Properties

Label 6825.2.a.bg.1.2
Level $6825$
Weight $2$
Character 6825.1
Self dual yes
Analytic conductor $54.498$
Analytic rank $1$
Dimension $4$
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] = [6825,2,Mod(1,6825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6825.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6825 = 3 \cdot 5^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6825.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: \(54.4978993795\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.17428.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 6x^{2} + 4x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 273)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.10710\) of defining polynomial
Character \(\chi\) \(=\) 6825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.43986 q^{2} -1.00000 q^{3} +0.0731828 q^{4} +1.43986 q^{6} -1.00000 q^{7} +2.77434 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.43986 q^{2} -1.00000 q^{3} +0.0731828 q^{4} +1.43986 q^{6} -1.00000 q^{7} +2.77434 q^{8} +1.00000 q^{9} +4.21419 q^{11} -0.0731828 q^{12} -1.00000 q^{13} +1.43986 q^{14} -4.14101 q^{16} +2.87971 q^{17} -1.43986 q^{18} -1.28738 q^{19} +1.00000 q^{21} -6.06783 q^{22} +8.02072 q^{23} -2.77434 q^{24} +1.43986 q^{26} -1.00000 q^{27} -0.0731828 q^{28} +3.28738 q^{29} -7.04680 q^{31} +0.413779 q^{32} -4.21419 q^{33} -4.14637 q^{34} +0.0731828 q^{36} -8.57475 q^{37} +1.85363 q^{38} +1.00000 q^{39} -12.0678 q^{41} -1.43986 q^{42} -7.14101 q^{43} +0.308407 q^{44} -11.5487 q^{46} -1.95289 q^{47} +4.14101 q^{48} +1.00000 q^{49} -2.87971 q^{51} -0.0731828 q^{52} +5.14101 q^{53} +1.43986 q^{54} -2.77434 q^{56} +1.28738 q^{57} -4.73334 q^{58} -7.33448 q^{59} +7.75942 q^{61} +10.1464 q^{62} -1.00000 q^{63} +7.68624 q^{64} +6.06783 q^{66} -12.0414 q^{67} +0.210745 q^{68} -8.02072 q^{69} +10.7889 q^{71} +2.77434 q^{72} +8.32568 q^{73} +12.3464 q^{74} -0.0942138 q^{76} -4.21419 q^{77} -1.43986 q^{78} +4.47204 q^{79} +1.00000 q^{81} +17.3759 q^{82} +3.80653 q^{83} +0.0731828 q^{84} +10.2820 q^{86} -3.28738 q^{87} +11.6916 q^{88} -5.64793 q^{89} +1.00000 q^{91} +0.586979 q^{92} +7.04680 q^{93} +2.81188 q^{94} -0.413779 q^{96} -6.90043 q^{97} -1.43986 q^{98} +4.21419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 4 q^{7} - 3 q^{8} + 4 q^{9} - 2 q^{11} - 7 q^{12} - 4 q^{13} + q^{14} + 9 q^{16} + 2 q^{17} - q^{18} + 7 q^{19} + 4 q^{21} + 8 q^{22} - 3 q^{23} + 3 q^{24} + q^{26} - 4 q^{27} - 7 q^{28} + q^{29} + 3 q^{31} - 7 q^{32} + 2 q^{33} - 30 q^{34} + 7 q^{36} - 10 q^{37} - 6 q^{38} + 4 q^{39} - 16 q^{41} - q^{42} - 3 q^{43} - 12 q^{44} - 18 q^{46} - 5 q^{47} - 9 q^{48} + 4 q^{49} - 2 q^{51} - 7 q^{52} - 5 q^{53} + q^{54} + 3 q^{56} - 7 q^{57} + 4 q^{58} - 20 q^{59} + 12 q^{61} + 54 q^{62} - 4 q^{63} + 5 q^{64} - 8 q^{66} + 22 q^{67} + 10 q^{68} + 3 q^{69} - 3 q^{72} + 13 q^{73} - 6 q^{74} - 6 q^{76} + 2 q^{77} - q^{78} + 11 q^{79} + 4 q^{81} - 10 q^{82} - q^{83} + 7 q^{84} - 10 q^{86} - q^{87} + 60 q^{88} - 5 q^{89} + 4 q^{91} - 34 q^{92} - 3 q^{93} + 34 q^{94} + 7 q^{96} + 17 q^{97} - q^{98} - 2 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.43986 −1.01813 −0.509066 0.860728i \(-0.670009\pi\)
−0.509066 + 0.860728i \(0.670009\pi\)
\(3\) −1.00000 −0.577350
\(4\) 0.0731828 0.0365914
\(5\) 0 0
\(6\) 1.43986 0.587818
\(7\) −1.00000 −0.377964
\(8\) 2.77434 0.980876
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.21419 1.27063 0.635313 0.772254i \(-0.280871\pi\)
0.635313 + 0.772254i \(0.280871\pi\)
\(12\) −0.0731828 −0.0211261
\(13\) −1.00000 −0.277350
\(14\) 1.43986 0.384817
\(15\) 0 0
\(16\) −4.14101 −1.03525
\(17\) 2.87971 0.698432 0.349216 0.937042i \(-0.386448\pi\)
0.349216 + 0.937042i \(0.386448\pi\)
\(18\) −1.43986 −0.339377
\(19\) −1.28738 −0.295344 −0.147672 0.989036i \(-0.547178\pi\)
−0.147672 + 0.989036i \(0.547178\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) −6.06783 −1.29367
\(23\) 8.02072 1.67244 0.836218 0.548397i \(-0.184762\pi\)
0.836218 + 0.548397i \(0.184762\pi\)
\(24\) −2.77434 −0.566309
\(25\) 0 0
\(26\) 1.43986 0.282379
\(27\) −1.00000 −0.192450
\(28\) −0.0731828 −0.0138303
\(29\) 3.28738 0.610450 0.305225 0.952280i \(-0.401268\pi\)
0.305225 + 0.952280i \(0.401268\pi\)
\(30\) 0 0
\(31\) −7.04680 −1.26564 −0.632821 0.774298i \(-0.718103\pi\)
−0.632821 + 0.774298i \(0.718103\pi\)
\(32\) 0.413779 0.0731465
\(33\) −4.21419 −0.733597
\(34\) −4.14637 −0.711096
\(35\) 0 0
\(36\) 0.0731828 0.0121971
\(37\) −8.57475 −1.40968 −0.704840 0.709366i \(-0.748981\pi\)
−0.704840 + 0.709366i \(0.748981\pi\)
\(38\) 1.85363 0.300699
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) −12.0678 −1.88468 −0.942339 0.334660i \(-0.891379\pi\)
−0.942339 + 0.334660i \(0.891379\pi\)
\(42\) −1.43986 −0.222174
\(43\) −7.14101 −1.08899 −0.544497 0.838763i \(-0.683279\pi\)
−0.544497 + 0.838763i \(0.683279\pi\)
\(44\) 0.308407 0.0464940
\(45\) 0 0
