Properties

Label 8526.2.a.bo.1.3
Level $8526$
Weight $2$
Character 8526.1
Self dual yes
Analytic conductor $68.080$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8526,2,Mod(1,8526)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8526.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8526, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8526 = 2 \cdot 3 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8526.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-3,-3,3,0,3,0,-3,3,0,1,-3,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.0804527633\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1218)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 8526.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.72161 q^{5} +1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -3.72161 q^{10} -6.04502 q^{11} -1.00000 q^{12} +2.32340 q^{13} -3.72161 q^{15} +1.00000 q^{16} -3.72161 q^{17} -1.00000 q^{18} -1.39821 q^{19} +3.72161 q^{20} +6.04502 q^{22} +7.24860 q^{23} +1.00000 q^{24} +8.85039 q^{25} -2.32340 q^{26} -1.00000 q^{27} -1.00000 q^{29} +3.72161 q^{30} -9.72161 q^{31} -1.00000 q^{32} +6.04502 q^{33} +3.72161 q^{34} +1.00000 q^{36} +6.19462 q^{37} +1.39821 q^{38} -2.32340 q^{39} -3.72161 q^{40} +4.92520 q^{41} +1.20359 q^{43} -6.04502 q^{44} +3.72161 q^{45} -7.24860 q^{46} +6.17380 q^{47} -1.00000 q^{48} -8.85039 q^{50} +3.72161 q^{51} +2.32340 q^{52} +4.60179 q^{53} +1.00000 q^{54} -22.4972 q^{55} +1.39821 q^{57} +1.00000 q^{58} -12.4134 q^{59} -3.72161 q^{60} -14.0900 q^{61} +9.72161 q^{62} +1.00000 q^{64} +8.64681 q^{65} -6.04502 q^{66} -7.89541 q^{67} -3.72161 q^{68} -7.24860 q^{69} +5.29362 q^{71} -1.00000 q^{72} +15.6170 q^{73} -6.19462 q^{74} -8.85039 q^{75} -1.39821 q^{76} +2.32340 q^{78} -5.29362 q^{79} +3.72161 q^{80} +1.00000 q^{81} -4.92520 q^{82} +17.6170 q^{83} -13.8504 q^{85} -1.20359 q^{86} +1.00000 q^{87} +6.04502 q^{88} -4.36842 q^{89} -3.72161 q^{90} +7.24860 q^{92} +9.72161 q^{93} -6.17380 q^{94} -5.20359 q^{95} +1.00000 q^{96} +3.26943 q^{97} -6.04502 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{6} - 3 q^{8} + 3 q^{9} + q^{11} - 3 q^{12} - q^{13} + 3 q^{16} - 3 q^{18} - q^{19} - q^{22} + 9 q^{23} + 3 q^{24} + 17 q^{25} + q^{26} - 3 q^{27} - 3 q^{29}+ \cdots + q^{99}+O(q^{100}) \) 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
\(5\) 3.72161 1.66436 0.832178 0.554509i \(-0.187094\pi\)
0.832178 + 0.554509i \(0.187094\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.72161 −1.17688
\(11\) −6.04502 −1.82264 −0.911320 0.411698i \(-0.864936\pi\)
−0.911320 + 0.411698i \(0.864936\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.32340 0.644396 0.322198 0.946672i \(-0.395578\pi\)
0.322198 + 0.946672i \(0.395578\pi\)
\(14\) 0 0
\(15\) −3.72161 −0.960916
\(16\) 1.00000 0.250000
\(17\) −3.72161 −0.902623 −0.451312 0.892366i \(-0.649044\pi\)
−0.451312 + 0.892366i \(0.649044\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.39821 −0.320771 −0.160385 0.987054i \(-0.551274\pi\)
−0.160385 + 0.987054i \(0.551274\pi\)
\(20\) 3.72161 0.832178
\(21\) 0 0
\(22\) 6.04502 1.28880
\(23\) 7.24860 1.51144 0.755719 0.654896i \(-0.227288\pi\)
0.755719 + 0.654896i \(0.227288\pi\)
\(24\) 1.00000 0.204124
\(25\) 8.85039 1.77008
\(26\) −2.32340 −0.455657
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 3.72161 0.679470
\(31\) −9.72161 −1.74605 −0.873027 0.487673i \(-0.837846\pi\)
−0.873027 + 0.487673i \(0.837846\pi\)
\(32\) −1.00000 −0.176777
\(33\) 6.04502 1.05230
\(34\) 3.72161 0.638251
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.19462 1.01839 0.509195 0.860651i \(-0.329943\pi\)
0.509195 + 0.860651i \(0.329943\pi\)
\(38\) 1.39821 0.226819
\(39\) −2.32340 −0.372042
\(40\) −3.72161 −0.588438
\(41\) 4.92520 0.769187 0.384593 0.923086i \(-0.374342\pi\)
0.384593 + 0.923086i \(0.374342\pi\)
\(42\) 0 0
\(43\) 1.20359 0.183545 0.0917725 0.995780i \(-0.470747\pi\)
0.0917725 + 0.995780i \(0.470747\pi\)
\(44\) −6.04502 −0.911320
\(45\) 3.72161 0.554785
\(46\) −7.24860 −1.06875
\(47\) 6.17380 0.900541 0.450270 0.892892i \(-0.351328\pi\)
0.450270 + 0.892892i \(0.351328\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −8.85039 −1.25163
\(51\) 3.72161 0.521130
\(52\) 2.32340 0.322198
\(53\) 4.60179 0.632105 0.316052 0.948742i \(-0.397642\pi\)
0.316052 + 0.948742i \(0.397642\pi\)
\(54\) 1.00000 0.136083
\(55\) −22.4972 −3.03352
\(56\) 0 0
\(57\) 1.39821 0.185197
\(58\) 1.00000 0.131306
\(59\) −12.4134 −1.61609 −0.808046 0.589120i \(-0.799475\pi\)
−0.808046 + 0.589120i \(0.799475\pi\)
\(60\) −3.72161 −0.480458
\(61\) −14.0900 −1.80404 −0.902022 0.431690i \(-0.857917\pi\)
−0.902022 + 0.431690i \(0.857917\pi\)
\(62\) 9.72161 1.23465
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 8.64681 1.07250
\(66\) −6.04502 −0.744090
\(67\) −7.89541 −0.964578 −0.482289 0.876012i \(-0.660194\pi\)
