41980 −1.15303 −0.576515 0.817087i \(-0.695588\pi\)
−0.576515 + 0.817087i \(0.695588\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.223482 0.0389033
\(34\) 2.06304 0.353809
\(35\) 3.45067 0.583269
\(36\) 1.00000 0.166667
\(37\) 5.05467 0.830983 0.415492 0.909597i \(-0.363610\pi\)
0.415492 + 0.909597i \(0.363610\pi\)
\(38\) 1.00000 0.162221
\(39\) −4.73778 −0.758652
\(40\) 2.02880 0.320781
\(41\) −5.36854 −0.838424 −0.419212 0.907888i \(-0.637694\pi\)
−0.419212 + 0.907888i \(0.637694\pi\)
\(42\) −1.70084 −0.262446
\(43\) −8.67694 −1.32322 −0.661611 0.749847i \(-0.730127\pi\)
−0.661611 + 0.749847i \(0.730127\pi\)
\(44\) −0.223482 −0.0336912
\(45\) −2.02880 −0.302435
\(46\) −4.59214 −0.677074
\(47\) 10.7858 1.57327 0.786633 0.617421i \(-0.211822\pi\)
0.786633 + 0.617421i \(0.211822\pi\)
\(48\) −1.00000 −0.144338
\(49\) −4.10714 −0.586734
\(50\) 0.883975 0.125013
\(51\) 2.06304 0.288884
\(52\) 4.73778 0.657012
\(53\) 1.00000 0.137361
\(54\) 1.00000 0.136083
\(55\) 0.453401 0.0611365
\(56\) 1.70084 0.227285
\(57\) 1.00000 0.132453
\(58\) −8.61390 −1.13106
\(59\) 9.34264 1.21631 0.608154 0.793819i \(-0.291910\pi\)
0.608154 + 0.793819i \(0.291910\pi\)
\(60\) 2.02880 0.261917
\(61\) −9.93398 −1.27192 −0.635958 0.771724i \(-0.719395\pi\)
−0.635958 + 0.771724i \(0.719395\pi\)
\(62\) 6.41980 0.815315
\(63\) −1.70084 −0.214286
\(64\) 1.00000 0.125000
\(65\) −9.61200 −1.19222
\(66\) −0.223482 −0.0275088
\(67\) −15.8980 −1.94225 −0.971127 0.238561i \(-0.923324\pi\)
−0.971127 + 0.238561i \(0.923324\pi\)
\(68\) −2.06304 −0.250181
\(69\) −4.59214 −0.552828
\(70\) −3.45067 −0.412434
\(71\) 7.18427 0.852616 0.426308 0.904578i \(-0.359814\pi\)
0.426308 + 0.904578i \(0.359814\pi\)
\(72\) −1.00000 −0.117851
\(73\) −7.93289 −0.928474 −0.464237 0.885711i \(-0.653671\pi\)
−0.464237 + 0.885711i \(0.653671\pi\)
\(74\) −5.05467 −0.587594
\(75\) 0.883975 0.102073
\(76\) −1.00000 −0.114708
\(77\) 0.380108 0.0433173
\(78\) 4.73778 0.536448
\(79\) 14.4968 1.63101 0.815506 0.578749i \(-0.196459\pi\)
0.815506 + 0.578749i \(0.196459\pi\)
\(80\) −2.02880 −0.226827
\(81\) 1.00000 0.111111
\(82\) 5.36854 0.592856
\(83\) 3.88043 0.425932 0.212966 0.977060i \(-0.431688\pi\)
0.212966 + 0.977060i \(0.431688\pi\)
\(84\) 1.70084 0.185577
\(85\) 4.18550 0.453981
\(86\) 8.67694 0.935659
\(87\) −8.61390 −0.923507
\(88\) 0.223482 0.0238233
\(89\) −2.21998 −0.235317 −0.117659 0.993054i \(-0.537539\pi\)
−0.117659 + 0.993054i \(0.537539\pi\)
\(90\) 2.02880 0.213854
\(91\) −8.05822 −0.844731
\(92\) 4.59214 0.478763
\(93\) 6.41980 0.665702
\(94\) −10.7858 −1.11247
\(95\) 2.02880 0.208150
\(96\) 1.00000 0.102062
\(97\) −3.71588 −0.377291 −0.188645 0.982045i \(-0.560410\pi\)
−0.188645 + 0.982045i \(0.560410\pi\)
\(98\) 4.10714 0.414883
\(99\) −0.223482 −0.0224608
\(100\) −0.883975 −0.0883975
\(101\) −17.5021 −1.74153 −0.870763 0.491704i \(-0.836374\pi\)
−0.870763 + 0.491704i \(0.836374\pi\)
\(102\) −2.06304 −0.204272
\(103\) 6.54700 0.645095 0.322547 0.946553i \(-0.395461\pi\)
0.322547 + 0.946553i \(0.395461\pi\)
\(104\) −4.73778 −0.464578
\(105\) −3.45067 −0.336751
\(106\) −1.00000 −0.0971286
\(107\) −8.87172 −0.857661 −0.428831 0.903385i \(-0.641074\pi\)
−0.428831 + 0.903385i \(0.641074\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.30195 −0.603617 −0.301809 0.953369i \(-0.597590\pi\)
−0.301809 + 0.953369i \(0.597590\pi\)
\(110\) −0.453401 −0.0432300
\(111\) −5.05467 −0.479768
\(112\) −1.70084 −0.160714
\(113\) −3.08374 −0.290094 −0.145047 0.989425i \(-0.546333\pi\)
−0.145047 + 0.989425i \(0.546333\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −9.31652 −0.868770
\(116\) 8.61390 0.799781
\(117\) 4.73778 0.438008
\(118\) −9.34264 −0.860059
\(119\) 3.50891 0.321661
\(120\) −2.02880 −0.185203
\(121\) −10.9501 −0.995460
\(122\) 9.93398 0.899380
\(123\) 5.36854 0.484065
\(124\) −6.41980 −0.576515
\(125\) 11.9374 1.06771
\(126\) 1.70084 0.151523
\(127\) 4.57655 0.406103 0.203051 0.979168i \(-0.434914\pi\)
0.203051 + 0.979168i \(0.434914\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 8.67694 0.763962
\(130\) 9.61200 0.843028
\(131\) 14.5165 1.26831 0.634155 0.773206i \(-0.281348\pi\)
0.634155 + 0.773206i \(0.281348\pi\)
\(132\) 0.223482 0.0194516
\(133\) 1.70084 0.147482
\(134\) 15.8980 1.37338
\(135\) 2.02880 0.174611
\(136\) 2.06304 0.176904
\(137\) −2.25805 −0.192918 −0.0964591 0.995337i \(-0.530752\pi\)
−0.0964591 + 0.995337i \(0.530752\pi\)
\(138\) 4.59214 0.390909
\(139\) −2.95933 −0.251007 −0.125504 0.992093i \(-0.540055\pi\)
−0.125504 + 0.992093i \(0.540055\pi\)
\(140\) 3.45067 0.291635
\(141\) −10.7858 −0.908325
\(142\) −7.18427 −0.602891
\(143\) −1.05881 −0.0885421
\(144\) 1.00000 0.0833333