\(46\) −11.5487 −1.70276
\(47\) −1.95289 −0.284859 −0.142429 0.989805i \(-0.545491\pi\)
−0.142429 + 0.989805i \(0.545491\pi\)
\(48\) 4.14101 0.597703
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −2.87971 −0.403240
\(52\) −0.0731828 −0.0101486
\(53\) 5.14101 0.706172 0.353086 0.935591i \(-0.385132\pi\)
0.353086 + 0.935591i \(0.385132\pi\)
\(54\) 1.43986 0.195939
\(55\) 0 0
\(56\) −2.77434 −0.370736
\(57\) 1.28738 0.170517
\(58\) −4.73334 −0.621519
\(59\) −7.33448 −0.954868 −0.477434 0.878668i \(-0.658433\pi\)
−0.477434 + 0.878668i \(0.658433\pi\)
\(60\) 0 0
\(61\) 7.75942 0.993492 0.496746 0.867896i \(-0.334528\pi\)
0.496746 + 0.867896i \(0.334528\pi\)
\(62\) 10.1464 1.28859
\(63\) −1.00000 −0.125988
\(64\) 7.68624 0.960780
\(65\) 0 0
\(66\) 6.06783 0.746898
\(67\) −12.0414 −1.47110 −0.735548 0.677473i \(-0.763075\pi\)
−0.735548 + 0.677473i \(0.763075\pi\)
\(68\) 0.210745 0.0255566
\(69\) −8.02072 −0.965581
\(70\) 0 0
\(71\) 10.7889 1.28041 0.640206 0.768203i \(-0.278849\pi\)
0.640206 + 0.768203i \(0.278849\pi\)
\(72\) 2.77434 0.326959
\(73\) 8.32568 0.974447 0.487224 0.873277i \(-0.338010\pi\)
0.487224 + 0.873277i \(0.338010\pi\)
\(74\) 12.3464 1.43524
\(75\) 0 0
\(76\) −0.0942138 −0.0108071
\(77\) −4.21419 −0.480252
\(78\) −1.43986 −0.163031
\(79\) 4.47204 0.503144 0.251572 0.967839i \(-0.419052\pi\)
0.251572 + 0.967839i \(0.419052\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 17.3759 1.91885
\(83\) 3.80653 0.417821 0.208910 0.977935i \(-0.433008\pi\)
0.208910 + 0.977935i \(0.433008\pi\)
\(84\) 0.0731828 0.00798490
\(85\) 0 0
\(86\) 10.2820 1.10874
\(87\) −3.28738 −0.352444
\(88\) 11.6916 1.24633
\(89\) −5.64793 −0.598680 −0.299340 0.954147i \(-0.596766\pi\)
−0.299340 + 0.954147i \(0.596766\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0.586979 0.0611968
\(93\) 7.04680 0.730719
\(94\) 2.81188 0.290024
\(95\) 0 0
\(96\) −0.413779 −0.0422312
\(97\) −6.90043 −0.700633 −0.350316 0.936631i \(-0.613926\pi\)
−0.350316 + 0.936631i \(0.613926\pi\)
\(98\) −1.43986 −0.145447
\(99\) 4.21419 0.423542
\(100\) 0 0
\(101\) −10.9739 −1.09195 −0.545973 0.837803i \(-0.683840\pi\)
−0.545973 + 0.837803i \(0.683840\pi\)
\(102\) 4.14637 0.410551
\(103\) −16.4284 −1.61874 −0.809368 0.587301i \(-0.800190\pi\)
−0.809368 + 0.587301i \(0.800190\pi\)
\(104\) −2.77434 −0.272046
\(105\) 0 0
\(106\) −7.40231 −0.718976
\(107\) 3.45446 0.333955 0.166978 0.985961i \(-0.446599\pi\)
0.166978 + 0.985961i \(0.446599\pi\)
\(108\) −0.0731828 −0.00704202
\(109\) 4.66896 0.447206 0.223603 0.974680i \(-0.428218\pi\)
0.223603 + 0.974680i \(0.428218\pi\)
\(110\) 0 0
\(111\) 8.57475 0.813879
\(112\) 4.14101 0.391289
\(113\) 3.28738 0.309250 0.154625 0.987973i \(-0.450583\pi\)
0.154625 + 0.987973i \(0.450583\pi\)
\(114\) −1.85363 −0.173609
\(115\) 0 0
\(116\) 0.240579 0.0223372
\(117\) −1.00000 −0.0924500
\(118\) 10.5606 0.972181
\(119\) −2.87971 −0.263983
\(120\) 0 0
\(121\) 6.75942 0.614493
\(122\) −11.1724 −1.01151
\(123\) 12.0678 1.08812
\(124\) −0.515704 −0.0463116
\(125\) 0 0
\(126\) 1.43986 0.128272
\(127\) 0.428386 0.0380131 0.0190065 0.999819i \(-0.493950\pi\)
0.0190065 + 0.999819i \(0.493950\pi\)
\(128\) −11.8946 −1.05135
\(129\) 7.14101 0.628731
\(130\) 0 0
\(131\) 14.1878 1.23959 0.619797 0.784762i \(-0.287215\pi\)
0.619797 + 0.784762i \(0.287215\pi\)
\(132\) −0.308407 −0.0268433
\(133\) 1.28738 0.111630
\(134\) 17.3379 1.49777
\(135\) 0 0
\(136\) 7.98929 0.685076
\(137\) −14.7070 −1.25650 −0.628250 0.778011i \(-0.716229\pi\)
−0.628250 + 0.778011i \(0.716229\pi\)
\(138\) 11.5487 0.983089
\(139\) 9.85363 0.835774 0.417887 0.908499i \(-0.362771\pi\)
0.417887 + 0.908499i \(0.362771\pi\)
\(140\) 0 0
\(141\) 1.95289 0.164463
\(142\) −15.5345 −1.30363
\(143\) −4.21419 −0.352409
\(144\) −4.14101 −0.345084
\(145\) 0 0
\(146\) −11.9878 −0.992115
\(147\) −1.00000 −0.0824786
\(148\) −0.627525 −0.0515822
\(149\) −3.38663 −0.277444 −0.138722 0.990331i \(-0.544299\pi\)
−0.138722 + 0.990331i \(0.544299\pi\)
\(150\) 0 0
\(151\) −21.8951 −1.78180 −0.890898 0.454204i \(-0.849924\pi\)
−0.890898 + 0.454204i \(0.849924\pi\)
\(152\) −3.57161 −0.289696
\(153\) 2.87971 0.232811
\(154\) 6.06783 0.488959
\(155\) 0 0
\(156\) 0.0731828 0.00585932
\(157\) 0.867482 0.0692326 0.0346163 0.999401i \(-0.488979\pi\)
0.0346163 + 0.999401i \(0.488979\pi\)
\(158\) −6.43910 −0.512267
\(159\) −5.14101 −0.407709
\(160\) 0 0
\(161\) −8.02072 −0.632121
\(162\) −1.43986 −0.113126
\(163\) −4.42839 −0.346858 −0.173429 0.984846i \(-0.555485\pi\)
−0.173429 + 0.984846i \(0.555485\pi\)
\(164\) −0.883158 −0.0689630
\(165\) 0 0
\(166\) −5.48085 −0.425396
\(167\) −20.5276 −1.58848 −0.794238 0.607606i \(-0.792130\pi\)
−0.794238 + 0.607606i \(0.792130\pi\)
\(168\) 2.77434 0.214045
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −1.28738 −0.0984481
\(172\) −0.522599 −0.0398478
\(173\) −15.0154 −1.14160 −0.570799 0.821090i \(-0.693366\pi\)