−0.482289 + 0.876012i \(0.660194\pi\)
\(68\) −3.72161 −0.451312
\(69\) −7.24860 −0.872629
\(70\) 0 0
\(71\) 5.29362 0.628237 0.314118 0.949384i \(-0.398291\pi\)
0.314118 + 0.949384i \(0.398291\pi\)
\(72\) −1.00000 −0.117851
\(73\) 15.6170 1.82783 0.913917 0.405901i \(-0.133042\pi\)
0.913917 + 0.405901i \(0.133042\pi\)
\(74\) −6.19462 −0.720110
\(75\) −8.85039 −1.02196
\(76\) −1.39821 −0.160385
\(77\) 0 0
\(78\) 2.32340 0.263074
\(79\) −5.29362 −0.595578 −0.297789 0.954632i \(-0.596249\pi\)
−0.297789 + 0.954632i \(0.596249\pi\)
\(80\) 3.72161 0.416089
\(81\) 1.00000 0.111111
\(82\) −4.92520 −0.543897
\(83\) 17.6170 1.93372 0.966860 0.255308i \(-0.0821770\pi\)
0.966860 + 0.255308i \(0.0821770\pi\)
\(84\) 0 0
\(85\) −13.8504 −1.50229
\(86\) −1.20359 −0.129786
\(87\) 1.00000 0.107211
\(88\) 6.04502 0.644401
\(89\) −4.36842 −0.463052 −0.231526 0.972829i \(-0.574372\pi\)
−0.231526 + 0.972829i \(0.574372\pi\)
\(90\) −3.72161 −0.392292
\(91\) 0 0
\(92\) 7.24860 0.755719
\(93\) 9.72161 1.00808
\(94\) −6.17380 −0.636779
\(95\) −5.20359 −0.533877
\(96\) 1.00000 0.102062
\(97\) 3.26943 0.331960 0.165980 0.986129i \(-0.446921\pi\)
0.165980 + 0.986129i \(0.446921\pi\)
\(98\) 0 0
\(99\) −6.04502 −0.607547
\(100\) 8.85039 0.885039
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) −3.72161 −0.368494
\(103\) −10.6018 −1.04463 −0.522313 0.852754i \(-0.674931\pi\)
−0.522313 + 0.852754i \(0.674931\pi\)
\(104\) −2.32340 −0.227829
\(105\) 0 0
\(106\) −4.60179 −0.446966
\(107\) 7.35319 0.710860 0.355430 0.934703i \(-0.384334\pi\)
0.355430 + 0.934703i \(0.384334\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −5.05398 −0.484083 −0.242042 0.970266i \(-0.577817\pi\)
−0.242042 + 0.970266i \(0.577817\pi\)
\(110\) 22.4972 2.14502
\(111\) −6.19462 −0.587968
\(112\) 0 0
\(113\) 8.99104 0.845806 0.422903 0.906175i \(-0.361011\pi\)
0.422903 + 0.906175i \(0.361011\pi\)
\(114\) −1.39821 −0.130954
\(115\) 26.9765 2.51557
\(116\) −1.00000 −0.0928477
\(117\) 2.32340 0.214799
\(118\) 12.4134 1.14275
\(119\) 0 0
\(120\) 3.72161 0.339735
\(121\) 25.5422 2.32202
\(122\) 14.0900 1.27565
\(123\) −4.92520 −0.444090
\(124\) −9.72161 −0.873027
\(125\) 14.3297 1.28168
\(126\) 0 0
\(127\) 7.89541 0.700604 0.350302 0.936637i \(-0.386079\pi\)
0.350302 + 0.936637i \(0.386079\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −1.20359 −0.105970
\(130\) −8.64681 −0.758375
\(131\) 0.946021 0.0826543 0.0413271 0.999146i \(-0.486841\pi\)
0.0413271 + 0.999146i \(0.486841\pi\)
\(132\) 6.04502 0.526151
\(133\) 0 0
\(134\) 7.89541 0.682060
\(135\) −3.72161 −0.320305
\(136\) 3.72161 0.319126
\(137\) 15.7458 1.34525 0.672627 0.739981i \(-0.265166\pi\)
0.672627 + 0.739981i \(0.265166\pi\)
\(138\) 7.24860 0.617042
\(139\) 19.0152 1.61285 0.806425 0.591336i \(-0.201399\pi\)
0.806425 + 0.591336i \(0.201399\pi\)
\(140\) 0 0
\(141\) −6.17380 −0.519928
\(142\) −5.29362 −0.444230
\(143\) −14.0450 −1.17450
\(144\) 1.00000 0.0833333
\(145\) −3.72161 −0.309063
\(146\) −15.6170 −1.29247
\(147\) 0 0
\(148\) 6.19462 0.509195
\(149\) 10.1946 0.835176 0.417588 0.908636i \(-0.362876\pi\)
0.417588 + 0.908636i \(0.362876\pi\)
\(150\) 8.85039 0.722632
\(151\) 18.4972 1.50528 0.752640 0.658432i \(-0.228780\pi\)
0.752640 + 0.658432i \(0.228780\pi\)
\(152\) 1.39821 0.113410
\(153\) −3.72161 −0.300874
\(154\) 0 0
\(155\) −36.1801 −2.90605
\(156\) −2.32340 −0.186021
\(157\) 1.44322 0.115182 0.0575909 0.998340i \(-0.481658\pi\)
0.0575909 + 0.998340i \(0.481658\pi\)
\(158\) 5.29362 0.421138
\(159\) −4.60179 −0.364946
\(160\) −3.72161 −0.294219
\(161\) 0 0
\(162\) −1.00000 −0.0785674
\(163\) 13.2036 1.03418 0.517092 0.855930i \(-0.327014\pi\)
0.517092 + 0.855930i \(0.327014\pi\)
\(164\) 4.92520 0.384593
\(165\) 22.4972 1.75140
\(166\) −17.6170 −1.36735
\(167\) −18.5872 −1.43832 −0.719162 0.694843i \(-0.755474\pi\)
−0.719162 + 0.694843i \(0.755474\pi\)
\(168\) 0 0
\(169\) −7.60179 −0.584753
\(170\) 13.8504 1.06228
\(171\) −1.39821 −0.106924
\(172\) 1.20359 0.0917725
\(173\) −4.27839 −0.325280 −0.162640 0.986685i \(-0.552001\pi\)
−0.162640 + 0.986685i \(0.552001\pi\)
\(174\) −1.00000 −0.0758098
\(175\) 0 0
\(176\) −6.04502 −0.455660
\(177\) 12.4134 0.933051
\(178\) 4.36842 0.327427
\(179\) −14.4972 −1.08357 −0.541786 0.840517i \(-0.682252\pi\)
−0.541786 + 0.840517i \(0.682252\pi\)
\(180\) 3.72161 0.277393
\(181\) −2.42799 −0.180471 −0.0902357 0.995920i \(-0.528762\pi\)
−0.0902357 + 0.995920i \(0.528762\pi\)
\(182\) 0 0
\(183\) 14.0900 1.04157
\(184\) −7.24860 −0.534374
\(185\) 23.0540 1.69496
\(186\) −9.72161 −0.712823
\(187\) 22.4972 1.64516
\(188\) 6.17380 0.450270
\(189\) 0 0
\(190\) 5.20359 0.377508
\(191\) 4.94602 0.357882 0.178941 0.983860i \(-0.442733\pi\)
0.178941 + 0.983860i \(0.442733\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 18.1801 1.30863 0.654315 0.756222i \(-0.272957\pi\)
