Properties

Label 1521.2.a.o.1.2
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
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] = [1521,2,Mod(1,1521)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1521, 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("1521.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.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: \(12.1452461474\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.24698\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.554958 q^{2} -1.69202 q^{4} -2.80194 q^{5} +2.69202 q^{7} +2.04892 q^{8} +O(q^{10})\) \(q-0.554958 q^{2} -1.69202 q^{4} -2.80194 q^{5} +2.69202 q^{7} +2.04892 q^{8} +1.55496 q^{10} -1.19806 q^{11} -1.49396 q^{14} +2.24698 q^{16} -1.13706 q^{17} -1.93900 q^{19} +4.74094 q^{20} +0.664874 q^{22} +4.60388 q^{23} +2.85086 q^{25} -4.55496 q^{28} +7.89977 q^{29} -5.89977 q^{31} -5.34481 q^{32} +0.631023 q^{34} -7.54288 q^{35} -0.951083 q^{37} +1.07606 q^{38} -5.74094 q^{40} +3.31767 q^{41} +7.15883 q^{43} +2.02715 q^{44} -2.55496 q^{46} -7.69202 q^{47} +0.246980 q^{49} -1.58211 q^{50} -5.87263 q^{53} +3.35690 q^{55} +5.51573 q^{56} -4.38404 q^{58} -0.0120816 q^{59} -8.03684 q^{61} +3.27413 q^{62} -1.52781 q^{64} +9.25667 q^{67} +1.92394 q^{68} +4.18598 q^{70} -13.7409 q^{71} -12.8170 q^{73} +0.527811 q^{74} +3.28083 q^{76} -3.22521 q^{77} +0.807315 q^{79} -6.29590 q^{80} -1.84117 q^{82} -16.3327 q^{83} +3.18598 q^{85} -3.97285 q^{86} -2.45473 q^{88} -14.7289 q^{89} -7.78986 q^{92} +4.26875 q^{94} +5.43296 q^{95} -3.13169 q^{97} -0.137063 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} - 4 q^{5} + 3 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} - 4 q^{5} + 3 q^{7} - 3 q^{8} + 5 q^{10} - 8 q^{11} + 5 q^{14} + 2 q^{16} + 2 q^{17} + 4 q^{19} + 3 q^{22} + 5 q^{23} - 5 q^{25} - 14 q^{28} + q^{29} + 5 q^{31} + 7 q^{32} - 13 q^{34} - 4 q^{35} - 12 q^{37} - 12 q^{38} - 3 q^{40} - 7 q^{41} + 13 q^{43} - 8 q^{46} - 18 q^{47} - 4 q^{49} + q^{50} - q^{53} + 6 q^{55} + 4 q^{56} - 3 q^{58} - 19 q^{59} + 4 q^{61} - q^{62} - 11 q^{64} + q^{67} + 21 q^{68} - 2 q^{70} - 27 q^{71} - 9 q^{73} + 8 q^{74} + 21 q^{76} - 8 q^{77} - 5 q^{79} - 5 q^{80} - 14 q^{82} - 7 q^{83} - 5 q^{85} - 18 q^{86} + 15 q^{88} - 11 q^{89} + 5 q^{94} - 3 q^{95} - 7 q^{97} + 5 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.554958 −0.392415 −0.196207 0.980562i \(-0.562863\pi\)
−0.196207 + 0.980562i \(0.562863\pi\)
\(3\) 0 0
\(4\) −1.69202 −0.846011
\(5\) −2.80194 −1.25306 −0.626532 0.779395i \(-0.715526\pi\)
−0.626532 + 0.779395i \(0.715526\pi\)
\(6\) 0 0
\(7\) 2.69202 1.01749 0.508744 0.860918i \(-0.330110\pi\)
0.508744 + 0.860918i \(0.330110\pi\)
\(8\) 2.04892 0.724402
\(9\) 0 0
\(10\) 1.55496 0.491721
\(11\) −1.19806 −0.361229 −0.180615 0.983554i \(-0.557809\pi\)
−0.180615 + 0.983554i \(0.557809\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) −1.49396 −0.399277
\(15\) 0 0
\(16\) 2.24698 0.561745
\(17\) −1.13706 −0.275778 −0.137889 0.990448i \(-0.544032\pi\)
−0.137889 + 0.990448i \(0.544032\pi\)
\(18\) 0 0
\(19\) −1.93900 −0.444837 −0.222419 0.974951i \(-0.571395\pi\)
−0.222419 + 0.974951i \(0.571395\pi\)
\(20\) 4.74094 1.06011
\(21\) 0 0
\(22\) 0.664874 0.141752
\(23\) 4.60388 0.959974 0.479987 0.877275i \(-0.340641\pi\)
0.479987 + 0.877275i \(0.340641\pi\)
\(24\) 0 0
\(25\) 2.85086 0.570171
\(26\) 0 0
\(27\) 0 0
\(28\) −4.55496 −0.860806
\(29\) 7.89977 1.46695 0.733475 0.679716i \(-0.237897\pi\)
0.733475 + 0.679716i \(0.237897\pi\)
\(30\) 0 0
\(31\) −5.89977 −1.05963 −0.529815 0.848113i \(-0.677739\pi\)
−0.529815 + 0.848113i \(0.677739\pi\)
\(32\) −5.34481 −0.944839
\(33\) 0 0
\(34\) 0.631023 0.108219
\(35\) −7.54288 −1.27498
\(36\) 0 0
\(37\) −0.951083 −0.156357 −0.0781785 0.996939i \(-0.524910\pi\)
−0.0781785 + 0.996939i \(0.524910\pi\)
\(38\) 1.07606 0.174561
\(39\) 0 0
\(40\) −5.74094 −0.907722
\(41\) 3.31767 0.518133 0.259066 0.965860i \(-0.416585\pi\)
0.259066 + 0.965860i \(0.416585\pi\)
\(42\) 0 0
\(43\) 7.15883 1.09171 0.545856 0.837879i \(-0.316205\pi\)
0.545856 + 0.837879i \(0.316205\pi\)
\(44\) 2.02715 0.305604
\(45\) 0 0
\(46\) −2.55496 −0.376708
\(47\) −7.69202 −1.12200 −0.560998 0.827817i \(-0.689583\pi\)
−0.560998 + 0.827817i \(0.689583\pi\)
\(48\) 0 0
\(49\) 0.246980 0.0352828
\(50\) −1.58211 −0.223743
\(51\) 0 0
\(52\) 0 0
\(53\) −5.87263 −0.806667 −0.403334 0.915053i \(-0.632149\pi\)
−0.403334 + 0.915053i \(0.632149\pi\)
\(54\) 0 0
\(55\) 3.35690 0.452644
\(56\) 5.51573 0.737070
\(57\) 0 0
\(58\) −4.38404 −0.575653
\(59\) −0.0120816 −0.00157289 −0.000786444 1.00000i \(-0.500250\pi\)
−0.000786444 1.00000i \(0.500250\pi\)
