Properties

Label 2166.2.a.s.1.3
Level $2166$
Weight $2$
Character 2166.1
Self dual yes
Analytic conductor $17.296$
Analytic rank $1$
Dimension $3$
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] = [2166,2,Mod(1,2166)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2166, 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("2166.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2166 = 2 \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2166.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: \(17.2955970778\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 114)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.53209\) of defining polynomial
Character \(\chi\) \(=\) 2166.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.41147 q^{5} -1.00000 q^{6} -4.87939 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +3.41147 q^{5} -1.00000 q^{6} -4.87939 q^{7} +1.00000 q^{8} +1.00000 q^{9} +3.41147 q^{10} -3.41147 q^{11} -1.00000 q^{12} -2.71688 q^{13} -4.87939 q^{14} -3.41147 q^{15} +1.00000 q^{16} +1.18479 q^{17} +1.00000 q^{18} +3.41147 q^{20} +4.87939 q^{21} -3.41147 q^{22} -3.41147 q^{23} -1.00000 q^{24} +6.63816 q^{25} -2.71688 q^{26} -1.00000 q^{27} -4.87939 q^{28} -3.77332 q^{29} -3.41147 q^{30} -6.61587 q^{31} +1.00000 q^{32} +3.41147 q^{33} +1.18479 q^{34} -16.6459 q^{35} +1.00000 q^{36} -6.75877 q^{37} +2.71688 q^{39} +3.41147 q^{40} -6.55438 q^{41} +4.87939 q^{42} +4.17024 q^{43} -3.41147 q^{44} +3.41147 q^{45} -3.41147 q^{46} +2.85710 q^{47} -1.00000 q^{48} +16.8084 q^{49} +6.63816 q^{50} -1.18479 q^{51} -2.71688 q^{52} -0.630415 q^{53} -1.00000 q^{54} -11.6382 q^{55} -4.87939 q^{56} -3.77332 q^{58} -11.6382 q^{59} -3.41147 q^{60} +6.90167 q^{61} -6.61587 q^{62} -4.87939 q^{63} +1.00000 q^{64} -9.26857 q^{65} +3.41147 q^{66} -0.709141 q^{67} +1.18479 q^{68} +3.41147 q^{69} -16.6459 q^{70} -7.95811 q^{71} +1.00000 q^{72} -12.7442 q^{73} -6.75877 q^{74} -6.63816 q^{75} +16.6459 q^{77} +2.71688 q^{78} -0.297667 q^{79} +3.41147 q^{80} +1.00000 q^{81} -6.55438 q^{82} +15.6040 q^{83} +4.87939 q^{84} +4.04189 q^{85} +4.17024 q^{86} +3.77332 q^{87} -3.41147 q^{88} -6.77332 q^{89} +3.41147 q^{90} +13.2567 q^{91} -3.41147 q^{92} +6.61587 q^{93} +2.85710 q^{94} -1.00000 q^{96} +7.12567 q^{97} +16.8084 q^{98} -3.41147 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 9 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{2} - 3 q^{3} + 3 q^{4} - 3 q^{6} - 9 q^{7} + 3 q^{8} + 3 q^{9} - 3 q^{12} - 9 q^{14} + 3 q^{16} + 3 q^{18} + 9 q^{21} - 3 q^{24} + 3 q^{25} - 3 q^{27} - 9 q^{28} - 18 q^{29} - 9 q^{31} + 3 q^{32} - 9 q^{35} + 3 q^{36} - 9 q^{37} - 9 q^{41} + 9 q^{42} - 9 q^{43} + 9 q^{47} - 3 q^{48} + 12 q^{49} + 3 q^{50} - 9 q^{53} - 3 q^{54} - 18 q^{55} - 9 q^{56} - 18 q^{58} - 18 q^{59} + 9 q^{61} - 9 q^{62} - 9 q^{63} + 3 q^{64} - 18 q^{65} - 18 q^{67} - 9 q^{70} - 27 q^{71} + 3 q^{72} - 9 q^{73} - 9 q^{74} - 3 q^{75} + 9 q^{77} - 27 q^{79} + 3 q^{81} - 9 q^{82} + 9 q^{83} + 9 q^{84} + 9 q^{85} - 9 q^{86} + 18 q^{87} - 27 q^{89} + 3 q^{91} + 9 q^{93} + 9 q^{94} - 3 q^{96} + 12 q^{97} + 12 q^{98}+O(q^{100}) \) 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
\(5\) 3.41147 1.52566 0.762829 0.646601i \(-0.223810\pi\)
0.762829 + 0.646601i \(0.223810\pi\)
\(6\) −1.00000 −0.408248
\(7\) −4.87939 −1.84423 −0.922117 0.386911i \(-0.873542\pi\)
−0.922117 + 0.386911i \(0.873542\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 3.41147 1.07880
\(11\) −3.41147 −1.02860 −0.514299 0.857611i \(-0.671948\pi\)
−0.514299 + 0.857611i \(0.671948\pi\)
\(12\) −1.00000 −0.288675
\(13\) −2.71688 −0.753527 −0.376764 0.926309i \(-0.622963\pi\)
−0.376764 + 0.926309i \(0.622963\pi\)
\(14\) −4.87939 −1.30407
\(15\) −3.41147 −0.880839
\(16\) 1.00000 0.250000
\(17\) 1.18479 0.287354 0.143677 0.989625i \(-0.454107\pi\)
0.143677 + 0.989625i \(0.454107\pi\)
\(18\) 1.00000 0.235702
\(19\) 0 0
\(20\) 3.41147 0.762829
\(21\) 4.87939 1.06477
\(22\) −3.41147 −0.727329
\(23\) −3.41147 −0.711342 −0.355671 0.934611i \(-0.615748\pi\)
−0.355671 + 0.934611i \(0.615748\pi\)
\(24\) −1.00000 −0.204124
\(25\) 6.63816 1.32763
\(26\) −2.71688 −0.532824
\(27\) −1.00000 −0.192450
\(28\) −4.87939 −0.922117
\(29\) −3.77332 −0.700688 −0.350344 0.936621i \(-0.613935\pi\)
−0.350344 + 0.936621i \(0.613935\pi\)
\(30\) −3.41147 −0.622847
\(31\) −6.61587 −1.18824 −0.594122 0.804375i \(-0.702501\pi\)
−0.594122 + 0.804375i \(0.702501\pi\)
\(32\) 1.00000 0.176777
\(33\) 3.41147 0.593861
\(34\) 1.18479 0.203190
\(35\) −16.6459 −2.81367
\(36\) 1.00000 0.166667
\(37\) −6.75877 −1.11114 −0.555568 0.831471i \(-0.687499\pi\)
−0.555568 + 0.831471i \(0.687499\pi\)
\(38\) 0 0
\(39\) 2.71688 0.435049
\(40\) 3.41147 0.539401
\(41\) −6.55438 −1.02362 −0.511811 0.859098i \(-0.671025\pi\)
−0.511811 + 0.859098i \(0.671025\pi\)
\(42\) 4.87939 0.752905
\(43\) 4.17024 0.635956 0.317978 0.948098i \(-0.396996\pi\)
0.317978 + 0.948098i \(0.396996\pi\)
\(44\) −3.41147 −0.514299
\(45\) 3.41147 0.508553
\(46\) −3.41147 −0.502994
\(47\) 2.85710 0.416750 0.208375 0.978049i \(-0.433182\pi\)
0.208375 + 0.978049i \(0.433182\pi\)
\(48\) −1.00000 −0.144338
\(49\) 16.8084 2.40120
\(50\) 6.63816 0.938777