\(145\) −17.4759 −1.45129
\(146\) 7.93289 0.656530
\(147\) 4.10714 0.338751
\(148\) 5.05467 0.415492
\(149\) −4.29566 −0.351914 −0.175957 0.984398i \(-0.556302\pi\)
−0.175957 + 0.984398i \(0.556302\pi\)
\(150\) −0.883975 −0.0721763
\(151\) 8.63019 0.702315 0.351158 0.936316i \(-0.385788\pi\)
0.351158 + 0.936316i \(0.385788\pi\)
\(152\) 1.00000 0.0811107
\(153\) −2.06304 −0.166787
\(154\) −0.380108 −0.0306300
\(155\) 13.0245 1.04615
\(156\) −4.73778 −0.379326
\(157\) 10.1567 0.810596 0.405298 0.914185i \(-0.367168\pi\)
0.405298 + 0.914185i \(0.367168\pi\)
\(158\) −14.4968 −1.15330
\(159\) −1.00000 −0.0793052
\(160\) 2.02880 0.160391
\(161\) −7.81050 −0.615554
\(162\) −1.00000 −0.0785674
\(163\) −18.0373 −1.41279 −0.706393 0.707819i \(-0.749679\pi\)
−0.706393 + 0.707819i \(0.749679\pi\)
\(164\) −5.36854 −0.419212
\(165\) −0.453401 −0.0352972
\(166\) −3.88043 −0.301180
\(167\) 5.49247 0.425020 0.212510 0.977159i \(-0.431836\pi\)
0.212510 + 0.977159i \(0.431836\pi\)
\(168\) −1.70084 −0.131223
\(169\) 9.44656 0.726659
\(170\) −4.18550 −0.321013
\(171\) −1.00000 −0.0764719
\(172\) −8.67694 −0.661611
\(173\) −5.96211 −0.453291 −0.226645 0.973977i \(-0.572776\pi\)
−0.226645 + 0.973977i \(0.572776\pi\)
\(174\) 8.61390 0.653018
\(175\) 1.50350 0.113654
\(176\) −0.223482 −0.0168456
\(177\) −9.34264 −0.702236
\(178\) 2.21998 0.166394
\(179\) −4.27562 −0.319575 −0.159788 0.987151i \(-0.551081\pi\)
−0.159788 + 0.987151i \(0.551081\pi\)
\(180\) −2.02880 −0.151218
\(181\) −6.32260 −0.469955 −0.234977 0.972001i \(-0.575502\pi\)
−0.234977 + 0.972001i \(0.575502\pi\)
\(182\) 8.05822 0.597315
\(183\) 9.93398 0.734341
\(184\) −4.59214 −0.338537
\(185\) −10.2549 −0.753956
\(186\) −6.41980 −0.470722
\(187\) 0.461053 0.0337155
\(188\) 10.7858 0.786633
\(189\) 1.70084 0.123718
\(190\) −2.02880 −0.147185
\(191\) −5.88433 −0.425775 −0.212888 0.977077i \(-0.568287\pi\)
−0.212888 + 0.977077i \(0.568287\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 19.1609 1.37923 0.689615 0.724176i \(-0.257780\pi\)
0.689615 + 0.724176i \(0.257780\pi\)
\(194\) 3.71588 0.266785
\(195\) 9.61200 0.688330
\(196\) −4.10714 −0.293367
\(197\) 4.17039 0.297128 0.148564 0.988903i \(-0.452535\pi\)
0.148564 + 0.988903i \(0.452535\pi\)
\(198\) 0.223482 0.0158822
\(199\) 18.9956 1.34656 0.673280 0.739388i \(-0.264885\pi\)
0.673280 + 0.739388i \(0.264885\pi\)
\(200\) 0.883975 0.0625065
\(201\) 15.8980 1.12136
\(202\) 17.5021 1.23144
\(203\) −14.6509 −1.02829
\(204\) 2.06304 0.144442
\(205\) 10.8917 0.760708
\(206\) −6.54700 −0.456151
\(207\) 4.59214 0.319176
\(208\) 4.73778 0.328506
\(209\) 0.223482 0.0154586
\(210\) 3.45067 0.238119
\(211\) −3.45387 −0.237775 −0.118887 0.992908i \(-0.537933\pi\)
−0.118887 + 0.992908i \(0.537933\pi\)
\(212\) 1.00000 0.0686803
\(213\) −7.18427 −0.492258
\(214\) 8.87172 0.606458
\(215\) 17.6038 1.20057
\(216\) 1.00000 0.0680414
\(217\) 10.9191 0.741234
\(218\) 6.30195 0.426822
\(219\) 7.93289 0.536055
\(220\) 0.453401 0.0305683
\(221\) −9.77424 −0.657486
\(222\) 5.05467 0.339247
\(223\) 12.6841 0.849392 0.424696 0.905336i \(-0.360381\pi\)
0.424696 + 0.905336i \(0.360381\pi\)
\(224\) 1.70084 0.113642
\(225\) −0.883975 −0.0589317
\(226\) 3.08374 0.205127
\(227\) 16.5026 1.09532 0.547658 0.836702i \(-0.315520\pi\)
0.547658 + 0.836702i \(0.315520\pi\)
\(228\) 1.00000 0.0662266
\(229\) 25.0432 1.65490 0.827450 0.561540i \(-0.189791\pi\)
0.827450 + 0.561540i \(0.189791\pi\)
\(230\) 9.31652 0.614313
\(231\) −0.380108 −0.0250093
\(232\) −8.61390 −0.565530
\(233\) 5.73384 0.375636 0.187818 0.982204i \(-0.439858\pi\)
0.187818 + 0.982204i \(0.439858\pi\)
\(234\) −4.73778 −0.309718
\(235\) −21.8822 −1.42743
\(236\) 9.34264 0.608154
\(237\) −14.4968 −0.941665
\(238\) −3.50891 −0.227449
\(239\) −1.76355 −0.114075 −0.0570373 0.998372i \(-0.518165\pi\)
−0.0570373 + 0.998372i \(0.518165\pi\)
\(240\) 2.02880 0.130958
\(241\) 20.9874 1.35192 0.675960 0.736938i \(-0.263729\pi\)
0.675960 + 0.736938i \(0.263729\pi\)
\(242\) 10.9501 0.703896
\(243\) −1.00000 −0.0641500
\(244\) −9.93398 −0.635958
\(245\) 8.33255 0.532347
\(246\) −5.36854 −0.342285
\(247\) −4.73778 −0.301458
\(248\) 6.41980 0.407657
\(249\) −3.88043 −0.245912
\(250\) −11.9374 −0.754988
\(251\) 14.7483 0.930903 0.465452 0.885073i \(-0.345892\pi\)
0.465452 + 0.885073i \(0.345892\pi\)
\(252\) −1.70084 −0.107143
\(253\) −1.02626 −0.0645205
\(254\) −4.57655 −0.287158
\(255\) −4.18550 −0.262106
\(256\) 1.00000 0.0625000
\(257\) −3.08991 −0.192743 −0.0963716 0.995345i \(-0.530724\pi\)
−0.0963716 + 0.995345i \(0.530724\pi\)
\(258\) −8.67694 −0.540203
\(259\) −8.59720 −0.534204
\(260\) −9.61200 −0.596111
\(261\) 8.61390 0.533187