0.654315 + 0.756222i \(0.272957\pi\)
\(194\) −3.26943 −0.234731
\(195\) −8.64681 −0.619211
\(196\) 0 0
\(197\) −2.55678 −0.182163 −0.0910814 0.995843i \(-0.529032\pi\)
−0.0910814 + 0.995843i \(0.529032\pi\)
\(198\) 6.04502 0.429601
\(199\) 5.39821 0.382669 0.191334 0.981525i \(-0.438719\pi\)
0.191334 + 0.981525i \(0.438719\pi\)
\(200\) −8.85039 −0.625817
\(201\) 7.89541 0.556899
\(202\) −6.00000 −0.422159
\(203\) 0 0
\(204\) 3.72161 0.260565
\(205\) 18.3297 1.28020
\(206\) 10.6018 0.738662
\(207\) 7.24860 0.503813
\(208\) 2.32340 0.161099
\(209\) 8.45219 0.584650
\(210\) 0 0
\(211\) 2.49720 0.171914 0.0859572 0.996299i \(-0.472605\pi\)
0.0859572 + 0.996299i \(0.472605\pi\)
\(212\) 4.60179 0.316052
\(213\) −5.29362 −0.362713
\(214\) −7.35319 −0.502654
\(215\) 4.47928 0.305484
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 5.05398 0.342299
\(219\) −15.6170 −1.05530
\(220\) −22.4972 −1.51676
\(221\) −8.64681 −0.581647
\(222\) 6.19462 0.415756
\(223\) −13.7875 −0.923276 −0.461638 0.887068i \(-0.652738\pi\)
−0.461638 + 0.887068i \(0.652738\pi\)
\(224\) 0 0
\(225\) 8.85039 0.590026
\(226\) −8.99104 −0.598075
\(227\) 4.42799 0.293896 0.146948 0.989144i \(-0.453055\pi\)
0.146948 + 0.989144i \(0.453055\pi\)
\(228\) 1.39821 0.0925985
\(229\) −17.1440 −1.13291 −0.566454 0.824093i \(-0.691685\pi\)
−0.566454 + 0.824093i \(0.691685\pi\)
\(230\) −26.9765 −1.77878
\(231\) 0 0
\(232\) 1.00000 0.0656532
\(233\) 22.0900 1.44717 0.723583 0.690237i \(-0.242494\pi\)
0.723583 + 0.690237i \(0.242494\pi\)
\(234\) −2.32340 −0.151886
\(235\) 22.9765 1.49882
\(236\) −12.4134 −0.808046
\(237\) 5.29362 0.343857
\(238\) 0 0
\(239\) 8.45219 0.546726 0.273363 0.961911i \(-0.411864\pi\)
0.273363 + 0.961911i \(0.411864\pi\)
\(240\) −3.72161 −0.240229
\(241\) 22.0000 1.41714 0.708572 0.705638i \(-0.249340\pi\)
0.708572 + 0.705638i \(0.249340\pi\)
\(242\) −25.5422 −1.64192
\(243\) −1.00000 −0.0641500
\(244\) −14.0900 −0.902022
\(245\) 0 0
\(246\) 4.92520 0.314019
\(247\) −3.24860 −0.206704
\(248\) 9.72161 0.617323
\(249\) −17.6170 −1.11643
\(250\) −14.3297 −0.906288
\(251\) 3.35319 0.211652 0.105826 0.994385i \(-0.466251\pi\)
0.105826 + 0.994385i \(0.466251\pi\)
\(252\) 0 0
\(253\) −43.8179 −2.75481
\(254\) −7.89541 −0.495402
\(255\) 13.8504 0.867345
\(256\) 1.00000 0.0625000
\(257\) 8.14961 0.508358 0.254179 0.967157i \(-0.418195\pi\)
0.254179 + 0.967157i \(0.418195\pi\)
\(258\) 1.20359 0.0749319
\(259\) 0 0
\(260\) 8.64681 0.536252
\(261\) −1.00000 −0.0618984
\(262\) −0.946021 −0.0584454
\(263\) 11.4432 0.705619 0.352810 0.935695i \(-0.385226\pi\)
0.352810 + 0.935695i \(0.385226\pi\)
\(264\) −6.04502 −0.372045
\(265\) 17.1261 1.05205
\(266\) 0 0
\(267\) 4.36842 0.267343
\(268\) −7.89541 −0.482289
\(269\) 7.95498 0.485024 0.242512 0.970148i \(-0.422029\pi\)
0.242512 + 0.970148i \(0.422029\pi\)
\(270\) 3.72161 0.226490
\(271\) 29.4224 1.78728 0.893642 0.448781i \(-0.148142\pi\)
0.893642 + 0.448781i \(0.148142\pi\)
\(272\) −3.72161 −0.225656
\(273\) 0 0
\(274\) −15.7458 −0.951239
\(275\) −53.5008 −3.22622
\(276\) −7.24860 −0.436315
\(277\) −7.20359 −0.432821 −0.216411 0.976302i \(-0.569435\pi\)
−0.216411 + 0.976302i \(0.569435\pi\)
\(278\) −19.0152 −1.14046
\(279\) −9.72161 −0.582018
\(280\) 0 0
\(281\) −2.64681 −0.157895 −0.0789477 0.996879i \(-0.525156\pi\)
−0.0789477 + 0.996879i \(0.525156\pi\)
\(282\) 6.17380 0.367644
\(283\) −30.7160 −1.82588 −0.912939 0.408096i \(-0.866193\pi\)
−0.912939 + 0.408096i \(0.866193\pi\)
\(284\) 5.29362 0.314118
\(285\) 5.20359 0.308234
\(286\) 14.0450 0.830499
\(287\) 0 0
\(288\) −1.00000 −0.0589256
\(289\) −3.14961 −0.185271
\(290\) 3.72161 0.218541
\(291\) −3.26943 −0.191657
\(292\) 15.6170 0.913917
\(293\) −4.60179 −0.268840 −0.134420 0.990924i \(-0.542917\pi\)
−0.134420 + 0.990924i \(0.542917\pi\)
\(294\) 0 0
\(295\) −46.1980 −2.68975
\(296\) −6.19462 −0.360055
\(297\) 6.04502 0.350767
\(298\) −10.1946 −0.590559
\(299\) 16.8414 0.973965
\(300\) −8.85039 −0.510978
\(301\) 0 0
\(302\) −18.4972 −1.06439
\(303\) −6.00000 −0.344691
\(304\) −1.39821 −0.0801927
\(305\) −52.4376 −3.00257
\(306\) 3.72161 0.212750
\(307\) −15.6378 −0.892499 −0.446249 0.894909i \(-0.647241\pi\)
−0.446249 + 0.894909i \(0.647241\pi\)
\(308\) 0 0
\(309\) 10.6018 0.603115
\(310\) 36.1801 2.05489
\(311\) 13.9792 0.792686 0.396343 0.918102i \(-0.370279\pi\)
0.396343 + 0.918102i \(0.370279\pi\)
\(312\) 2.32340 0.131537
\(313\) 11.5928 0.655265 0.327633 0.944805i \(-0.393749\pi\)
0.327633 + 0.944805i \(0.393749\pi\)
\(314\) −1.44322 −0.0814458
\(315\) 0 0
\(316\) −5.29362 −0.297789
\(317\) 19.2936 1.08364 0.541819 0.840495i \(-0.317736\pi\)
0.541819 + 0.840495i \(0.317736\pi\)
\(318\) 4.60179 0.258056
\(319\) 6.04502 0.338456
\(320\) 3.72161 0.208044
\(321\) −7.35319 −0.410415
\(322\) 0 0
\(323\) 5.20359 0.289535
\(324\) 1.00000 0.0555556