−0.570799 + 0.821090i \(0.693366\pi\)
\(174\) 4.73334 0.358834
\(175\) 0 0
\(176\) −17.4510 −1.31542
\(177\) 7.33448 0.551293
\(178\) 8.13221 0.609535
\(179\) −5.35176 −0.400009 −0.200004 0.979795i \(-0.564096\pi\)
−0.200004 + 0.979795i \(0.564096\pi\)
\(180\) 0 0
\(181\) 11.4667 0.852312 0.426156 0.904650i \(-0.359867\pi\)
0.426156 + 0.904650i \(0.359867\pi\)
\(182\) −1.43986 −0.106729
\(183\) −7.75942 −0.573593
\(184\) 22.2522 1.64045
\(185\) 0 0
\(186\) −10.1464 −0.743968
\(187\) 12.1357 0.887447
\(188\) −0.142918 −0.0104234
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 20.1756 1.45985 0.729927 0.683525i \(-0.239554\pi\)
0.729927 + 0.683525i \(0.239554\pi\)
\(192\) −7.68624 −0.554706
\(193\) 26.3756 1.89856 0.949279 0.314435i \(-0.101815\pi\)
0.949279 + 0.314435i \(0.101815\pi\)
\(194\) 9.93562 0.713336
\(195\) 0 0
\(196\) 0.0731828 0.00522734
\(197\) 20.1322 1.43436 0.717180 0.696888i \(-0.245432\pi\)
0.717180 + 0.696888i \(0.245432\pi\)
\(198\) −6.06783 −0.431222
\(199\) 1.75942 0.124722 0.0623610 0.998054i \(-0.480137\pi\)
0.0623610 + 0.998054i \(0.480137\pi\)
\(200\) 0 0
\(201\) 12.0414 0.849338
\(202\) 15.8009 1.11174
\(203\) −3.28738 −0.230729
\(204\) −0.210745 −0.0147551
\(205\) 0 0
\(206\) 23.6545 1.64809
\(207\) 8.02072 0.557479
\(208\) 4.14101 0.287127
\(209\) −5.42525 −0.375272
\(210\) 0 0
\(211\) 15.5694 1.07184 0.535921 0.844268i \(-0.319965\pi\)
0.535921 + 0.844268i \(0.319965\pi\)
\(212\) 0.376234 0.0258398
\(213\) −10.7889 −0.739246
\(214\) −4.97392 −0.340010
\(215\) 0 0
\(216\) −2.77434 −0.188770
\(217\) 7.04680 0.478368
\(218\) −6.72263 −0.455314
\(219\) −8.32568 −0.562597
\(220\) 0 0
\(221\) −2.87971 −0.193710
\(222\) −12.3464 −0.828636
\(223\) 11.5694 0.774744 0.387372 0.921923i \(-0.373383\pi\)
0.387372 + 0.921923i \(0.373383\pi\)
\(224\) −0.413779 −0.0276468
\(225\) 0 0
\(226\) −4.73334 −0.314857
\(227\) 11.0418 0.732867 0.366433 0.930444i \(-0.380579\pi\)
0.366433 + 0.930444i \(0.380579\pi\)
\(228\) 0.0942138 0.00623946
\(229\) −13.7073 −0.905802 −0.452901 0.891561i \(-0.649611\pi\)
−0.452901 + 0.891561i \(0.649611\pi\)
\(230\) 0 0
\(231\) 4.21419 0.277274
\(232\) 9.12029 0.598776
\(233\) −1.23522 −0.0809222 −0.0404611 0.999181i \(-0.512883\pi\)
−0.0404611 + 0.999181i \(0.512883\pi\)
\(234\) 1.43986 0.0941263
\(235\) 0 0
\(236\) −0.536758 −0.0349400
\(237\) −4.47204 −0.290491
\(238\) 4.14637 0.268769
\(239\) 19.1061 1.23587 0.617936 0.786228i \(-0.287969\pi\)
0.617936 + 0.786228i \(0.287969\pi\)
\(240\) 0 0
\(241\) −12.0329 −0.775110 −0.387555 0.921847i \(-0.626680\pi\)
−0.387555 + 0.921847i \(0.626680\pi\)
\(242\) −9.73259 −0.625634
\(243\) −1.00000 −0.0641500
\(244\) 0.567856 0.0363533
\(245\) 0 0
\(246\) −17.3759 −1.10785
\(247\) 1.28738 0.0819137
\(248\) −19.5502 −1.24144
\(249\) −3.80653 −0.241229
\(250\) 0 0
\(251\) −15.0031 −0.946990 −0.473495 0.880797i \(-0.657008\pi\)
−0.473495 + 0.880797i \(0.657008\pi\)
\(252\) −0.0731828 −0.00461008
\(253\) 33.8009 2.12504
\(254\) −0.616813 −0.0387023
\(255\) 0 0
\(256\) 1.75406 0.109629
\(257\) 13.0261 0.812544 0.406272 0.913752i \(-0.366828\pi\)
0.406272 + 0.913752i \(0.366828\pi\)
\(258\) −10.2820 −0.640131
\(259\) 8.57475 0.532809
\(260\) 0 0
\(261\) 3.28738 0.203483
\(262\) −20.4284 −1.26207
\(263\) 2.59547 0.160044 0.0800218 0.996793i \(-0.474501\pi\)
0.0800218 + 0.996793i \(0.474501\pi\)
\(264\) −11.6916 −0.719568
\(265\) 0 0
\(266\) −1.85363 −0.113654
\(267\) 5.64793 0.345648
\(268\) −0.881227 −0.0538295
\(269\) −23.3081 −1.42112 −0.710560 0.703637i \(-0.751558\pi\)
−0.710560 + 0.703637i \(0.751558\pi\)
\(270\) 0 0
\(271\) −6.05215 −0.367642 −0.183821 0.982960i \(-0.558847\pi\)
−0.183821 + 0.982960i \(0.558847\pi\)
\(272\) −11.9249 −0.723054
\(273\) −1.00000 −0.0605228
\(274\) 21.1759 1.27928
\(275\) 0 0
\(276\) −0.586979 −0.0353320
\(277\) −15.1932 −0.912869 −0.456434 0.889757i \(-0.650874\pi\)
−0.456434 + 0.889757i \(0.650874\pi\)
\(278\) −14.1878 −0.850928
\(279\) −7.04680 −0.421881
\(280\) 0 0
\(281\) −7.57506 −0.451890 −0.225945 0.974140i \(-0.572547\pi\)
−0.225945 + 0.974140i \(0.572547\pi\)
\(282\) −2.81188 −0.167445
\(283\) −19.0974 −1.13522 −0.567610 0.823298i \(-0.692132\pi\)
−0.567610 + 0.823298i \(0.692132\pi\)
\(284\) 0.789565 0.0468521
\(285\) 0 0
\(286\) 6.06783 0.358798
\(287\) 12.0678 0.712341
\(288\) 0.413779 0.0243822
\(289\) −8.70727 −0.512192
\(290\) 0 0
\(291\) 6.90043 0.404510
\(292\) 0.609297 0.0356564
\(293\) 3.79430 0.221665 0.110833 0.993839i \(-0.464648\pi\)
0.110833 + 0.993839i \(0.464648\pi\)
\(294\) 1.43986 0.0839741
\(295\) 0 0
\(296\) −23.7893 −1.38272
\(297\) −4.21419 −0.244532
\(298\) 4.87626 0.282474
\(299\) −8.02072 −0.463850
\(300\) 0 0
\(301\) 7.14101 0.411601
\(302\) 31.5257 1.81410
\(303\) 10.9739 0.630435
\(304\) 5.33104 0.305756
\(305\) 0 0
\(306\) −4.14637 −0.237032
\(307\) 5.19316 0.296389 0.148195 0.988958i \(-0.452654\pi\)
0.148195 + 0.988958i \(0.452654\pi\)
\(308\) −0.308407 −0.0175731