\(60\) 0 0
\(61\) −8.03684 −1.02901 −0.514506 0.857487i \(-0.672025\pi\)
−0.514506 + 0.857487i \(0.672025\pi\)
\(62\) 3.27413 0.415815
\(63\) 0 0
\(64\) −1.52781 −0.190976
\(65\) 0 0
\(66\) 0 0
\(67\) 9.25667 1.13088 0.565441 0.824789i \(-0.308706\pi\)
0.565441 + 0.824789i \(0.308706\pi\)
\(68\) 1.92394 0.233311
\(69\) 0 0
\(70\) 4.18598 0.500320
\(71\) −13.7409 −1.63075 −0.815375 0.578934i \(-0.803469\pi\)
−0.815375 + 0.578934i \(0.803469\pi\)
\(72\) 0 0
\(73\) −12.8170 −1.50012 −0.750058 0.661372i \(-0.769975\pi\)
−0.750058 + 0.661372i \(0.769975\pi\)
\(74\) 0.527811 0.0613568
\(75\) 0 0
\(76\) 3.28083 0.376337
\(77\) −3.22521 −0.367547
\(78\) 0 0
\(79\) 0.807315 0.0908300 0.0454150 0.998968i \(-0.485539\pi\)
0.0454150 + 0.998968i \(0.485539\pi\)
\(80\) −6.29590 −0.703903
\(81\) 0 0
\(82\) −1.84117 −0.203323
\(83\) −16.3327 −1.79275 −0.896375 0.443296i \(-0.853809\pi\)
−0.896375 + 0.443296i \(0.853809\pi\)
\(84\) 0 0
\(85\) 3.18598 0.345568
\(86\) −3.97285 −0.428404
\(87\) 0 0
\(88\) −2.45473 −0.261675
\(89\) −14.7289 −1.56126 −0.780628 0.624996i \(-0.785101\pi\)
−0.780628 + 0.624996i \(0.785101\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −7.78986 −0.812149
\(93\) 0 0
\(94\) 4.26875 0.440288
\(95\) 5.43296 0.557410
\(96\) 0 0
\(97\) −3.13169 −0.317975 −0.158987 0.987281i \(-0.550823\pi\)
−0.158987 + 0.987281i \(0.550823\pi\)
\(98\) −0.137063 −0.0138455
\(99\) 0 0
\(100\) −4.82371 −0.482371
\(101\) −5.29052 −0.526426 −0.263213 0.964738i \(-0.584782\pi\)
−0.263213 + 0.964738i \(0.584782\pi\)
\(102\) 0 0
\(103\) −13.5308 −1.33323 −0.666614 0.745403i \(-0.732257\pi\)
−0.666614 + 0.745403i \(0.732257\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 3.25906 0.316548
\(107\) −5.63102 −0.544371 −0.272186 0.962245i \(-0.587747\pi\)
−0.272186 + 0.962245i \(0.587747\pi\)
\(108\) 0 0
\(109\) −4.17629 −0.400016 −0.200008 0.979794i \(-0.564097\pi\)
−0.200008 + 0.979794i \(0.564097\pi\)
\(110\) −1.86294 −0.177624
\(111\) 0 0
\(112\) 6.04892 0.571569
\(113\) −7.64310 −0.719003 −0.359501 0.933145i \(-0.617053\pi\)
−0.359501 + 0.933145i \(0.617053\pi\)
\(114\) 0 0
\(115\) −12.8998 −1.20291
\(116\) −13.3666 −1.24106
\(117\) 0 0
\(118\) 0.00670477 0.000617224 0
\(119\) −3.06100 −0.280601
\(120\) 0 0
\(121\) −9.56465 −0.869513
\(122\) 4.46011 0.403799
\(123\) 0 0
\(124\) 9.98254 0.896459
\(125\) 6.02177 0.538604
\(126\) 0 0
\(127\) 6.77777 0.601430 0.300715 0.953714i \(-0.402775\pi\)
0.300715 + 0.953714i \(0.402775\pi\)
\(128\) 11.5375 1.01978
\(129\) 0 0
\(130\) 0 0
\(131\) 13.6799 1.19522 0.597611 0.801786i \(-0.296117\pi\)
0.597611 + 0.801786i \(0.296117\pi\)
\(132\) 0 0
\(133\) −5.21983 −0.452617
\(134\) −5.13706 −0.443775
\(135\) 0 0
\(136\) −2.32975 −0.199774
\(137\) 12.9879 1.10963 0.554816 0.831973i \(-0.312789\pi\)
0.554816 + 0.831973i \(0.312789\pi\)
\(138\) 0 0
\(139\) 12.0465 1.02177 0.510886 0.859648i \(-0.329317\pi\)
0.510886 + 0.859648i \(0.329317\pi\)
\(140\) 12.7627 1.07865
\(141\) 0 0
\(142\) 7.62565 0.639930
\(143\) 0 0
\(144\) 0 0
\(145\) −22.1347 −1.83818
\(146\) 7.11290 0.588668
\(147\) 0 0
\(148\) 1.60925 0.132280
\(149\) −0.740939 −0.0607001 −0.0303500 0.999539i \(-0.509662\pi\)
−0.0303500 + 0.999539i \(0.509662\pi\)
\(150\) 0 0
\(151\) 19.0737 1.55219 0.776097 0.630614i \(-0.217197\pi\)
0.776097 + 0.630614i \(0.217197\pi\)
\(152\) −3.97285 −0.322241
\(153\) 0 0
\(154\) 1.78986 0.144231
\(155\) 16.5308 1.32779
\(156\) 0 0
\(157\) −4.02177 −0.320972 −0.160486 0.987038i \(-0.551306\pi\)
−0.160486 + 0.987038i \(0.551306\pi\)
\(158\) −0.448026 −0.0356430
\(159\) 0 0
\(160\) 14.9758 1.18394
\(161\) 12.3937 0.976763
\(162\) 0 0
\(163\) −15.1371 −1.18563 −0.592813 0.805340i \(-0.701983\pi\)
−0.592813 + 0.805340i \(0.701983\pi\)
\(164\) −5.61356 −0.438346
\(165\) 0 0
\(166\) 9.06398 0.703502
\(167\) −6.26337 −0.484674 −0.242337 0.970192i \(-0.577914\pi\)
−0.242337 + 0.970192i \(0.577914\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −1.76809 −0.135606
\(171\) 0 0
\(172\) −12.1129 −0.923600
\(173\) −16.3913 −1.24621 −0.623105 0.782138i \(-0.714129\pi\)
−0.623105 + 0.782138i \(0.714129\pi\)
\(174\) 0 0
\(175\) 7.67456 0.580142
\(176\) −2.69202 −0.202919
\(177\) 0 0
\(178\) 8.17390 0.612660
\(179\) 2.45473 0.183475 0.0917376 0.995783i \(-0.470758\pi\)
0.0917376 + 0.995783i \(0.470758\pi\)
\(180\) 0 0
\(181\) 11.8073 0.877631 0.438815 0.898577i \(-0.355398\pi\)
0.438815 + 0.898577i \(0.355398\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 9.43296 0.695407
\(185\) 2.66487 0.195925
\(186\) 0 0
\(187\) 1.36227 0.0996192
\(188\) 13.0151 0.949221
\(189\) 0 0
\(190\) −3.01507 −0.218736
\(191\) 8.99330 0.650732 0.325366 0.945588i \(-0.394512\pi\)