\(325\) 20.5630 1.14063
\(326\) −13.2036 −0.731279
\(327\) 5.05398 0.279486
\(328\) −4.92520 −0.271949
\(329\) 0 0
\(330\) −22.4972 −1.23843
\(331\) 18.4972 1.01670 0.508349 0.861151i \(-0.330256\pi\)
0.508349 + 0.861151i \(0.330256\pi\)
\(332\) 17.6170 0.966860
\(333\) 6.19462 0.339463
\(334\) 18.5872 1.01705
\(335\) −29.3836 −1.60540
\(336\) 0 0
\(337\) −25.8325 −1.40718 −0.703592 0.710604i \(-0.748422\pi\)
−0.703592 + 0.710604i \(0.748422\pi\)
\(338\) 7.60179 0.413483
\(339\) −8.99104 −0.488326
\(340\) −13.8504 −0.751143
\(341\) 58.7673 3.18243
\(342\) 1.39821 0.0756064
\(343\) 0 0
\(344\) −1.20359 −0.0648930
\(345\) −26.9765 −1.45236
\(346\) 4.27839 0.230008
\(347\) 17.6829 0.949266 0.474633 0.880184i \(-0.342581\pi\)
0.474633 + 0.880184i \(0.342581\pi\)
\(348\) 1.00000 0.0536056
\(349\) 21.3144 1.14094 0.570468 0.821320i \(-0.306762\pi\)
0.570468 + 0.821320i \(0.306762\pi\)
\(350\) 0 0
\(351\) −2.32340 −0.124014
\(352\) 6.04502 0.322200
\(353\) −18.1801 −0.967627 −0.483814 0.875171i \(-0.660749\pi\)
−0.483814 + 0.875171i \(0.660749\pi\)
\(354\) −12.4134 −0.659767
\(355\) 19.7008 1.04561
\(356\) −4.36842 −0.231526
\(357\) 0 0
\(358\) 14.4972 0.766201
\(359\) −13.5928 −0.717402 −0.358701 0.933452i \(-0.616780\pi\)
−0.358701 + 0.933452i \(0.616780\pi\)
\(360\) −3.72161 −0.196146
\(361\) −17.0450 −0.897106
\(362\) 2.42799 0.127613
\(363\) −25.5422 −1.34062
\(364\) 0 0
\(365\) 58.1205 3.04217
\(366\) −14.0900 −0.736498
\(367\) 4.12878 0.215521 0.107760 0.994177i \(-0.465632\pi\)
0.107760 + 0.994177i \(0.465632\pi\)
\(368\) 7.24860 0.377859
\(369\) 4.92520 0.256396
\(370\) −23.0540 −1.19852
\(371\) 0 0
\(372\) 9.72161 0.504042
\(373\) −37.8809 −1.96140 −0.980698 0.195528i \(-0.937358\pi\)
−0.980698 + 0.195528i \(0.937358\pi\)
\(374\) −22.4972 −1.16330
\(375\) −14.3297 −0.739981
\(376\) −6.17380 −0.318389
\(377\) −2.32340 −0.119661
\(378\) 0 0
\(379\) 26.8864 1.38106 0.690532 0.723302i \(-0.257376\pi\)
0.690532 + 0.723302i \(0.257376\pi\)
\(380\) −5.20359 −0.266938
\(381\) −7.89541 −0.404494
\(382\) −4.94602 −0.253060
\(383\) −22.8864 −1.16944 −0.584721 0.811234i \(-0.698796\pi\)
−0.584721 + 0.811234i \(0.698796\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −18.1801 −0.925341
\(387\) 1.20359 0.0611817
\(388\) 3.26943 0.165980
\(389\) 1.05398 0.0534388 0.0267194 0.999643i \(-0.491494\pi\)
0.0267194 + 0.999643i \(0.491494\pi\)
\(390\) 8.64681 0.437848
\(391\) −26.9765 −1.36426
\(392\) 0 0
\(393\) −0.946021 −0.0477205
\(394\) 2.55678 0.128809
\(395\) −19.7008 −0.991254
\(396\) −6.04502 −0.303773
\(397\) −6.58097 −0.330289 −0.165145 0.986269i \(-0.552809\pi\)
−0.165145 + 0.986269i \(0.552809\pi\)
\(398\) −5.39821 −0.270588
\(399\) 0 0
\(400\) 8.85039 0.442520
\(401\) −8.58723 −0.428826 −0.214413 0.976743i \(-0.568784\pi\)
−0.214413 + 0.976743i \(0.568784\pi\)
\(402\) −7.89541 −0.393787
\(403\) −22.5872 −1.12515
\(404\) 6.00000 0.298511
\(405\) 3.72161 0.184928
\(406\) 0 0
\(407\) −37.4466 −1.85616
\(408\) −3.72161 −0.184247
\(409\) −26.0513 −1.28815 −0.644076 0.764961i \(-0.722758\pi\)
−0.644076 + 0.764961i \(0.722758\pi\)
\(410\) −18.3297 −0.905238
\(411\) −15.7458 −0.776683
\(412\) −10.6018 −0.522313
\(413\) 0 0
\(414\) −7.24860 −0.356249
\(415\) 65.5637 3.21840
\(416\) −2.32340 −0.113914
\(417\) −19.0152 −0.931180
\(418\) −8.45219 −0.413410
\(419\) 26.4585 1.29258 0.646290 0.763092i \(-0.276320\pi\)
0.646290 + 0.763092i \(0.276320\pi\)
\(420\) 0 0
\(421\) −32.6289 −1.59023 −0.795117 0.606456i \(-0.792591\pi\)
−0.795117 + 0.606456i \(0.792591\pi\)
\(422\) −2.49720 −0.121562
\(423\) 6.17380 0.300180
\(424\) −4.60179 −0.223483
\(425\) −32.9377 −1.59771
\(426\) 5.29362 0.256477
\(427\) 0 0
\(428\) 7.35319 0.355430
\(429\) 14.0450 0.678100
\(430\) −4.47928 −0.216010
\(431\) 36.5422 1.76018 0.880088 0.474810i \(-0.157483\pi\)
0.880088 + 0.474810i \(0.157483\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 2.68556 0.129060 0.0645299 0.997916i \(-0.479445\pi\)
0.0645299 + 0.997916i \(0.479445\pi\)
\(434\) 0 0
\(435\) 3.72161 0.178438
\(436\) −5.05398 −0.242042
\(437\) −10.1350 −0.484825
\(438\) 15.6170 0.746210
\(439\) −19.2486 −0.918686 −0.459343 0.888259i \(-0.651915\pi\)
−0.459343 + 0.888259i \(0.651915\pi\)
\(440\) 22.4972 1.07251
\(441\) 0 0
\(442\) 8.64681 0.411287
\(443\) −1.95498 −0.0928841 −0.0464420 0.998921i \(-0.514788\pi\)
−0.0464420 + 0.998921i \(0.514788\pi\)
\(444\) −6.19462 −0.293984
\(445\) −16.2576 −0.770682
\(446\) 13.7875 0.652855
\(447\) −10.1946 −0.482189
\(448\) 0 0
\(449\) 22.1946 1.04743 0.523714 0.851894i \(-0.324546\pi\)
0.523714 + 0.851894i \(0.324546\pi\)
\(450\) −8.85039 −0.417212
\(451\) −29.7729 −1.40195
\(452\) 8.99104 0.422903
\(453\) −18.4972 −0.869074
\(454\) −4.42799 −0.207816
\(455\) 0 0
\(456\) −1.39821 −0.0654771
\(457\) −12.6018 −0.589487 −0.294743 0.955576i \(-0.595234\pi\)