\(309\) 16.4284 0.934578
\(310\) 0 0
\(311\) −13.8536 −0.785568 −0.392784 0.919631i \(-0.628488\pi\)
−0.392784 + 0.919631i \(0.628488\pi\)
\(312\) 2.77434 0.157066
\(313\) 32.6162 1.84358 0.921788 0.387694i \(-0.126728\pi\)
0.921788 + 0.387694i \(0.126728\pi\)
\(314\) −1.24905 −0.0704879
\(315\) 0 0
\(316\) 0.327277 0.0184108
\(317\) −12.0380 −0.676121 −0.338061 0.941124i \(-0.609771\pi\)
−0.338061 + 0.941124i \(0.609771\pi\)
\(318\) 7.40231 0.415101
\(319\) 13.8536 0.775655
\(320\) 0 0
\(321\) −3.45446 −0.192809
\(322\) 11.5487 0.643583
\(323\) −3.70727 −0.206278
\(324\) 0.0731828 0.00406571
\(325\) 0 0
\(326\) 6.37623 0.353147
\(327\) −4.66896 −0.258194
\(328\) −33.4802 −1.84864
\(329\) 1.95289 0.107666
\(330\) 0 0
\(331\) −26.2820 −1.44459 −0.722295 0.691585i \(-0.756913\pi\)
−0.722295 + 0.691585i \(0.756913\pi\)
\(332\) 0.278572 0.0152886
\(333\) −8.57475 −0.469893
\(334\) 29.5568 1.61728
\(335\) 0 0
\(336\) −4.14101 −0.225911
\(337\) −26.2905 −1.43214 −0.716068 0.698031i \(-0.754060\pi\)
−0.716068 + 0.698031i \(0.754060\pi\)
\(338\) −1.43986 −0.0783178
\(339\) −3.28738 −0.178546
\(340\) 0 0
\(341\) −29.6966 −1.60816
\(342\) 1.85363 0.100233
\(343\) −1.00000 −0.0539949
\(344\) −19.8116 −1.06817
\(345\) 0 0
\(346\) 21.6199 1.16230
\(347\) −16.4928 −0.885378 −0.442689 0.896675i \(-0.645975\pi\)
−0.442689 + 0.896675i \(0.645975\pi\)
\(348\) −0.240579 −0.0128964
\(349\) 14.1793 0.759001 0.379501 0.925191i \(-0.376096\pi\)
0.379501 + 0.925191i \(0.376096\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 1.74375 0.0929419
\(353\) 29.8272 1.58754 0.793772 0.608215i \(-0.208114\pi\)
0.793772 + 0.608215i \(0.208114\pi\)
\(354\) −10.5606 −0.561289
\(355\) 0 0
\(356\) −0.413332 −0.0219065
\(357\) 2.87971 0.152410
\(358\) 7.70575 0.407262
\(359\) 1.34671 0.0710767 0.0355383 0.999368i \(-0.488685\pi\)
0.0355383 + 0.999368i \(0.488685\pi\)
\(360\) 0 0
\(361\) −17.3427 −0.912772
\(362\) −16.5104 −0.867766
\(363\) −6.75942 −0.354778
\(364\) 0.0731828 0.00383582
\(365\) 0 0
\(366\) 11.1724 0.583993
\(367\) −36.1771 −1.88843 −0.944214 0.329331i \(-0.893177\pi\)
−0.944214 + 0.329331i \(0.893177\pi\)
\(368\) −33.2139 −1.73139
\(369\) −12.0678 −0.628226
\(370\) 0 0
\(371\) −5.14101 −0.266908
\(372\) 0.515704 0.0267380
\(373\) 16.9089 0.875511 0.437755 0.899094i \(-0.355774\pi\)
0.437755 + 0.899094i \(0.355774\pi\)
\(374\) −17.4736 −0.903538
\(375\) 0 0
\(376\) −5.41798 −0.279411
\(377\) −3.28738 −0.169308
\(378\) −1.43986 −0.0740582
\(379\) −11.6131 −0.596523 −0.298261 0.954484i \(-0.596407\pi\)
−0.298261 + 0.954484i \(0.596407\pi\)
\(380\) 0 0
\(381\) −0.428386 −0.0219469
\(382\) −29.0499 −1.48632
\(383\) −14.6134 −0.746708 −0.373354 0.927689i \(-0.621792\pi\)
−0.373354 + 0.927689i \(0.621792\pi\)
\(384\) 11.8946 0.606995
\(385\) 0 0
\(386\) −37.9771 −1.93298
\(387\) −7.14101 −0.362998
\(388\) −0.504993 −0.0256371
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 23.0974 1.16808
\(392\) 2.77434 0.140125
\(393\) −14.1878 −0.715680
\(394\) −28.9875 −1.46037
\(395\) 0 0
\(396\) 0.308407 0.0154980
\(397\) −25.6982 −1.28975 −0.644877 0.764287i \(-0.723091\pi\)
−0.644877 + 0.764287i \(0.723091\pi\)
\(398\) −2.53331 −0.126983
\(399\) −1.28738 −0.0644494
\(400\) 0 0
\(401\) −19.5637 −0.976966 −0.488483 0.872573i \(-0.662450\pi\)
−0.488483 + 0.872573i \(0.662450\pi\)
\(402\) −17.3379 −0.864737
\(403\) 7.04680 0.351026
\(404\) −0.803103 −0.0399559
\(405\) 0 0
\(406\) 4.73334 0.234912
\(407\) −36.1357 −1.79118
\(408\) −7.98929 −0.395529
\(409\) 3.19316 0.157892 0.0789458 0.996879i \(-0.474845\pi\)
0.0789458 + 0.996879i \(0.474845\pi\)
\(410\) 0 0
\(411\) 14.7070 0.725441
\(412\) −1.20228 −0.0592319
\(413\) 7.33448 0.360906
\(414\) −11.5487 −0.567586
\(415\) 0 0
\(416\) −0.413779 −0.0202872
\(417\) −9.85363 −0.482535
\(418\) 7.81157 0.382076
\(419\) 30.2292 1.47680 0.738398 0.674366i \(-0.235583\pi\)
0.738398 + 0.674366i \(0.235583\pi\)
\(420\) 0 0
\(421\) 26.9510 1.31351 0.656755 0.754104i \(-0.271928\pi\)
0.656755 + 0.754104i \(0.271928\pi\)
\(422\) −22.4177 −1.09128
\(423\) −1.95289 −0.0949529
\(424\) 14.2629 0.692668
\(425\) 0 0
\(426\) 15.5345 0.752650
\(427\) −7.75942 −0.375505
\(428\) 0.252807 0.0122199
\(429\) 4.21419 0.203463
\(430\) 0 0
\(431\) 16.7713 0.807847 0.403923 0.914793i \(-0.367646\pi\)
0.403923 + 0.914793i \(0.367646\pi\)
\(432\) 4.14101 0.199234
\(433\) −25.8530 −1.24242 −0.621208 0.783646i \(-0.713358\pi\)
−0.621208 + 0.783646i \(0.713358\pi\)
\(434\) −10.1464 −0.487041
\(435\) 0 0
\(436\) 0.341688 0.0163639
\(437\) −10.3257 −0.493944
\(438\) 11.9878 0.572798
\(439\) 25.0553 1.19582 0.597912 0.801562i \(-0.295997\pi\)
0.597912 + 0.801562i \(0.295997\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 4.14637 0.197223
\(443\) −4.64762 −0.220815 −0.110408 0.993886i \(-0.535216\pi\)
−0.110408 + 0.993886i \(0.535216\pi\)
\(444\) 0.627525 0.0297810
\(445\) 0 0
\(446\) −16.6583 −0.788791