\(262\) −14.5165 −0.896830
\(263\) −2.74031 −0.168975 −0.0844873 0.996425i \(-0.526925\pi\)
−0.0844873 + 0.996425i \(0.526925\pi\)
\(264\) −0.223482 −0.0137544
\(265\) −2.02880 −0.124628
\(266\) −1.70084 −0.104285
\(267\) 2.21998 0.135860
\(268\) −15.8980 −0.971127
\(269\) 9.03286 0.550743 0.275372 0.961338i \(-0.411199\pi\)
0.275372 + 0.961338i \(0.411199\pi\)
\(270\) −2.02880 −0.123469
\(271\) 0.501586 0.0304692 0.0152346 0.999884i \(-0.495150\pi\)
0.0152346 + 0.999884i \(0.495150\pi\)
\(272\) −2.06304 −0.125090
\(273\) 8.05822 0.487705
\(274\) 2.25805 0.136414
\(275\) 0.197553 0.0119129
\(276\) −4.59214 −0.276414
\(277\) −16.0326 −0.963308 −0.481654 0.876362i \(-0.659964\pi\)
−0.481654 + 0.876362i \(0.659964\pi\)
\(278\) 2.95933 0.177489
\(279\) −6.41980 −0.384343
\(280\) −3.45067 −0.206217
\(281\) −10.0444 −0.599201 −0.299600 0.954065i \(-0.596853\pi\)
−0.299600 + 0.954065i \(0.596853\pi\)
\(282\) 10.7858 0.642283
\(283\) 4.04982 0.240737 0.120368 0.992729i \(-0.461592\pi\)
0.120368 + 0.992729i \(0.461592\pi\)
\(284\) 7.18427 0.426308
\(285\) −2.02880 −0.120176
\(286\) 1.05881 0.0626087
\(287\) 9.13103 0.538988
\(288\) −1.00000 −0.0589256
\(289\) −12.7439 −0.749639
\(290\) 17.4759 1.02622
\(291\) 3.71588 0.217829
\(292\) −7.93289 −0.464237
\(293\) 18.6011 1.08669 0.543345 0.839509i \(-0.317157\pi\)
0.543345 + 0.839509i \(0.317157\pi\)
\(294\) −4.10714 −0.239533
\(295\) −18.9543 −1.10356
\(296\) −5.05467 −0.293797
\(297\) 0.223482 0.0129678
\(298\) 4.29566 0.248841
\(299\) 21.7565 1.25821
\(300\) 0.883975 0.0510363
\(301\) 14.7581 0.850643
\(302\) −8.63019 −0.496612
\(303\) 17.5021 1.00547
\(304\) −1.00000 −0.0573539
\(305\) 20.1540 1.15402
\(306\) 2.06304 0.117936
\(307\) 32.9887 1.88277 0.941384 0.337338i \(-0.109526\pi\)
0.941384 + 0.337338i \(0.109526\pi\)
\(308\) 0.380108 0.0216587
\(309\) −6.54700 −0.372446
\(310\) −13.0245 −0.739741
\(311\) −5.42378 −0.307554 −0.153777 0.988106i \(-0.549144\pi\)
−0.153777 + 0.988106i \(0.549144\pi\)
\(312\) 4.73778 0.268224
\(313\) −6.21772 −0.351446 −0.175723 0.984440i \(-0.556226\pi\)
−0.175723 + 0.984440i \(0.556226\pi\)
\(314\) −10.1567 −0.573178
\(315\) 3.45067 0.194423
\(316\) 14.4968 0.815506
\(317\) −15.1750 −0.852313 −0.426156 0.904650i \(-0.640133\pi\)
−0.426156 + 0.904650i \(0.640133\pi\)
\(318\) 1.00000 0.0560772
\(319\) −1.92505 −0.107782
\(320\) −2.02880 −0.113413
\(321\) 8.87172 0.495171
\(322\) 7.81050 0.435262
\(323\) 2.06304 0.114791
\(324\) 1.00000 0.0555556
\(325\) −4.18808 −0.232313
\(326\) 18.0373 0.998991
\(327\) 6.30195 0.348499
\(328\) 5.36854 0.296428
\(329\) −18.3449 −1.01139
\(330\) 0.453401 0.0249589
\(331\) −28.0327 −1.54082 −0.770408 0.637552i \(-0.779947\pi\)
−0.770408 + 0.637552i \(0.779947\pi\)
\(332\) 3.88043 0.212966
\(333\) 5.05467 0.276994
\(334\) −5.49247 −0.300535
\(335\) 32.2539 1.76222
\(336\) 1.70084 0.0927886
\(337\) 21.2001 1.15484 0.577421 0.816447i \(-0.304059\pi\)
0.577421 + 0.816447i \(0.304059\pi\)
\(338\) −9.44656 −0.513825
\(339\) 3.08374 0.167486
\(340\) 4.18550 0.226990
\(341\) 1.43471 0.0776939
\(342\) 1.00000 0.0540738
\(343\) 18.8915 1.02004
\(344\) 8.67694 0.467829
\(345\) 9.31652 0.501585
\(346\) 5.96211 0.320525
\(347\) 12.4103 0.666218 0.333109 0.942888i \(-0.391902\pi\)
0.333109 + 0.942888i \(0.391902\pi\)
\(348\) −8.61390 −0.461754
\(349\) 16.0095 0.856969 0.428485 0.903549i \(-0.359048\pi\)
0.428485 + 0.903549i \(0.359048\pi\)
\(350\) −1.50350 −0.0803656
\(351\) −4.73778 −0.252884
\(352\) 0.223482 0.0119116
\(353\) −22.0225 −1.17214 −0.586069 0.810261i \(-0.699325\pi\)
−0.586069 + 0.810261i \(0.699325\pi\)
\(354\) 9.34264 0.496556
\(355\) −14.5754 −0.773584
\(356\) −2.21998 −0.117659
\(357\) −3.50891 −0.185711
\(358\) 4.27562 0.225974
\(359\) 14.2394 0.751526 0.375763 0.926716i \(-0.377381\pi\)
0.375763 + 0.926716i \(0.377381\pi\)
\(360\) 2.02880 0.106927
\(361\) 1.00000 0.0526316
\(362\) 6.32260 0.332308
\(363\) 10.9501 0.574729
\(364\) −8.05822 −0.422365
\(365\) 16.0942 0.842410
\(366\) −9.93398 −0.519257
\(367\) 33.7719 1.76288 0.881440 0.472296i \(-0.156575\pi\)
0.881440 + 0.472296i \(0.156575\pi\)
\(368\) 4.59214 0.239382
\(369\) −5.36854 −0.279475
\(370\) 10.2549 0.533128
\(371\) −1.70084 −0.0883033
\(372\) 6.41980 0.332851
\(373\) 21.7568 1.12652 0.563262 0.826278i \(-0.309546\pi\)
0.563262 + 0.826278i \(0.309546\pi\)
\(374\) −0.461053 −0.0238405
\(375\) −11.9374 −0.616445
\(376\) −10.7858 −0.556233
\(377\) 40.8108 2.10186
\(378\) −1.70084 −0.0874819
\(379\) −8.08932 −0.415520 −0.207760 0.978180i \(-0.566617\pi\)
−0.207760 + 0.978180i \(0.566617\pi\)
\(380\) 2.02880 0.104075
\(381\) −4.57655 −0.234464
\(382\) 5.88433 0.301069
\(383\) 5.30508 0.271077 0.135539 0.990772i \(-0.456724\pi\)
