Properties

Label 819.2.a.k.1.2
Level $819$
Weight $2$
Character 819.1
Self dual yes
Analytic conductor $6.540$
Analytic rank $0$
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] = [819,2,Mod(1,819)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(819, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("819.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 819 = 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 819.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: \(6.53974792554\)
Analytic rank: \(0\)
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 \(\chi\) \(=\) 819.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.43986 q^{2} +0.0731828 q^{4} -0.926817 q^{5} +1.00000 q^{7} +2.77434 q^{8} +O(q^{10})\) \(q-1.43986 q^{2} +0.0731828 q^{4} -0.926817 q^{5} +1.00000 q^{7} +2.77434 q^{8} +1.33448 q^{10} -4.21419 q^{11} +1.00000 q^{13} -1.43986 q^{14} -4.14101 q^{16} +2.87971 q^{17} -1.28738 q^{19} -0.0678271 q^{20} +6.06783 q^{22} +8.02072 q^{23} -4.14101 q^{25} -1.43986 q^{26} +0.0731828 q^{28} -3.28738 q^{29} -7.04680 q^{31} +0.413779 q^{32} -4.14637 q^{34} -0.926817 q^{35} +8.57475 q^{37} +1.85363 q^{38} -2.57130 q^{40} +12.0678 q^{41} +7.14101 q^{43} -0.308407 q^{44} -11.5487 q^{46} -1.95289 q^{47} +1.00000 q^{49} +5.96245 q^{50} +0.0731828 q^{52} +5.14101 q^{53} +3.90579 q^{55} +2.77434 q^{56} +4.73334 q^{58} +7.33448 q^{59} +7.75942 q^{61} +10.1464 q^{62} +7.68624 q^{64} -0.926817 q^{65} +12.0414 q^{67} +0.210745 q^{68} +1.33448 q^{70} -10.7889 q^{71} -8.32568 q^{73} -12.3464 q^{74} -0.0942138 q^{76} -4.21419 q^{77} +4.47204 q^{79} +3.83796 q^{80} -17.3759 q^{82} +3.80653 q^{83} -2.66896 q^{85} -10.2820 q^{86} -11.6916 q^{88} +5.64793 q^{89} +1.00000 q^{91} +0.586979 q^{92} +2.81188 q^{94} +1.19316 q^{95} +6.90043 q^{97} -1.43986 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 7 q^{4} + 3 q^{5} + 4 q^{7} - 3 q^{8} - 4 q^{10} + 2 q^{11} + 4 q^{13} - q^{14} + 9 q^{16} + 2 q^{17} + 7 q^{19} + 32 q^{20} - 8 q^{22} - 3 q^{23} + 9 q^{25} - q^{26} + 7 q^{28} - q^{29} + 3 q^{31} - 7 q^{32} - 30 q^{34} + 3 q^{35} + 10 q^{37} - 6 q^{38} - 14 q^{40} + 16 q^{41} + 3 q^{43} + 12 q^{44} - 18 q^{46} - 5 q^{47} + 4 q^{49} - 13 q^{50} + 7 q^{52} - 5 q^{53} + 10 q^{55} - 3 q^{56} - 4 q^{58} + 20 q^{59} + 12 q^{61} + 54 q^{62} + 5 q^{64} + 3 q^{65} - 22 q^{67} + 10 q^{68} - 4 q^{70} - 13 q^{73} + 6 q^{74} - 6 q^{76} + 2 q^{77} + 11 q^{79} + 42 q^{80} + 10 q^{82} - q^{83} + 8 q^{85} + 10 q^{86} - 60 q^{88} + 5 q^{89} + 4 q^{91} - 34 q^{92} + 34 q^{94} - 13 q^{95} - 17 q^{97} - q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.43986 −1.01813 −0.509066 0.860728i \(-0.670009\pi\)
−0.509066 + 0.860728i \(0.670009\pi\)
\(3\) 0 0
\(4\) 0.0731828 0.0365914
\(5\) −0.926817 −0.414485 −0.207243 0.978290i \(-0.566449\pi\)
−0.207243 + 0.978290i \(0.566449\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 2.77434 0.980876
\(9\) 0 0
\(10\) 1.33448 0.422000
\(11\) −4.21419 −1.27063 −0.635313 0.772254i \(-0.719129\pi\)
−0.635313 + 0.772254i \(0.719129\pi\)
\(12\) 0 0
\(13\) 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 \(0.386448\pi\)
\(18\) 0 0
\(19\) −1.28738 −0.295344 −0.147672 0.989036i \(-0.547178\pi\)
−0.147672 + 0.989036i \(0.547178\pi\)
\(20\) −0.0678271 −0.0151666
\(21\) 0 0
\(22\) 6.06783 1.29367
\(23\) 8.02072 1.67244 0.836218 0.548397i \(-0.184762\pi\)
0.836218 + 0.548397i \(0.184762\pi\)
\(24\) 0 0
\(25\) −4.14101 −0.828202
\(26\) −1.43986 −0.282379
\(27\) 0 0
\(28\) 0.0731828 0.0138303
\(29\) −3.28738 −0.610450 −0.305225 0.952280i \(-0.598732\pi\)
−0.305225 + 0.952280i \(0.598732\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −4.14637 −0.711096
\(35\) −0.926817 −0.156661
\(36\) 0 0
\(37\) 8.57475 1.40968 0.704840 0.709366i \(-0.251019\pi\)
0.704840 + 0.709366i \(0.251019\pi\)
\(38\) 1.85363 0.300699
\(39\) 0 0
\(40\) −2.57130 −0.406559
\(41\) 12.0678 1.88468 0.942339 0.334660i \(-0.108621\pi\)
0.942339 + 0.334660i \(0.108621\pi\)
\(42\) 0 0
\(43\) 7.14101 1.08899 0.544497 0.838763i \(-0.316721\pi\)