\(51\) −1.18479 −0.165904
\(52\) −2.71688 −0.376764
\(53\) −0.630415 −0.0865942 −0.0432971 0.999062i \(-0.513786\pi\)
−0.0432971 + 0.999062i \(0.513786\pi\)
\(54\) −1.00000 −0.136083
\(55\) −11.6382 −1.56929
\(56\) −4.87939 −0.652035
\(57\) 0 0
\(58\) −3.77332 −0.495461
\(59\) −11.6382 −1.51516 −0.757579 0.652743i \(-0.773618\pi\)
−0.757579 + 0.652743i \(0.773618\pi\)
\(60\) −3.41147 −0.440419
\(61\) 6.90167 0.883669 0.441834 0.897097i \(-0.354328\pi\)
0.441834 + 0.897097i \(0.354328\pi\)
\(62\) −6.61587 −0.840216
\(63\) −4.87939 −0.614745
\(64\) 1.00000 0.125000
\(65\) −9.26857 −1.14962
\(66\) 3.41147 0.419923
\(67\) −0.709141 −0.0866353 −0.0433177 0.999061i \(-0.513793\pi\)
−0.0433177 + 0.999061i \(0.513793\pi\)
\(68\) 1.18479 0.143677
\(69\) 3.41147 0.410693
\(70\) −16.6459 −1.98957
\(71\) −7.95811 −0.944454 −0.472227 0.881477i \(-0.656550\pi\)
−0.472227 + 0.881477i \(0.656550\pi\)
\(72\) 1.00000 0.117851
\(73\) −12.7442 −1.49160 −0.745799 0.666171i \(-0.767932\pi\)
−0.745799 + 0.666171i \(0.767932\pi\)
\(74\) −6.75877 −0.785691
\(75\) −6.63816 −0.766508
\(76\) 0 0
\(77\) 16.6459 1.89698
\(78\) 2.71688 0.307626
\(79\) −0.297667 −0.0334901 −0.0167450 0.999860i \(-0.505330\pi\)
−0.0167450 + 0.999860i \(0.505330\pi\)
\(80\) 3.41147 0.381414
\(81\) 1.00000 0.111111
\(82\) −6.55438 −0.723810
\(83\) 15.6040 1.71276 0.856381 0.516344i \(-0.172707\pi\)
0.856381 + 0.516344i \(0.172707\pi\)
\(84\) 4.87939 0.532385
\(85\) 4.04189 0.438404
\(86\) 4.17024 0.449689
\(87\) 3.77332 0.404542
\(88\) −3.41147 −0.363664
\(89\) −6.77332 −0.717970 −0.358985 0.933343i \(-0.616877\pi\)
−0.358985 + 0.933343i \(0.616877\pi\)
\(90\) 3.41147 0.359601
\(91\) 13.2567 1.38968
\(92\) −3.41147 −0.355671
\(93\) 6.61587 0.686033
\(94\) 2.85710 0.294687
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 7.12567 0.723502 0.361751 0.932275i \(-0.382179\pi\)
0.361751 + 0.932275i \(0.382179\pi\)
\(98\) 16.8084 1.69790
\(99\) −3.41147 −0.342866
\(100\) 6.63816 0.663816
\(101\) −4.64590 −0.462284 −0.231142 0.972920i \(-0.574246\pi\)
−0.231142 + 0.972920i \(0.574246\pi\)
\(102\) −1.18479 −0.117312
\(103\) 12.3824 1.22007 0.610036 0.792374i \(-0.291155\pi\)
0.610036 + 0.792374i \(0.291155\pi\)
\(104\) −2.71688 −0.266412
\(105\) 16.6459 1.62447
\(106\) −0.630415 −0.0612313
\(107\) −3.89899 −0.376929 −0.188465 0.982080i \(-0.560351\pi\)
−0.188465 + 0.982080i \(0.560351\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 11.5963 1.11072 0.555360 0.831610i \(-0.312580\pi\)
0.555360 + 0.831610i \(0.312580\pi\)
\(110\) −11.6382 −1.10965
\(111\) 6.75877 0.641514
\(112\) −4.87939 −0.461059
\(113\) −1.94087 −0.182582 −0.0912911 0.995824i \(-0.529099\pi\)
−0.0912911 + 0.995824i \(0.529099\pi\)
\(114\) 0 0
\(115\) −11.6382 −1.08526
\(116\) −3.77332 −0.350344
\(117\) −2.71688 −0.251176
\(118\) −11.6382 −1.07138
\(119\) −5.78106 −0.529949
\(120\) −3.41147 −0.311424
\(121\) 0.638156 0.0580142
\(122\) 6.90167 0.624848
\(123\) 6.55438 0.590988
\(124\) −6.61587 −0.594122
\(125\) 5.58853 0.499853
\(126\) −4.87939 −0.434690
\(127\) 6.08378 0.539848 0.269924 0.962882i \(-0.413001\pi\)
0.269924 + 0.962882i \(0.413001\pi\)
\(128\) 1.00000 0.0883883
\(129\) −4.17024 −0.367170
\(130\) −9.26857 −0.812907
\(131\) −18.6040 −1.62544 −0.812720 0.582655i \(-0.802014\pi\)
−0.812720 + 0.582655i \(0.802014\pi\)
\(132\) 3.41147 0.296931
\(133\) 0 0
\(134\) −0.709141 −0.0612604
\(135\) −3.41147 −0.293613
\(136\) 1.18479 0.101595
\(137\) 4.54664 0.388445 0.194223 0.980957i \(-0.437782\pi\)
0.194223 + 0.980957i \(0.437782\pi\)
\(138\) 3.41147 0.290404
\(139\) −3.01455 −0.255691 −0.127845 0.991794i \(-0.540806\pi\)
−0.127845 + 0.991794i \(0.540806\pi\)
\(140\) −16.6459 −1.40684
\(141\) −2.85710 −0.240611
\(142\) −7.95811 −0.667830
\(143\) 9.26857 0.775077
\(144\) 1.00000 0.0833333
\(145\) −12.8726 −1.06901
\(146\) −12.7442 −1.05472
\(147\) −16.8084 −1.38633
\(148\) −6.75877 −0.555568
\(149\) 1.18479 0.0970620 0.0485310 0.998822i \(-0.484546\pi\)
0.0485310 + 0.998822i \(0.484546\pi\)
\(150\) −6.63816 −0.542003
\(151\) 3.63816 0.296069 0.148034 0.988982i \(-0.452705\pi\)
0.148034 + 0.988982i \(0.452705\pi\)
\(152\) 0 0
\(153\) 1.18479 0.0957848
\(154\) 16.6459 1.34136
\(155\) −22.5699 −1.81285
\(156\) 2.71688 0.217525
\(157\) 6.27126 0.500501 0.250250 0.968181i \(-0.419487\pi\)
0.250250 + 0.968181i \(0.419487\pi\)
\(158\) −0.297667 −0.0236811
\(159\) 0.630415 0.0499952
\(160\) 3.41147 0.269701
\(161\) 16.6459 1.31188
\(162\) 1.00000 0.0785674
\(163\) −19.2986 −1.51158 −0.755792 0.654812i \(-0.772748\pi\)
−0.755792 + 0.654812i \(0.772748\pi\)
\(164\) −6.55438 −0.511811
\(165\) 11.6382 0.906029
\(166\) 15.6040 1.21111
\(167\) 7.72193 0.597541 0.298771 0.954325i \(-0.403423\pi\)
0.298771 + 0.954325i \(0.403423\pi\)
\(168\) 4.87939 0.376453
\(169\) −5.61856 −0.432197
\(170\) 4.04189 0.309999
\(171\) 0 0
\(172\) 4.17024 0.317978
\(173\) 14.4953 1.10205 0.551027 0.834488i \(-0.314236\pi\)
0.551027 + 0.834488i \(0.314236\pi\)
\(174\) 3.77332 0.286055
\(175\) −32.3901 −2.44846
\(176\) −3.41147 −0.257150
\(177\) 11.6382 0.874777
\(178\) −6.77332 −0.507682
\(179\) 9.82295 0.734202 0.367101 0.930181i \(-0.380350\pi\)
0.367101 + 0.930181i \(0.380350\pi\)