−0.294743 + 0.955576i \(0.595234\pi\)
\(458\) 17.1440 0.801087
\(459\) 3.72161 0.173710
\(460\) 26.9765 1.25778
\(461\) 16.3442 0.761227 0.380613 0.924734i \(-0.375713\pi\)
0.380613 + 0.924734i \(0.375713\pi\)
\(462\) 0 0
\(463\) 0.299213 0.0139056 0.00695279 0.999976i \(-0.497787\pi\)
0.00695279 + 0.999976i \(0.497787\pi\)
\(464\) −1.00000 −0.0464238
\(465\) 36.1801 1.67781
\(466\) −22.0900 −1.02330
\(467\) 26.0305 1.20455 0.602273 0.798290i \(-0.294262\pi\)
0.602273 + 0.798290i \(0.294262\pi\)
\(468\) 2.32340 0.107399
\(469\) 0 0
\(470\) −22.9765 −1.05983
\(471\) −1.44322 −0.0665002
\(472\) 12.4134 0.571375
\(473\) −7.27569 −0.334537
\(474\) −5.29362 −0.243144
\(475\) −12.3747 −0.567790
\(476\) 0 0
\(477\) 4.60179 0.210702
\(478\) −8.45219 −0.386594
\(479\) −28.2043 −1.28869 −0.644343 0.764737i \(-0.722869\pi\)
−0.644343 + 0.764737i \(0.722869\pi\)
\(480\) 3.72161 0.169868
\(481\) 14.3926 0.656247
\(482\) −22.0000 −1.00207
\(483\) 0 0
\(484\) 25.5422 1.16101
\(485\) 12.1675 0.552499
\(486\) 1.00000 0.0453609
\(487\) 32.1801 1.45822 0.729109 0.684398i \(-0.239935\pi\)
0.729109 + 0.684398i \(0.239935\pi\)
\(488\) 14.0900 0.637826
\(489\) −13.2036 −0.597087
\(490\) 0 0
\(491\) 12.8414 0.579526 0.289763 0.957098i \(-0.406424\pi\)
0.289763 + 0.957098i \(0.406424\pi\)
\(492\) −4.92520 −0.222045
\(493\) 3.72161 0.167613
\(494\) 3.24860 0.146161
\(495\) −22.4972 −1.01117
\(496\) −9.72161 −0.436513
\(497\) 0 0
\(498\) 17.6170 0.789438
\(499\) −6.13505 −0.274642 −0.137321 0.990527i \(-0.543849\pi\)
−0.137321 + 0.990527i \(0.543849\pi\)
\(500\) 14.3297 0.640842
\(501\) 18.5872 0.830416
\(502\) −3.35319 −0.149660
\(503\) 6.82061 0.304116 0.152058 0.988372i \(-0.451410\pi\)
0.152058 + 0.988372i \(0.451410\pi\)
\(504\) 0 0
\(505\) 22.3297 0.993657
\(506\) 43.8179 1.94794
\(507\) 7.60179 0.337607
\(508\) 7.89541 0.350302
\(509\) 3.97918 0.176374 0.0881869 0.996104i \(-0.471893\pi\)
0.0881869 + 0.996104i \(0.471893\pi\)
\(510\) −13.8504 −0.613306
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 1.39821 0.0617324
\(514\) −8.14961 −0.359464
\(515\) −39.4558 −1.73863
\(516\) −1.20359 −0.0529849
\(517\) −37.3207 −1.64136
\(518\) 0 0
\(519\) 4.27839 0.187800
\(520\) −8.64681 −0.379188
\(521\) 41.1711 1.80374 0.901869 0.432009i \(-0.142195\pi\)
0.901869 + 0.432009i \(0.142195\pi\)
\(522\) 1.00000 0.0437688
\(523\) −28.7756 −1.25827 −0.629134 0.777297i \(-0.716590\pi\)
−0.629134 + 0.777297i \(0.716590\pi\)
\(524\) 0.946021 0.0413271
\(525\) 0 0
\(526\) −11.4432 −0.498948
\(527\) 36.1801 1.57603
\(528\) 6.04502 0.263076
\(529\) 29.5422 1.28444
\(530\) −17.1261 −0.743910
\(531\) −12.4134 −0.538697
\(532\) 0 0
\(533\) 11.4432 0.495661
\(534\) −4.36842 −0.189040
\(535\) 27.3657 1.18312
\(536\) 7.89541 0.341030
\(537\) 14.4972 0.625600
\(538\) −7.95498 −0.342964
\(539\) 0 0
\(540\) −3.72161 −0.160153
\(541\) −3.86495 −0.166167 −0.0830836 0.996543i \(-0.526477\pi\)
−0.0830836 + 0.996543i \(0.526477\pi\)
\(542\) −29.4224 −1.26380
\(543\) 2.42799 0.104195
\(544\) 3.72161 0.159563
\(545\) −18.8089 −0.805687
\(546\) 0 0
\(547\) 35.7763 1.52968 0.764841 0.644219i \(-0.222817\pi\)
0.764841 + 0.644219i \(0.222817\pi\)
\(548\) 15.7458 0.672627
\(549\) −14.0900 −0.601348
\(550\) 53.5008 2.28128
\(551\) 1.39821 0.0595656
\(552\) 7.24860 0.308521
\(553\) 0 0
\(554\) 7.20359 0.306051
\(555\) −23.0540 −0.978587
\(556\) 19.0152 0.806425
\(557\) 21.5062 0.911245 0.455623 0.890173i \(-0.349417\pi\)
0.455623 + 0.890173i \(0.349417\pi\)
\(558\) 9.72161 0.411549
\(559\) 2.79641 0.118276
\(560\) 0 0
\(561\) −22.4972 −0.949833
\(562\) 2.64681 0.111649
\(563\) 20.0484 0.844939 0.422469 0.906377i \(-0.361163\pi\)
0.422469 + 0.906377i \(0.361163\pi\)
\(564\) −6.17380 −0.259964
\(565\) 33.4611 1.40772
\(566\) 30.7160 1.29109
\(567\) 0 0
\(568\) −5.29362 −0.222115
\(569\) 5.93706 0.248894 0.124447 0.992226i \(-0.460284\pi\)
0.124447 + 0.992226i \(0.460284\pi\)
\(570\) −5.20359 −0.217954
\(571\) −6.13505 −0.256744 −0.128372 0.991726i \(-0.540975\pi\)
−0.128372 + 0.991726i \(0.540975\pi\)
\(572\) −14.0450 −0.587252
\(573\) −4.94602 −0.206623
\(574\) 0 0
\(575\) 64.1530 2.67536
\(576\) 1.00000 0.0416667
\(577\) 22.1142 0.920627 0.460314 0.887756i \(-0.347737\pi\)
0.460314 + 0.887756i \(0.347737\pi\)
\(578\) 3.14961 0.131006
\(579\) −18.1801 −0.755538
\(580\) −3.72161 −0.154532
\(581\) 0 0
\(582\) 3.26943 0.135522
\(583\) −27.8179 −1.15210
\(584\) −15.6170 −0.646237
\(585\) 8.64681 0.357502
\(586\) 4.60179 0.190098
\(587\) −42.4585 −1.75245 −0.876224 0.481904i \(-0.839945\pi\)
−0.876224 + 0.481904i \(0.839945\pi\)
\(588\) 0 0
\(589\) 13.5928 0.560083
\(590\) 46.1980 1.90194
\(591\) 2.55678 0.105172
\(592\) 6.19462 0.254597
\(593\) −12.1350 −0.498327 −0.249163 0.968461i \(-0.580156\pi\)
−0.249163 + 0.968461i \(0.580156\pi\)
\(594\) −6.04502 −0.248030
\(595\) 0 0
\(596\) 10.1946 0.417588