0.544497 + 0.838763i \(0.316721\pi\)
\(44\) −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 \(0.545491\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 5.96245 0.843218
\(51\) 0 0
\(52\) 0.0731828 0.0101486
\(53\) 5.14101 0.706172 0.353086 0.935591i \(-0.385132\pi\)
0.353086 + 0.935591i \(0.385132\pi\)
\(54\) 0 0
\(55\) 3.90579 0.526656
\(56\) 2.77434 0.370736
\(57\) 0 0
\(58\) 4.73334 0.621519
\(59\) 7.33448 0.954868 0.477434 0.878668i \(-0.341567\pi\)
0.477434 + 0.878668i \(0.341567\pi\)
\(60\) 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
\(63\) 0 0
\(64\) 7.68624 0.960780
\(65\) −0.926817 −0.114958
\(66\) 0 0
\(67\) 12.0414 1.47110 0.735548 0.677473i \(-0.236925\pi\)
0.735548 + 0.677473i \(0.236925\pi\)
\(68\) 0.210745 0.0255566
\(69\) 0 0
\(70\) 1.33448 0.159501
\(71\) −10.7889 −1.28041 −0.640206 0.768203i \(-0.721151\pi\)
−0.640206 + 0.768203i \(0.721151\pi\)
\(72\) 0 0
\(73\) −8.32568 −0.974447 −0.487224 0.873277i \(-0.661990\pi\)
−0.487224 + 0.873277i \(0.661990\pi\)
\(74\) −12.3464 −1.43524
\(75\) 0 0
\(76\) −0.0942138 −0.0108071
\(77\) −4.21419 −0.480252
\(78\) 0 0
\(79\) 4.47204 0.503144 0.251572 0.967839i \(-0.419052\pi\)
0.251572 + 0.967839i \(0.419052\pi\)
\(80\) 3.83796 0.429097
\(81\) 0 0
\(82\) −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 0
\(85\) −2.66896 −0.289490
\(86\) −10.2820 −1.10874
\(87\) 0 0
\(88\) −11.6916 −1.24633
\(89\) 5.64793 0.598680 0.299340 0.954147i \(-0.403234\pi\)
0.299340 + 0.954147i \(0.403234\pi\)
\(90\) 0 0
\(91\) 1.00000 0.104828
\(92\) 0.586979 0.0611968
\(93\) 0 0
\(94\) 2.81188 0.290024
\(95\) 1.19316 0.122416
\(96\) 0 0
\(97\) 6.90043 0.700633 0.350316 0.936631i \(-0.386074\pi\)
0.350316 + 0.936631i \(0.386074\pi\)
\(98\) −1.43986 −0.145447
\(99\) 0 0
\(100\) −0.303051 −0.0303051
\(101\) 10.9739 1.09195 0.545973 0.837803i \(-0.316160\pi\)
0.545973 + 0.837803i \(0.316160\pi\)
\(102\) 0 0
\(103\) 16.4284 1.61874 0.809368 0.587301i \(-0.199810\pi\)
0.809368 + 0.587301i \(0.199810\pi\)
\(104\) 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 0
\(109\) 4.66896 0.447206 0.223603 0.974680i \(-0.428218\pi\)
0.223603 + 0.974680i \(0.428218\pi\)
\(110\) −5.62377 −0.536205
\(111\) 0 0
\(112\) −4.14101 −0.391289
\(113\) 3.28738 0.309250 0.154625 0.987973i \(-0.450583\pi\)
0.154625 + 0.987973i \(0.450583\pi\)
\(114\) 0 0
\(115\) −7.43374 −0.693200
\(116\) −0.240579 −0.0223372
\(117\) 0 0
\(118\) −10.5606 −0.972181
\(119\) 2.87971 0.263983
\(120\) 0 0
\(121\) 6.75942 0.614493
\(122\) −11.1724 −1.01151
\(123\) 0 0
\(124\) −0.515704 −0.0463116
\(125\) 8.47204 0.757763
\(126\) 0 0
\(127\) −0.428386 −0.0380131 −0.0190065 0.999819i \(-0.506050\pi\)
−0.0190065 + 0.999819i \(0.506050\pi\)
\(128\) −11.8946 −1.05135
\(129\) 0 0
\(130\) 1.33448 0.117042
\(131\) −14.1878 −1.23959 −0.619797 0.784762i \(-0.712785\pi\)
−0.619797 + 0.784762i \(0.712785\pi\)
\(132\) 0 0
\(133\) −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 \(0.716229\pi\)
\(138\) 0 0
\(139\) 9.85363 0.835774 0.417887 0.908499i \(-0.362771\pi\)
0.417887 + 0.908499i \(0.362771\pi\)
\(140\) −0.0678271 −0.00573244
\(141\) 0 0
\(142\) 15.5345 1.30363
\(143\) −4.21419 −0.352409
\(144\) 0 0
\(145\) 3.04680 0.253023
\(146\) 11.9878 0.992115
\(147\) 0 0
\(148\) 0.627525 0.0515822
\(149\) 3.38663 0.277444 0.138722 0.990331i \(-0.455701\pi\)
0.138722 + 0.990331i \(0.455701\pi\)
\(150\) 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
\(153\) 0 0
\(154\) 6.06783 0.488959
\(155\) 6.53109 0.524590
\(156\) 0 0
\(157\) −0.867482 −0.0692326 −0.0346163 0.999401i \(-0.511021\pi\)
−0.0346163 + 0.999401i \(0.511021\pi\)
\(158\) −6.43910 −0.512267
\(159\) 0 0
\(160\) −0.383498 −0.0303182
\(161\) 8.02072 0.632121
\(162\) 0 0
\(163\) 4.42839 0.346858 0.173429 0.984846i \(-0.444515\pi\)
0.173429 + 0.984846i \(0.444515\pi\)
\(164\) 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 \(0.792130\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 3.84292 0.294739
\(171\) 0 0
\(172\) 0.522599 0.0398478