\(180\) 3.41147 0.254276
\(181\) −18.9145 −1.40590 −0.702951 0.711239i \(-0.748135\pi\)
−0.702951 + 0.711239i \(0.748135\pi\)
\(182\) 13.2567 0.982653
\(183\) −6.90167 −0.510186
\(184\) −3.41147 −0.251497
\(185\) −23.0574 −1.69521
\(186\) 6.61587 0.485099
\(187\) −4.04189 −0.295572
\(188\) 2.85710 0.208375
\(189\) 4.87939 0.354923
\(190\) 0 0
\(191\) −10.0915 −0.730197 −0.365098 0.930969i \(-0.618965\pi\)
−0.365098 + 0.930969i \(0.618965\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −2.52435 −0.181707 −0.0908533 0.995864i \(-0.528959\pi\)
−0.0908533 + 0.995864i \(0.528959\pi\)
\(194\) 7.12567 0.511593
\(195\) 9.26857 0.663736
\(196\) 16.8084 1.20060
\(197\) 14.8571 1.05852 0.529262 0.848458i \(-0.322469\pi\)
0.529262 + 0.848458i \(0.322469\pi\)
\(198\) −3.41147 −0.242443
\(199\) 8.06923 0.572013 0.286006 0.958228i \(-0.407672\pi\)
0.286006 + 0.958228i \(0.407672\pi\)
\(200\) 6.63816 0.469388
\(201\) 0.709141 0.0500189
\(202\) −4.64590 −0.326884
\(203\) 18.4115 1.29223
\(204\) −1.18479 −0.0829521
\(205\) −22.3601 −1.56170
\(206\) 12.3824 0.862721
\(207\) −3.41147 −0.237114
\(208\) −2.71688 −0.188382
\(209\) 0 0
\(210\) 16.6459 1.14868
\(211\) −7.94356 −0.546857 −0.273429 0.961892i \(-0.588158\pi\)
−0.273429 + 0.961892i \(0.588158\pi\)
\(212\) −0.630415 −0.0432971
\(213\) 7.95811 0.545281
\(214\) −3.89899 −0.266529
\(215\) 14.2267 0.970252
\(216\) −1.00000 −0.0680414
\(217\) 32.2814 2.19140
\(218\) 11.5963 0.785398
\(219\) 12.7442 0.861175
\(220\) −11.6382 −0.784644
\(221\) −3.21894 −0.216529
\(222\) 6.75877 0.453619
\(223\) −11.9682 −0.801451 −0.400726 0.916198i \(-0.631242\pi\)
−0.400726 + 0.916198i \(0.631242\pi\)
\(224\) −4.87939 −0.326018
\(225\) 6.63816 0.442544
\(226\) −1.94087 −0.129105
\(227\) −26.5945 −1.76514 −0.882570 0.470181i \(-0.844189\pi\)
−0.882570 + 0.470181i \(0.844189\pi\)
\(228\) 0 0
\(229\) −19.9026 −1.31520 −0.657601 0.753367i \(-0.728429\pi\)
−0.657601 + 0.753367i \(0.728429\pi\)
\(230\) −11.6382 −0.767397
\(231\) −16.6459 −1.09522
\(232\) −3.77332 −0.247730
\(233\) −4.31046 −0.282388 −0.141194 0.989982i \(-0.545094\pi\)
−0.141194 + 0.989982i \(0.545094\pi\)
\(234\) −2.71688 −0.177608
\(235\) 9.74691 0.635818
\(236\) −11.6382 −0.757579
\(237\) 0.297667 0.0193355
\(238\) −5.78106 −0.374730
\(239\) 0.285807 0.0184873 0.00924366 0.999957i \(-0.497058\pi\)
0.00924366 + 0.999957i \(0.497058\pi\)
\(240\) −3.41147 −0.220210
\(241\) −9.56624 −0.616216 −0.308108 0.951351i \(-0.599696\pi\)
−0.308108 + 0.951351i \(0.599696\pi\)
\(242\) 0.638156 0.0410222
\(243\) −1.00000 −0.0641500
\(244\) 6.90167 0.441834
\(245\) 57.3414 3.66341
\(246\) 6.55438 0.417892
\(247\) 0 0
\(248\) −6.61587 −0.420108
\(249\) −15.6040 −0.988864
\(250\) 5.58853 0.353449
\(251\) 29.6878 1.87388 0.936938 0.349495i \(-0.113647\pi\)
0.936938 + 0.349495i \(0.113647\pi\)
\(252\) −4.87939 −0.307372
\(253\) 11.6382 0.731685
\(254\) 6.08378 0.381730
\(255\) −4.04189 −0.253113
\(256\) 1.00000 0.0625000
\(257\) −15.7965 −0.985361 −0.492681 0.870210i \(-0.663983\pi\)
−0.492681 + 0.870210i \(0.663983\pi\)
\(258\) −4.17024 −0.259628
\(259\) 32.9786 2.04919
\(260\) −9.26857 −0.574812
\(261\) −3.77332 −0.233563
\(262\) −18.6040 −1.14936
\(263\) 10.8648 0.669955 0.334977 0.942226i \(-0.391271\pi\)
0.334977 + 0.942226i \(0.391271\pi\)
\(264\) 3.41147 0.209962
\(265\) −2.15064 −0.132113
\(266\) 0 0
\(267\) 6.77332 0.414520
\(268\) −0.709141 −0.0433177
\(269\) 17.2935 1.05441 0.527203 0.849739i \(-0.323241\pi\)
0.527203 + 0.849739i \(0.323241\pi\)
\(270\) −3.41147 −0.207616
\(271\) 1.44562 0.0878153 0.0439077 0.999036i \(-0.486019\pi\)
0.0439077 + 0.999036i \(0.486019\pi\)
\(272\) 1.18479 0.0718386
\(273\) −13.2567 −0.802333
\(274\) 4.54664 0.274672
\(275\) −22.6459 −1.36560
\(276\) 3.41147 0.205347
\(277\) 21.1908 1.27323 0.636615 0.771182i \(-0.280334\pi\)
0.636615 + 0.771182i \(0.280334\pi\)
\(278\) −3.01455 −0.180801
\(279\) −6.61587 −0.396082
\(280\) −16.6459 −0.994783
\(281\) 24.3678 1.45366 0.726831 0.686816i \(-0.240992\pi\)
0.726831 + 0.686816i \(0.240992\pi\)
\(282\) −2.85710 −0.170138
\(283\) −4.12567 −0.245245 −0.122623 0.992453i \(-0.539131\pi\)
−0.122623 + 0.992453i \(0.539131\pi\)
\(284\) −7.95811 −0.472227
\(285\) 0 0
\(286\) 9.26857 0.548062
\(287\) 31.9813 1.88780
\(288\) 1.00000 0.0589256
\(289\) −15.5963 −0.917427
\(290\) −12.8726 −0.755904
\(291\) −7.12567 −0.417714
\(292\) −12.7442 −0.745799
\(293\) −2.78106 −0.162471 −0.0812356 0.996695i \(-0.525887\pi\)
−0.0812356 + 0.996695i \(0.525887\pi\)
\(294\) −16.8084 −0.980286
\(295\) −39.7033 −2.31161
\(296\) −6.75877 −0.392846
\(297\) 3.41147 0.197954
\(298\) 1.18479 0.0686332
\(299\) 9.26857 0.536015
\(300\) −6.63816 −0.383254
\(301\) −20.3482 −1.17285
\(302\) 3.63816 0.209352
\(303\) 4.64590 0.266900
\(304\) 0 0
\(305\) 23.5449 1.34818
\(306\) 1.18479 0.0677301
\(307\) 25.0232 1.42815 0.714075 0.700069i \(-0.246847\pi\)
0.714075 + 0.700069i \(0.246847\pi\)
\(308\) 16.6459 0.948488
\(309\) −12.3824 −0.704409
\(310\) −22.5699 −1.28188
\(311\) 23.9736 1.35942 0.679709 0.733482i \(-0.262106\pi\)
0.679709 + 0.733482i \(0.262106\pi\)
\(312\) 2.71688 0.153813
\(313\) −12.6382 −0.714351 −0.357175 0.934037i \(-0.616260\pi\)