0.325366 + 0.945588i \(0.394512\pi\)
\(192\) 0 0
\(193\) −13.5254 −0.973581 −0.486790 0.873519i \(-0.661832\pi\)
−0.486790 + 0.873519i \(0.661832\pi\)
\(194\) 1.73795 0.124778
\(195\) 0 0
\(196\) −0.417895 −0.0298496
\(197\) 12.9758 0.924490 0.462245 0.886752i \(-0.347044\pi\)
0.462245 + 0.886752i \(0.347044\pi\)
\(198\) 0 0
\(199\) −13.5864 −0.963116 −0.481558 0.876414i \(-0.659929\pi\)
−0.481558 + 0.876414i \(0.659929\pi\)
\(200\) 5.84117 0.413033
\(201\) 0 0
\(202\) 2.93602 0.206577
\(203\) 21.2664 1.49261
\(204\) 0 0
\(205\) −9.29590 −0.649254
\(206\) 7.50902 0.523179
\(207\) 0 0
\(208\) 0 0
\(209\) 2.32304 0.160688
\(210\) 0 0
\(211\) 10.4601 0.720103 0.360052 0.932932i \(-0.382759\pi\)
0.360052 + 0.932932i \(0.382759\pi\)
\(212\) 9.93661 0.682449
\(213\) 0 0
\(214\) 3.12498 0.213619
\(215\) −20.0586 −1.36799
\(216\) 0 0
\(217\) −15.8823 −1.07816
\(218\) 2.31767 0.156972
\(219\) 0 0
\(220\) −5.67994 −0.382941
\(221\) 0 0
\(222\) 0 0
\(223\) 11.4058 0.763790 0.381895 0.924206i \(-0.375272\pi\)
0.381895 + 0.924206i \(0.375272\pi\)
\(224\) −14.3884 −0.961362
\(225\) 0 0
\(226\) 4.24160 0.282147
\(227\) 10.6407 0.706249 0.353124 0.935576i \(-0.385119\pi\)
0.353124 + 0.935576i \(0.385119\pi\)
\(228\) 0 0
\(229\) 1.13946 0.0752974 0.0376487 0.999291i \(-0.488013\pi\)
0.0376487 + 0.999291i \(0.488013\pi\)
\(230\) 7.15883 0.472040
\(231\) 0 0
\(232\) 16.1860 1.06266
\(233\) 10.8509 0.710863 0.355432 0.934702i \(-0.384334\pi\)
0.355432 + 0.934702i \(0.384334\pi\)
\(234\) 0 0
\(235\) 21.5526 1.40593
\(236\) 0.0204423 0.00133068
\(237\) 0 0
\(238\) 1.69873 0.110112
\(239\) −11.9293 −0.771643 −0.385822 0.922573i \(-0.626082\pi\)
−0.385822 + 0.922573i \(0.626082\pi\)
\(240\) 0 0
\(241\) 3.64848 0.235019 0.117510 0.993072i \(-0.462509\pi\)
0.117510 + 0.993072i \(0.462509\pi\)
\(242\) 5.30798 0.341210
\(243\) 0 0
\(244\) 13.5985 0.870555
\(245\) −0.692021 −0.0442116
\(246\) 0 0
\(247\) 0 0
\(248\) −12.0881 −0.767598
\(249\) 0 0
\(250\) −3.34183 −0.211356
\(251\) −1.37329 −0.0866813 −0.0433406 0.999060i \(-0.513800\pi\)
−0.0433406 + 0.999060i \(0.513800\pi\)
\(252\) 0 0
\(253\) −5.51573 −0.346771
\(254\) −3.76138 −0.236010
\(255\) 0 0
\(256\) −3.34721 −0.209200
\(257\) −29.4359 −1.83616 −0.918082 0.396391i \(-0.870263\pi\)
−0.918082 + 0.396391i \(0.870263\pi\)
\(258\) 0 0
\(259\) −2.56033 −0.159091
\(260\) 0 0
\(261\) 0 0
\(262\) −7.59179 −0.469023
\(263\) −10.6963 −0.659564 −0.329782 0.944057i \(-0.606975\pi\)
−0.329782 + 0.944057i \(0.606975\pi\)
\(264\) 0 0
\(265\) 16.4547 1.01081
\(266\) 2.89679 0.177613
\(267\) 0 0
\(268\) −15.6625 −0.956738
\(269\) 10.1860 0.621050 0.310525 0.950565i \(-0.399495\pi\)
0.310525 + 0.950565i \(0.399495\pi\)
\(270\) 0 0
\(271\) 29.4523 1.78910 0.894551 0.446966i \(-0.147495\pi\)
0.894551 + 0.446966i \(0.147495\pi\)
\(272\) −2.55496 −0.154917
\(273\) 0 0
\(274\) −7.20775 −0.435436
\(275\) −3.41550 −0.205963
\(276\) 0 0
\(277\) −10.2446 −0.615538 −0.307769 0.951461i \(-0.599582\pi\)
−0.307769 + 0.951461i \(0.599582\pi\)
\(278\) −6.68532 −0.400959
\(279\) 0 0
\(280\) −15.4547 −0.923597
\(281\) −11.5646 −0.689889 −0.344944 0.938623i \(-0.612102\pi\)
−0.344944 + 0.938623i \(0.612102\pi\)
\(282\) 0 0
\(283\) −30.7090 −1.82546 −0.912730 0.408562i \(-0.866030\pi\)
−0.912730 + 0.408562i \(0.866030\pi\)
\(284\) 23.2500 1.37963
\(285\) 0 0
\(286\) 0 0
\(287\) 8.93123 0.527194
\(288\) 0 0
\(289\) −15.7071 −0.923946
\(290\) 12.2838 0.721330
\(291\) 0 0
\(292\) 21.6866 1.26911
\(293\) −18.6082 −1.08710 −0.543551 0.839376i \(-0.682921\pi\)
−0.543551 + 0.839376i \(0.682921\pi\)
\(294\) 0 0
\(295\) 0.0338518 0.00197093
\(296\) −1.94869 −0.113265
\(297\) 0 0
\(298\) 0.411190 0.0238196
\(299\) 0 0
\(300\) 0 0
\(301\) 19.2717 1.11080
\(302\) −10.5851 −0.609103
\(303\) 0 0
\(304\) −4.35690 −0.249885
\(305\) 22.5187 1.28942
\(306\) 0 0
\(307\) −8.94438 −0.510483 −0.255241 0.966877i \(-0.582155\pi\)
−0.255241 + 0.966877i \(0.582155\pi\)
\(308\) 5.45712 0.310948
\(309\) 0 0
\(310\) −9.17390 −0.521042
\(311\) −21.0398 −1.19306 −0.596529 0.802591i \(-0.703454\pi\)
−0.596529 + 0.802591i \(0.703454\pi\)
\(312\) 0 0
\(313\) −7.12737 −0.402863 −0.201432 0.979503i \(-0.564559\pi\)
−0.201432 + 0.979503i \(0.564559\pi\)
\(314\) 2.23191 0.125954
\(315\) 0 0
\(316\) −1.36599 −0.0768431
\(317\) 23.9651 1.34601 0.673007 0.739636i \(-0.265003\pi\)
0.673007 + 0.739636i \(0.265003\pi\)
\(318\) 0 0
\(319\) −9.46442 −0.529906
\(320\) 4.28083 0.239306
\(321\) 0 0
\(322\) −6.87800 −0.383296
\(323\) 2.20477 0.122677
\(324\) 0 0
\(325\) 0 0
\(326\) 8.40044 0.465257
\(327\) 0 0
\(328\) 6.79763 0.375336
\(329\) −20.7071 −1.14162