\(173\) −15.0154 −1.14160 −0.570799 0.821090i \(-0.693366\pi\)
−0.570799 + 0.821090i \(0.693366\pi\)
\(174\) 0 0
\(175\) −4.14101 −0.313031
\(176\) 17.4510 1.31542
\(177\) 0 0
\(178\) −8.13221 −0.609535
\(179\) 5.35176 0.400009 0.200004 0.979795i \(-0.435904\pi\)
0.200004 + 0.979795i \(0.435904\pi\)
\(180\) 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
\(183\) 0 0
\(184\) 22.2522 1.64045
\(185\) −7.94723 −0.584292
\(186\) 0 0
\(187\) −12.1357 −0.887447
\(188\) −0.142918 −0.0104234
\(189\) 0 0
\(190\) −1.71798 −0.124635
\(191\) −20.1756 −1.45985 −0.729927 0.683525i \(-0.760446\pi\)
−0.729927 + 0.683525i \(0.760446\pi\)
\(192\) 0 0
\(193\) −26.3756 −1.89856 −0.949279 0.314435i \(-0.898185\pi\)
−0.949279 + 0.314435i \(0.898185\pi\)
\(194\) −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 \(0.245432\pi\)
\(198\) 0 0
\(199\) 1.75942 0.124722 0.0623610 0.998054i \(-0.480137\pi\)
0.0623610 + 0.998054i \(0.480137\pi\)
\(200\) −11.4886 −0.812364
\(201\) 0 0
\(202\) −15.8009 −1.11174
\(203\) −3.28738 −0.230729
\(204\) 0 0
\(205\) −11.1847 −0.781171
\(206\) −23.6545 −1.64809
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) −4.97392 −0.340010
\(215\) −6.61841 −0.451372
\(216\) 0 0
\(217\) −7.04680 −0.478368
\(218\) −6.72263 −0.455314
\(219\) 0 0
\(220\) 0.285836 0.0192711
\(221\) 2.87971 0.193710
\(222\) 0 0
\(223\) −11.5694 −0.774744 −0.387372 0.921923i \(-0.626617\pi\)
−0.387372 + 0.921923i \(0.626617\pi\)
\(224\) 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 0
\(229\) −13.7073 −0.905802 −0.452901 0.891561i \(-0.649611\pi\)
−0.452901 + 0.891561i \(0.649611\pi\)
\(230\) 10.7035 0.705769
\(231\) 0 0
\(232\) −9.12029 −0.598776
\(233\) −1.23522 −0.0809222 −0.0404611 0.999181i \(-0.512883\pi\)
−0.0404611 + 0.999181i \(0.512883\pi\)
\(234\) 0 0
\(235\) 1.80997 0.118070
\(236\) 0.536758 0.0349400
\(237\) 0 0
\(238\) −4.14637 −0.268769
\(239\) −19.1061 −1.23587 −0.617936 0.786228i \(-0.712031\pi\)
−0.617936 + 0.786228i \(0.712031\pi\)
\(240\) 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
\(243\) 0 0
\(244\) 0.567856 0.0363533
\(245\) −0.926817 −0.0592122
\(246\) 0 0
\(247\) −1.28738 −0.0819137
\(248\) −19.5502 −1.24144
\(249\) 0 0
\(250\) −12.1985 −0.771502
\(251\) 15.0031 0.946990 0.473495 0.880797i \(-0.342992\pi\)
0.473495 + 0.880797i \(0.342992\pi\)
\(252\) 0 0
\(253\) −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 \(0.366828\pi\)
\(258\) 0 0
\(259\) 8.57475 0.532809
\(260\) −0.0678271 −0.00420646
\(261\) 0 0
\(262\) 20.4284 1.26207
\(263\) 2.59547 0.160044 0.0800218 0.996793i \(-0.474501\pi\)
0.0800218 + 0.996793i \(0.474501\pi\)
\(264\) 0 0
\(265\) −4.76478 −0.292698
\(266\) 1.85363 0.113654
\(267\) 0 0
\(268\) 0.881227 0.0538295
\(269\) 23.3081 1.42112 0.710560 0.703637i \(-0.248442\pi\)
0.710560 + 0.703637i \(0.248442\pi\)
\(270\) 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
\(273\) 0 0
\(274\) 21.1759 1.27928
\(275\) 17.4510 1.05234
\(276\) 0 0
\(277\) 15.1932 0.912869 0.456434 0.889757i \(-0.349126\pi\)
0.456434 + 0.889757i \(0.349126\pi\)
\(278\) −14.1878 −0.850928
\(279\) 0 0
\(280\) −2.57130 −0.153665
\(281\) 7.57506 0.451890 0.225945 0.974140i \(-0.427453\pi\)
0.225945 + 0.974140i \(0.427453\pi\)
\(282\) 0 0
\(283\) 19.0974 1.13522 0.567610 0.823298i \(-0.307868\pi\)
0.567610 + 0.823298i \(0.307868\pi\)
\(284\) −0.789565 −0.0468521
\(285\) 0 0
\(286\) 6.06783 0.358798
\(287\) 12.0678 0.712341
\(288\) 0 0
\(289\) −8.70727 −0.512192
\(290\) −4.38695 −0.257610
\(291\) 0 0
\(292\) −0.609297 −0.0356564
\(293\) 3.79430 0.221665 0.110833 0.993839i \(-0.464648\pi\)
0.110833 + 0.993839i \(0.464648\pi\)
\(294\) 0 0
\(295\) −6.79772 −0.395779
\(296\) 23.7893 1.38272
\(297\) 0 0
\(298\) −4.87626 −0.282474
\(299\) 8.02072 0.463850
\(300\) 0 0
\(301\) 7.14101 0.411601
\(302\) 31.5257 1.81410
\(303\) 0 0
\(304\) 5.33104 0.305756
\(305\) −7.19156 −0.411788
\(306\) 0 0
\(307\) −5.19316 −0.296389 −0.148195 0.988958i \(-0.547346\pi\)
−0.148195 + 0.988958i \(0.547346\pi\)
\(308\) −0.308407 −0.0175731
\(309\) 0 0
\(310\) −9.40383 −0.534101