−0.357175 + 0.934037i \(0.616260\pi\)
\(314\) 6.27126 0.353908
\(315\) −16.6459 −0.937890
\(316\) −0.297667 −0.0167450
\(317\) 14.1334 0.793811 0.396906 0.917859i \(-0.370084\pi\)
0.396906 + 0.917859i \(0.370084\pi\)
\(318\) 0.630415 0.0353519
\(319\) 12.8726 0.720726
\(320\) 3.41147 0.190707
\(321\) 3.89899 0.217620
\(322\) 16.6459 0.927640
\(323\) 0 0
\(324\) 1.00000 0.0555556
\(325\) −18.0351 −1.00041
\(326\) −19.2986 −1.06885
\(327\) −11.5963 −0.641275
\(328\) −6.55438 −0.361905
\(329\) −13.9409 −0.768585
\(330\) 11.6382 0.640659
\(331\) −29.4979 −1.62135 −0.810677 0.585494i \(-0.800901\pi\)
−0.810677 + 0.585494i \(0.800901\pi\)
\(332\) 15.6040 0.856381
\(333\) −6.75877 −0.370378
\(334\) 7.72193 0.422525
\(335\) −2.41921 −0.132176
\(336\) 4.87939 0.266192
\(337\) 22.2490 1.21198 0.605989 0.795473i \(-0.292777\pi\)
0.605989 + 0.795473i \(0.292777\pi\)
\(338\) −5.61856 −0.305609
\(339\) 1.94087 0.105414
\(340\) 4.04189 0.219202
\(341\) 22.5699 1.22223
\(342\) 0 0
\(343\) −47.8590 −2.58414
\(344\) 4.17024 0.224845
\(345\) 11.6382 0.626577
\(346\) 14.4953 0.779270
\(347\) −14.0250 −0.752900 −0.376450 0.926437i \(-0.622855\pi\)
−0.376450 + 0.926437i \(0.622855\pi\)
\(348\) 3.77332 0.202271
\(349\) −34.1121 −1.82598 −0.912988 0.407986i \(-0.866231\pi\)
−0.912988 + 0.407986i \(0.866231\pi\)
\(350\) −32.3901 −1.73132
\(351\) 2.71688 0.145016
\(352\) −3.41147 −0.181832
\(353\) 30.4439 1.62036 0.810182 0.586179i \(-0.199368\pi\)
0.810182 + 0.586179i \(0.199368\pi\)
\(354\) 11.6382 0.618561
\(355\) −27.1489 −1.44091
\(356\) −6.77332 −0.358985
\(357\) 5.78106 0.305966
\(358\) 9.82295 0.519159
\(359\) 13.6459 0.720203 0.360101 0.932913i \(-0.382742\pi\)
0.360101 + 0.932913i \(0.382742\pi\)
\(360\) 3.41147 0.179800
\(361\) 0 0
\(362\) −18.9145 −0.994122
\(363\) −0.638156 −0.0334945
\(364\) 13.2567 0.694840
\(365\) −43.4766 −2.27567
\(366\) −6.90167 −0.360756
\(367\) 23.7665 1.24060 0.620301 0.784364i \(-0.287010\pi\)
0.620301 + 0.784364i \(0.287010\pi\)
\(368\) −3.41147 −0.177835
\(369\) −6.55438 −0.341207
\(370\) −23.0574 −1.19870
\(371\) 3.07604 0.159700
\(372\) 6.61587 0.343017
\(373\) 23.2344 1.20303 0.601516 0.798860i \(-0.294563\pi\)
0.601516 + 0.798860i \(0.294563\pi\)
\(374\) −4.04189 −0.209001
\(375\) −5.58853 −0.288590
\(376\) 2.85710 0.147344
\(377\) 10.2517 0.527987
\(378\) 4.87939 0.250968
\(379\) 4.50030 0.231165 0.115583 0.993298i \(-0.463127\pi\)
0.115583 + 0.993298i \(0.463127\pi\)
\(380\) 0 0
\(381\) −6.08378 −0.311681
\(382\) −10.0915 −0.516327
\(383\) −4.36009 −0.222790 −0.111395 0.993776i \(-0.535532\pi\)
−0.111395 + 0.993776i \(0.535532\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 56.7870 2.89414
\(386\) −2.52435 −0.128486
\(387\) 4.17024 0.211985
\(388\) 7.12567 0.361751
\(389\) −1.31046 −0.0664429 −0.0332215 0.999448i \(-0.510577\pi\)
−0.0332215 + 0.999448i \(0.510577\pi\)
\(390\) 9.26857 0.469332
\(391\) −4.04189 −0.204407
\(392\) 16.8084 0.848952
\(393\) 18.6040 0.938448
\(394\) 14.8571 0.748490
\(395\) −1.01548 −0.0510944
\(396\) −3.41147 −0.171433
\(397\) −19.0128 −0.954225 −0.477112 0.878842i \(-0.658317\pi\)
−0.477112 + 0.878842i \(0.658317\pi\)
\(398\) 8.06923 0.404474
\(399\) 0 0
\(400\) 6.63816 0.331908
\(401\) −35.8631 −1.79092 −0.895458 0.445145i \(-0.853152\pi\)
−0.895458 + 0.445145i \(0.853152\pi\)
\(402\) 0.709141 0.0353687
\(403\) 17.9745 0.895375
\(404\) −4.64590 −0.231142
\(405\) 3.41147 0.169518
\(406\) 18.4115 0.913746
\(407\) 23.0574 1.14291
\(408\) −1.18479 −0.0586560
\(409\) 29.1088 1.43934 0.719668 0.694319i \(-0.244294\pi\)
0.719668 + 0.694319i \(0.244294\pi\)
\(410\) −22.3601 −1.10429
\(411\) −4.54664 −0.224269
\(412\) 12.3824 0.610036
\(413\) 56.7870 2.79431
\(414\) −3.41147 −0.167665
\(415\) 53.2327 2.61309
\(416\) −2.71688 −0.133206
\(417\) 3.01455 0.147623
\(418\) 0 0
\(419\) 13.1584 0.642829 0.321415 0.946939i \(-0.395842\pi\)
0.321415 + 0.946939i \(0.395842\pi\)
\(420\) 16.6459 0.812237
\(421\) 0.150644 0.00734195 0.00367098 0.999993i \(-0.498831\pi\)
0.00367098 + 0.999993i \(0.498831\pi\)
\(422\) −7.94356 −0.386687
\(423\) 2.85710 0.138917
\(424\) −0.630415 −0.0306157
\(425\) 7.86484 0.381501
\(426\) 7.95811 0.385572
\(427\) −33.6759 −1.62969
\(428\) −3.89899 −0.188465
\(429\) −9.26857 −0.447491
\(430\) 14.2267 0.686072
\(431\) 30.7965 1.48342 0.741709 0.670722i \(-0.234016\pi\)
0.741709 + 0.670722i \(0.234016\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 29.4338 1.41450 0.707248 0.706965i \(-0.249936\pi\)
0.707248 + 0.706965i \(0.249936\pi\)
\(434\) 32.2814 1.54956
\(435\) 12.8726 0.617193
\(436\) 11.5963 0.555360
\(437\) 0 0
\(438\) 12.7442 0.608943
\(439\) −19.8007 −0.945034 −0.472517 0.881322i \(-0.656654\pi\)
−0.472517 + 0.881322i \(0.656654\pi\)
\(440\) −11.6382 −0.554827
\(441\) 16.8084 0.800400
\(442\) −3.21894 −0.153109
\(443\) −18.5868 −0.883084 −0.441542 0.897241i \(-0.645568\pi\)
−0.441542 + 0.897241i \(0.645568\pi\)
\(444\) 6.75877 0.320757
\(445\) −23.1070 −1.09538
\(446\) −11.9682 −0.566711
\(447\) −1.18479 −0.0560388
\(448\) −4.87939 −0.230529
\(449\) −3.37908 −0.159469 −0.0797343 0.996816i \(-0.525407\pi\)
−0.0797343 + 0.996816i \(0.525407\pi\)
\(450\) 6.63816 0.312926
\(451\) 22.3601 1.05290