\(330\) 0 0
\(331\) −2.89546 −0.159149 −0.0795745 0.996829i \(-0.525356\pi\)
−0.0795745 + 0.996829i \(0.525356\pi\)
\(332\) 27.6353 1.51669
\(333\) 0 0
\(334\) 3.47591 0.190193
\(335\) −25.9366 −1.41707
\(336\) 0 0
\(337\) −3.10560 −0.169173 −0.0845865 0.996416i \(-0.526957\pi\)
−0.0845865 + 0.996416i \(0.526957\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −5.39075 −0.292354
\(341\) 7.06829 0.382770
\(342\) 0 0
\(343\) −18.1793 −0.981589
\(344\) 14.6679 0.790838
\(345\) 0 0
\(346\) 9.09651 0.489031
\(347\) 11.3787 0.610839 0.305419 0.952218i \(-0.401203\pi\)
0.305419 + 0.952218i \(0.401203\pi\)
\(348\) 0 0
\(349\) 3.34721 0.179172 0.0895859 0.995979i \(-0.471446\pi\)
0.0895859 + 0.995979i \(0.471446\pi\)
\(350\) −4.25906 −0.227656
\(351\) 0 0
\(352\) 6.40342 0.341303
\(353\) 0.637727 0.0339428 0.0169714 0.999856i \(-0.494598\pi\)
0.0169714 + 0.999856i \(0.494598\pi\)
\(354\) 0 0
\(355\) 38.5013 2.04343
\(356\) 24.9215 1.32084
\(357\) 0 0
\(358\) −1.36227 −0.0719983
\(359\) −21.4590 −1.13256 −0.566282 0.824211i \(-0.691619\pi\)
−0.566282 + 0.824211i \(0.691619\pi\)
\(360\) 0 0
\(361\) −15.2403 −0.802120
\(362\) −6.55257 −0.344395
\(363\) 0 0
\(364\) 0 0
\(365\) 35.9124 1.87974
\(366\) 0 0
\(367\) −9.38703 −0.489999 −0.244999 0.969523i \(-0.578788\pi\)
−0.244999 + 0.969523i \(0.578788\pi\)
\(368\) 10.3448 0.539261
\(369\) 0 0
\(370\) −1.47889 −0.0768840
\(371\) −15.8092 −0.820775
\(372\) 0 0
\(373\) 27.7265 1.43562 0.717811 0.696238i \(-0.245144\pi\)
0.717811 + 0.696238i \(0.245144\pi\)
\(374\) −0.756004 −0.0390921
\(375\) 0 0
\(376\) −15.7603 −0.812776
\(377\) 0 0
\(378\) 0 0
\(379\) −35.8702 −1.84253 −0.921265 0.388935i \(-0.872843\pi\)
−0.921265 + 0.388935i \(0.872843\pi\)
\(380\) −9.19269 −0.471575
\(381\) 0 0
\(382\) −4.99090 −0.255357
\(383\) 4.85517 0.248087 0.124044 0.992277i \(-0.460414\pi\)
0.124044 + 0.992277i \(0.460414\pi\)
\(384\) 0 0
\(385\) 9.03684 0.460560
\(386\) 7.50604 0.382047
\(387\) 0 0
\(388\) 5.29888 0.269010
\(389\) −2.38537 −0.120943 −0.0604716 0.998170i \(-0.519260\pi\)
−0.0604716 + 0.998170i \(0.519260\pi\)
\(390\) 0 0
\(391\) −5.23490 −0.264740
\(392\) 0.506041 0.0255589
\(393\) 0 0
\(394\) −7.20105 −0.362783
\(395\) −2.26205 −0.113816
\(396\) 0 0
\(397\) 15.2664 0.766196 0.383098 0.923708i \(-0.374857\pi\)
0.383098 + 0.923708i \(0.374857\pi\)
\(398\) 7.53989 0.377941
\(399\) 0 0
\(400\) 6.40581 0.320291
\(401\) 12.7584 0.637124 0.318562 0.947902i \(-0.396800\pi\)
0.318562 + 0.947902i \(0.396800\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 8.95167 0.445362
\(405\) 0 0
\(406\) −11.8019 −0.585720
\(407\) 1.13946 0.0564807
\(408\) 0 0
\(409\) 25.3588 1.25391 0.626956 0.779054i \(-0.284300\pi\)
0.626956 + 0.779054i \(0.284300\pi\)
\(410\) 5.15883 0.254777
\(411\) 0 0
\(412\) 22.8944 1.12793
\(413\) −0.0325239 −0.00160040
\(414\) 0 0
\(415\) 45.7633 2.24643
\(416\) 0 0
\(417\) 0 0
\(418\) −1.28919 −0.0630565
\(419\) 11.6673 0.569983 0.284992 0.958530i \(-0.408009\pi\)
0.284992 + 0.958530i \(0.408009\pi\)
\(420\) 0 0
\(421\) 8.29291 0.404172 0.202086 0.979368i \(-0.435228\pi\)
0.202086 + 0.979368i \(0.435228\pi\)
\(422\) −5.80492 −0.282579
\(423\) 0 0
\(424\) −12.0325 −0.584351
\(425\) −3.24160 −0.157241
\(426\) 0 0
\(427\) −21.6353 −1.04701
\(428\) 9.52781 0.460544
\(429\) 0 0
\(430\) 11.1317 0.536818
\(431\) 0.932296 0.0449071 0.0224536 0.999748i \(-0.492852\pi\)
0.0224536 + 0.999748i \(0.492852\pi\)
\(432\) 0 0
\(433\) −13.3502 −0.641569 −0.320785 0.947152i \(-0.603947\pi\)
−0.320785 + 0.947152i \(0.603947\pi\)
\(434\) 8.81402 0.423086
\(435\) 0 0
\(436\) 7.06638 0.338418
\(437\) −8.92692 −0.427032
\(438\) 0 0
\(439\) 13.9922 0.667813 0.333906 0.942606i \(-0.391633\pi\)
0.333906 + 0.942606i \(0.391633\pi\)
\(440\) 6.87800 0.327896
\(441\) 0 0
\(442\) 0 0
\(443\) −23.7017 −1.12610 −0.563051 0.826422i \(-0.690373\pi\)
−0.563051 + 0.826422i \(0.690373\pi\)
\(444\) 0 0
\(445\) 41.2693 1.95635
\(446\) −6.32975 −0.299722
\(447\) 0 0
\(448\) −4.11290 −0.194316
\(449\) −12.5864 −0.593990 −0.296995 0.954879i \(-0.595984\pi\)
−0.296995 + 0.954879i \(0.595984\pi\)
\(450\) 0 0
\(451\) −3.97477 −0.187165
\(452\) 12.9323 0.608284
\(453\) 0 0
\(454\) −5.90515 −0.277142
\(455\) 0 0
\(456\) 0 0
\(457\) 33.6383 1.57353 0.786767 0.617250i \(-0.211753\pi\)
0.786767 + 0.617250i \(0.211753\pi\)
\(458\) −0.632351 −0.0295478
\(459\) 0 0
\(460\) 21.8267 1.01767
\(461\) −1.40283 −0.0653363 −0.0326681 0.999466i \(-0.510400\pi\)
−0.0326681 + 0.999466i \(0.510400\pi\)
\(462\) 0 0
\(463\) 15.2010 0.706453 0.353226 0.935538i \(-0.385085\pi\)
0.353226 + 0.935538i \(0.385085\pi\)
\(464\) 17.7506 0.824052
\(465\) 0 0