0.135539 + 0.990772i \(0.456724\pi\)
\(384\) 1.00000 0.0510310
\(385\) −0.771163 −0.0393021
\(386\) −19.1609 −0.975263
\(387\) −8.67694 −0.441074
\(388\) −3.71588 −0.188645
\(389\) 11.9606 0.606424 0.303212 0.952923i \(-0.401941\pi\)
0.303212 + 0.952923i \(0.401941\pi\)
\(390\) −9.61200 −0.486723
\(391\) −9.47377 −0.479109
\(392\) 4.10714 0.207442
\(393\) −14.5165 −0.732259
\(394\) −4.17039 −0.210101
\(395\) −29.4110 −1.47983
\(396\) −0.223482 −0.0112304
\(397\) −2.38568 −0.119734 −0.0598669 0.998206i \(-0.519068\pi\)
−0.0598669 + 0.998206i \(0.519068\pi\)
\(398\) −18.9956 −0.952161
\(399\) −1.70084 −0.0851486
\(400\) −0.883975 −0.0441988
\(401\) 12.3413 0.616293 0.308146 0.951339i \(-0.400291\pi\)
0.308146 + 0.951339i \(0.400291\pi\)
\(402\) −15.8980 −0.792922
\(403\) −30.4156 −1.51511
\(404\) −17.5021 −0.870763
\(405\) −2.02880 −0.100812
\(406\) 14.6509 0.727111
\(407\) −1.12963 −0.0559937
\(408\) −2.06304 −0.102136
\(409\) −0.0414247 −0.00204832 −0.00102416 0.999999i \(-0.500326\pi\)
−0.00102416 + 0.999999i \(0.500326\pi\)
\(410\) −10.8917 −0.537902
\(411\) 2.25805 0.111381
\(412\) 6.54700 0.322547
\(413\) −15.8903 −0.781913
\(414\) −4.59214 −0.225691
\(415\) −7.87261 −0.386451
\(416\) −4.73778 −0.232289
\(417\) 2.95933 0.144919
\(418\) −0.223482 −0.0109309
\(419\) −22.9342 −1.12041 −0.560205 0.828354i \(-0.689278\pi\)
−0.560205 + 0.828354i \(0.689278\pi\)
\(420\) −3.45067 −0.168375
\(421\) −6.38045 −0.310964 −0.155482 0.987839i \(-0.549693\pi\)
−0.155482 + 0.987839i \(0.549693\pi\)
\(422\) 3.45387 0.168132
\(423\) 10.7858 0.524422
\(424\) −1.00000 −0.0485643
\(425\) 1.82368 0.0884614
\(426\) 7.18427 0.348079
\(427\) 16.8961 0.817661
\(428\) −8.87172 −0.428831
\(429\) 1.05881 0.0511198
\(430\) −17.6038 −0.848929
\(431\) −26.9070 −1.29607 −0.648033 0.761613i \(-0.724408\pi\)
−0.648033 + 0.761613i \(0.724408\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −0.497344 −0.0239008 −0.0119504 0.999929i \(-0.503804\pi\)
−0.0119504 + 0.999929i \(0.503804\pi\)
\(434\) −10.9191 −0.524132
\(435\) 17.4759 0.837904
\(436\) −6.30195 −0.301809
\(437\) −4.59214 −0.219672
\(438\) −7.93289 −0.379048
\(439\) −17.0256 −0.812590 −0.406295 0.913742i \(-0.633179\pi\)
−0.406295 + 0.913742i \(0.633179\pi\)
\(440\) −0.453401 −0.0216150
\(441\) −4.10714 −0.195578
\(442\) 9.77424 0.464913
\(443\) 24.4326 1.16083 0.580414 0.814322i \(-0.302891\pi\)
0.580414 + 0.814322i \(0.302891\pi\)
\(444\) −5.05467 −0.239884
\(445\) 4.50389 0.213505
\(446\) −12.6841 −0.600611
\(447\) 4.29566 0.203178
\(448\) −1.70084 −0.0803572
\(449\) 11.6675 0.550625 0.275313 0.961355i \(-0.411219\pi\)
0.275313 + 0.961355i \(0.411219\pi\)
\(450\) 0.883975 0.0416710
\(451\) 1.19977 0.0564951
\(452\) −3.08374 −0.145047
\(453\) −8.63019 −0.405482
\(454\) −16.5026 −0.774505
\(455\) 16.3485 0.766430
\(456\) −1.00000 −0.0468293
\(457\) 4.47028 0.209111 0.104555 0.994519i \(-0.466658\pi\)
0.104555 + 0.994519i \(0.466658\pi\)
\(458\) −25.0432 −1.17019
\(459\) 2.06304 0.0962945
\(460\) −9.31652 −0.434385
\(461\) 17.0418 0.793717 0.396858 0.917880i \(-0.370100\pi\)
0.396858 + 0.917880i \(0.370100\pi\)
\(462\) 0.380108 0.0176842
\(463\) −29.2509 −1.35940 −0.679702 0.733488i \(-0.737891\pi\)
−0.679702 + 0.733488i \(0.737891\pi\)
\(464\) 8.61390 0.399890
\(465\) −13.0245 −0.603996
\(466\) −5.73384 −0.265615
\(467\) 28.3787 1.31321 0.656604 0.754236i \(-0.271992\pi\)
0.656604 + 0.754236i \(0.271992\pi\)
\(468\) 4.73778 0.219004
\(469\) 27.0401 1.24859
\(470\) 21.8822 1.00935
\(471\) −10.1567 −0.467998
\(472\) −9.34264 −0.430030
\(473\) 1.93914 0.0891619
\(474\) 14.4968 0.665858
\(475\) 0.883975 0.0405596
\(476\) 3.50891 0.160831
\(477\) 1.00000 0.0457869
\(478\) 1.76355 0.0806629
\(479\) −5.30183 −0.242247 −0.121123 0.992637i \(-0.538650\pi\)
−0.121123 + 0.992637i \(0.538650\pi\)
\(480\) −2.02880 −0.0926016
\(481\) 23.9479 1.09193
\(482\) −20.9874 −0.955952
\(483\) 7.81050 0.355390
\(484\) −10.9501 −0.497730
\(485\) 7.53878 0.342318
\(486\) 1.00000 0.0453609
\(487\) −12.5119 −0.566969 −0.283484 0.958977i \(-0.591490\pi\)
−0.283484 + 0.958977i \(0.591490\pi\)
\(488\) 9.93398 0.449690
\(489\) 18.0373 0.815673
\(490\) −8.33255 −0.376426
\(491\) −42.2114 −1.90497 −0.952487 0.304580i \(-0.901484\pi\)
−0.952487 + 0.304580i \(0.901484\pi\)
\(492\) 5.36854 0.242032
\(493\) −17.7708 −0.800358
\(494\) 4.73778 0.213163
\(495\) 0.453401 0.0203788
\(496\) −6.41980 −0.288257
\(497\) −12.2193 −0.548111
\(498\) 3.88043 0.173886
\(499\) −0.579520 −0.0259429 −0.0129714 0.999916i \(-0.504129\pi\)
−0.0129714 + 0.999916i \(0.504129\pi\)
\(500\) 11.9374 0.533857
\(501\) −5.49247 −0.245386
\(502\) −14.7483 −0.658248