\(452\) −1.94087 −0.0912911
\(453\) −3.63816 −0.170935
\(454\) −26.5945 −1.24814
\(455\) 45.2249 2.12018
\(456\) 0 0
\(457\) 21.7912 1.01935 0.509674 0.860368i \(-0.329766\pi\)
0.509674 + 0.860368i \(0.329766\pi\)
\(458\) −19.9026 −0.929988
\(459\) −1.18479 −0.0553014
\(460\) −11.6382 −0.542632
\(461\) 11.7638 0.547896 0.273948 0.961745i \(-0.411670\pi\)
0.273948 + 0.961745i \(0.411670\pi\)
\(462\) −16.6459 −0.774437
\(463\) 16.9932 0.789741 0.394870 0.918737i \(-0.370790\pi\)
0.394870 + 0.918737i \(0.370790\pi\)
\(464\) −3.77332 −0.175172
\(465\) 22.5699 1.04665
\(466\) −4.31046 −0.199678
\(467\) −11.7050 −0.541644 −0.270822 0.962629i \(-0.587295\pi\)
−0.270822 + 0.962629i \(0.587295\pi\)
\(468\) −2.71688 −0.125588
\(469\) 3.46017 0.159776
\(470\) 9.74691 0.449591
\(471\) −6.27126 −0.288964
\(472\) −11.6382 −0.535690
\(473\) −14.2267 −0.654144
\(474\) 0.297667 0.0136723
\(475\) 0 0
\(476\) −5.78106 −0.264974
\(477\) −0.630415 −0.0288647
\(478\) 0.285807 0.0130725
\(479\) 29.9796 1.36980 0.684901 0.728636i \(-0.259845\pi\)
0.684901 + 0.728636i \(0.259845\pi\)
\(480\) −3.41147 −0.155712
\(481\) 18.3628 0.837271
\(482\) −9.56624 −0.435730
\(483\) −16.6459 −0.757415
\(484\) 0.638156 0.0290071
\(485\) 24.3090 1.10382
\(486\) −1.00000 −0.0453609
\(487\) −14.5107 −0.657544 −0.328772 0.944409i \(-0.606635\pi\)
−0.328772 + 0.944409i \(0.606635\pi\)
\(488\) 6.90167 0.312424
\(489\) 19.2986 0.872713
\(490\) 57.3414 2.59042
\(491\) −11.6554 −0.526000 −0.263000 0.964796i \(-0.584712\pi\)
−0.263000 + 0.964796i \(0.584712\pi\)
\(492\) 6.55438 0.295494
\(493\) −4.47060 −0.201346
\(494\) 0 0
\(495\) −11.6382 −0.523096
\(496\) −6.61587 −0.297061
\(497\) 38.8307 1.74179
\(498\) −15.6040 −0.699232
\(499\) −1.68685 −0.0755139 −0.0377569 0.999287i \(-0.512021\pi\)
−0.0377569 + 0.999287i \(0.512021\pi\)
\(500\) 5.58853 0.249926
\(501\) −7.72193 −0.344991
\(502\) 29.6878 1.32503
\(503\) 17.2935 0.771081 0.385541 0.922691i \(-0.374015\pi\)
0.385541 + 0.922691i \(0.374015\pi\)
\(504\) −4.87939 −0.217345
\(505\) −15.8494 −0.705287
\(506\) 11.6382 0.517379
\(507\) 5.61856 0.249529
\(508\) 6.08378 0.269924
\(509\) −32.4424 −1.43799 −0.718993 0.695017i \(-0.755397\pi\)
−0.718993 + 0.695017i \(0.755397\pi\)
\(510\) −4.04189 −0.178978
\(511\) 62.1840 2.75086
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −15.7965 −0.696756
\(515\) 42.2422 1.86141
\(516\) −4.17024 −0.183585
\(517\) −9.74691 −0.428669
\(518\) 32.9786 1.44900
\(519\) −14.4953 −0.636271
\(520\) −9.26857 −0.406454
\(521\) −31.1644 −1.36534 −0.682668 0.730729i \(-0.739180\pi\)
−0.682668 + 0.730729i \(0.739180\pi\)
\(522\) −3.77332 −0.165154
\(523\) 16.6604 0.728510 0.364255 0.931299i \(-0.381324\pi\)
0.364255 + 0.931299i \(0.381324\pi\)
\(524\) −18.6040 −0.812720
\(525\) 32.3901 1.41362
\(526\) 10.8648 0.473729
\(527\) −7.83843 −0.341447
\(528\) 3.41147 0.148465
\(529\) −11.3618 −0.493993
\(530\) −2.15064 −0.0934180
\(531\) −11.6382 −0.505053
\(532\) 0 0
\(533\) 17.8075 0.771327
\(534\) 6.77332 0.293110
\(535\) −13.3013 −0.575065
\(536\) −0.709141 −0.0306302
\(537\) −9.82295 −0.423892
\(538\) 17.2935 0.745578
\(539\) −57.3414 −2.46987
\(540\) −3.41147 −0.146806
\(541\) 36.2449 1.55829 0.779144 0.626845i \(-0.215654\pi\)
0.779144 + 0.626845i \(0.215654\pi\)
\(542\) 1.44562 0.0620948
\(543\) 18.9145 0.811697
\(544\) 1.18479 0.0507976
\(545\) 39.5604 1.69458
\(546\) −13.2567 −0.567335
\(547\) 17.1976 0.735316 0.367658 0.929961i \(-0.380160\pi\)
0.367658 + 0.929961i \(0.380160\pi\)
\(548\) 4.54664 0.194223
\(549\) 6.90167 0.294556
\(550\) −22.6459 −0.965624
\(551\) 0 0
\(552\) 3.41147 0.145202
\(553\) 1.45243 0.0617636
\(554\) 21.1908 0.900310
\(555\) 23.0574 0.978731
\(556\) −3.01455 −0.127845
\(557\) −16.5699 −0.702087 −0.351044 0.936359i \(-0.614173\pi\)
−0.351044 + 0.936359i \(0.614173\pi\)
\(558\) −6.61587 −0.280072
\(559\) −11.3301 −0.479210
\(560\) −16.6459 −0.703418
\(561\) 4.04189 0.170649
\(562\) 24.3678 1.02789
\(563\) 30.6973 1.29374 0.646868 0.762602i \(-0.276078\pi\)
0.646868 + 0.762602i \(0.276078\pi\)
\(564\) −2.85710 −0.120305
\(565\) −6.62124 −0.278558
\(566\) −4.12567 −0.173415
\(567\) −4.87939 −0.204915
\(568\) −7.95811 −0.333915
\(569\) −13.4534 −0.563994 −0.281997 0.959415i \(-0.590997\pi\)
−0.281997 + 0.959415i \(0.590997\pi\)
\(570\) 0 0
\(571\) −9.47565 −0.396544 −0.198272 0.980147i \(-0.563533\pi\)
−0.198272 + 0.980147i \(0.563533\pi\)
\(572\) 9.26857 0.387538
\(573\) 10.0915 0.421579
\(574\) 31.9813 1.33488
\(575\) −22.6459 −0.944399
\(576\) 1.00000 0.0416667
\(577\) −18.7638 −0.781148 −0.390574 0.920571i \(-0.627723\pi\)
−0.390574 + 0.920571i \(0.627723\pi\)
\(578\) −15.5963 −0.648719
\(579\) 2.52435 0.104908
\(580\) −12.8726 −0.534505
\(581\) −76.1380 −3.15873
\(582\) −7.12567 −0.295368
\(583\) 2.15064 0.0890706
\(584\) −12.7442 −0.527360
\(585\) −9.26857 −0.383208
\(586\) −2.78106 −0.114884
\(587\) −18.1753 −0.750175 −0.375087 0.926989i \(-0.622387\pi\)
−0.375087 + 0.926989i \(0.622387\pi\)
\(588\) −16.8084 −0.693167
\(589\) 0 0
\(590\) −39.7033 −1.63456
\(591\) −14.8571 −0.611139
\(592\) −6.75877 −0.277784
\(593\) −32.6938 −1.34257 −0.671286 0.741198i \(-0.734258\pi\)
−0.671286 + 0.741198i \(0.734258\pi\)