\(597\) −5.39821 −0.220934
\(598\) −16.8414 −0.688697
\(599\) 7.74244 0.316347 0.158174 0.987411i \(-0.449439\pi\)
0.158174 + 0.987411i \(0.449439\pi\)
\(600\) 8.85039 0.361316
\(601\) 48.1863 1.96556 0.982781 0.184776i \(-0.0591559\pi\)
0.982781 + 0.184776i \(0.0591559\pi\)
\(602\) 0 0
\(603\) −7.89541 −0.321526
\(604\) 18.4972 0.752640
\(605\) 95.0582 3.86467
\(606\) 6.00000 0.243733
\(607\) 0.817239 0.0331707 0.0165854 0.999862i \(-0.494720\pi\)
0.0165854 + 0.999862i \(0.494720\pi\)
\(608\) 1.39821 0.0567048
\(609\) 0 0
\(610\) 52.4376 2.12314
\(611\) 14.3442 0.580305
\(612\) −3.72161 −0.150437
\(613\) 14.6468 0.591579 0.295790 0.955253i \(-0.404417\pi\)
0.295790 + 0.955253i \(0.404417\pi\)
\(614\) 15.6378 0.631092
\(615\) −18.3297 −0.739124
\(616\) 0 0
\(617\) 25.8809 1.04192 0.520962 0.853580i \(-0.325573\pi\)
0.520962 + 0.853580i \(0.325573\pi\)
\(618\) −10.6018 −0.426467
\(619\) −34.8719 −1.40162 −0.700810 0.713348i \(-0.747178\pi\)
−0.700810 + 0.713348i \(0.747178\pi\)
\(620\) −36.1801 −1.45303
\(621\) −7.24860 −0.290876
\(622\) −13.9792 −0.560514
\(623\) 0 0
\(624\) −2.32340 −0.0930106
\(625\) 9.07750 0.363100
\(626\) −11.5928 −0.463343
\(627\) −8.45219 −0.337548
\(628\) 1.44322 0.0575909
\(629\) −23.0540 −0.919222
\(630\) 0 0
\(631\) −41.4737 −1.65104 −0.825521 0.564372i \(-0.809118\pi\)
−0.825521 + 0.564372i \(0.809118\pi\)
\(632\) 5.29362 0.210569
\(633\) −2.49720 −0.0992549
\(634\) −19.2936 −0.766247
\(635\) 29.3836 1.16605
\(636\) −4.60179 −0.182473
\(637\) 0 0
\(638\) −6.04502 −0.239324
\(639\) 5.29362 0.209412
\(640\) −3.72161 −0.147110
\(641\) −4.40717 −0.174073 −0.0870364 0.996205i \(-0.527740\pi\)
−0.0870364 + 0.996205i \(0.527740\pi\)
\(642\) 7.35319 0.290207
\(643\) −27.2728 −1.07553 −0.537767 0.843094i \(-0.680732\pi\)
−0.537767 + 0.843094i \(0.680732\pi\)
\(644\) 0 0
\(645\) −4.47928 −0.176371
\(646\) −5.20359 −0.204732
\(647\) 12.3476 0.485434 0.242717 0.970097i \(-0.421961\pi\)
0.242717 + 0.970097i \(0.421961\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 75.0394 2.94555
\(650\) −20.5630 −0.806549
\(651\) 0 0
\(652\) 13.2036 0.517092
\(653\) 14.0900 0.551386 0.275693 0.961246i \(-0.411093\pi\)
0.275693 + 0.961246i \(0.411093\pi\)
\(654\) −5.05398 −0.197626
\(655\) 3.52072 0.137566
\(656\) 4.92520 0.192297
\(657\) 15.6170 0.609278
\(658\) 0 0
\(659\) −41.4737 −1.61559 −0.807793 0.589467i \(-0.799338\pi\)
−0.807793 + 0.589467i \(0.799338\pi\)
\(660\) 22.4972 0.875702
\(661\) 3.82620 0.148822 0.0744110 0.997228i \(-0.476292\pi\)
0.0744110 + 0.997228i \(0.476292\pi\)
\(662\) −18.4972 −0.718914
\(663\) 8.64681 0.335814
\(664\) −17.6170 −0.683673
\(665\) 0 0
\(666\) −6.19462 −0.240037
\(667\) −7.24860 −0.280667
\(668\) −18.5872 −0.719162
\(669\) 13.7875 0.533054
\(670\) 29.3836 1.13519
\(671\) 85.1745 3.28812
\(672\) 0 0
\(673\) 10.4938 0.404508 0.202254 0.979333i \(-0.435173\pi\)
0.202254 + 0.979333i \(0.435173\pi\)
\(674\) 25.8325 0.995029
\(675\) −8.85039 −0.340652
\(676\) −7.60179 −0.292377
\(677\) −31.1890 −1.19869 −0.599346 0.800490i \(-0.704573\pi\)
−0.599346 + 0.800490i \(0.704573\pi\)
\(678\) 8.99104 0.345299
\(679\) 0 0
\(680\) 13.8504 0.531138
\(681\) −4.42799 −0.169681
\(682\) −58.7673 −2.25032
\(683\) 40.5693 1.55234 0.776171 0.630523i \(-0.217159\pi\)
0.776171 + 0.630523i \(0.217159\pi\)
\(684\) −1.39821 −0.0534618
\(685\) 58.5998 2.23898
\(686\) 0 0
\(687\) 17.1440 0.654085
\(688\) 1.20359 0.0458863
\(689\) 10.6918 0.407326
\(690\) 26.9765 1.02698
\(691\) −24.8656 −0.945933 −0.472966 0.881080i \(-0.656817\pi\)
−0.472966 + 0.881080i \(0.656817\pi\)
\(692\) −4.27839 −0.162640
\(693\) 0 0
\(694\) −17.6829 −0.671232
\(695\) 70.7673 2.68436
\(696\) −1.00000 −0.0379049
\(697\) −18.3297 −0.694286
\(698\) −21.3144 −0.806764
\(699\) −22.0900 −0.835522
\(700\) 0 0
\(701\) 28.0721 1.06027 0.530134 0.847914i \(-0.322141\pi\)
0.530134 + 0.847914i \(0.322141\pi\)
\(702\) 2.32340 0.0876912
\(703\) −8.66137 −0.326670
\(704\) −6.04502 −0.227830
\(705\) −22.9765 −0.865344
\(706\) 18.1801 0.684216
\(707\) 0 0
\(708\) 12.4134 0.466526
\(709\) −6.73684 −0.253007 −0.126504 0.991966i \(-0.540376\pi\)
−0.126504 + 0.991966i \(0.540376\pi\)
\(710\) −19.7008 −0.739357
\(711\) −5.29362 −0.198526
\(712\) 4.36842 0.163713
\(713\) −70.4681 −2.63905
\(714\) 0 0
\(715\) −52.2701 −1.95479
\(716\) −14.4972 −0.541786
\(717\) −8.45219 −0.315653
\(718\) 13.5928 0.507280
\(719\) −50.8448 −1.89619 −0.948096 0.317986i \(-0.896994\pi\)
−0.948096 + 0.317986i \(0.896994\pi\)
\(720\) 3.72161 0.138696
\(721\) 0 0
\(722\) 17.0450 0.634350
\(723\) −22.0000 −0.818189
\(724\) −2.42799 −0.0902357
\(725\) −8.85039 −0.328695
\(726\) 25.5422 0.947961
\(727\) 34.2909 1.27178 0.635890 0.771780i \(-0.280633\pi\)
0.635890 + 0.771780i \(0.280633\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −58.1205 −2.15114
\(731\) −4.47928 −0.165672