\(466\) −6.02177 −0.278953
\(467\) 39.3414 1.82050 0.910250 0.414058i \(-0.135889\pi\)
0.910250 + 0.414058i \(0.135889\pi\)
\(468\) 0 0
\(469\) 24.9191 1.15066
\(470\) −11.9608 −0.551709
\(471\) 0 0
\(472\) −0.0247542 −0.00113940
\(473\) −8.57673 −0.394358
\(474\) 0 0
\(475\) −5.52781 −0.253633
\(476\) 5.17928 0.237392
\(477\) 0 0
\(478\) 6.62027 0.302804
\(479\) −22.3690 −1.02206 −0.511032 0.859561i \(-0.670737\pi\)
−0.511032 + 0.859561i \(0.670737\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −2.02475 −0.0922250
\(483\) 0 0
\(484\) 16.1836 0.735618
\(485\) 8.77479 0.398443
\(486\) 0 0
\(487\) −22.9205 −1.03863 −0.519313 0.854584i \(-0.673812\pi\)
−0.519313 + 0.854584i \(0.673812\pi\)
\(488\) −16.4668 −0.745418
\(489\) 0 0
\(490\) 0.384043 0.0173493
\(491\) −1.84356 −0.0831987 −0.0415993 0.999134i \(-0.513245\pi\)
−0.0415993 + 0.999134i \(0.513245\pi\)
\(492\) 0 0
\(493\) −8.98254 −0.404553
\(494\) 0 0
\(495\) 0 0
\(496\) −13.2567 −0.595242
\(497\) −36.9909 −1.65927
\(498\) 0 0
\(499\) 12.0344 0.538736 0.269368 0.963037i \(-0.413185\pi\)
0.269368 + 0.963037i \(0.413185\pi\)
\(500\) −10.1890 −0.455664
\(501\) 0 0
\(502\) 0.762118 0.0340150
\(503\) 30.5056 1.36018 0.680088 0.733130i \(-0.261942\pi\)
0.680088 + 0.733130i \(0.261942\pi\)
\(504\) 0 0
\(505\) 14.8237 0.659646
\(506\) 3.06100 0.136078
\(507\) 0 0
\(508\) −11.4681 −0.508816
\(509\) −1.51142 −0.0669924 −0.0334962 0.999439i \(-0.510664\pi\)
−0.0334962 + 0.999439i \(0.510664\pi\)
\(510\) 0 0
\(511\) −34.5036 −1.52635
\(512\) −21.2174 −0.937687
\(513\) 0 0
\(514\) 16.3357 0.720538
\(515\) 37.9124 1.67062
\(516\) 0 0
\(517\) 9.21552 0.405298
\(518\) 1.42088 0.0624298
\(519\) 0 0
\(520\) 0 0
\(521\) 5.64012 0.247098 0.123549 0.992338i \(-0.460572\pi\)
0.123549 + 0.992338i \(0.460572\pi\)
\(522\) 0 0
\(523\) −31.7506 −1.38836 −0.694179 0.719802i \(-0.744232\pi\)
−0.694179 + 0.719802i \(0.744232\pi\)
\(524\) −23.1468 −1.01117
\(525\) 0 0
\(526\) 5.93602 0.258823
\(527\) 6.70841 0.292223
\(528\) 0 0
\(529\) −1.80433 −0.0784492
\(530\) −9.13169 −0.396655
\(531\) 0 0
\(532\) 8.83207 0.382919
\(533\) 0 0
\(534\) 0 0
\(535\) 15.7778 0.682133
\(536\) 18.9661 0.819213
\(537\) 0 0
\(538\) −5.65279 −0.243709
\(539\) −0.295897 −0.0127452
\(540\) 0 0
\(541\) −24.3297 −1.04602 −0.523009 0.852327i \(-0.675191\pi\)
−0.523009 + 0.852327i \(0.675191\pi\)
\(542\) −16.3448 −0.702070
\(543\) 0 0
\(544\) 6.07739 0.260566
\(545\) 11.7017 0.501246
\(546\) 0 0
\(547\) −8.18896 −0.350135 −0.175067 0.984556i \(-0.556014\pi\)
−0.175067 + 0.984556i \(0.556014\pi\)
\(548\) −21.9758 −0.938761
\(549\) 0 0
\(550\) 1.89546 0.0808227
\(551\) −15.3177 −0.652555
\(552\) 0 0
\(553\) 2.17331 0.0924185
\(554\) 5.68532 0.241546
\(555\) 0 0
\(556\) −20.3830 −0.864431
\(557\) 25.3327 1.07338 0.536691 0.843779i \(-0.319674\pi\)
0.536691 + 0.843779i \(0.319674\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −16.9487 −0.716213
\(561\) 0 0
\(562\) 6.41789 0.270723
\(563\) 25.3937 1.07022 0.535109 0.844783i \(-0.320270\pi\)
0.535109 + 0.844783i \(0.320270\pi\)
\(564\) 0 0
\(565\) 21.4155 0.900957
\(566\) 17.0422 0.716338
\(567\) 0 0
\(568\) −28.1540 −1.18132
\(569\) 31.1347 1.30523 0.652617 0.757688i \(-0.273671\pi\)
0.652617 + 0.757688i \(0.273671\pi\)
\(570\) 0 0
\(571\) −20.5090 −0.858276 −0.429138 0.903239i \(-0.641183\pi\)
−0.429138 + 0.903239i \(0.641183\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −4.95646 −0.206879
\(575\) 13.1250 0.547350
\(576\) 0 0
\(577\) −15.6890 −0.653143 −0.326572 0.945172i \(-0.605893\pi\)
−0.326572 + 0.945172i \(0.605893\pi\)
\(578\) 8.71678 0.362570
\(579\) 0 0
\(580\) 37.4523 1.55512
\(581\) −43.9681 −1.82410
\(582\) 0 0
\(583\) 7.03577 0.291392
\(584\) −26.2610 −1.08669
\(585\) 0 0
\(586\) 10.3268 0.426595
\(587\) 30.5687 1.26171 0.630853 0.775903i \(-0.282705\pi\)
0.630853 + 0.775903i \(0.282705\pi\)
\(588\) 0 0
\(589\) 11.4397 0.471363
\(590\) −0.0187864 −0.000773422 0
\(591\) 0 0
\(592\) −2.13706 −0.0878328
\(593\) −29.6883 −1.21915 −0.609576 0.792727i \(-0.708660\pi\)
−0.609576 + 0.792727i \(0.708660\pi\)
\(594\) 0 0
\(595\) 8.57673 0.351612
\(596\) 1.25368 0.0513529
\(597\) 0 0
\(598\) 0 0
\(599\) −24.2325 −0.990113 −0.495057 0.868861i \(-0.664853\pi\)
−0.495057 + 0.868861i \(0.664853\pi\)
\(600\) 0 0
\(601\) 16.4819 0.672310 0.336155 0.941807i \(-0.390873\pi\)
0.336155 + 0.941807i \(0.390873\pi\)
\(602\) −10.6950 −0.435896
\(603\) 0 0
\(604\) −32.2731 −1.31317
\(605\) 26.7995 1.08956
\(606\) 0 0
\(607\) 1.43190 0.0581188 0.0290594 0.999578i \(-0.490749\pi\)
0.0290594 + 0.999578i \(0.490749\pi\)
\(608\) 10.3636 0.420300
\(609\) 0 0
\(610\) −12.4969 −0.505986
\(611\) 0 0
\(612\) 0 0