\(503\) 33.0715 1.47459 0.737293 0.675573i \(-0.236104\pi\)
0.737293 + 0.675573i \(0.236104\pi\)
\(504\) 1.70084 0.0757615
\(505\) 35.5083 1.58010
\(506\) 1.02626 0.0456229
\(507\) −9.44656 −0.419537
\(508\) 4.57655 0.203051
\(509\) −2.70935 −0.120090 −0.0600448 0.998196i \(-0.519124\pi\)
−0.0600448 + 0.998196i \(0.519124\pi\)
\(510\) 4.18550 0.185337
\(511\) 13.4926 0.596877
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) 3.08991 0.136290
\(515\) −13.2825 −0.585299
\(516\) 8.67694 0.381981
\(517\) −2.41043 −0.106010
\(518\) 8.59720 0.377739
\(519\) 5.96211 0.261708
\(520\) 9.61200 0.421514
\(521\) 7.25875 0.318012 0.159006 0.987278i \(-0.449171\pi\)
0.159006 + 0.987278i \(0.449171\pi\)
\(522\) −8.61390 −0.377020
\(523\) 33.8400 1.47972 0.739860 0.672761i \(-0.234892\pi\)
0.739860 + 0.672761i \(0.234892\pi\)
\(524\) 14.5165 0.634155
\(525\) −1.50350 −0.0656182
\(526\) 2.74031 0.119483
\(527\) 13.2443 0.576931
\(528\) 0.223482 0.00972582
\(529\) −1.91228 −0.0831427
\(530\) 2.02880 0.0881254
\(531\) 9.34264 0.405436
\(532\) 1.70084 0.0737409
\(533\) −25.4349 −1.10171
\(534\) −2.21998 −0.0960679
\(535\) 17.9989 0.778162
\(536\) 15.8980 0.686691
\(537\) 4.27562 0.184507
\(538\) −9.03286 −0.389434
\(539\) 0.917872 0.0395355
\(540\) 2.02880 0.0873056
\(541\) 9.47507 0.407365 0.203683 0.979037i \(-0.434709\pi\)
0.203683 + 0.979037i \(0.434709\pi\)
\(542\) −0.501586 −0.0215450
\(543\) 6.32260 0.271329
\(544\) 2.06304 0.0884522
\(545\) 12.7854 0.547666
\(546\) −8.05822 −0.344860
\(547\) 25.5433 1.09215 0.546076 0.837736i \(-0.316121\pi\)
0.546076 + 0.837736i \(0.316121\pi\)
\(548\) −2.25805 −0.0964591
\(549\) −9.93398 −0.423972
\(550\) −0.197553 −0.00842368
\(551\) −8.61390 −0.366965
\(552\) 4.59214 0.195454
\(553\) −24.6567 −1.04851
\(554\) 16.0326 0.681161
\(555\) 10.2549 0.435297
\(556\) −2.95933 −0.125504
\(557\) 32.8585 1.39226 0.696130 0.717916i \(-0.254904\pi\)
0.696130 + 0.717916i \(0.254904\pi\)
\(558\) 6.41980 0.271772
\(559\) −41.1095 −1.73874
\(560\) 3.45067 0.145817
\(561\) −0.461053 −0.0194657
\(562\) 10.0444 0.423699
\(563\) 25.5273 1.07585 0.537923 0.842994i \(-0.319209\pi\)
0.537923 + 0.842994i \(0.319209\pi\)
\(564\) −10.7858 −0.454163
\(565\) 6.25629 0.263204
\(566\) −4.04982 −0.170227
\(567\) −1.70084 −0.0714287
\(568\) −7.18427 −0.301445
\(569\) −3.30396 −0.138509 −0.0692547 0.997599i \(-0.522062\pi\)
−0.0692547 + 0.997599i \(0.522062\pi\)
\(570\) 2.02880 0.0849770
\(571\) 39.3805 1.64802 0.824011 0.566574i \(-0.191731\pi\)
0.824011 + 0.566574i \(0.191731\pi\)
\(572\) −1.05881 −0.0442711
\(573\) 5.88433 0.245821
\(574\) −9.13103 −0.381122
\(575\) −4.05934 −0.169286
\(576\) 1.00000 0.0416667
\(577\) −10.1944 −0.424397 −0.212199 0.977227i \(-0.568062\pi\)
−0.212199 + 0.977227i \(0.568062\pi\)
\(578\) 12.7439 0.530075
\(579\) −19.1609 −0.796299
\(580\) −17.4759 −0.725646
\(581\) −6.60000 −0.273814
\(582\) −3.71588 −0.154028
\(583\) −0.223482 −0.00925569
\(584\) 7.93289 0.328265
\(585\) −9.61200 −0.397407
\(586\) −18.6011 −0.768406
\(587\) 21.5564 0.889729 0.444864 0.895598i \(-0.353252\pi\)
0.444864 + 0.895598i \(0.353252\pi\)
\(588\) 4.10714 0.169375
\(589\) 6.41980 0.264523
\(590\) 18.9543 0.780337
\(591\) −4.17039 −0.171547
\(592\) 5.05467 0.207746
\(593\) 21.0705 0.865263 0.432632 0.901571i \(-0.357585\pi\)
0.432632 + 0.901571i \(0.357585\pi\)
\(594\) −0.223482 −0.00916959
\(595\) −7.11887 −0.291845
\(596\) −4.29566 −0.175957
\(597\) −18.9956 −0.777436
\(598\) −21.7565 −0.889691
\(599\) 29.3944 1.20102 0.600511 0.799616i \(-0.294964\pi\)
0.600511 + 0.799616i \(0.294964\pi\)
\(600\) −0.883975 −0.0360881
\(601\) −27.4704 −1.12054 −0.560270 0.828310i \(-0.689303\pi\)
−0.560270 + 0.828310i \(0.689303\pi\)
\(602\) −14.7581 −0.601496
\(603\) −15.8980 −0.647418
\(604\) 8.63019 0.351158
\(605\) 22.2155 0.903187
\(606\) −17.5021 −0.710975
\(607\) 24.1580 0.980544 0.490272 0.871569i \(-0.336897\pi\)
0.490272 + 0.871569i \(0.336897\pi\)
\(608\) 1.00000 0.0405554
\(609\) 14.6509 0.593684
\(610\) −20.1540 −0.816013
\(611\) 51.1006 2.06731
\(612\) −2.06304 −0.0833935
\(613\) −12.7610 −0.515413 −0.257707 0.966223i \(-0.582967\pi\)
−0.257707 + 0.966223i \(0.582967\pi\)
\(614\) −32.9887 −1.33132
\(615\) −10.8917 −0.439195
\(616\) −0.380108 −0.0153150
\(617\) −18.5230 −0.745706 −0.372853 0.927890i \(-0.621620\pi\)
−0.372853 + 0.927890i \(0.621620\pi\)
\(618\) 6.54700 0.263359
\(619\) −35.1184 −1.41153 −0.705764 0.708447i \(-0.749396\pi\)
−0.705764 + 0.708447i \(0.749396\pi\)
\(620\) 13.0245 0.523076
\(621\) −4.59214 −0.184276
\(622\) 5.42378 0.217474
\(623\) 3.77583 0.151276
\(624\) −4.73778 −0.189663
\(625\) −19.7987 −0.791948
\(626\) 6.21772 0.248510