\(594\) 3.41147 0.139974
\(595\) −19.7219 −0.808520
\(596\) 1.18479 0.0485310
\(597\) −8.06923 −0.330252
\(598\) 9.26857 0.379020
\(599\) −8.83069 −0.360812 −0.180406 0.983592i \(-0.557741\pi\)
−0.180406 + 0.983592i \(0.557741\pi\)
\(600\) −6.63816 −0.271002
\(601\) −33.0523 −1.34823 −0.674116 0.738625i \(-0.735475\pi\)
−0.674116 + 0.738625i \(0.735475\pi\)
\(602\) −20.3482 −0.829332
\(603\) −0.709141 −0.0288784
\(604\) 3.63816 0.148034
\(605\) 2.17705 0.0885097
\(606\) 4.64590 0.188727
\(607\) −5.16250 −0.209540 −0.104770 0.994497i \(-0.533411\pi\)
−0.104770 + 0.994497i \(0.533411\pi\)
\(608\) 0 0
\(609\) −18.4115 −0.746071
\(610\) 23.5449 0.953304
\(611\) −7.76239 −0.314033
\(612\) 1.18479 0.0478924
\(613\) −47.0883 −1.90188 −0.950940 0.309376i \(-0.899880\pi\)
−0.950940 + 0.309376i \(0.899880\pi\)
\(614\) 25.0232 1.00986
\(615\) 22.3601 0.901646
\(616\) 16.6459 0.670682
\(617\) −42.3851 −1.70636 −0.853179 0.521618i \(-0.825329\pi\)
−0.853179 + 0.521618i \(0.825329\pi\)
\(618\) −12.3824 −0.498092
\(619\) −1.85616 −0.0746055 −0.0373027 0.999304i \(-0.511877\pi\)
−0.0373027 + 0.999304i \(0.511877\pi\)
\(620\) −22.5699 −0.906427
\(621\) 3.41147 0.136898
\(622\) 23.9736 0.961253
\(623\) 33.0496 1.32411
\(624\) 2.71688 0.108762
\(625\) −14.1257 −0.565027
\(626\) −12.6382 −0.505122
\(627\) 0 0
\(628\) 6.27126 0.250250
\(629\) −8.00774 −0.319290
\(630\) −16.6459 −0.663188
\(631\) 24.6483 0.981232 0.490616 0.871376i \(-0.336772\pi\)
0.490616 + 0.871376i \(0.336772\pi\)
\(632\) −0.297667 −0.0118405
\(633\) 7.94356 0.315728
\(634\) 14.1334 0.561309
\(635\) 20.7547 0.823623
\(636\) 0.630415 0.0249976
\(637\) −45.6664 −1.80937
\(638\) 12.8726 0.509630
\(639\) −7.95811 −0.314818
\(640\) 3.41147 0.134850
\(641\) −17.6382 −0.696665 −0.348333 0.937371i \(-0.613252\pi\)
−0.348333 + 0.937371i \(0.613252\pi\)
\(642\) 3.89899 0.153881
\(643\) 29.8485 1.17711 0.588556 0.808457i \(-0.299697\pi\)
0.588556 + 0.808457i \(0.299697\pi\)
\(644\) 16.6459 0.655940
\(645\) −14.2267 −0.560175
\(646\) 0 0
\(647\) 23.9391 0.941144 0.470572 0.882362i \(-0.344048\pi\)
0.470572 + 0.882362i \(0.344048\pi\)
\(648\) 1.00000 0.0392837
\(649\) 39.7033 1.55849
\(650\) −18.0351 −0.707394
\(651\) −32.2814 −1.26521
\(652\) −19.2986 −0.755792
\(653\) −26.4861 −1.03648 −0.518240 0.855235i \(-0.673413\pi\)
−0.518240 + 0.855235i \(0.673413\pi\)
\(654\) −11.5963 −0.453450
\(655\) −63.4671 −2.47986
\(656\) −6.55438 −0.255905
\(657\) −12.7442 −0.497199
\(658\) −13.9409 −0.543472
\(659\) 5.29355 0.206207 0.103104 0.994671i \(-0.467123\pi\)
0.103104 + 0.994671i \(0.467123\pi\)
\(660\) 11.6382 0.453015
\(661\) −46.1070 −1.79335 −0.896677 0.442685i \(-0.854026\pi\)
−0.896677 + 0.442685i \(0.854026\pi\)
\(662\) −29.4979 −1.14647
\(663\) 3.21894 0.125013
\(664\) 15.6040 0.605553
\(665\) 0 0
\(666\) −6.75877 −0.261897
\(667\) 12.8726 0.498428
\(668\) 7.72193 0.298771
\(669\) 11.9682 0.462718
\(670\) −2.41921 −0.0934624
\(671\) −23.5449 −0.908940
\(672\) 4.87939 0.188226
\(673\) −2.38144 −0.0917979 −0.0458990 0.998946i \(-0.514615\pi\)
−0.0458990 + 0.998946i \(0.514615\pi\)
\(674\) 22.2490 0.856998
\(675\) −6.63816 −0.255503
\(676\) −5.61856 −0.216098
\(677\) −15.9486 −0.612955 −0.306478 0.951878i \(-0.599150\pi\)
−0.306478 + 0.951878i \(0.599150\pi\)
\(678\) 1.94087 0.0745388
\(679\) −34.7689 −1.33431
\(680\) 4.04189 0.154999
\(681\) 26.5945 1.01910
\(682\) 22.5699 0.864245
\(683\) 1.26083 0.0482443 0.0241222 0.999709i \(-0.492321\pi\)
0.0241222 + 0.999709i \(0.492321\pi\)
\(684\) 0 0
\(685\) 15.5107 0.592635
\(686\) −47.8590 −1.82726
\(687\) 19.9026 0.759332
\(688\) 4.17024 0.158989
\(689\) 1.71276 0.0652511
\(690\) 11.6382 0.443057
\(691\) −8.10876 −0.308472 −0.154236 0.988034i \(-0.549292\pi\)
−0.154236 + 0.988034i \(0.549292\pi\)
\(692\) 14.4953 0.551027
\(693\) 16.6459 0.632325
\(694\) −14.0250 −0.532381
\(695\) −10.2841 −0.390096
\(696\) 3.77332 0.143027
\(697\) −7.76558 −0.294142
\(698\) −34.1121 −1.29116
\(699\) 4.31046 0.163037
\(700\) −32.3901 −1.22423
\(701\) 10.2284 0.386323 0.193161 0.981167i \(-0.438126\pi\)
0.193161 + 0.981167i \(0.438126\pi\)
\(702\) 2.71688 0.102542
\(703\) 0 0
\(704\) −3.41147 −0.128575
\(705\) −9.74691 −0.367090
\(706\) 30.4439 1.14577
\(707\) 22.6691 0.852560
\(708\) 11.6382 0.437389
\(709\) 0.204393 0.00767614 0.00383807 0.999993i \(-0.498778\pi\)
0.00383807 + 0.999993i \(0.498778\pi\)
\(710\) −27.1489 −1.01888
\(711\) −0.297667 −0.0111634
\(712\) −6.77332 −0.253841
\(713\) 22.5699 0.845248
\(714\) 5.78106 0.216351
\(715\) 31.6195 1.18250
\(716\) 9.82295 0.367101
\(717\) −0.285807 −0.0106737
\(718\) 13.6459 0.509260
\(719\) 15.9394 0.594441 0.297220 0.954809i \(-0.403940\pi\)
0.297220 + 0.954809i \(0.403940\pi\)
\(720\) 3.41147 0.127138
\(721\) −60.4184 −2.25010
\(722\) 0 0
\(723\) 9.56624 0.355772
\(724\) −18.9145 −0.702951
\(725\) −25.0479 −0.930255
\(726\) −0.638156 −0.0236842
\(727\) −30.9632 −1.14836 −0.574180 0.818729i \(-0.694679\pi\)
−0.574180 + 0.818729i \(0.694679\pi\)
\(728\) 13.2567 0.491326
\(729\) 1.00000 0.0370370
\(730\) −43.4766 −1.60914
\(731\) 4.94087 0.182745
\(732\) −6.90167 −0.255093
\(733\) 3.95811 0.146196 0.0730981 0.997325i \(-0.476711\pi\)