\(613\) −3.84846 −0.155438 −0.0777190 0.996975i \(-0.524764\pi\)
−0.0777190 + 0.996975i \(0.524764\pi\)
\(614\) 4.96376 0.200321
\(615\) 0 0
\(616\) −6.60819 −0.266251
\(617\) −15.0388 −0.605437 −0.302719 0.953080i \(-0.597894\pi\)
−0.302719 + 0.953080i \(0.597894\pi\)
\(618\) 0 0
\(619\) −12.8170 −0.515159 −0.257579 0.966257i \(-0.582925\pi\)
−0.257579 + 0.966257i \(0.582925\pi\)
\(620\) −27.9705 −1.12332
\(621\) 0 0
\(622\) 11.6762 0.468174
\(623\) −39.6504 −1.58856
\(624\) 0 0
\(625\) −31.1269 −1.24508
\(626\) 3.95539 0.158089
\(627\) 0 0
\(628\) 6.80492 0.271546
\(629\) 1.08144 0.0431199
\(630\) 0 0
\(631\) −25.7517 −1.02516 −0.512579 0.858640i \(-0.671310\pi\)
−0.512579 + 0.858640i \(0.671310\pi\)
\(632\) 1.65412 0.0657974
\(633\) 0 0
\(634\) −13.2996 −0.528195
\(635\) −18.9909 −0.753631
\(636\) 0 0
\(637\) 0 0
\(638\) 5.25236 0.207943
\(639\) 0 0
\(640\) −32.3274 −1.27785
\(641\) −24.4571 −0.965998 −0.482999 0.875621i \(-0.660453\pi\)
−0.482999 + 0.875621i \(0.660453\pi\)
\(642\) 0 0
\(643\) 9.97344 0.393314 0.196657 0.980472i \(-0.436991\pi\)
0.196657 + 0.980472i \(0.436991\pi\)
\(644\) −20.9705 −0.826352
\(645\) 0 0
\(646\) −1.22355 −0.0481401
\(647\) −11.8431 −0.465600 −0.232800 0.972525i \(-0.574789\pi\)
−0.232800 + 0.972525i \(0.574789\pi\)
\(648\) 0 0
\(649\) 0.0144745 0.000568173 0
\(650\) 0 0
\(651\) 0 0
\(652\) 25.6122 1.00305
\(653\) 7.47411 0.292484 0.146242 0.989249i \(-0.453282\pi\)
0.146242 + 0.989249i \(0.453282\pi\)
\(654\) 0 0
\(655\) −38.3303 −1.49769
\(656\) 7.45473 0.291058
\(657\) 0 0
\(658\) 11.4916 0.447988
\(659\) −34.1739 −1.33123 −0.665613 0.746297i \(-0.731830\pi\)
−0.665613 + 0.746297i \(0.731830\pi\)
\(660\) 0 0
\(661\) −33.6088 −1.30723 −0.653615 0.756827i \(-0.726748\pi\)
−0.653615 + 0.756827i \(0.726748\pi\)
\(662\) 1.60686 0.0624524
\(663\) 0 0
\(664\) −33.4644 −1.29867
\(665\) 14.6256 0.567158
\(666\) 0 0
\(667\) 36.3696 1.40824
\(668\) 10.5978 0.410040
\(669\) 0 0
\(670\) 14.3937 0.556078
\(671\) 9.62863 0.371709
\(672\) 0 0
\(673\) 48.0320 1.85150 0.925750 0.378137i \(-0.123435\pi\)
0.925750 + 0.378137i \(0.123435\pi\)
\(674\) 1.72348 0.0663860
\(675\) 0 0
\(676\) 0 0
\(677\) 33.6582 1.29359 0.646794 0.762665i \(-0.276109\pi\)
0.646794 + 0.762665i \(0.276109\pi\)
\(678\) 0 0
\(679\) −8.43057 −0.323535
\(680\) 6.52781 0.250330
\(681\) 0 0
\(682\) −3.92261 −0.150204
\(683\) 15.9041 0.608553 0.304276 0.952584i \(-0.401585\pi\)
0.304276 + 0.952584i \(0.401585\pi\)
\(684\) 0 0
\(685\) −36.3913 −1.39044
\(686\) 10.0887 0.385190
\(687\) 0 0
\(688\) 16.0858 0.613264
\(689\) 0 0
\(690\) 0 0
\(691\) −33.1903 −1.26262 −0.631309 0.775531i \(-0.717482\pi\)
−0.631309 + 0.775531i \(0.717482\pi\)
\(692\) 27.7345 1.05431
\(693\) 0 0
\(694\) −6.31468 −0.239702
\(695\) −33.7536 −1.28035
\(696\) 0 0
\(697\) −3.77240 −0.142890
\(698\) −1.85756 −0.0703097
\(699\) 0 0
\(700\) −12.9855 −0.490807
\(701\) 14.9129 0.563253 0.281627 0.959524i \(-0.409126\pi\)
0.281627 + 0.959524i \(0.409126\pi\)
\(702\) 0 0
\(703\) 1.84415 0.0695534
\(704\) 1.83041 0.0689863
\(705\) 0 0
\(706\) −0.353912 −0.0133197
\(707\) −14.2422 −0.535633
\(708\) 0 0
\(709\) −38.4312 −1.44331 −0.721656 0.692252i \(-0.756619\pi\)
−0.721656 + 0.692252i \(0.756619\pi\)
\(710\) −21.3666 −0.801874
\(711\) 0 0
\(712\) −30.1782 −1.13098
\(713\) −27.1618 −1.01722
\(714\) 0 0
\(715\) 0 0
\(716\) −4.15346 −0.155222
\(717\) 0 0
\(718\) 11.9089 0.444435
\(719\) 11.4373 0.426538 0.213269 0.976993i \(-0.431589\pi\)
0.213269 + 0.976993i \(0.431589\pi\)
\(720\) 0 0
\(721\) −36.4252 −1.35654
\(722\) 8.45771 0.314764
\(723\) 0 0
\(724\) −19.9782 −0.742485
\(725\) 22.5211 0.836413
\(726\) 0 0
\(727\) 3.63640 0.134867 0.0674333 0.997724i \(-0.478519\pi\)
0.0674333 + 0.997724i \(0.478519\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −19.9299 −0.737639
\(731\) −8.14005 −0.301071
\(732\) 0 0
\(733\) 3.52217 0.130094 0.0650472 0.997882i \(-0.479280\pi\)
0.0650472 + 0.997882i \(0.479280\pi\)
\(734\) 5.20941 0.192283
\(735\) 0 0
\(736\) −24.6069 −0.907021
\(737\) −11.0901 −0.408508
\(738\) 0 0
\(739\) −0.420288 −0.0154605 −0.00773027 0.999970i \(-0.502461\pi\)
−0.00773027 + 0.999970i \(0.502461\pi\)
\(740\) −4.50902 −0.165755
\(741\) 0 0
\(742\) 8.77346 0.322084
\(743\) −25.3623 −0.930452 −0.465226 0.885192i \(-0.654027\pi\)
−0.465226 + 0.885192i \(0.654027\pi\)
\(744\) 0 0
\(745\) 2.07606 0.0760611
\(746\) −15.3870 −0.563359
\(747\) 0 0
\(748\) −2.30499 −0.0842790
\(749\) −15.1588 −0.553892
\(750\) 0 0
\(751\) 0.650874 0.0237507 0.0118754 0.999929i \(-0.496220\pi\)
0.0118754 + 0.999929i \(0.496220\pi\)
\(752\) −17.2838 −0.630276
\(753\) 0 0
\(754\) 0 0