\(627\) −0.223482 −0.00892502
\(628\) 10.1567 0.405298
\(629\) −10.4280 −0.415792
\(630\) −3.45067 −0.137478
\(631\) 30.7829 1.22545 0.612724 0.790297i \(-0.290074\pi\)
0.612724 + 0.790297i \(0.290074\pi\)
\(632\) −14.4968 −0.576650
\(633\) 3.45387 0.137279
\(634\) 15.1750 0.602676
\(635\) −9.28490 −0.368460
\(636\) −1.00000 −0.0396526
\(637\) −19.4587 −0.770982
\(638\) 1.92505 0.0762136
\(639\) 7.18427 0.284205
\(640\) 2.02880 0.0801953
\(641\) 33.7703 1.33385 0.666924 0.745126i \(-0.267611\pi\)
0.666924 + 0.745126i \(0.267611\pi\)
\(642\) −8.87172 −0.350139
\(643\) 15.9466 0.628871 0.314435 0.949279i \(-0.398185\pi\)
0.314435 + 0.949279i \(0.398185\pi\)
\(644\) −7.81050 −0.307777
\(645\) −17.6038 −0.693148
\(646\) −2.06304 −0.0811693
\(647\) 23.3905 0.919574 0.459787 0.888029i \(-0.347926\pi\)
0.459787 + 0.888029i \(0.347926\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −2.08791 −0.0819578
\(650\) 4.18808 0.164270
\(651\) −10.9191 −0.427952
\(652\) −18.0373 −0.706393
\(653\) 15.9056 0.622433 0.311217 0.950339i \(-0.399264\pi\)
0.311217 + 0.950339i \(0.399264\pi\)
\(654\) −6.30195 −0.246426
\(655\) −29.4510 −1.15074
\(656\) −5.36854 −0.209606
\(657\) −7.93289 −0.309491
\(658\) 18.3449 0.715158
\(659\) 3.30773 0.128851 0.0644255 0.997923i \(-0.479479\pi\)
0.0644255 + 0.997923i \(0.479479\pi\)
\(660\) −0.453401 −0.0176486
\(661\) −6.39916 −0.248898 −0.124449 0.992226i \(-0.539716\pi\)
−0.124449 + 0.992226i \(0.539716\pi\)
\(662\) 28.0327 1.08952
\(663\) 9.77424 0.379600
\(664\) −3.88043 −0.150590
\(665\) −3.45067 −0.133811
\(666\) −5.05467 −0.195865
\(667\) 39.5562 1.53162
\(668\) 5.49247 0.212510
\(669\) −12.6841 −0.490397
\(670\) −32.2539 −1.24608
\(671\) 2.22007 0.0857048
\(672\) −1.70084 −0.0656114
\(673\) 17.0437 0.656986 0.328493 0.944506i \(-0.393459\pi\)
0.328493 + 0.944506i \(0.393459\pi\)
\(674\) −21.2001 −0.816596
\(675\) 0.883975 0.0340242
\(676\) 9.44656 0.363329
\(677\) 11.2277 0.431516 0.215758 0.976447i \(-0.430778\pi\)
0.215758 + 0.976447i \(0.430778\pi\)
\(678\) −3.08374 −0.118430
\(679\) 6.32013 0.242544
\(680\) −4.18550 −0.160506
\(681\) −16.5026 −0.632381
\(682\) −1.43471 −0.0549379
\(683\) −4.10099 −0.156920 −0.0784600 0.996917i \(-0.525000\pi\)
−0.0784600 + 0.996917i \(0.525000\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 4.58113 0.175036
\(686\) −18.8915 −0.721280
\(687\) −25.0432 −0.955457
\(688\) −8.67694 −0.330805
\(689\) 4.73778 0.180495
\(690\) −9.31652 −0.354674
\(691\) 7.45659 0.283662 0.141831 0.989891i \(-0.454701\pi\)
0.141831 + 0.989891i \(0.454701\pi\)
\(692\) −5.96211 −0.226645
\(693\) 0.380108 0.0144391
\(694\) −12.4103 −0.471087
\(695\) 6.00389 0.227741
\(696\) 8.61390 0.326509
\(697\) 11.0755 0.419515
\(698\) −16.0095 −0.605969
\(699\) −5.73384 −0.216874
\(700\) 1.50350 0.0568271
\(701\) 11.7003 0.441914 0.220957 0.975284i \(-0.429082\pi\)
0.220957 + 0.975284i \(0.429082\pi\)
\(702\) 4.73778 0.178816
\(703\) −5.05467 −0.190641
\(704\) −0.223482 −0.00842280
\(705\) 21.8822 0.824130
\(706\) 22.0225 0.828827
\(707\) 29.7683 1.11955
\(708\) −9.34264 −0.351118
\(709\) −11.6337 −0.436914 −0.218457 0.975847i \(-0.570102\pi\)
−0.218457 + 0.975847i \(0.570102\pi\)
\(710\) 14.5754 0.547007
\(711\) 14.4968 0.543671
\(712\) 2.21998 0.0831972
\(713\) −29.4806 −1.10406
\(714\) 3.50891 0.131318
\(715\) 2.14811 0.0803348
\(716\) −4.27562 −0.159788
\(717\) 1.76355 0.0658610
\(718\) −14.2394 −0.531409
\(719\) 46.1040 1.71939 0.859694 0.510809i \(-0.170654\pi\)
0.859694 + 0.510809i \(0.170654\pi\)
\(720\) −2.02880 −0.0756089
\(721\) −11.1354 −0.414704
\(722\) −1.00000 −0.0372161
\(723\) −20.9874 −0.780531
\(724\) −6.32260 −0.234977
\(725\) −7.61448 −0.282795
\(726\) −10.9501 −0.406395
\(727\) 17.1220 0.635019 0.317510 0.948255i \(-0.397153\pi\)
0.317510 + 0.948255i \(0.397153\pi\)
\(728\) 8.05822 0.298657
\(729\) 1.00000 0.0370370
\(730\) −16.0942 −0.595674
\(731\) 17.9009 0.662088
\(732\) 9.93398 0.367170
\(733\) 5.23908 0.193510 0.0967549 0.995308i \(-0.469154\pi\)
0.0967549 + 0.995308i \(0.469154\pi\)
\(734\) −33.7719 −1.24654
\(735\) −8.33255 −0.307351
\(736\) −4.59214 −0.169268
\(737\) 3.55293 0.130874
\(738\) 5.36854 0.197619
\(739\) −2.05156 −0.0754678 −0.0377339 0.999288i \(-0.512014\pi\)
−0.0377339 + 0.999288i \(0.512014\pi\)
\(740\) −10.2549 −0.376978
\(741\) 4.73778 0.174047
\(742\) 1.70084 0.0624399
\(743\) −21.0817 −0.773412 −0.386706 0.922203i \(-0.626387\pi\)
−0.386706 + 0.922203i \(0.626387\pi\)
\(744\) −6.41980 −0.235361
\(745\) 8.71503 0.319294
\(746\) −21.7568 −0.796573
\(747\) 3.88043 0.141977
\(748\) 0.461053 0.0168578
\(749\) 15.0894 0.551354
\(750\) 11.9374 0.435892
\(751\) 50.6645 1.84877 0.924387 0.381456i \(-0.124577\pi\)