\(447\) 3.38663 0.160182
\(448\) −7.68624 −0.363141
\(449\) 0.494696 0.0233462 0.0116731 0.999932i \(-0.496284\pi\)
0.0116731 + 0.999932i \(0.496284\pi\)
\(450\) 0 0
\(451\) −50.8561 −2.39472
\(452\) 0.240579 0.0113159
\(453\) 21.8951 1.02872
\(454\) −15.8985 −0.746155
\(455\) 0 0
\(456\) 3.57161 0.167256
\(457\) −26.4698 −1.23821 −0.619103 0.785310i \(-0.712504\pi\)
−0.619103 + 0.785310i \(0.712504\pi\)
\(458\) 19.7365 0.922225
\(459\) −2.87971 −0.134413
\(460\) 0 0
\(461\) −1.08168 −0.0503786 −0.0251893 0.999683i \(-0.508019\pi\)
−0.0251893 + 0.999683i \(0.508019\pi\)
\(462\) −6.06783 −0.282301
\(463\) 32.7098 1.52015 0.760076 0.649834i \(-0.225162\pi\)
0.760076 + 0.649834i \(0.225162\pi\)
\(464\) −13.6131 −0.631970
\(465\) 0 0
\(466\) 1.77854 0.0823894
\(467\) 17.0797 0.790356 0.395178 0.918605i \(-0.370683\pi\)
0.395178 + 0.918605i \(0.370683\pi\)
\(468\) −0.0731828 −0.00338288
\(469\) 12.0414 0.556022
\(470\) 0 0
\(471\) −0.867482 −0.0399715
\(472\) −20.3483 −0.936608
\(473\) −30.0936 −1.38370
\(474\) 6.43910 0.295758
\(475\) 0 0
\(476\) −0.210745 −0.00965950
\(477\) 5.14101 0.235391
\(478\) −27.5101 −1.25828
\(479\) −17.1790 −0.784929 −0.392464 0.919767i \(-0.628377\pi\)
−0.392464 + 0.919767i \(0.628377\pi\)
\(480\) 0 0
\(481\) 8.57475 0.390975
\(482\) 17.3257 0.789164
\(483\) 8.02072 0.364955
\(484\) 0.494674 0.0224852
\(485\) 0 0
\(486\) 1.43986 0.0653132
\(487\) 4.28264 0.194065 0.0970325 0.995281i \(-0.469065\pi\)
0.0970325 + 0.995281i \(0.469065\pi\)
\(488\) 21.5273 0.974493
\(489\) 4.42839 0.200259
\(490\) 0 0
\(491\) 27.4545 1.23900 0.619501 0.784996i \(-0.287335\pi\)
0.619501 + 0.784996i \(0.287335\pi\)
\(492\) 0.883158 0.0398158
\(493\) 9.46669 0.426358
\(494\) −1.85363 −0.0833990
\(495\) 0 0
\(496\) 29.1809 1.31026
\(497\) −10.7889 −0.483950
\(498\) 5.48085 0.245603
\(499\) −37.0452 −1.65837 −0.829185 0.558974i \(-0.811195\pi\)
−0.829185 + 0.558974i \(0.811195\pi\)
\(500\) 0 0
\(501\) 20.5276 0.917108
\(502\) 21.6023 0.964160
\(503\) −3.23744 −0.144350 −0.0721752 0.997392i \(-0.522994\pi\)
−0.0721752 + 0.997392i \(0.522994\pi\)
\(504\) −2.77434 −0.123579
\(505\) 0 0
\(506\) −48.6683 −2.16357
\(507\) −1.00000 −0.0444116
\(508\) 0.0313505 0.00139095
\(509\) −30.4877 −1.35134 −0.675672 0.737202i \(-0.736147\pi\)
−0.675672 + 0.737202i \(0.736147\pi\)
\(510\) 0 0
\(511\) −8.32568 −0.368306
\(512\) 21.2637 0.939730
\(513\) 1.28738 0.0568390
\(514\) −18.7557 −0.827277
\(515\) 0 0
\(516\) 0.522599 0.0230062
\(517\) −8.22987 −0.361949
\(518\) −12.3464 −0.542470
\(519\) 15.0154 0.659101
\(520\) 0 0
\(521\) −25.3602 −1.11105 −0.555526 0.831499i \(-0.687483\pi\)
−0.555526 + 0.831499i \(0.687483\pi\)
\(522\) −4.73334 −0.207173
\(523\) 7.00314 0.306226 0.153113 0.988209i \(-0.451070\pi\)
0.153113 + 0.988209i \(0.451070\pi\)
\(524\) 1.03830 0.0453585
\(525\) 0 0
\(526\) −3.73710 −0.162945
\(527\) −20.2927 −0.883965
\(528\) 17.4510 0.759458
\(529\) 41.3320 1.79704
\(530\) 0 0
\(531\) −7.33448 −0.318289
\(532\) 0.0942138 0.00408469
\(533\) 12.0678 0.522716
\(534\) −8.13221 −0.351915
\(535\) 0 0
\(536\) −33.4070 −1.44296
\(537\) 5.35176 0.230945
\(538\) 33.5603 1.44689
\(539\) 4.21419 0.181518
\(540\) 0 0
\(541\) −1.42463 −0.0612495 −0.0306248 0.999531i \(-0.509750\pi\)
−0.0306248 + 0.999531i \(0.509750\pi\)
\(542\) 8.71422 0.374308
\(543\) −11.4667 −0.492083
\(544\) 1.19156 0.0510879
\(545\) 0 0
\(546\) 1.43986 0.0616201
\(547\) 22.0499 0.942787 0.471394 0.881923i \(-0.343751\pi\)
0.471394 + 0.881923i \(0.343751\pi\)
\(548\) −1.07630 −0.0459771
\(549\) 7.75942 0.331164
\(550\) 0 0
\(551\) −4.23209 −0.180293
\(552\) −22.2522 −0.947116
\(553\) −4.47204 −0.190171
\(554\) 21.8760 0.929420
\(555\) 0 0
\(556\) 0.721117 0.0305822
\(557\) 9.08319 0.384867 0.192434 0.981310i \(-0.438362\pi\)
0.192434 + 0.981310i \(0.438362\pi\)
\(558\) 10.1464 0.429530
\(559\) 7.14101 0.302033
\(560\) 0 0
\(561\) −12.1357 −0.512368
\(562\) 10.9070 0.460084
\(563\) −34.1350 −1.43862 −0.719310 0.694689i \(-0.755542\pi\)
−0.719310 + 0.694689i \(0.755542\pi\)
\(564\) 0.142918 0.00601794
\(565\) 0 0
\(566\) 27.4974 1.15580
\(567\) −1.00000 −0.0419961
\(568\) 29.9322 1.25593
\(569\) 1.23522 0.0517833 0.0258916 0.999665i \(-0.491758\pi\)
0.0258916 + 0.999665i \(0.491758\pi\)
\(570\) 0 0
\(571\) −26.4613 −1.10737 −0.553686 0.832725i \(-0.686779\pi\)
−0.553686 + 0.832725i \(0.686779\pi\)
\(572\) −0.308407 −0.0128951
\(573\) −20.1756 −0.842847
\(574\) −17.3759 −0.725257
\(575\) 0 0
\(576\) 7.68624 0.320260
\(577\) 23.2852 0.969374 0.484687 0.874688i \(-0.338934\pi\)
0.484687 + 0.874688i \(0.338934\pi\)
\(578\) 12.5372 0.521479
\(579\) −26.3756 −1.09613
\(580\) 0 0
\(581\) −3.80653 −0.157921
\(582\) −9.93562 −0.411845
\(583\) 21.6652 0.897281
\(584\) 23.0982 0.955812
\(585\) 0 0
\(586\) −5.46324 −0.225684
\(587\) 10.7155 0.442274 0.221137 0.975243i \(-0.429023\pi\)
0.221137 + 0.975243i \(0.429023\pi\)
\(588\) −0.0731828 −0.00301801