0.0730981 + 0.997325i \(0.476711\pi\)
\(734\) 23.7665 0.877238
\(735\) −57.3414 −2.11507
\(736\) −3.41147 −0.125749
\(737\) 2.41921 0.0891129
\(738\) −6.55438 −0.241270
\(739\) −25.8033 −0.949191 −0.474596 0.880204i \(-0.657406\pi\)
−0.474596 + 0.880204i \(0.657406\pi\)
\(740\) −23.0574 −0.847606
\(741\) 0 0
\(742\) 3.07604 0.112925
\(743\) −31.9145 −1.17083 −0.585414 0.810734i \(-0.699068\pi\)
−0.585414 + 0.810734i \(0.699068\pi\)
\(744\) 6.61587 0.242549
\(745\) 4.04189 0.148083
\(746\) 23.2344 0.850673
\(747\) 15.6040 0.570921
\(748\) −4.04189 −0.147786
\(749\) 19.0247 0.695146
\(750\) −5.58853 −0.204064
\(751\) 0.508045 0.0185388 0.00926942 0.999957i \(-0.497049\pi\)
0.00926942 + 0.999957i \(0.497049\pi\)
\(752\) 2.85710 0.104188
\(753\) −29.6878 −1.08188
\(754\) 10.2517 0.373343
\(755\) 12.4115 0.451700
\(756\) 4.87939 0.177462
\(757\) −48.1694 −1.75075 −0.875374 0.483447i \(-0.839385\pi\)
−0.875374 + 0.483447i \(0.839385\pi\)
\(758\) 4.50030 0.163458
\(759\) −11.6382 −0.422438
\(760\) 0 0
\(761\) 5.15064 0.186711 0.0933554 0.995633i \(-0.470241\pi\)
0.0933554 + 0.995633i \(0.470241\pi\)
\(762\) −6.08378 −0.220392
\(763\) −56.5827 −2.04843
\(764\) −10.0915 −0.365098
\(765\) 4.04189 0.146135
\(766\) −4.36009 −0.157536
\(767\) 31.6195 1.14171
\(768\) −1.00000 −0.0360844
\(769\) 19.5725 0.705804 0.352902 0.935660i \(-0.385195\pi\)
0.352902 + 0.935660i \(0.385195\pi\)
\(770\) 56.7870 2.04646
\(771\) 15.7965 0.568899
\(772\) −2.52435 −0.0908533
\(773\) −22.3946 −0.805476 −0.402738 0.915315i \(-0.631941\pi\)
−0.402738 + 0.915315i \(0.631941\pi\)
\(774\) 4.17024 0.149896
\(775\) −43.9172 −1.57755
\(776\) 7.12567 0.255797
\(777\) −32.9786 −1.18310
\(778\) −1.31046 −0.0469823
\(779\) 0 0
\(780\) 9.26857 0.331868
\(781\) 27.1489 0.971464
\(782\) −4.04189 −0.144538
\(783\) 3.77332 0.134847
\(784\) 16.8084 0.600300
\(785\) 21.3942 0.763593
\(786\) 18.6040 0.663583
\(787\) −33.6887 −1.20087 −0.600437 0.799672i \(-0.705007\pi\)
−0.600437 + 0.799672i \(0.705007\pi\)
\(788\) 14.8571 0.529262
\(789\) −10.8648 −0.386798
\(790\) −1.01548 −0.0361292
\(791\) 9.47028 0.336724
\(792\) −3.41147 −0.121221
\(793\) −18.7510 −0.665869
\(794\) −19.0128 −0.674739
\(795\) 2.15064 0.0762755
\(796\) 8.06923 0.286006
\(797\) 47.8631 1.69540 0.847699 0.530478i \(-0.177988\pi\)
0.847699 + 0.530478i \(0.177988\pi\)
\(798\) 0 0
\(799\) 3.38507 0.119755
\(800\) 6.63816 0.234694
\(801\) −6.77332 −0.239323
\(802\) −35.8631 −1.26637
\(803\) 43.4766 1.53426
\(804\) 0.709141 0.0250095
\(805\) 56.7870 2.00148
\(806\) 17.9745 0.633126
\(807\) −17.2935 −0.608762
\(808\) −4.64590 −0.163442
\(809\) −3.67406 −0.129173 −0.0645865 0.997912i \(-0.520573\pi\)
−0.0645865 + 0.997912i \(0.520573\pi\)
\(810\) 3.41147 0.119867
\(811\) 18.7050 0.656822 0.328411 0.944535i \(-0.393487\pi\)
0.328411 + 0.944535i \(0.393487\pi\)
\(812\) 18.4115 0.646116
\(813\) −1.44562 −0.0507002
\(814\) 23.0574 0.808160
\(815\) −65.8367 −2.30616
\(816\) −1.18479 −0.0414760
\(817\) 0 0
\(818\) 29.1088 1.01776
\(819\) 13.2567 0.463227
\(820\) −22.3601 −0.780848
\(821\) −22.3164 −0.778849 −0.389425 0.921058i \(-0.627326\pi\)
−0.389425 + 0.921058i \(0.627326\pi\)
\(822\) −4.54664 −0.158582
\(823\) −26.4037 −0.920376 −0.460188 0.887821i \(-0.652218\pi\)
−0.460188 + 0.887821i \(0.652218\pi\)
\(824\) 12.3824 0.431361
\(825\) 22.6459 0.788429
\(826\) 56.7870 1.97587
\(827\) −29.6705 −1.03175 −0.515873 0.856665i \(-0.672532\pi\)
−0.515873 + 0.856665i \(0.672532\pi\)
\(828\) −3.41147 −0.118557
\(829\) −1.76827 −0.0614144 −0.0307072 0.999528i \(-0.509776\pi\)
−0.0307072 + 0.999528i \(0.509776\pi\)
\(830\) 53.2327 1.84773
\(831\) −21.1908 −0.735100
\(832\) −2.71688 −0.0941909
\(833\) 19.9145 0.689995
\(834\) 3.01455 0.104385
\(835\) 26.3432 0.911643
\(836\) 0 0
\(837\) 6.61587 0.228678
\(838\) 13.1584 0.454549
\(839\) −26.7142 −0.922276 −0.461138 0.887328i \(-0.652559\pi\)
−0.461138 + 0.887328i \(0.652559\pi\)
\(840\) 16.6459 0.574338
\(841\) −14.7621 −0.509037
\(842\) 0.150644 0.00519154
\(843\) −24.3678 −0.839273
\(844\) −7.94356 −0.273429
\(845\) −19.1676 −0.659384
\(846\) 2.85710 0.0982290
\(847\) −3.11381 −0.106992
\(848\) −0.630415 −0.0216485
\(849\) 4.12567 0.141593
\(850\) 7.86484 0.269762
\(851\) 23.0574 0.790396
\(852\) 7.95811 0.272640
\(853\) 13.9263 0.476828 0.238414 0.971164i \(-0.423372\pi\)
0.238414 + 0.971164i \(0.423372\pi\)
\(854\) −33.6759 −1.15237
\(855\) 0 0
\(856\) −3.89899 −0.133265
\(857\) 19.6691 0.671884 0.335942 0.941883i \(-0.390945\pi\)
0.335942 + 0.941883i \(0.390945\pi\)
\(858\) −9.26857 −0.316424
\(859\) −43.3979 −1.48072 −0.740358 0.672213i \(-0.765344\pi\)
−0.740358 + 0.672213i \(0.765344\pi\)
\(860\) 14.2267 0.485126
\(861\) −31.9813 −1.08992
\(862\) 30.7965 1.04893
\(863\) 29.4192 1.00144 0.500721 0.865609i \(-0.333068\pi\)
0.500721 + 0.865609i \(0.333068\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 49.4502 1.68136
\(866\) 29.4338 1.00020
\(867\) 15.5963 0.529677
\(868\) 32.2814 1.09570
\(869\) 1.01548 0.0344479
\(870\) 12.8726 0.436421
\(871\) 1.92665 0.0652821
\(872\) 11.5963 0.392699
\(873\) 7.12567 0.241167
\(874\) 0 0
\(875\) −27.2686 −0.921846
\(876\) 12.7442 0.430587