\(311\) 13.8536 0.785568 0.392784 0.919631i \(-0.371512\pi\)
0.392784 + 0.919631i \(0.371512\pi\)
\(312\) 0 0
\(313\) −32.6162 −1.84358 −0.921788 0.387694i \(-0.873272\pi\)
−0.921788 + 0.387694i \(0.873272\pi\)
\(314\) 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 \(0.609771\pi\)
\(318\) 0 0
\(319\) 13.8536 0.775655
\(320\) −7.12374 −0.398229
\(321\) 0 0
\(322\) −11.5487 −0.643583
\(323\) −3.70727 −0.206278
\(324\) 0 0
\(325\) −4.14101 −0.229702
\(326\) −6.37623 −0.353147
\(327\) 0 0
\(328\) 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
\(333\) 0 0
\(334\) 29.5568 1.61728
\(335\) −11.1602 −0.609748
\(336\) 0 0
\(337\) 26.2905 1.43214 0.716068 0.698031i \(-0.245940\pi\)
0.716068 + 0.698031i \(0.245940\pi\)
\(338\) −1.43986 −0.0783178
\(339\) 0 0
\(340\) −0.195322 −0.0105928
\(341\) 29.6966 1.60816
\(342\) 0 0
\(343\) 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 0
\(349\) 14.1793 0.759001 0.379501 0.925191i \(-0.376096\pi\)
0.379501 + 0.925191i \(0.376096\pi\)
\(350\) 5.96245 0.318707
\(351\) 0 0
\(352\) −1.74375 −0.0929419
\(353\) 29.8272 1.58754 0.793772 0.608215i \(-0.208114\pi\)
0.793772 + 0.608215i \(0.208114\pi\)
\(354\) 0 0
\(355\) 9.99938 0.530712
\(356\) 0.413332 0.0219065
\(357\) 0 0
\(358\) −7.70575 −0.407262
\(359\) −1.34671 −0.0710767 −0.0355383 0.999368i \(-0.511315\pi\)
−0.0355383 + 0.999368i \(0.511315\pi\)
\(360\) 0 0
\(361\) −17.3427 −0.912772
\(362\) −16.5104 −0.867766
\(363\) 0 0
\(364\) 0.0731828 0.00383582
\(365\) 7.71638 0.403894
\(366\) 0 0
\(367\) 36.1771 1.88843 0.944214 0.329331i \(-0.106823\pi\)
0.944214 + 0.329331i \(0.106823\pi\)
\(368\) −33.2139 −1.73139
\(369\) 0 0
\(370\) 11.4429 0.594886
\(371\) 5.14101 0.266908
\(372\) 0 0
\(373\) −16.9089 −0.875511 −0.437755 0.899094i \(-0.644226\pi\)
−0.437755 + 0.899094i \(0.644226\pi\)
\(374\) 17.4736 0.903538
\(375\) 0 0
\(376\) −5.41798 −0.279411
\(377\) −3.28738 −0.169308
\(378\) 0 0
\(379\) −11.6131 −0.596523 −0.298261 0.954484i \(-0.596407\pi\)
−0.298261 + 0.954484i \(0.596407\pi\)
\(380\) 0.0873190 0.00447937
\(381\) 0 0
\(382\) 29.0499 1.48632
\(383\) −14.6134 −0.746708 −0.373354 0.927689i \(-0.621792\pi\)
−0.373354 + 0.927689i \(0.621792\pi\)
\(384\) 0 0
\(385\) 3.90579 0.199057
\(386\) 37.9771 1.93298
\(387\) 0 0
\(388\) 0.504993 0.0256371
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 23.0974 1.16808
\(392\) 2.77434 0.140125
\(393\) 0 0
\(394\) −28.9875 −1.46037
\(395\) −4.14477 −0.208546
\(396\) 0 0
\(397\) 25.6982 1.28975 0.644877 0.764287i \(-0.276909\pi\)
0.644877 + 0.764287i \(0.276909\pi\)
\(398\) −2.53331 −0.126983
\(399\) 0 0
\(400\) 17.1480 0.857398
\(401\) 19.5637 0.976966 0.488483 0.872573i \(-0.337550\pi\)
0.488483 + 0.872573i \(0.337550\pi\)
\(402\) 0 0
\(403\) −7.04680 −0.351026
\(404\) 0.803103 0.0399559
\(405\) 0 0
\(406\) 4.73334 0.234912
\(407\) −36.1357 −1.79118
\(408\) 0 0
\(409\) 3.19316 0.157892 0.0789458 0.996879i \(-0.474845\pi\)
0.0789458 + 0.996879i \(0.474845\pi\)
\(410\) 16.1043 0.795335
\(411\) 0 0
\(412\) 1.20228 0.0592319
\(413\) 7.33448 0.360906
\(414\) 0 0
\(415\) −3.52796 −0.173181
\(416\) 0.413779 0.0202872
\(417\) 0 0
\(418\) −7.81157 −0.382076
\(419\) −30.2292 −1.47680 −0.738398 0.674366i \(-0.764417\pi\)
−0.738398 + 0.674366i \(0.764417\pi\)
\(420\) 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
\(423\) 0 0
\(424\) 14.2629 0.692668
\(425\) −11.9249 −0.578443
\(426\) 0 0
\(427\) 7.75942 0.375505
\(428\) 0.252807 0.0122199
\(429\) 0 0
\(430\) 9.52955 0.459556
\(431\) −16.7713 −0.807847 −0.403923 0.914793i \(-0.632354\pi\)
−0.403923 + 0.914793i \(0.632354\pi\)
\(432\) 0 0
\(433\) 25.8530 1.24242 0.621208 0.783646i \(-0.286642\pi\)
0.621208 + 0.783646i \(0.286642\pi\)
\(434\) 10.1464 0.487041
\(435\) 0 0
\(436\) 0.341688 0.0163639
\(437\) −10.3257 −0.493944
\(438\) 0 0
\(439\) 25.0553 1.19582 0.597912 0.801562i \(-0.295997\pi\)
0.597912 + 0.801562i \(0.295997\pi\)
\(440\) 10.8360 0.516585
\(441\) 0 0
\(442\) −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 0
\(445\) −5.23460 −0.248144