\(589\) 9.07187 0.373800
\(590\) 0 0
\(591\) −20.1322 −0.828128
\(592\) 35.5081 1.45938
\(593\) −14.1008 −0.579049 −0.289525 0.957171i \(-0.593497\pi\)
−0.289525 + 0.957171i \(0.593497\pi\)
\(594\) 6.06783 0.248966
\(595\) 0 0
\(596\) −0.247843 −0.0101521
\(597\) −1.75942 −0.0720083
\(598\) 11.5487 0.472260
\(599\) −38.9296 −1.59062 −0.795311 0.606202i \(-0.792692\pi\)
−0.795311 + 0.606202i \(0.792692\pi\)
\(600\) 0 0
\(601\) −32.6162 −1.33044 −0.665221 0.746646i \(-0.731663\pi\)
−0.665221 + 0.746646i \(0.731663\pi\)
\(602\) −10.2820 −0.419064
\(603\) −12.0414 −0.490365
\(604\) −1.60234 −0.0651984
\(605\) 0 0
\(606\) −15.8009 −0.641866
\(607\) −19.3203 −0.784188 −0.392094 0.919925i \(-0.628249\pi\)
−0.392094 + 0.919925i \(0.628249\pi\)
\(608\) −0.532689 −0.0216034
\(609\) 3.28738 0.133211
\(610\) 0 0
\(611\) 1.95289 0.0790056
\(612\) 0.210745 0.00851888
\(613\) −35.3197 −1.42655 −0.713275 0.700885i \(-0.752789\pi\)
−0.713275 + 0.700885i \(0.752789\pi\)
\(614\) −7.47740 −0.301763
\(615\) 0 0
\(616\) −11.6916 −0.471068
\(617\) 24.4318 0.983589 0.491794 0.870711i \(-0.336341\pi\)
0.491794 + 0.870711i \(0.336341\pi\)
\(618\) −23.6545 −0.951523
\(619\) 2.90892 0.116919 0.0584597 0.998290i \(-0.481381\pi\)
0.0584597 + 0.998290i \(0.481381\pi\)
\(620\) 0 0
\(621\) −8.02072 −0.321860
\(622\) 19.9472 0.799811
\(623\) 5.64793 0.226280
\(624\) −4.14101 −0.165773
\(625\) 0 0
\(626\) −46.9626 −1.87700
\(627\) 5.42525 0.216664
\(628\) 0.0634848 0.00253332
\(629\) −24.6928 −0.984566
\(630\) 0 0
\(631\) 5.66521 0.225528 0.112764 0.993622i \(-0.464030\pi\)
0.112764 + 0.993622i \(0.464030\pi\)
\(632\) 12.4070 0.493522
\(633\) −15.5694 −0.618828
\(634\) 17.3330 0.688380
\(635\) 0 0
\(636\) −0.376234 −0.0149186
\(637\) −1.00000 −0.0396214
\(638\) −19.9472 −0.789718
\(639\) 10.7889 0.426804
\(640\) 0 0
\(641\) 12.2842 0.485198 0.242599 0.970127i \(-0.422000\pi\)
0.242599 + 0.970127i \(0.422000\pi\)
\(642\) 4.97392 0.196305
\(643\) −9.14950 −0.360821 −0.180411 0.983591i \(-0.557743\pi\)
−0.180411 + 0.983591i \(0.557743\pi\)
\(644\) −0.586979 −0.0231302
\(645\) 0 0
\(646\) 5.33793 0.210018
\(647\) −38.4108 −1.51008 −0.755042 0.655677i \(-0.772383\pi\)
−0.755042 + 0.655677i \(0.772383\pi\)
\(648\) 2.77434 0.108986
\(649\) −30.9089 −1.21328
\(650\) 0 0
\(651\) −7.04680 −0.276186
\(652\) −0.324082 −0.0126920
\(653\) −28.4805 −1.11453 −0.557265 0.830335i \(-0.688149\pi\)
−0.557265 + 0.830335i \(0.688149\pi\)
\(654\) 6.72263 0.262876
\(655\) 0 0
\(656\) 49.9730 1.95112
\(657\) 8.32568 0.324816
\(658\) −2.81188 −0.109619
\(659\) −30.4384 −1.18571 −0.592856 0.805309i \(-0.702000\pi\)
−0.592856 + 0.805309i \(0.702000\pi\)
\(660\) 0 0
\(661\) 36.7710 1.43023 0.715114 0.699008i \(-0.246375\pi\)
0.715114 + 0.699008i \(0.246375\pi\)
\(662\) 37.8423 1.47078
\(663\) 2.87971 0.111839
\(664\) 10.5606 0.409830
\(665\) 0 0
\(666\) 12.3464 0.478413
\(667\) 26.3671 1.02094
\(668\) −1.50227 −0.0581246
\(669\) −11.5694 −0.447299
\(670\) 0 0
\(671\) 32.6997 1.26236
\(672\) 0.413779 0.0159619
\(673\) −26.7961 −1.03291 −0.516457 0.856313i \(-0.672750\pi\)
−0.516457 + 0.856313i \(0.672750\pi\)
\(674\) 37.8545 1.45810
\(675\) 0 0
\(676\) 0.0731828 0.00281472
\(677\) 20.3050 0.780383 0.390191 0.920734i \(-0.372409\pi\)
0.390191 + 0.920734i \(0.372409\pi\)
\(678\) 4.73334 0.181783
\(679\) 6.90043 0.264814
\(680\) 0 0
\(681\) −11.0418 −0.423121
\(682\) 42.7587 1.63732
\(683\) −44.9240 −1.71897 −0.859484 0.511162i \(-0.829215\pi\)
−0.859484 + 0.511162i \(0.829215\pi\)
\(684\) −0.0942138 −0.00360235
\(685\) 0 0
\(686\) 1.43986 0.0549739
\(687\) 13.7073 0.522965
\(688\) 29.5710 1.12738
\(689\) −5.14101 −0.195857
\(690\) 0 0
\(691\) 22.0085 0.837243 0.418621 0.908161i \(-0.362513\pi\)
0.418621 + 0.908161i \(0.362513\pi\)
\(692\) −1.09887 −0.0417726
\(693\) −4.21419 −0.160084
\(694\) 23.7472 0.901431
\(695\) 0 0
\(696\) −9.12029 −0.345704
\(697\) −34.7518 −1.31632
\(698\) −20.4162 −0.772763
\(699\) 1.23522 0.0467205
\(700\) 0 0
\(701\) 31.4645 1.18840 0.594198 0.804319i \(-0.297469\pi\)
0.594198 + 0.804319i \(0.297469\pi\)
\(702\) −1.43986 −0.0543438
\(703\) 11.0389 0.416341
\(704\) 32.3913 1.22079
\(705\) 0 0
\(706\) −42.9469 −1.61633
\(707\) 10.9739 0.412717
\(708\) 0.536758 0.0201726
\(709\) −25.3373 −0.951563 −0.475781 0.879564i \(-0.657835\pi\)
−0.475781 + 0.879564i \(0.657835\pi\)
\(710\) 0 0
\(711\) 4.47204 0.167715
\(712\) −15.6693 −0.587231
\(713\) −56.5204 −2.11670
\(714\) −4.14637 −0.155174
\(715\) 0 0
\(716\) −0.391657 −0.0146369
\(717\) −19.1061 −0.713532
\(718\) −1.93907 −0.0723654
\(719\) −43.8002 −1.63347 −0.816737 0.577011i \(-0.804219\pi\)
−0.816737 + 0.577011i \(0.804219\pi\)
\(720\) 0 0
\(721\) 16.4284 0.611825
\(722\) 24.9709 0.929322
\(723\) 12.0329 0.447510
\(724\) 0.839165 0.0311873
\(725\) 0 0
\(726\) 9.73259 0.361210
\(727\) −0.944090 −0.0350144 −0.0175072 0.999847i \(-0.505573\pi\)
−0.0175072 + 0.999847i \(0.505573\pi\)