0.924387 + 0.381456i \(0.124577\pi\)
\(752\) 10.7858 0.393316
\(753\) −14.7483 −0.537457
\(754\) −40.8108 −1.48624
\(755\) −17.5089 −0.637215
\(756\) 1.70084 0.0618590
\(757\) 21.2177 0.771172 0.385586 0.922672i \(-0.373999\pi\)
0.385586 + 0.922672i \(0.373999\pi\)
\(758\) 8.08932 0.293817
\(759\) 1.02626 0.0372509
\(760\) −2.02880 −0.0735923
\(761\) 17.6053 0.638193 0.319096 0.947722i \(-0.396621\pi\)
0.319096 + 0.947722i \(0.396621\pi\)
\(762\) 4.57655 0.165791
\(763\) 10.7186 0.388040
\(764\) −5.88433 −0.212888
\(765\) 4.18550 0.151327
\(766\) −5.30508 −0.191680
\(767\) 44.2634 1.59826
\(768\) −1.00000 −0.0360844
\(769\) −24.0275 −0.866452 −0.433226 0.901285i \(-0.642625\pi\)
−0.433226 + 0.901285i \(0.642625\pi\)
\(770\) 0.771163 0.0277908
\(771\) 3.08991 0.111280
\(772\) 19.1609 0.689615
\(773\) −5.20207 −0.187105 −0.0935527 0.995614i \(-0.529822\pi\)
−0.0935527 + 0.995614i \(0.529822\pi\)
\(774\) 8.67694 0.311886
\(775\) 5.67494 0.203850
\(776\) 3.71588 0.133392
\(777\) 8.59720 0.308423
\(778\) −11.9606 −0.428807
\(779\) 5.36854 0.192348
\(780\) 9.61200 0.344165
\(781\) −1.60556 −0.0574514
\(782\) 9.47377 0.338781
\(783\) −8.61390 −0.307836
\(784\) −4.10714 −0.146683
\(785\) −20.6060 −0.735459
\(786\) 14.5165 0.517785
\(787\) −7.79610 −0.277901 −0.138951 0.990299i \(-0.544373\pi\)
−0.138951 + 0.990299i \(0.544373\pi\)
\(788\) 4.17039 0.148564
\(789\) 2.74031 0.0975576
\(790\) 29.4110 1.04640
\(791\) 5.24496 0.186489
\(792\) 0.223482 0.00794110
\(793\) −47.0650 −1.67133
\(794\) 2.38568 0.0846646
\(795\) 2.02880 0.0719541
\(796\) 18.9956 0.673280
\(797\) 31.1103 1.10198 0.550992 0.834511i \(-0.314250\pi\)
0.550992 + 0.834511i \(0.314250\pi\)
\(798\) 1.70084 0.0602092
\(799\) −22.2515 −0.787201
\(800\) 0.883975 0.0312533
\(801\) −2.21998 −0.0784391
\(802\) −12.3413 −0.435785
\(803\) 1.77286 0.0625628
\(804\) 15.8980 0.560681
\(805\) 15.8459 0.558496
\(806\) 30.4156 1.07134
\(807\) −9.03286 −0.317972
\(808\) 17.5021 0.615722
\(809\) 3.95472 0.139040 0.0695202 0.997581i \(-0.477853\pi\)
0.0695202 + 0.997581i \(0.477853\pi\)
\(810\) 2.02880 0.0712847
\(811\) 7.90285 0.277507 0.138753 0.990327i \(-0.455690\pi\)
0.138753 + 0.990327i \(0.455690\pi\)
\(812\) −14.6509 −0.514145
\(813\) −0.501586 −0.0175914
\(814\) 1.12963 0.0395935
\(815\) 36.5940 1.28183
\(816\) 2.06304 0.0722209
\(817\) 8.67694 0.303568
\(818\) 0.0414247 0.00144838
\(819\) −8.05822 −0.281577
\(820\) 10.8917 0.380354
\(821\) 8.33085 0.290749 0.145374 0.989377i \(-0.453561\pi\)
0.145374 + 0.989377i \(0.453561\pi\)
\(822\) −2.25805 −0.0787585
\(823\) 1.02874 0.0358598 0.0179299 0.999839i \(-0.494292\pi\)
0.0179299 + 0.999839i \(0.494292\pi\)
\(824\) −6.54700 −0.228075
\(825\) −0.197553 −0.00687791
\(826\) 15.8903 0.552896
\(827\) 4.57843 0.159208 0.0796038 0.996827i \(-0.474634\pi\)
0.0796038 + 0.996827i \(0.474634\pi\)
\(828\) 4.59214 0.159588
\(829\) −50.9301 −1.76887 −0.884437 0.466660i \(-0.845457\pi\)
−0.884437 + 0.466660i \(0.845457\pi\)
\(830\) 7.87261 0.273262
\(831\) 16.0326 0.556166
\(832\) 4.73778 0.164253
\(833\) 8.47319 0.293579
\(834\) −2.95933 −0.102473
\(835\) −11.1431 −0.385624
\(836\) 0.223482 0.00772930
\(837\) 6.41980 0.221901
\(838\) 22.9342 0.792249
\(839\) −27.6799 −0.955618 −0.477809 0.878464i \(-0.658569\pi\)
−0.477809 + 0.878464i \(0.658569\pi\)
\(840\) 3.45067 0.119059
\(841\) 45.1993 1.55860
\(842\) 6.38045 0.219885
\(843\) 10.0444 0.345949
\(844\) −3.45387 −0.118887
\(845\) −19.1652 −0.659302
\(846\) −10.7858 −0.370822
\(847\) 18.6243 0.639939
\(848\) 1.00000 0.0343401
\(849\) −4.04982 −0.138989
\(850\) −1.82368 −0.0625516
\(851\) 23.2117 0.795688
\(852\) −7.18427 −0.246129
\(853\) −11.3815 −0.389694 −0.194847 0.980834i \(-0.562421\pi\)
−0.194847 + 0.980834i \(0.562421\pi\)
\(854\) −16.8961 −0.578174
\(855\) 2.02880 0.0693835
\(856\) 8.87172 0.303229
\(857\) −48.1413 −1.64448 −0.822238 0.569144i \(-0.807275\pi\)
−0.822238 + 0.569144i \(0.807275\pi\)
\(858\) −1.05881 −0.0361472
\(859\) 36.9543 1.26087 0.630433 0.776244i \(-0.282877\pi\)
0.630433 + 0.776244i \(0.282877\pi\)
\(860\) 17.6038 0.600284
\(861\) −9.13103 −0.311185
\(862\) 26.9070 0.916457
\(863\) 15.8799 0.540557 0.270278 0.962782i \(-0.412884\pi\)
0.270278 + 0.962782i \(0.412884\pi\)
\(864\) 1.00000 0.0340207
\(865\) 12.0959 0.411274
\(866\) 0.497344 0.0169004
\(867\) 12.7439 0.432804
\(868\) 10.9191 0.370617
\(869\) −3.23977 −0.109902
\(870\) −17.4759 −0.592488
\(871\) −75.3214 −2.55217
\(872\) 6.30195 0.213411
\(873\) −3.71588 −0.125764
\(874\) 4.59214 0.155331
\(875\) −20.3036 −0.686388
\(876\) 7.93289 0.268027
\(877\) 36.0829 1.21843 0.609216 0.793004i \(-0.291484\pi\)
0.609216 + 0.793004i \(0.291484\pi\)