\(446\) 16.6583 0.788791
\(447\) 0 0
\(448\) 7.68624 0.363141
\(449\) −0.494696 −0.0233462 −0.0116731 0.999932i \(-0.503716\pi\)
−0.0116731 + 0.999932i \(0.503716\pi\)
\(450\) 0 0
\(451\) −50.8561 −2.39472
\(452\) 0.240579 0.0113159
\(453\) 0 0
\(454\) −15.8985 −0.746155
\(455\) −0.926817 −0.0434499
\(456\) 0 0
\(457\) 26.4698 1.23821 0.619103 0.785310i \(-0.287496\pi\)
0.619103 + 0.785310i \(0.287496\pi\)
\(458\) 19.7365 0.922225
\(459\) 0 0
\(460\) −0.544022 −0.0253652
\(461\) 1.08168 0.0503786 0.0251893 0.999683i \(-0.491981\pi\)
0.0251893 + 0.999683i \(0.491981\pi\)
\(462\) 0 0
\(463\) −32.7098 −1.52015 −0.760076 0.649834i \(-0.774838\pi\)
−0.760076 + 0.649834i \(0.774838\pi\)
\(464\) 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 0
\(469\) 12.0414 0.556022
\(470\) −2.60610 −0.120211
\(471\) 0 0
\(472\) 20.3483 0.936608
\(473\) −30.0936 −1.38370
\(474\) 0 0
\(475\) 5.33104 0.244605
\(476\) 0.210745 0.00965950
\(477\) 0 0
\(478\) 27.5101 1.25828
\(479\) 17.1790 0.784929 0.392464 0.919767i \(-0.371623\pi\)
0.392464 + 0.919767i \(0.371623\pi\)
\(480\) 0 0
\(481\) 8.57475 0.390975
\(482\) 17.3257 0.789164
\(483\) 0 0
\(484\) 0.494674 0.0224852
\(485\) −6.39544 −0.290402
\(486\) 0 0
\(487\) −4.28264 −0.194065 −0.0970325 0.995281i \(-0.530935\pi\)
−0.0970325 + 0.995281i \(0.530935\pi\)
\(488\) 21.5273 0.974493
\(489\) 0 0
\(490\) 1.33448 0.0602858
\(491\) −27.4545 −1.23900 −0.619501 0.784996i \(-0.712665\pi\)
−0.619501 + 0.784996i \(0.712665\pi\)
\(492\) 0 0
\(493\) −9.46669 −0.426358
\(494\) 1.85363 0.0833990
\(495\) 0 0
\(496\) 29.1809 1.31026
\(497\) −10.7889 −0.483950
\(498\) 0 0
\(499\) −37.0452 −1.65837 −0.829185 0.558974i \(-0.811195\pi\)
−0.829185 + 0.558974i \(0.811195\pi\)
\(500\) 0.620008 0.0277276
\(501\) 0 0
\(502\) −21.6023 −0.964160
\(503\) −3.23744 −0.144350 −0.0721752 0.997392i \(-0.522994\pi\)
−0.0721752 + 0.997392i \(0.522994\pi\)
\(504\) 0 0
\(505\) −10.1708 −0.452596
\(506\) 48.6683 2.16357
\(507\) 0 0
\(508\) −0.0313505 −0.00139095
\(509\) 30.4877 1.35134 0.675672 0.737202i \(-0.263853\pi\)
0.675672 + 0.737202i \(0.263853\pi\)
\(510\) 0 0
\(511\) −8.32568 −0.368306
\(512\) 21.2637 0.939730
\(513\) 0 0
\(514\) −18.7557 −0.827277
\(515\) −15.2261 −0.670943
\(516\) 0 0
\(517\) 8.22987 0.361949
\(518\) −12.3464 −0.542470
\(519\) 0 0
\(520\) −2.57130 −0.112759
\(521\) 25.3602 1.11105 0.555526 0.831499i \(-0.312517\pi\)
0.555526 + 0.831499i \(0.312517\pi\)
\(522\) 0 0
\(523\) −7.00314 −0.306226 −0.153113 0.988209i \(-0.548930\pi\)
−0.153113 + 0.988209i \(0.548930\pi\)
\(524\) −1.03830 −0.0453585
\(525\) 0 0
\(526\) −3.73710 −0.162945
\(527\) −20.2927 −0.883965
\(528\) 0 0
\(529\) 41.3320 1.79704
\(530\) 6.86059 0.298005
\(531\) 0 0
\(532\) −0.0942138 −0.00408469
\(533\) 12.0678 0.522716
\(534\) 0 0
\(535\) −3.20165 −0.138420
\(536\) 33.4070 1.44296
\(537\) 0 0
\(538\) −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
\(543\) 0 0
\(544\) 1.19156 0.0510879
\(545\) −4.32728 −0.185360
\(546\) 0 0
\(547\) −22.0499 −0.942787 −0.471394 0.881923i \(-0.656249\pi\)
−0.471394 + 0.881923i \(0.656249\pi\)
\(548\) −1.07630 −0.0459771
\(549\) 0 0
\(550\) −25.1269 −1.07142
\(551\) 4.23209 0.180293
\(552\) 0 0
\(553\) 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 \(0.438362\pi\)
\(558\) 0 0
\(559\) 7.14101 0.302033
\(560\) 3.83796 0.162183
\(561\) 0 0
\(562\) −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 0
\(565\) −3.04680 −0.128180
\(566\) −27.4974 −1.15580
\(567\) 0 0
\(568\) −29.9322 −1.25593
\(569\) −1.23522 −0.0517833 −0.0258916 0.999665i \(-0.508242\pi\)
−0.0258916 + 0.999665i \(0.508242\pi\)
\(570\) 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
\(573\) 0 0
\(574\) −17.3759 −0.725257
\(575\) −33.2139 −1.38511
\(576\) 0 0
\(577\) −23.2852 −0.969374 −0.484687 0.874688i \(-0.661066\pi\)
−0.484687 + 0.874688i \(0.661066\pi\)
\(578\) 12.5372 0.521479
\(579\) 0 0
\(580\) 0.222973 0.00925846
\(581\) 3.80653 0.157921
\(582\) 0 0
\(583\) −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 0