\(728\) 2.77434 0.102824
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −20.5640 −0.760588
\(732\) −0.567856 −0.0209886
\(733\) 5.09957 0.188357 0.0941784 0.995555i \(-0.469978\pi\)
0.0941784 + 0.995555i \(0.469978\pi\)
\(734\) 52.0898 1.92267
\(735\) 0 0
\(736\) 3.31881 0.122333
\(737\) −50.7450 −1.86921
\(738\) 17.3759 0.639617
\(739\) −8.25129 −0.303529 −0.151764 0.988417i \(-0.548495\pi\)
−0.151764 + 0.988417i \(0.548495\pi\)
\(740\) 0 0
\(741\) −1.28738 −0.0472929
\(742\) 7.40231 0.271747
\(743\) −38.4962 −1.41229 −0.706145 0.708068i \(-0.749567\pi\)
−0.706145 + 0.708068i \(0.749567\pi\)
\(744\) 19.5502 0.716745
\(745\) 0 0
\(746\) −24.3464 −0.891385
\(747\) 3.80653 0.139274
\(748\) 0.888121 0.0324729
\(749\) −3.45446 −0.126223
\(750\) 0 0
\(751\) −41.3175 −1.50770 −0.753848 0.657049i \(-0.771805\pi\)
−0.753848 + 0.657049i \(0.771805\pi\)
\(752\) 8.08695 0.294901
\(753\) 15.0031 0.546745
\(754\) 4.73334 0.172378
\(755\) 0 0
\(756\) 0.0731828 0.00266163
\(757\) 39.7158 1.44349 0.721747 0.692157i \(-0.243339\pi\)
0.721747 + 0.692157i \(0.243339\pi\)
\(758\) 16.7211 0.607338
\(759\) −33.8009 −1.22689
\(760\) 0 0
\(761\) 10.7390 0.389289 0.194644 0.980874i \(-0.437645\pi\)
0.194644 + 0.980874i \(0.437645\pi\)
\(762\) 0.616813 0.0223448
\(763\) −4.66896 −0.169028
\(764\) 1.47651 0.0534181
\(765\) 0 0
\(766\) 21.0411 0.760247
\(767\) 7.33448 0.264833
\(768\) −1.75406 −0.0632944
\(769\) −50.5549 −1.82306 −0.911529 0.411237i \(-0.865097\pi\)
−0.911529 + 0.411237i \(0.865097\pi\)
\(770\) 0 0
\(771\) −13.0261 −0.469123
\(772\) 1.93024 0.0694709
\(773\) −8.91770 −0.320747 −0.160374 0.987056i \(-0.551270\pi\)
−0.160374 + 0.987056i \(0.551270\pi\)
\(774\) 10.2820 0.369580
\(775\) 0 0
\(776\) −19.1441 −0.687234
\(777\) −8.57475 −0.307617
\(778\) −8.63913 −0.309728
\(779\) 15.5358 0.556629
\(780\) 0 0
\(781\) 45.4667 1.62693
\(782\) −33.2568 −1.18926
\(783\) −3.28738 −0.117481
\(784\) −4.14101 −0.147893
\(785\) 0 0
\(786\) 20.4284 0.728656
\(787\) −22.3671 −0.797302 −0.398651 0.917103i \(-0.630521\pi\)
−0.398651 + 0.917103i \(0.630521\pi\)
\(788\) 1.47333 0.0524853
\(789\) −2.59547 −0.0924012
\(790\) 0 0
\(791\) −3.28738 −0.116886
\(792\) 11.6916 0.415443
\(793\) −7.75942 −0.275545
\(794\) 37.0016 1.31314
\(795\) 0 0
\(796\) 0.128759 0.00456375
\(797\) −20.5518 −0.727983 −0.363991 0.931402i \(-0.618586\pi\)
−0.363991 + 0.931402i \(0.618586\pi\)
\(798\) 1.85363 0.0656179
\(799\) −5.62377 −0.198955
\(800\) 0 0
\(801\) −5.64793 −0.199560
\(802\) 28.1689 0.994680
\(803\) 35.0860 1.23816
\(804\) 0.881227 0.0310785
\(805\) 0 0
\(806\) −10.1464 −0.357390
\(807\) 23.3081 0.820484
\(808\) −30.4454 −1.07106
\(809\) 7.83443 0.275444 0.137722 0.990471i \(-0.456022\pi\)
0.137722 + 0.990471i \(0.456022\pi\)
\(810\) 0 0
\(811\) −47.5502 −1.66971 −0.834857 0.550468i \(-0.814449\pi\)
−0.834857 + 0.550468i \(0.814449\pi\)
\(812\) −0.240579 −0.00844268
\(813\) 6.05215 0.212258
\(814\) 52.0301 1.82365
\(815\) 0 0
\(816\) 11.9249 0.417455
\(817\) 9.19316 0.321628
\(818\) −4.59769 −0.160754
\(819\) 1.00000 0.0349428
\(820\) 0 0
\(821\) 4.42494 0.154431 0.0772157 0.997014i \(-0.475397\pi\)
0.0772157 + 0.997014i \(0.475397\pi\)
\(822\) −21.1759 −0.738594
\(823\) 28.7525 1.00225 0.501124 0.865375i \(-0.332920\pi\)
0.501124 + 0.865375i \(0.332920\pi\)
\(824\) −45.5779 −1.58778
\(825\) 0 0
\(826\) −10.5606 −0.367450
\(827\) 5.58667 0.194267 0.0971337 0.995271i \(-0.469033\pi\)
0.0971337 + 0.995271i \(0.469033\pi\)
\(828\) 0.586979 0.0203989
\(829\) 53.4974 1.85804 0.929021 0.370027i \(-0.120652\pi\)
0.929021 + 0.370027i \(0.120652\pi\)
\(830\) 0 0
\(831\) 15.1932 0.527045
\(832\) −7.68624 −0.266472
\(833\) 2.87971 0.0997760
\(834\) 14.1878 0.491284
\(835\) 0 0
\(836\) −0.397035 −0.0137317
\(837\) 7.04680 0.243573
\(838\) −43.5257 −1.50357
\(839\) −43.0405 −1.48592 −0.742962 0.669334i \(-0.766580\pi\)
−0.742962 + 0.669334i \(0.766580\pi\)
\(840\) 0 0
\(841\) −18.1932 −0.627350
\(842\) −38.8055 −1.33733
\(843\) 7.57506 0.260899
\(844\) 1.13941 0.0392202
\(845\) 0 0
\(846\) 2.81188 0.0966745
\(847\) −6.75942 −0.232256
\(848\) −21.2890 −0.731066
\(849\) 19.0974 0.655419
\(850\) 0 0
\(851\) −68.7757 −2.35760
\(852\) −0.789565 −0.0270501
\(853\) 30.7434 1.05263 0.526316 0.850289i \(-0.323573\pi\)
0.526316 + 0.850289i \(0.323573\pi\)
\(854\) 11.1724 0.382313
\(855\) 0 0
\(856\) 9.58384 0.327569
\(857\) 24.5449 0.838438 0.419219 0.907885i \(-0.362304\pi\)
0.419219 + 0.907885i \(0.362304\pi\)
\(858\) −6.06783 −0.207152
\(859\) −52.1243 −1.77846 −0.889229 0.457461i \(-0.848759\pi\)
−0.889229 + 0.457461i \(0.848759\pi\)
\(860\) 0 0
\(861\) −12.0678 −0.411270
\(862\) −24.1483 −0.822494
\(863\) −22.2563 −0.757612 −0.378806 0.925476i \(-0.623665\pi\)
−0.378806 + 0.925476i \(0.623665\pi\)
\(864\) −0.413779 −0.0140771
\(865\) 0 0
\(866\) 37.2246 1.26494
\(867\) 8.70727 0.295714
\(868\) 0.515704 0.0175041
\(869\) 18.8461 0.639309