\(732\) 14.0900 0.520783
\(733\) 45.2757 1.67230 0.836148 0.548504i \(-0.184802\pi\)
0.836148 + 0.548504i \(0.184802\pi\)
\(734\) −4.12878 −0.152396
\(735\) 0 0
\(736\) −7.24860 −0.267187
\(737\) 47.7279 1.75808
\(738\) −4.92520 −0.181299
\(739\) 39.8809 1.46704 0.733520 0.679667i \(-0.237876\pi\)
0.733520 + 0.679667i \(0.237876\pi\)
\(740\) 23.0540 0.847481
\(741\) 3.24860 0.119340
\(742\) 0 0
\(743\) 38.6289 1.41716 0.708578 0.705632i \(-0.249337\pi\)
0.708578 + 0.705632i \(0.249337\pi\)
\(744\) −9.72161 −0.356412
\(745\) 37.9404 1.39003
\(746\) 37.8809 1.38692
\(747\) 17.6170 0.644573
\(748\) 22.4972 0.822579
\(749\) 0 0
\(750\) 14.3297 0.523246
\(751\) 8.75140 0.319343 0.159672 0.987170i \(-0.448956\pi\)
0.159672 + 0.987170i \(0.448956\pi\)
\(752\) 6.17380 0.225135
\(753\) −3.35319 −0.122197
\(754\) 2.32340 0.0846134
\(755\) 68.8394 2.50532
\(756\) 0 0
\(757\) −2.25756 −0.0820526 −0.0410263 0.999158i \(-0.513063\pi\)
−0.0410263 + 0.999158i \(0.513063\pi\)
\(758\) −26.8864 −0.976560
\(759\) 43.8179 1.59049
\(760\) 5.20359 0.188754
\(761\) 0.733473 0.0265884 0.0132942 0.999912i \(-0.495768\pi\)
0.0132942 + 0.999912i \(0.495768\pi\)
\(762\) 7.89541 0.286021
\(763\) 0 0
\(764\) 4.94602 0.178941
\(765\) −13.8504 −0.500762
\(766\) 22.8864 0.826921
\(767\) −28.8414 −1.04140
\(768\) −1.00000 −0.0360844
\(769\) 6.72721 0.242589 0.121295 0.992617i \(-0.461295\pi\)
0.121295 + 0.992617i \(0.461295\pi\)
\(770\) 0 0
\(771\) −8.14961 −0.293501
\(772\) 18.1801 0.654315
\(773\) −14.7368 −0.530047 −0.265024 0.964242i \(-0.585380\pi\)
−0.265024 + 0.964242i \(0.585380\pi\)
\(774\) −1.20359 −0.0432620
\(775\) −86.0401 −3.09065
\(776\) −3.26943 −0.117366
\(777\) 0 0
\(778\) −1.05398 −0.0377870
\(779\) −6.88645 −0.246733
\(780\) −8.64681 −0.309605
\(781\) −32.0000 −1.14505
\(782\) 26.9765 0.964677
\(783\) 1.00000 0.0357371
\(784\) 0 0
\(785\) 5.37112 0.191703
\(786\) 0.946021 0.0337435
\(787\) 14.0692 0.501513 0.250757 0.968050i \(-0.419321\pi\)
0.250757 + 0.968050i \(0.419321\pi\)
\(788\) −2.55678 −0.0910814
\(789\) −11.4432 −0.407390
\(790\) 19.7008 0.700923
\(791\) 0 0
\(792\) 6.04502 0.214800
\(793\) −32.7368 −1.16252
\(794\) 6.58097 0.233550
\(795\) −17.1261 −0.607400
\(796\) 5.39821 0.191334
\(797\) 34.7368 1.23044 0.615221 0.788355i \(-0.289067\pi\)
0.615221 + 0.788355i \(0.289067\pi\)
\(798\) 0 0
\(799\) −22.9765 −0.812849
\(800\) −8.85039 −0.312909
\(801\) −4.36842 −0.154351
\(802\) 8.58723 0.303226
\(803\) −94.4051 −3.33149
\(804\) 7.89541 0.278450
\(805\) 0 0
\(806\) 22.5872 0.795601
\(807\) −7.95498 −0.280029
\(808\) −6.00000 −0.211079
\(809\) 16.1496 0.567790 0.283895 0.958855i \(-0.408373\pi\)
0.283895 + 0.958855i \(0.408373\pi\)
\(810\) −3.72161 −0.130764
\(811\) 35.6620 1.25226 0.626132 0.779717i \(-0.284637\pi\)
0.626132 + 0.779717i \(0.284637\pi\)
\(812\) 0 0
\(813\) −29.4224 −1.03189
\(814\) 37.4466 1.31250
\(815\) 49.1386 1.72125
\(816\) 3.72161 0.130282
\(817\) −1.68286 −0.0588759
\(818\) 26.0513 0.910862
\(819\) 0 0
\(820\) 18.3297 0.640100
\(821\) −37.5962 −1.31212 −0.656058 0.754710i \(-0.727777\pi\)
−0.656058 + 0.754710i \(0.727777\pi\)
\(822\) 15.7458 0.549198
\(823\) −48.2555 −1.68208 −0.841041 0.540971i \(-0.818057\pi\)
−0.841041 + 0.540971i \(0.818057\pi\)
\(824\) 10.6018 0.369331
\(825\) 53.5008 1.86266
\(826\) 0 0
\(827\) −33.7458 −1.17346 −0.586728 0.809784i \(-0.699584\pi\)
−0.586728 + 0.809784i \(0.699584\pi\)
\(828\) 7.24860 0.251906
\(829\) −16.1080 −0.559452 −0.279726 0.960080i \(-0.590244\pi\)
−0.279726 + 0.960080i \(0.590244\pi\)
\(830\) −65.5637 −2.27575
\(831\) 7.20359 0.249890
\(832\) 2.32340 0.0805496
\(833\) 0 0
\(834\) 19.0152 0.658444
\(835\) −69.1745 −2.39388
\(836\) 8.45219 0.292325
\(837\) 9.72161 0.336028
\(838\) −26.4585 −0.913992
\(839\) −16.9073 −0.583704 −0.291852 0.956464i \(-0.594271\pi\)
−0.291852 + 0.956464i \(0.594271\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 32.6289 1.12447
\(843\) 2.64681 0.0911609
\(844\) 2.49720 0.0859572
\(845\) −28.2909 −0.973237
\(846\) −6.17380 −0.212260
\(847\) 0 0
\(848\) 4.60179 0.158026
\(849\) 30.7160 1.05417
\(850\) 32.9377 1.12975
\(851\) 44.9023 1.53923
\(852\) −5.29362 −0.181356
\(853\) −27.6829 −0.947843 −0.473922 0.880567i \(-0.657162\pi\)
−0.473922 + 0.880567i \(0.657162\pi\)
\(854\) 0 0
\(855\) −5.20359 −0.177959
\(856\) −7.35319 −0.251327
\(857\) −3.74580 −0.127954 −0.0639771 0.997951i \(-0.520378\pi\)
−0.0639771 + 0.997951i \(0.520378\pi\)
\(858\) −14.0450 −0.479489
\(859\) 37.5366 1.28073 0.640367 0.768069i \(-0.278782\pi\)
0.640367 + 0.768069i \(0.278782\pi\)
\(860\) 4.47928 0.152742
\(861\) 0 0
\(862\) −36.5422 −1.24463
\(863\) 23.2486 0.791392 0.395696 0.918382i \(-0.370503\pi\)
0.395696 + 0.918382i \(0.370503\pi\)
\(864\) 1.00000 0.0340207
\(865\) −15.9225 −0.541381
\(866\) −2.68556 −0.0912590
\(867\) 3.14961 0.106966
\(868\) 0 0