\(589\) 9.07187 0.373800
\(590\) 9.78774 0.402955
\(591\) 0 0
\(592\) −35.5081 −1.45938
\(593\) −14.1008 −0.579049 −0.289525 0.957171i \(-0.593497\pi\)
−0.289525 + 0.957171i \(0.593497\pi\)
\(594\) 0 0
\(595\) −2.66896 −0.109417
\(596\) 0.247843 0.0101521
\(597\) 0 0
\(598\) −11.5487 −0.472260
\(599\) 38.9296 1.59062 0.795311 0.606202i \(-0.207308\pi\)
0.795311 + 0.606202i \(0.207308\pi\)
\(600\) 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
\(603\) 0 0
\(604\) −1.60234 −0.0651984
\(605\) −6.26475 −0.254698
\(606\) 0 0
\(607\) 19.3203 0.784188 0.392094 0.919925i \(-0.371751\pi\)
0.392094 + 0.919925i \(0.371751\pi\)
\(608\) −0.532689 −0.0216034
\(609\) 0 0
\(610\) 10.3548 0.419254
\(611\) −1.95289 −0.0790056
\(612\) 0 0
\(613\) 35.3197 1.42655 0.713275 0.700885i \(-0.247211\pi\)
0.713275 + 0.700885i \(0.247211\pi\)
\(614\) 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 \(0.336341\pi\)
\(618\) 0 0
\(619\) 2.90892 0.116919 0.0584597 0.998290i \(-0.481381\pi\)
0.0584597 + 0.998290i \(0.481381\pi\)
\(620\) 0.477964 0.0191955
\(621\) 0 0
\(622\) −19.9472 −0.799811
\(623\) 5.64793 0.226280
\(624\) 0 0
\(625\) 12.8530 0.514121
\(626\) 46.9626 1.87700
\(627\) 0 0
\(628\) −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
\(633\) 0 0
\(634\) 17.3330 0.688380
\(635\) 0.397035 0.0157559
\(636\) 0 0
\(637\) 1.00000 0.0396214
\(638\) −19.9472 −0.789718
\(639\) 0 0
\(640\) 11.0241 0.435768
\(641\) −12.2842 −0.485198 −0.242599 0.970127i \(-0.578000\pi\)
−0.242599 + 0.970127i \(0.578000\pi\)
\(642\) 0 0
\(643\) 9.14950 0.360821 0.180411 0.983591i \(-0.442257\pi\)
0.180411 + 0.983591i \(0.442257\pi\)
\(644\) 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 \(0.772383\pi\)
\(648\) 0 0
\(649\) −30.9089 −1.21328
\(650\) 5.96245 0.233867
\(651\) 0 0
\(652\) 0.324082 0.0126920
\(653\) −28.4805 −1.11453 −0.557265 0.830335i \(-0.688149\pi\)
−0.557265 + 0.830335i \(0.688149\pi\)
\(654\) 0 0
\(655\) 13.1495 0.513794
\(656\) −49.9730 −1.95112
\(657\) 0 0
\(658\) 2.81188 0.109619
\(659\) 30.4384 1.18571 0.592856 0.805309i \(-0.298000\pi\)
0.592856 + 0.805309i \(0.298000\pi\)
\(660\) 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
\(663\) 0 0
\(664\) 10.5606 0.409830
\(665\) 1.19316 0.0462688
\(666\) 0 0
\(667\) −26.3671 −1.02094
\(668\) −1.50227 −0.0581246
\(669\) 0 0
\(670\) 16.0691 0.620803
\(671\) −32.6997 −1.26236
\(672\) 0 0
\(673\) 26.7961 1.03291 0.516457 0.856313i \(-0.327250\pi\)
0.516457 + 0.856313i \(0.327250\pi\)
\(674\) −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 \(0.372409\pi\)
\(678\) 0 0
\(679\) 6.90043 0.264814
\(680\) −7.40461 −0.283954
\(681\) 0 0
\(682\) −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 0
\(685\) 13.6307 0.520801
\(686\) −1.43986 −0.0549739
\(687\) 0 0
\(688\) −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
\(693\) 0 0
\(694\) 23.7472 0.901431
\(695\) −9.13252 −0.346416
\(696\) 0 0
\(697\) 34.7518 1.31632
\(698\) −20.4162 −0.772763
\(699\) 0 0
\(700\) −0.303051 −0.0114542
\(701\) −31.4645 −1.18840 −0.594198 0.804319i \(-0.702531\pi\)
−0.594198 + 0.804319i \(0.702531\pi\)
\(702\) 0 0
\(703\) −11.0389 −0.416341
\(704\) −32.3913 −1.22079
\(705\) 0 0
\(706\) −42.9469 −1.61633
\(707\) 10.9739 0.412717
\(708\) 0 0
\(709\) −25.3373 −0.951563 −0.475781 0.879564i \(-0.657835\pi\)
−0.475781 + 0.879564i \(0.657835\pi\)
\(710\) −14.3977 −0.540334
\(711\) 0 0
\(712\) 15.6693 0.587231
\(713\) −56.5204 −2.11670
\(714\) 0 0
\(715\) 3.90579 0.146068
\(716\) 0.391657 0.0146369
\(717\) 0 0
\(718\) 1.93907 0.0723654
\(719\) 43.8002 1.63347 0.816737 0.577011i \(-0.195781\pi\)
0.816737 + 0.577011i \(0.195781\pi\)
\(720\) 0 0
\(721\) 16.4284 0.611825
\(722\) 24.9709 0.929322
\(723\) 0 0
\(724\) 0.839165 0.0311873
\(725\) 13.6131 0.505576
\(726\) 0 0
\(727\) 0.944090 0.0350144 0.0175072 0.999847i \(-0.494427\pi\)
0.0175072 + 0.999847i \(0.494427\pi\)
\(728\) 2.77434 0.102824
\(729\) 0 0
\(730\) −11.1105 −0.411217
\(731\) 20.5640 0.760588
\(732\) 0 0
\(733\) −5.09957 −0.188357 −0.0941784 0.995555i \(-0.530022\pi\)