\(755\) −53.4432 −1.94500
\(756\) 0 0
\(757\) −16.7909 −0.610276 −0.305138 0.952308i \(-0.598703\pi\)
−0.305138 + 0.952308i \(0.598703\pi\)
\(758\) 19.9065 0.723036
\(759\) 0 0
\(760\) 11.1317 0.403789
\(761\) 30.9221 1.12093 0.560463 0.828179i \(-0.310623\pi\)
0.560463 + 0.828179i \(0.310623\pi\)
\(762\) 0 0
\(763\) −11.2427 −0.407012
\(764\) −15.2168 −0.550526
\(765\) 0 0
\(766\) −2.69441 −0.0973531
\(767\) 0 0
\(768\) 0 0
\(769\) 43.7689 1.57835 0.789174 0.614169i \(-0.210509\pi\)
0.789174 + 0.614169i \(0.210509\pi\)
\(770\) −5.01507 −0.180730
\(771\) 0 0
\(772\) 22.8853 0.823660
\(773\) −42.4209 −1.52577 −0.762886 0.646532i \(-0.776218\pi\)
−0.762886 + 0.646532i \(0.776218\pi\)
\(774\) 0 0
\(775\) −16.8194 −0.604171
\(776\) −6.41657 −0.230341
\(777\) 0 0
\(778\) 1.32378 0.0474598
\(779\) −6.43296 −0.230485
\(780\) 0 0
\(781\) 16.4625 0.589075
\(782\) 2.90515 0.103888
\(783\) 0 0
\(784\) 0.554958 0.0198199
\(785\) 11.2687 0.402199
\(786\) 0 0
\(787\) 36.0116 1.28368 0.641838 0.766841i \(-0.278172\pi\)
0.641838 + 0.766841i \(0.278172\pi\)
\(788\) −21.9554 −0.782129
\(789\) 0 0
\(790\) 1.25534 0.0446630
\(791\) −20.5754 −0.731577
\(792\) 0 0
\(793\) 0 0
\(794\) −8.47219 −0.300667
\(795\) 0 0
\(796\) 22.9885 0.814806
\(797\) 31.7101 1.12323 0.561614 0.827399i \(-0.310181\pi\)
0.561614 + 0.827399i \(0.310181\pi\)
\(798\) 0 0
\(799\) 8.74632 0.309422
\(800\) −15.2373 −0.538720
\(801\) 0 0
\(802\) −7.08038 −0.250017
\(803\) 15.3556 0.541886
\(804\) 0 0
\(805\) −34.7265 −1.22395
\(806\) 0 0
\(807\) 0 0
\(808\) −10.8398 −0.381344
\(809\) 45.2814 1.59201 0.796005 0.605290i \(-0.206943\pi\)
0.796005 + 0.605290i \(0.206943\pi\)
\(810\) 0 0
\(811\) 42.8635 1.50514 0.752571 0.658511i \(-0.228813\pi\)
0.752571 + 0.658511i \(0.228813\pi\)
\(812\) −35.9831 −1.26276
\(813\) 0 0
\(814\) −0.632351 −0.0221639
\(815\) 42.4131 1.48567
\(816\) 0 0
\(817\) −13.8810 −0.485634
\(818\) −14.0731 −0.492054
\(819\) 0 0
\(820\) 15.7289 0.549276
\(821\) −7.82776 −0.273191 −0.136595 0.990627i \(-0.543616\pi\)
−0.136595 + 0.990627i \(0.543616\pi\)
\(822\) 0 0
\(823\) −36.7754 −1.28191 −0.640955 0.767579i \(-0.721461\pi\)
−0.640955 + 0.767579i \(0.721461\pi\)
\(824\) −27.7235 −0.965793
\(825\) 0 0
\(826\) 0.0180494 0.000628019 0
\(827\) 47.3293 1.64580 0.822900 0.568186i \(-0.192355\pi\)
0.822900 + 0.568186i \(0.192355\pi\)
\(828\) 0 0
\(829\) 25.2687 0.877620 0.438810 0.898580i \(-0.355400\pi\)
0.438810 + 0.898580i \(0.355400\pi\)
\(830\) −25.3967 −0.881533
\(831\) 0 0
\(832\) 0 0
\(833\) −0.280831 −0.00973023
\(834\) 0 0
\(835\) 17.5496 0.607328
\(836\) −3.93064 −0.135944
\(837\) 0 0
\(838\) −6.47484 −0.223670
\(839\) −37.6883 −1.30114 −0.650572 0.759444i \(-0.725471\pi\)
−0.650572 + 0.759444i \(0.725471\pi\)
\(840\) 0 0
\(841\) 33.4064 1.15194
\(842\) −4.60222 −0.158603
\(843\) 0 0
\(844\) −17.6987 −0.609215
\(845\) 0 0
\(846\) 0 0
\(847\) −25.7482 −0.884720
\(848\) −13.1957 −0.453141
\(849\) 0 0
\(850\) 1.79895 0.0617036
\(851\) −4.37867 −0.150099
\(852\) 0 0
\(853\) 31.0121 1.06183 0.530917 0.847424i \(-0.321848\pi\)
0.530917 + 0.847424i \(0.321848\pi\)
\(854\) 12.0067 0.410861
\(855\) 0 0
\(856\) −11.5375 −0.394344
\(857\) −12.4692 −0.425940 −0.212970 0.977059i \(-0.568314\pi\)
−0.212970 + 0.977059i \(0.568314\pi\)
\(858\) 0 0
\(859\) −17.3163 −0.590826 −0.295413 0.955370i \(-0.595457\pi\)
−0.295413 + 0.955370i \(0.595457\pi\)
\(860\) 33.9396 1.15733
\(861\) 0 0
\(862\) −0.517385 −0.0176222
\(863\) −3.46383 −0.117910 −0.0589550 0.998261i \(-0.518777\pi\)
−0.0589550 + 0.998261i \(0.518777\pi\)
\(864\) 0 0
\(865\) 45.9275 1.56158
\(866\) 7.40880 0.251761
\(867\) 0 0
\(868\) 26.8732 0.912136
\(869\) −0.967213 −0.0328105
\(870\) 0 0
\(871\) 0 0
\(872\) −8.55688 −0.289772
\(873\) 0 0
\(874\) 4.95407 0.167574
\(875\) 16.2107 0.548023
\(876\) 0 0
\(877\) −57.2549 −1.93336 −0.966680 0.255989i \(-0.917599\pi\)
−0.966680 + 0.255989i \(0.917599\pi\)
\(878\) −7.76510 −0.262059
\(879\) 0 0
\(880\) 7.54288 0.254270
\(881\) 43.1782 1.45471 0.727355 0.686261i \(-0.240749\pi\)
0.727355 + 0.686261i \(0.240749\pi\)
\(882\) 0 0
\(883\) 49.9560 1.68115 0.840576 0.541693i \(-0.182216\pi\)
0.840576 + 0.541693i \(0.182216\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 13.1535 0.441899
\(887\) −17.6746 −0.593454 −0.296727 0.954962i \(-0.595895\pi\)
−0.296727 + 0.954962i \(0.595895\pi\)
\(888\) 0 0
\(889\) 18.2459 0.611948
\(890\) −22.9028 −0.767702
\(891\) 0 0
\(892\) −19.2989 −0.646174
\(893\) 14.9148 0.499106
\(894\) 0 0
\(895\) −6.87800 −0.229906
\(896\) 31.0592 1.03761
\(897\) 0 0
\(898\) 6.98493 0.233090
\(899\) −46.6069 −1.55443
\(900\) 0 0
\(901\) 6.67755 0.222461
\(902\) 2.20583 0.0734462