\(869\) 32.0000 1.08553
\(870\) −3.72161 −0.126174
\(871\) −18.3442 −0.621570
\(872\) 5.05398 0.171149
\(873\) 3.26943 0.110653
\(874\) 10.1350 0.342823
\(875\) 0 0
\(876\) −15.6170 −0.527650
\(877\) 48.1205 1.62491 0.812457 0.583021i \(-0.198130\pi\)
0.812457 + 0.583021i \(0.198130\pi\)
\(878\) 19.2486 0.649609
\(879\) 4.60179 0.155215
\(880\) −22.4972 −0.758381
\(881\) 26.2605 0.884737 0.442369 0.896833i \(-0.354138\pi\)
0.442369 + 0.896833i \(0.354138\pi\)
\(882\) 0 0
\(883\) 28.1655 0.947845 0.473922 0.880567i \(-0.342838\pi\)
0.473922 + 0.880567i \(0.342838\pi\)
\(884\) −8.64681 −0.290824
\(885\) 46.1980 1.55293
\(886\) 1.95498 0.0656790
\(887\) −0.921830 −0.0309520 −0.0154760 0.999880i \(-0.504926\pi\)
−0.0154760 + 0.999880i \(0.504926\pi\)
\(888\) 6.19462 0.207878
\(889\) 0 0
\(890\) 16.2576 0.544955
\(891\) −6.04502 −0.202516
\(892\) −13.7875 −0.461638
\(893\) −8.63225 −0.288867
\(894\) 10.1946 0.340959
\(895\) −53.9530 −1.80345
\(896\) 0 0
\(897\) −16.8414 −0.562319
\(898\) −22.1946 −0.740644
\(899\) 9.72161 0.324234
\(900\) 8.85039 0.295013
\(901\) −17.1261 −0.570553
\(902\) 29.7729 0.991329
\(903\) 0 0
\(904\) −8.99104 −0.299037
\(905\) −9.03605 −0.300369
\(906\) 18.4972 0.614528
\(907\) −45.3836 −1.50694 −0.753470 0.657483i \(-0.771621\pi\)
−0.753470 + 0.657483i \(0.771621\pi\)
\(908\) 4.42799 0.146948
\(909\) 6.00000 0.199007
\(910\) 0 0
\(911\) 11.7008 0.387664 0.193832 0.981035i \(-0.437908\pi\)
0.193832 + 0.981035i \(0.437908\pi\)
\(912\) 1.39821 0.0462993
\(913\) −106.495 −3.52448
\(914\) 12.6018 0.416830
\(915\) 52.4376 1.73353
\(916\) −17.1440 −0.566454
\(917\) 0 0
\(918\) −3.72161 −0.122831
\(919\) −8.09003 −0.266866 −0.133433 0.991058i \(-0.542600\pi\)
−0.133433 + 0.991058i \(0.542600\pi\)
\(920\) −26.9765 −0.889388
\(921\) 15.6378 0.515285
\(922\) −16.3442 −0.538269
\(923\) 12.2992 0.404834
\(924\) 0 0
\(925\) 54.8248 1.80263
\(926\) −0.299213 −0.00983274
\(927\) −10.6018 −0.348209
\(928\) 1.00000 0.0328266
\(929\) −23.7729 −0.779963 −0.389982 0.920823i \(-0.627519\pi\)
−0.389982 + 0.920823i \(0.627519\pi\)
\(930\) −36.1801 −1.18639
\(931\) 0 0
\(932\) 22.0900 0.723583
\(933\) −13.9792 −0.457658
\(934\) −26.0305 −0.851743
\(935\) 83.7259 2.73813
\(936\) −2.32340 −0.0759428
\(937\) 0.407170 0.0133017 0.00665084 0.999978i \(-0.497883\pi\)
0.00665084 + 0.999978i \(0.497883\pi\)
\(938\) 0 0
\(939\) −11.5928 −0.378318
\(940\) 22.9765 0.749410
\(941\) 17.3144 0.564435 0.282217 0.959350i \(-0.408930\pi\)
0.282217 + 0.959350i \(0.408930\pi\)
\(942\) 1.44322 0.0470228
\(943\) 35.7008 1.16258
\(944\) −12.4134 −0.404023
\(945\) 0 0
\(946\) 7.27569 0.236553
\(947\) −60.1801 −1.95559 −0.977795 0.209565i \(-0.932795\pi\)
−0.977795 + 0.209565i \(0.932795\pi\)
\(948\) 5.29362 0.171929
\(949\) 36.2847 1.17785
\(950\) 12.3747 0.401488
\(951\) −19.2936 −0.625638
\(952\) 0 0
\(953\) 40.3297 1.30641 0.653203 0.757183i \(-0.273425\pi\)
0.653203 + 0.757183i \(0.273425\pi\)
\(954\) −4.60179 −0.148989
\(955\) 18.4072 0.595642
\(956\) 8.45219 0.273363
\(957\) −6.04502 −0.195408
\(958\) 28.2043 0.911238
\(959\) 0 0
\(960\) −3.72161 −0.120115
\(961\) 63.5097 2.04870
\(962\) −14.3926 −0.464036
\(963\) 7.35319 0.236953
\(964\) 22.0000 0.708572
\(965\) 67.6591 2.17802
\(966\) 0 0
\(967\) 8.15297 0.262182 0.131091 0.991370i \(-0.458152\pi\)
0.131091 + 0.991370i \(0.458152\pi\)
\(968\) −25.5422 −0.820958
\(969\) −5.20359 −0.167163
\(970\) −12.1675 −0.390676
\(971\) −9.85039 −0.316114 −0.158057 0.987430i \(-0.550523\pi\)
−0.158057 + 0.987430i \(0.550523\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) −32.1801 −1.03112
\(975\) −20.5630 −0.658544
\(976\) −14.0900 −0.451011
\(977\) −35.7312 −1.14314 −0.571572 0.820552i \(-0.693666\pi\)
−0.571572 + 0.820552i \(0.693666\pi\)
\(978\) 13.2036 0.422204
\(979\) 26.4072 0.843977
\(980\) 0 0
\(981\) −5.05398 −0.161361
\(982\) −12.8414 −0.409787
\(983\) 22.7577 0.725857 0.362928 0.931817i \(-0.381777\pi\)
0.362928 + 0.931817i \(0.381777\pi\)
\(984\) 4.92520 0.157010
\(985\) −9.51533 −0.303184
\(986\) −3.72161 −0.118520
\(987\) 0 0
\(988\) −3.24860 −0.103352
\(989\) 8.72431 0.277417
\(990\) 22.4972 0.715008
\(991\) 22.4972 0.714647 0.357324 0.933981i \(-0.383689\pi\)
0.357324 + 0.933981i \(0.383689\pi\)
\(992\) 9.72161 0.308661
\(993\) −18.4972 −0.586991
\(994\) 0 0
\(995\) 20.0900 0.636897
\(996\) −17.6170 −0.558217
\(997\) −3.80875 −0.120624 −0.0603121 0.998180i \(-0.519210\pi\)
−0.0603121 + 0.998180i \(0.519210\pi\)
\(998\) 6.13505 0.194202
\(999\) −6.19462 −0.195989
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 8526.2.a.bo.1.3 3
7.6 odd 2 1218.2.a.p.1.1 3
21.20 even 2 3654.2.a.be.1.3 3
28.27 even 2 9744.2.a.bh.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1218.2.a.p.1.1 3 7.6 odd 2
3654.2.a.be.1.3 3 21.20 even 2
8526.2.a.bo.1.3 3 1.1 even 1 trivial
9744.2.a.bh.1.1 3 28.27 even 2