−0.0941784 + 0.995555i \(0.530022\pi\)
\(734\) −52.0898 −1.92267
\(735\) 0 0
\(736\) 3.31881 0.122333
\(737\) −50.7450 −1.86921
\(738\) 0 0
\(739\) −8.25129 −0.303529 −0.151764 0.988417i \(-0.548495\pi\)
−0.151764 + 0.988417i \(0.548495\pi\)
\(740\) −0.581601 −0.0213801
\(741\) 0 0
\(742\) −7.40231 −0.271747
\(743\) −38.4962 −1.41229 −0.706145 0.708068i \(-0.749567\pi\)
−0.706145 + 0.708068i \(0.749567\pi\)
\(744\) 0 0
\(745\) −3.13879 −0.114996
\(746\) 24.3464 0.891385
\(747\) 0 0
\(748\) −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
\(753\) 0 0
\(754\) 4.73334 0.172378
\(755\) 20.2927 0.738528
\(756\) 0 0
\(757\) −39.7158 −1.44349 −0.721747 0.692157i \(-0.756661\pi\)
−0.721747 + 0.692157i \(0.756661\pi\)
\(758\) 16.7211 0.607338
\(759\) 0 0
\(760\) 3.31023 0.120075
\(761\) −10.7390 −0.389289 −0.194644 0.980874i \(-0.562355\pi\)
−0.194644 + 0.980874i \(0.562355\pi\)
\(762\) 0 0
\(763\) 4.66896 0.169028
\(764\) −1.47651 −0.0534181
\(765\) 0 0
\(766\) 21.0411 0.760247
\(767\) 7.33448 0.264833
\(768\) 0 0
\(769\) −50.5549 −1.82306 −0.911529 0.411237i \(-0.865097\pi\)
−0.911529 + 0.411237i \(0.865097\pi\)
\(770\) −5.62377 −0.202666
\(771\) 0 0
\(772\) −1.93024 −0.0694709
\(773\) −8.91770 −0.320747 −0.160374 0.987056i \(-0.551270\pi\)
−0.160374 + 0.987056i \(0.551270\pi\)
\(774\) 0 0
\(775\) 29.1809 1.04821
\(776\) 19.1441 0.687234
\(777\) 0 0
\(778\) 8.63913 0.309728
\(779\) −15.5358 −0.556629
\(780\) 0 0
\(781\) 45.4667 1.62693
\(782\) −33.2568 −1.18926
\(783\) 0 0
\(784\) −4.14101 −0.147893
\(785\) 0.803998 0.0286959
\(786\) 0 0
\(787\) 22.3671 0.797302 0.398651 0.917103i \(-0.369479\pi\)
0.398651 + 0.917103i \(0.369479\pi\)
\(788\) 1.47333 0.0524853
\(789\) 0 0
\(790\) 5.96787 0.212327
\(791\) 3.28738 0.116886
\(792\) 0 0
\(793\) 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 \(0.618586\pi\)
\(798\) 0 0
\(799\) −5.62377 −0.198955
\(800\) −1.71346 −0.0605801
\(801\) 0 0
\(802\) −28.1689 −0.994680
\(803\) 35.0860 1.23816
\(804\) 0 0
\(805\) −7.43374 −0.262005
\(806\) 10.1464 0.357390
\(807\) 0 0
\(808\) 30.4454 1.07106
\(809\) −7.83443 −0.275444 −0.137722 0.990471i \(-0.543978\pi\)
−0.137722 + 0.990471i \(0.543978\pi\)
\(810\) 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
\(813\) 0 0
\(814\) 52.0301 1.82365
\(815\) −4.10430 −0.143767
\(816\) 0 0
\(817\) −9.19316 −0.321628
\(818\) −4.59769 −0.160754
\(819\) 0 0
\(820\) −0.818526 −0.0285842
\(821\) −4.42494 −0.154431 −0.0772157 0.997014i \(-0.524603\pi\)
−0.0772157 + 0.997014i \(0.524603\pi\)
\(822\) 0 0
\(823\) −28.7525 −1.00225 −0.501124 0.865375i \(-0.667080\pi\)
−0.501124 + 0.865375i \(0.667080\pi\)
\(824\) 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 0
\(829\) 53.4974 1.85804 0.929021 0.370027i \(-0.120652\pi\)
0.929021 + 0.370027i \(0.120652\pi\)
\(830\) 5.07974 0.176320
\(831\) 0 0
\(832\) 7.68624 0.266472
\(833\) 2.87971 0.0997760
\(834\) 0 0
\(835\) 19.0254 0.658400
\(836\) 0.397035 0.0137317
\(837\) 0 0
\(838\) 43.5257 1.50357
\(839\) 43.0405 1.48592 0.742962 0.669334i \(-0.233420\pi\)
0.742962 + 0.669334i \(0.233420\pi\)
\(840\) 0 0
\(841\) −18.1932 −0.627350
\(842\) −38.8055 −1.33733
\(843\) 0 0
\(844\) 1.13941 0.0392202
\(845\) −0.926817 −0.0318835
\(846\) 0 0
\(847\) 6.75942 0.232256
\(848\) −21.2890 −0.731066
\(849\) 0 0
\(850\) 17.1701 0.588931
\(851\) 68.7757 2.35760
\(852\) 0 0
\(853\) −30.7434 −1.05263 −0.526316 0.850289i \(-0.676427\pi\)
−0.526316 + 0.850289i \(0.676427\pi\)
\(854\) −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 \(0.362304\pi\)
\(858\) 0 0
\(859\) −52.1243 −1.77846 −0.889229 0.457461i \(-0.848759\pi\)
−0.889229 + 0.457461i \(0.848759\pi\)
\(860\) −0.484354 −0.0165163
\(861\) 0 0
\(862\) 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 0
\(865\) 13.9165 0.473175
\(866\) −37.2246 −1.26494
\(867\) 0 0
\(868\) −0.515704 −0.0175041
\(869\) −18.8461 −0.639309
\(870\) 0 0
\(871\) 12.0414 0.408009
\(872\) 12.9533 0.438654
\(873\) 0 0
\(874\) 14.8675 0.502900
\(875\) 8.47204 0.286407
\(876\) 0 0