\(870\) 0 0
\(871\) 12.0414 0.408009
\(872\) 12.9533 0.438654
\(873\) −6.90043 −0.233544
\(874\) 14.8675 0.502900
\(875\) 0 0
\(876\) −0.609297 −0.0205862
\(877\) −41.0547 −1.38632 −0.693159 0.720785i \(-0.743782\pi\)
−0.693159 + 0.720785i \(0.743782\pi\)
\(878\) −36.0760 −1.21751
\(879\) −3.79430 −0.127979
\(880\) 0 0
\(881\) −31.0568 −1.04633 −0.523165 0.852231i \(-0.675249\pi\)
−0.523165 + 0.852231i \(0.675249\pi\)
\(882\) −1.43986 −0.0484824
\(883\) 56.5357 1.90258 0.951289 0.308300i \(-0.0997600\pi\)
0.951289 + 0.308300i \(0.0997600\pi\)
\(884\) −0.210745 −0.00708813
\(885\) 0 0
\(886\) 6.69190 0.224819
\(887\) −9.94785 −0.334016 −0.167008 0.985956i \(-0.553411\pi\)
−0.167008 + 0.985956i \(0.553411\pi\)
\(888\) 23.7893 0.798315
\(889\) −0.428386 −0.0143676
\(890\) 0 0
\(891\) 4.21419 0.141181
\(892\) 0.846681 0.0283490
\(893\) 2.51411 0.0841314
\(894\) −4.87626 −0.163087
\(895\) 0 0
\(896\) 11.8946 0.397372
\(897\) 8.02072 0.267804
\(898\) −0.712291 −0.0237695
\(899\) −23.1655 −0.772612
\(900\) 0 0
\(901\) 14.8046 0.493213
\(902\) 73.2255 2.43814
\(903\) −7.14101 −0.237638
\(904\) 9.12029 0.303336
\(905\) 0 0
\(906\) −31.5257 −1.04737
\(907\) 31.2936 1.03909 0.519544 0.854444i \(-0.326102\pi\)
0.519544 + 0.854444i \(0.326102\pi\)
\(908\) 0.808067 0.0268166
\(909\) −10.9739 −0.363982
\(910\) 0 0
\(911\) −9.39320 −0.311210 −0.155605 0.987819i \(-0.549733\pi\)
−0.155605 + 0.987819i \(0.549733\pi\)
\(912\) −5.33104 −0.176528
\(913\) 16.0414 0.530894
\(914\) 38.1127 1.26066
\(915\) 0 0
\(916\) −1.00314 −0.0331446
\(917\) −14.1878 −0.468523
\(918\) 4.14637 0.136850
\(919\) 21.7066 0.716036 0.358018 0.933715i \(-0.383453\pi\)
0.358018 + 0.933715i \(0.383453\pi\)
\(920\) 0 0
\(921\) −5.19316 −0.171120
\(922\) 1.55746 0.0512921
\(923\) −10.7889 −0.355122
\(924\) 0.308407 0.0101458
\(925\) 0 0
\(926\) −47.0974 −1.54771
\(927\) −16.4284 −0.539579
\(928\) 1.36025 0.0446523
\(929\) 35.6787 1.17058 0.585289 0.810824i \(-0.300981\pi\)
0.585289 + 0.810824i \(0.300981\pi\)
\(930\) 0 0
\(931\) −1.28738 −0.0421920
\(932\) −0.0903972 −0.00296106
\(933\) 13.8536 0.453548
\(934\) −24.5924 −0.804687
\(935\) 0 0
\(936\) −2.77434 −0.0906821
\(937\) 47.0866 1.53825 0.769127 0.639096i \(-0.220691\pi\)
0.769127 + 0.639096i \(0.220691\pi\)
\(938\) −17.3379 −0.566103
\(939\) −32.6162 −1.06439
\(940\) 0 0
\(941\) 45.2923 1.47649 0.738244 0.674534i \(-0.235655\pi\)
0.738244 + 0.674534i \(0.235655\pi\)
\(942\) 1.24905 0.0406962
\(943\) −96.7927 −3.15200
\(944\) 30.3722 0.988530
\(945\) 0 0
\(946\) 43.3304 1.40879
\(947\) −4.77134 −0.155048 −0.0775238 0.996991i \(-0.524701\pi\)
−0.0775238 + 0.996991i \(0.524701\pi\)
\(948\) −0.327277 −0.0106295
\(949\) −8.32568 −0.270263
\(950\) 0 0
\(951\) 12.0380 0.390359
\(952\) −7.98929 −0.258934
\(953\) 3.95572 0.128138 0.0640692 0.997945i \(-0.479592\pi\)
0.0640692 + 0.997945i \(0.479592\pi\)
\(954\) −7.40231 −0.239659
\(955\) 0 0
\(956\) 1.39824 0.0452223
\(957\) −13.8536 −0.447824
\(958\) 24.7353 0.799160
\(959\) 14.7070 0.474912
\(960\) 0 0
\(961\) 18.6573 0.601850
\(962\) −12.3464 −0.398064
\(963\) 3.45446 0.111318
\(964\) −0.880605 −0.0283624
\(965\) 0 0
\(966\) −11.5487 −0.371573
\(967\) −50.2292 −1.61526 −0.807632 0.589687i \(-0.799251\pi\)
−0.807632 + 0.589687i \(0.799251\pi\)
\(968\) 18.7529 0.602742
\(969\) 3.70727 0.119095
\(970\) 0 0
\(971\) 38.6576 1.24058 0.620291 0.784372i \(-0.287014\pi\)
0.620291 + 0.784372i \(0.287014\pi\)
\(972\) −0.0731828 −0.00234734
\(973\) −9.85363 −0.315893
\(974\) −6.16638 −0.197584
\(975\) 0 0
\(976\) −32.1318 −1.02852
\(977\) 20.6134 0.659480 0.329740 0.944072i \(-0.393039\pi\)
0.329740 + 0.944072i \(0.393039\pi\)
\(978\) −6.37623 −0.203889
\(979\) −23.8015 −0.760699
\(980\) 0 0
\(981\) 4.66896 0.149069
\(982\) −39.5304 −1.26147
\(983\) 4.48620 0.143088 0.0715438 0.997437i \(-0.477207\pi\)
0.0715438 + 0.997437i \(0.477207\pi\)
\(984\) 33.4802 1.06731
\(985\) 0 0
\(986\) −13.6307 −0.434089
\(987\) −1.95289 −0.0621613
\(988\) 0.0942138 0.00299734
\(989\) −57.2760 −1.82127
\(990\) 0 0
\(991\) −32.7971 −1.04183 −0.520917 0.853607i \(-0.674410\pi\)
−0.520917 + 0.853607i \(0.674410\pi\)
\(992\) −2.91582 −0.0925773
\(993\) 26.2820 0.834035
\(994\) 15.5345 0.492725
\(995\) 0 0
\(996\) −0.278572 −0.00882691
\(997\) −0.491870 −0.0155777 −0.00778884 0.999970i \(-0.502479\pi\)
−0.00778884 + 0.999970i \(0.502479\pi\)
\(998\) 53.3397 1.68844
\(999\) 8.57475 0.271293
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 6825.2.a.bg.1.2 4
5.4 even 2 273.2.a.e.1.3 4
15.14 odd 2 819.2.a.k.1.2 4
20.19 odd 2 4368.2.a.br.1.3 4
35.34 odd 2 1911.2.a.s.1.3 4
65.64 even 2 3549.2.a.w.1.2 4
105.104 even 2 5733.2.a.bf.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.e.1.3 4 5.4 even 2
819.2.a.k.1.2 4 15.14 odd 2
1911.2.a.s.1.3 4 35.34 odd 2
3549.2.a.w.1.2 4 65.64 even 2
4368.2.a.br.1.3 4 20.19 odd 2
5733.2.a.bf.1.2 4 105.104 even 2
6825.2.a.bg.1.2 4 1.1 even 1 trivial