\(877\) 26.1652 0.883536 0.441768 0.897129i \(-0.354351\pi\)
0.441768 + 0.897129i \(0.354351\pi\)
\(878\) −19.8007 −0.668240
\(879\) 2.78106 0.0938028
\(880\) −11.6382 −0.392322
\(881\) 22.6287 0.762379 0.381189 0.924497i \(-0.375515\pi\)
0.381189 + 0.924497i \(0.375515\pi\)
\(882\) 16.8084 0.565968
\(883\) −8.28169 −0.278701 −0.139350 0.990243i \(-0.544501\pi\)
−0.139350 + 0.990243i \(0.544501\pi\)
\(884\) −3.21894 −0.108265
\(885\) 39.7033 1.33461
\(886\) −18.5868 −0.624435
\(887\) −40.3569 −1.35505 −0.677526 0.735499i \(-0.736948\pi\)
−0.677526 + 0.735499i \(0.736948\pi\)
\(888\) 6.75877 0.226809
\(889\) −29.6851 −0.995606
\(890\) −23.1070 −0.774548
\(891\) −3.41147 −0.114289
\(892\) −11.9682 −0.400726
\(893\) 0 0
\(894\) −1.18479 −0.0396254
\(895\) 33.5107 1.12014
\(896\) −4.87939 −0.163009
\(897\) −9.26857 −0.309469
\(898\) −3.37908 −0.112761
\(899\) 24.9638 0.832588
\(900\) 6.63816 0.221272
\(901\) −0.746911 −0.0248832
\(902\) 22.3601 0.744510
\(903\) 20.3482 0.677147
\(904\) −1.94087 −0.0645525
\(905\) −64.5262 −2.14492
\(906\) −3.63816 −0.120870
\(907\) 19.2513 0.639230 0.319615 0.947547i \(-0.396446\pi\)
0.319615 + 0.947547i \(0.396446\pi\)
\(908\) −26.5945 −0.882570
\(909\) −4.64590 −0.154095
\(910\) 45.2249 1.49919
\(911\) 58.7684 1.94708 0.973542 0.228510i \(-0.0733853\pi\)
0.973542 + 0.228510i \(0.0733853\pi\)
\(912\) 0 0
\(913\) −53.2327 −1.76174
\(914\) 21.7912 0.720788
\(915\) −23.5449 −0.778370
\(916\) −19.9026 −0.657601
\(917\) 90.7761 2.99769
\(918\) −1.18479 −0.0391040
\(919\) −53.7428 −1.77281 −0.886406 0.462909i \(-0.846806\pi\)
−0.886406 + 0.462909i \(0.846806\pi\)
\(920\) −11.6382 −0.383699
\(921\) −25.0232 −0.824543
\(922\) 11.7638 0.387421
\(923\) 21.6212 0.711672
\(924\) −16.6459 −0.547610
\(925\) −44.8658 −1.47518
\(926\) 16.9932 0.558431
\(927\) 12.3824 0.406691
\(928\) −3.77332 −0.123865
\(929\) −31.0387 −1.01835 −0.509173 0.860664i \(-0.670049\pi\)
−0.509173 + 0.860664i \(0.670049\pi\)
\(930\) 22.5699 0.740095
\(931\) 0 0
\(932\) −4.31046 −0.141194
\(933\) −23.9736 −0.784860
\(934\) −11.7050 −0.383000
\(935\) −13.7888 −0.450942
\(936\) −2.71688 −0.0888040
\(937\) −20.7428 −0.677637 −0.338819 0.940852i \(-0.610027\pi\)
−0.338819 + 0.940852i \(0.610027\pi\)
\(938\) 3.46017 0.112979
\(939\) 12.6382 0.412431
\(940\) 9.74691 0.317909
\(941\) 14.8631 0.484523 0.242261 0.970211i \(-0.422111\pi\)
0.242261 + 0.970211i \(0.422111\pi\)
\(942\) −6.27126 −0.204329
\(943\) 22.3601 0.728145
\(944\) −11.6382 −0.378790
\(945\) 16.6459 0.541491
\(946\) −14.2267 −0.462549
\(947\) −25.1999 −0.818888 −0.409444 0.912335i \(-0.634277\pi\)
−0.409444 + 0.912335i \(0.634277\pi\)
\(948\) 0.297667 0.00966776
\(949\) 34.6245 1.12396
\(950\) 0 0
\(951\) −14.1334 −0.458307
\(952\) −5.78106 −0.187365
\(953\) 12.3851 0.401192 0.200596 0.979674i \(-0.435712\pi\)
0.200596 + 0.979674i \(0.435712\pi\)
\(954\) −0.630415 −0.0204104
\(955\) −34.4270 −1.11403
\(956\) 0.285807 0.00924366
\(957\) −12.8726 −0.416111
\(958\) 29.9796 0.968596
\(959\) −22.1848 −0.716384
\(960\) −3.41147 −0.110105
\(961\) 12.7697 0.411926
\(962\) 18.3628 0.592040
\(963\) −3.89899 −0.125643
\(964\) −9.56624 −0.308108
\(965\) −8.61175 −0.277222
\(966\) −16.6459 −0.535573
\(967\) −3.40879 −0.109619 −0.0548096 0.998497i \(-0.517455\pi\)
−0.0548096 + 0.998497i \(0.517455\pi\)
\(968\) 0.638156 0.0205111
\(969\) 0 0
\(970\) 24.3090 0.780516
\(971\) 13.8972 0.445983 0.222992 0.974820i \(-0.428418\pi\)
0.222992 + 0.974820i \(0.428418\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 14.7091 0.471553
\(974\) −14.5107 −0.464954
\(975\) 18.0351 0.577585
\(976\) 6.90167 0.220917
\(977\) −12.8402 −0.410794 −0.205397 0.978679i \(-0.565849\pi\)
−0.205397 + 0.978679i \(0.565849\pi\)
\(978\) 19.2986 0.617101
\(979\) 23.1070 0.738503
\(980\) 57.3414 1.83170
\(981\) 11.5963 0.370240
\(982\) −11.6554 −0.371939
\(983\) 0.866592 0.0276400 0.0138200 0.999904i \(-0.495601\pi\)
0.0138200 + 0.999904i \(0.495601\pi\)
\(984\) 6.55438 0.208946
\(985\) 50.6846 1.61495
\(986\) −4.47060 −0.142373
\(987\) 13.9409 0.443743
\(988\) 0 0
\(989\) −14.2267 −0.452382
\(990\) −11.6382 −0.369885
\(991\) 20.9709 0.666163 0.333081 0.942898i \(-0.391912\pi\)
0.333081 + 0.942898i \(0.391912\pi\)
\(992\) −6.61587 −0.210054
\(993\) 29.4979 0.936089
\(994\) 38.8307 1.23163
\(995\) 27.5280 0.872695
\(996\) −15.6040 −0.494432
\(997\) 39.7475 1.25882 0.629408 0.777075i \(-0.283297\pi\)
0.629408 + 0.777075i \(0.283297\pi\)
\(998\) −1.68685 −0.0533964
\(999\) 6.75877 0.213838
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 2166.2.a.s.1.3 3
3.2 odd 2 6498.2.a.bm.1.1 3
19.3 odd 18 114.2.i.a.85.1 yes 6
19.13 odd 18 114.2.i.a.55.1 6
19.18 odd 2 2166.2.a.q.1.3 3
57.32 even 18 342.2.u.e.55.1 6
57.41 even 18 342.2.u.e.199.1 6
57.56 even 2 6498.2.a.br.1.1 3
76.3 even 18 912.2.bo.a.769.1 6
76.51 even 18 912.2.bo.a.625.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.2.i.a.55.1 6 19.13 odd 18
114.2.i.a.85.1 yes 6 19.3 odd 18
342.2.u.e.55.1 6 57.32 even 18
342.2.u.e.199.1 6 57.41 even 18
912.2.bo.a.625.1 6 76.51 even 18
912.2.bo.a.769.1 6 76.3 even 18
2166.2.a.q.1.3 3 19.18 odd 2
2166.2.a.s.1.3 3 1.1 even 1 trivial
6498.2.a.bm.1.1 3 3.2 odd 2
6498.2.a.br.1.1 3 57.56 even 2