\(878\) 17.0256 0.574588
\(879\) −18.6011 −0.627401
\(880\) 0.453401 0.0152841
\(881\) 14.1218 0.475774 0.237887 0.971293i \(-0.423545\pi\)
0.237887 + 0.971293i \(0.423545\pi\)
\(882\) 4.10714 0.138294
\(883\) 10.0116 0.336918 0.168459 0.985709i \(-0.446121\pi\)
0.168459 + 0.985709i \(0.446121\pi\)
\(884\) −9.77424 −0.328743
\(885\) 18.9543 0.637143
\(886\) −24.4326 −0.820829
\(887\) 24.9380 0.837338 0.418669 0.908139i \(-0.362497\pi\)
0.418669 + 0.908139i \(0.362497\pi\)
\(888\) 5.05467 0.169624
\(889\) −7.78399 −0.261067
\(890\) −4.50389 −0.150971
\(891\) −0.223482 −0.00748694
\(892\) 12.6841 0.424696
\(893\) −10.7858 −0.360932
\(894\) −4.29566 −0.143668
\(895\) 8.67438 0.289953
\(896\) 1.70084 0.0568212
\(897\) −21.7565 −0.726430
\(898\) −11.6675 −0.389351
\(899\) −55.2995 −1.84434
\(900\) −0.883975 −0.0294658
\(901\) −2.06304 −0.0687299
\(902\) −1.19977 −0.0399481
\(903\) −14.7581 −0.491119
\(904\) 3.08374 0.102564
\(905\) 12.8273 0.426393
\(906\) 8.63019 0.286719
\(907\) −0.977827 −0.0324682 −0.0162341 0.999868i \(-0.505168\pi\)
−0.0162341 + 0.999868i \(0.505168\pi\)
\(908\) 16.5026 0.547658
\(909\) −17.5021 −0.580508
\(910\) −16.3485 −0.541948
\(911\) −36.6570 −1.21450 −0.607251 0.794510i \(-0.707728\pi\)
−0.607251 + 0.794510i \(0.707728\pi\)
\(912\) 1.00000 0.0331133
\(913\) −0.867207 −0.0287004
\(914\) −4.47028 −0.147864
\(915\) −20.1540 −0.666272
\(916\) 25.0432 0.827450
\(917\) −24.6902 −0.815343
\(918\) −2.06304 −0.0680905
\(919\) −46.4054 −1.53077 −0.765387 0.643570i \(-0.777452\pi\)
−0.765387 + 0.643570i \(0.777452\pi\)
\(920\) 9.31652 0.307157
\(921\) −32.9887 −1.08702
\(922\) −17.0418 −0.561242
\(923\) 34.0375 1.12036
\(924\) −0.380108 −0.0125046
\(925\) −4.46821 −0.146914
\(926\) 29.2509 0.961244
\(927\) 6.54700 0.215032
\(928\) −8.61390 −0.282765
\(929\) −40.7275 −1.33623 −0.668113 0.744060i \(-0.732898\pi\)
−0.668113 + 0.744060i \(0.732898\pi\)
\(930\) 13.0245 0.427089
\(931\) 4.10714 0.134606
\(932\) 5.73384 0.187818
\(933\) 5.42378 0.177567
\(934\) −28.3787 −0.928578
\(935\) −0.935384 −0.0305903
\(936\) −4.73778 −0.154859
\(937\) −33.1944 −1.08441 −0.542207 0.840245i \(-0.682411\pi\)
−0.542207 + 0.840245i \(0.682411\pi\)
\(938\) −27.0401 −0.882889
\(939\) 6.21772 0.202908
\(940\) −21.8822 −0.713717
\(941\) −27.4054 −0.893391 −0.446695 0.894686i \(-0.647399\pi\)
−0.446695 + 0.894686i \(0.647399\pi\)
\(942\) 10.1567 0.330925
\(943\) −24.6531 −0.802814
\(944\) 9.34264 0.304077
\(945\) −3.45067 −0.112250
\(946\) −1.93914 −0.0630470
\(947\) 30.2622 0.983389 0.491694 0.870768i \(-0.336378\pi\)
0.491694 + 0.870768i \(0.336378\pi\)
\(948\) −14.4968 −0.470833
\(949\) −37.5843 −1.22004
\(950\) −0.883975 −0.0286800
\(951\) 15.1750 0.492083
\(952\) −3.50891 −0.113724
\(953\) −1.25115 −0.0405287 −0.0202643 0.999795i \(-0.506451\pi\)
−0.0202643 + 0.999795i \(0.506451\pi\)
\(954\) −1.00000 −0.0323762
\(955\) 11.9381 0.386309
\(956\) −1.76355 −0.0570373
\(957\) 1.92505 0.0622282
\(958\) 5.30183 0.171294
\(959\) 3.84059 0.124019
\(960\) 2.02880 0.0654792
\(961\) 10.2138 0.329477
\(962\) −23.9479 −0.772112
\(963\) −8.87172 −0.285887
\(964\) 20.9874 0.675960
\(965\) −38.8736 −1.25138
\(966\) −7.81050 −0.251299
\(967\) −8.68626 −0.279331 −0.139666 0.990199i \(-0.544603\pi\)
−0.139666 + 0.990199i \(0.544603\pi\)
\(968\) 10.9501 0.351948
\(969\) −2.06304 −0.0662744
\(970\) −7.53878 −0.242056
\(971\) −24.2896 −0.779490 −0.389745 0.920923i \(-0.627437\pi\)
−0.389745 + 0.920923i \(0.627437\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 5.03336 0.161362
\(974\) 12.5119 0.400908
\(975\) 4.18808 0.134126
\(976\) −9.93398 −0.317979
\(977\) −22.0269 −0.704703 −0.352352 0.935868i \(-0.614618\pi\)
−0.352352 + 0.935868i \(0.614618\pi\)
\(978\) −18.0373 −0.576768
\(979\) 0.496126 0.0158562
\(980\) 8.33255 0.266174
\(981\) −6.30195 −0.201206
\(982\) 42.2114 1.34702
\(983\) 7.37854 0.235339 0.117669 0.993053i \(-0.462458\pi\)
0.117669 + 0.993053i \(0.462458\pi\)
\(984\) −5.36854 −0.171143
\(985\) −8.46088 −0.269586
\(986\) 17.7708 0.565939
\(987\) 18.3449 0.583924
\(988\) −4.73778 −0.150729
\(989\) −39.8457 −1.26702
\(990\) −0.453401 −0.0144100
\(991\) −46.2751 −1.46998 −0.734988 0.678080i \(-0.762812\pi\)
−0.734988 + 0.678080i \(0.762812\pi\)
\(992\) 6.41980 0.203829
\(993\) 28.0327 0.889590
\(994\) 12.2193 0.387573
\(995\) −38.5382 −1.22174
\(996\) −3.88043 −0.122956
\(997\) 12.7776 0.404670 0.202335 0.979316i \(-0.435147\pi\)
0.202335 + 0.979316i \(0.435147\pi\)
\(998\) 0.579520 0.0183444
\(999\) −5.05467 −0.159923
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 6042.2.a.z.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.z.1.3 9 1.1 even 1 trivial