\(877\) 41.0547 1.38632 0.693159 0.720785i \(-0.256218\pi\)
0.693159 + 0.720785i \(0.256218\pi\)
\(878\) −36.0760 −1.21751
\(879\) 0 0
\(880\) −16.1739 −0.545222
\(881\) 31.0568 1.04633 0.523165 0.852231i \(-0.324751\pi\)
0.523165 + 0.852231i \(0.324751\pi\)
\(882\) 0 0
\(883\) −56.5357 −1.90258 −0.951289 0.308300i \(-0.900240\pi\)
−0.951289 + 0.308300i \(0.900240\pi\)
\(884\) 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 \(0.553411\pi\)
\(888\) 0 0
\(889\) −0.428386 −0.0143676
\(890\) 7.53707 0.252643
\(891\) 0 0
\(892\) −0.846681 −0.0283490
\(893\) 2.51411 0.0841314
\(894\) 0 0
\(895\) −4.96010 −0.165798
\(896\) −11.8946 −0.397372
\(897\) 0 0
\(898\) 0.712291 0.0237695
\(899\) 23.1655 0.772612
\(900\) 0 0
\(901\) 14.8046 0.493213
\(902\) 73.2255 2.43814
\(903\) 0 0
\(904\) 9.12029 0.303336
\(905\) −10.6275 −0.353271
\(906\) 0 0
\(907\) −31.2936 −1.03909 −0.519544 0.854444i \(-0.673898\pi\)
−0.519544 + 0.854444i \(0.673898\pi\)
\(908\) 0.808067 0.0268166
\(909\) 0 0
\(910\) 1.33448 0.0442377
\(911\) 9.39320 0.311210 0.155605 0.987819i \(-0.450267\pi\)
0.155605 + 0.987819i \(0.450267\pi\)
\(912\) 0 0
\(913\) −16.0414 −0.530894
\(914\) −38.1127 −1.26066
\(915\) 0 0
\(916\) −1.00314 −0.0331446
\(917\) −14.1878 −0.468523
\(918\) 0 0
\(919\) 21.7066 0.716036 0.358018 0.933715i \(-0.383453\pi\)
0.358018 + 0.933715i \(0.383453\pi\)
\(920\) −20.6237 −0.679944
\(921\) 0 0
\(922\) −1.55746 −0.0512921
\(923\) −10.7889 −0.355122
\(924\) 0 0
\(925\) −35.5081 −1.16750
\(926\) 47.0974 1.54771
\(927\) 0 0
\(928\) −1.36025 −0.0446523
\(929\) −35.6787 −1.17058 −0.585289 0.810824i \(-0.699019\pi\)
−0.585289 + 0.810824i \(0.699019\pi\)
\(930\) 0 0
\(931\) −1.28738 −0.0421920
\(932\) −0.0903972 −0.00296106
\(933\) 0 0
\(934\) −24.5924 −0.804687
\(935\) 11.2475 0.367834
\(936\) 0 0
\(937\) −47.0866 −1.53825 −0.769127 0.639096i \(-0.779309\pi\)
−0.769127 + 0.639096i \(0.779309\pi\)
\(938\) −17.3379 −0.566103
\(939\) 0 0
\(940\) 0.132459 0.00432034
\(941\) −45.2923 −1.47649 −0.738244 0.674534i \(-0.764345\pi\)
−0.738244 + 0.674534i \(0.764345\pi\)
\(942\) 0 0
\(943\) 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 0
\(949\) −8.32568 −0.270263
\(950\) −7.67592 −0.249040
\(951\) 0 0
\(952\) 7.98929 0.258934
\(953\) 3.95572 0.128138 0.0640692 0.997945i \(-0.479592\pi\)
0.0640692 + 0.997945i \(0.479592\pi\)
\(954\) 0 0
\(955\) 18.6991 0.605088
\(956\) −1.39824 −0.0452223
\(957\) 0 0
\(958\) −24.7353 −0.799160
\(959\) −14.7070 −0.474912
\(960\) 0 0
\(961\) 18.6573 0.601850
\(962\) −12.3464 −0.398064
\(963\) 0 0
\(964\) −0.880605 −0.0283624
\(965\) 24.4454 0.786924
\(966\) 0 0
\(967\) 50.2292 1.61526 0.807632 0.589687i \(-0.200749\pi\)
0.807632 + 0.589687i \(0.200749\pi\)
\(968\) 18.7529 0.602742
\(969\) 0 0
\(970\) 9.20850 0.295667
\(971\) −38.6576 −1.24058 −0.620291 0.784372i \(-0.712986\pi\)
−0.620291 + 0.784372i \(0.712986\pi\)
\(972\) 0 0
\(973\) 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 \(0.393039\pi\)
\(978\) 0 0
\(979\) −23.8015 −0.760699
\(980\) −0.0678271 −0.00216666
\(981\) 0 0
\(982\) 39.5304 1.26147
\(983\) 4.48620 0.143088 0.0715438 0.997437i \(-0.477207\pi\)
0.0715438 + 0.997437i \(0.477207\pi\)
\(984\) 0 0
\(985\) −18.6589 −0.594521
\(986\) 13.6307 0.434089
\(987\) 0 0
\(988\) −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
\(993\) 0 0
\(994\) 15.5345 0.492725
\(995\) −1.63066 −0.0516954
\(996\) 0 0
\(997\) 0.491870 0.0155777 0.00778884 0.999970i \(-0.497521\pi\)
0.00778884 + 0.999970i \(0.497521\pi\)
\(998\) 53.3397 1.68844
\(999\) 0 0
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 819.2.a.k.1.2 4
3.2 odd 2 273.2.a.e.1.3 4
7.6 odd 2 5733.2.a.bf.1.2 4
12.11 even 2 4368.2.a.br.1.3 4
15.14 odd 2 6825.2.a.bg.1.2 4
21.20 even 2 1911.2.a.s.1.3 4
39.38 odd 2 3549.2.a.w.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
273.2.a.e.1.3 4 3.2 odd 2
819.2.a.k.1.2 4 1.1 even 1 trivial
1911.2.a.s.1.3 4 21.20 even 2
3549.2.a.w.1.2 4 39.38 odd 2
4368.2.a.br.1.3 4 12.11 even 2
5733.2.a.bf.1.2 4 7.6 odd 2
6825.2.a.bg.1.2 4 15.14 odd 2