\(903\) 0 0
\(904\) −15.6601 −0.520847
\(905\) −33.0834 −1.09973
\(906\) 0 0
\(907\) 7.73423 0.256811 0.128406 0.991722i \(-0.459014\pi\)
0.128406 + 0.991722i \(0.459014\pi\)
\(908\) −18.0043 −0.597494
\(909\) 0 0
\(910\) 0 0
\(911\) −39.6179 −1.31260 −0.656299 0.754501i \(-0.727879\pi\)
−0.656299 + 0.754501i \(0.727879\pi\)
\(912\) 0 0
\(913\) 19.5676 0.647594
\(914\) −18.6679 −0.617478
\(915\) 0 0
\(916\) −1.92798 −0.0637024
\(917\) 36.8267 1.21612
\(918\) 0 0
\(919\) 14.6213 0.482313 0.241157 0.970486i \(-0.422473\pi\)
0.241157 + 0.970486i \(0.422473\pi\)
\(920\) −26.4306 −0.871390
\(921\) 0 0
\(922\) 0.778512 0.0256389
\(923\) 0 0
\(924\) 0 0
\(925\) −2.71140 −0.0891502
\(926\) −8.43594 −0.277222
\(927\) 0 0
\(928\) −42.2228 −1.38603
\(929\) 3.55735 0.116713 0.0583565 0.998296i \(-0.481414\pi\)
0.0583565 + 0.998296i \(0.481414\pi\)
\(930\) 0 0
\(931\) −0.478894 −0.0156951
\(932\) −18.3599 −0.601398
\(933\) 0 0
\(934\) −21.8328 −0.714391
\(935\) −3.81700 −0.124829
\(936\) 0 0
\(937\) 34.5526 1.12878 0.564392 0.825507i \(-0.309111\pi\)
0.564392 + 0.825507i \(0.309111\pi\)
\(938\) −13.8291 −0.451536
\(939\) 0 0
\(940\) −36.4674 −1.18944
\(941\) 20.6233 0.672299 0.336149 0.941809i \(-0.390875\pi\)
0.336149 + 0.941809i \(0.390875\pi\)
\(942\) 0 0
\(943\) 15.2741 0.497394
\(944\) −0.0271471 −0.000883562 0
\(945\) 0 0
\(946\) 4.75973 0.154752
\(947\) 29.4999 0.958619 0.479309 0.877646i \(-0.340887\pi\)
0.479309 + 0.877646i \(0.340887\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 3.06770 0.0995295
\(951\) 0 0
\(952\) −6.27173 −0.203268
\(953\) −26.2389 −0.849963 −0.424981 0.905202i \(-0.639719\pi\)
−0.424981 + 0.905202i \(0.639719\pi\)
\(954\) 0 0
\(955\) −25.1987 −0.815409
\(956\) 20.1847 0.652818
\(957\) 0 0
\(958\) 12.4138 0.401073
\(959\) 34.9638 1.12904
\(960\) 0 0
\(961\) 3.80731 0.122817
\(962\) 0 0
\(963\) 0 0
\(964\) −6.17331 −0.198829
\(965\) 37.8974 1.21996
\(966\) 0 0
\(967\) −17.5176 −0.563330 −0.281665 0.959513i \(-0.590887\pi\)
−0.281665 + 0.959513i \(0.590887\pi\)
\(968\) −19.5972 −0.629877
\(969\) 0 0
\(970\) −4.86964 −0.156355
\(971\) −20.5120 −0.658262 −0.329131 0.944284i \(-0.606756\pi\)
−0.329131 + 0.944284i \(0.606756\pi\)
\(972\) 0 0
\(973\) 32.4295 1.03964
\(974\) 12.7199 0.407572
\(975\) 0 0
\(976\) −18.0586 −0.578042
\(977\) 25.4450 0.814059 0.407030 0.913415i \(-0.366565\pi\)
0.407030 + 0.913415i \(0.366565\pi\)
\(978\) 0 0
\(979\) 17.6461 0.563971
\(980\) 1.17092 0.0374035
\(981\) 0 0
\(982\) 1.02310 0.0326484
\(983\) −39.5244 −1.26063 −0.630316 0.776339i \(-0.717074\pi\)
−0.630316 + 0.776339i \(0.717074\pi\)
\(984\) 0 0
\(985\) −36.3575 −1.15845
\(986\) 4.98493 0.158753
\(987\) 0 0
\(988\) 0 0
\(989\) 32.9584 1.04802
\(990\) 0 0
\(991\) −29.8377 −0.947826 −0.473913 0.880572i \(-0.657159\pi\)
−0.473913 + 0.880572i \(0.657159\pi\)
\(992\) 31.5332 1.00118
\(993\) 0 0
\(994\) 20.5284 0.651121
\(995\) 38.0683 1.20685
\(996\) 0 0
\(997\) −4.93123 −0.156174 −0.0780868 0.996947i \(-0.524881\pi\)
−0.0780868 + 0.996947i \(0.524881\pi\)
\(998\) −6.67861 −0.211408
\(999\) 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 1521.2.a.o.1.2 3
3.2 odd 2 169.2.a.c.1.2 yes 3
12.11 even 2 2704.2.a.ba.1.1 3
13.5 odd 4 1521.2.b.l.1351.4 6
13.8 odd 4 1521.2.b.l.1351.3 6
13.12 even 2 1521.2.a.r.1.2 3
15.14 odd 2 4225.2.a.bb.1.2 3
21.20 even 2 8281.2.a.bj.1.2 3
39.2 even 12 169.2.e.b.147.4 12
39.5 even 4 169.2.b.b.168.3 6
39.8 even 4 169.2.b.b.168.4 6
39.11 even 12 169.2.e.b.147.3 12
39.17 odd 6 169.2.c.c.146.2 6
39.20 even 12 169.2.e.b.23.4 12
39.23 odd 6 169.2.c.c.22.2 6
39.29 odd 6 169.2.c.b.22.2 6
39.32 even 12 169.2.e.b.23.3 12
39.35 odd 6 169.2.c.b.146.2 6
39.38 odd 2 169.2.a.b.1.2 3
156.47 odd 4 2704.2.f.o.337.2 6
156.83 odd 4 2704.2.f.o.337.1 6
156.155 even 2 2704.2.a.z.1.1 3
195.194 odd 2 4225.2.a.bg.1.2 3
273.272 even 2 8281.2.a.bf.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.2 3 39.38 odd 2
169.2.a.c.1.2 yes 3 3.2 odd 2
169.2.b.b.168.3 6 39.5 even 4
169.2.b.b.168.4 6 39.8 even 4
169.2.c.b.22.2 6 39.29 odd 6
169.2.c.b.146.2 6 39.35 odd 6
169.2.c.c.22.2 6 39.23 odd 6
169.2.c.c.146.2 6 39.17 odd 6
169.2.e.b.23.3 12 39.32 even 12
169.2.e.b.23.4 12 39.20 even 12
169.2.e.b.147.3 12 39.11 even 12
169.2.e.b.147.4 12 39.2 even 12
1521.2.a.o.1.2 3 1.1 even 1 trivial
1521.2.a.r.1.2 3 13.12 even 2
1521.2.b.l.1351.3 6 13.8 odd 4
1521.2.b.l.1351.4 6 13.5 odd 4
2704.2.a.z.1.1 3 156.155 even 2
2704.2.a.ba.1.1 3 12.11 even 2
2704.2.f.o.337.1 6 156.83 odd 4
2704.2.f.o.337.2 6 156.47 odd 4
4225.2.a.bb.1.2 3 15.14 odd 2
4225.2.a.bg.1.2 3 195.194 odd 2
8281.2.a.bf.1.2 3 273.272 even 2
8281.2.a.bj.1.2 3 21.20 even 2