Properties

Label 1521.2.b.l.1351.3
Level $1521$
Weight $2$
Character 1521.1351
Analytic conductor $12.145$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1521,2,Mod(1351,1521)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1521.1351");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(12.1452461474\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 169)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1351.3
Root \(-1.80194i\) of defining polynomial
Character \(\chi\) \(=\) 1521.1351
Dual form 1521.2.b.l.1351.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.554958i q^{2} +1.69202 q^{4} -2.80194i q^{5} -2.69202i q^{7} -2.04892i q^{8} +O(q^{10})\) \(q-0.554958i q^{2} +1.69202 q^{4} -2.80194i q^{5} -2.69202i q^{7} -2.04892i q^{8} -1.55496 q^{10} +1.19806i q^{11} -1.49396 q^{14} +2.24698 q^{16} +1.13706 q^{17} -1.93900i q^{19} -4.74094i q^{20} +0.664874 q^{22} -4.60388 q^{23} -2.85086 q^{25} -4.55496i q^{28} +7.89977 q^{29} -5.89977i q^{31} -5.34481i q^{32} -0.631023i q^{34} -7.54288 q^{35} +0.951083i q^{37} -1.07606 q^{38} -5.74094 q^{40} +3.31767i q^{41} -7.15883 q^{43} +2.02715i q^{44} +2.55496i q^{46} +7.69202i q^{47} -0.246980 q^{49} +1.58211i q^{50} -5.87263 q^{53} +3.35690 q^{55} -5.51573 q^{56} -4.38404i q^{58} +0.0120816i q^{59} -8.03684 q^{61} -3.27413 q^{62} +1.52781 q^{64} +9.25667i q^{67} +1.92394 q^{68} +4.18598i q^{70} -13.7409i q^{71} +12.8170i q^{73} +0.527811 q^{74} -3.28083i q^{76} +3.22521 q^{77} +0.807315 q^{79} -6.29590i q^{80} +1.84117 q^{82} -16.3327i q^{83} -3.18598i q^{85} +3.97285i q^{86} +2.45473 q^{88} +14.7289i q^{89} -7.78986 q^{92} +4.26875 q^{94} -5.43296 q^{95} -3.13169i q^{97} +0.137063i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 10 q^{10} + 10 q^{14} + 4 q^{16} - 4 q^{17} + 6 q^{22} - 10 q^{23} + 10 q^{25} + 2 q^{29} - 8 q^{35} + 24 q^{38} - 6 q^{40} - 26 q^{43} + 8 q^{49} - 2 q^{53} + 12 q^{55} - 8 q^{56} + 8 q^{61} + 2 q^{62} + 22 q^{64} + 42 q^{68} + 16 q^{74} + 16 q^{77} - 10 q^{79} + 28 q^{82} - 30 q^{88} + 10 q^{94} + 6 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1521\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(847\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 0.554958i − 0.392415i −0.980562 0.196207i \(-0.937137\pi\)
0.980562 0.196207i \(-0.0628626\pi\)
\(3\) 0 0
\(4\) 1.69202 0.846011
\(5\) − 2.80194i − 1.25306i −0.779395 0.626532i \(-0.784474\pi\)
0.779395 0.626532i \(-0.215526\pi\)
\(6\) 0 0
\(7\) − 2.69202i − 1.01749i −0.860918 0.508744i \(-0.830110\pi\)
0.860918 0.508744i \(-0.169890\pi\)
\(8\) − 2.04892i − 0.724402i
\(9\) 0 0
\(10\) −1.55496 −0.491721
\(11\) 1.19806i 0.361229i 0.983554 + 0.180615i \(0.0578087\pi\)
−0.983554 + 0.180615i \(0.942191\pi\)
\(12\) 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 \(-0.455968\pi\)
0.137889 + 0.990448i \(0.455968\pi\)
\(18\) 0 0
\(19\) − 1.93900i − 0.444837i −0.974951 0.222419i \(-0.928605\pi\)
0.974951 0.222419i \(-0.0713952\pi\)
\(20\) − 4.74094i − 1.06011i
\(21\) 0 0
\(22\) 0.664874 0.141752
\(23\) −4.60388 −0.959974 −0.479987 0.877275i \(-0.659359\pi\)
−0.479987 + 0.877275i \(0.659359\pi\)
\(24\) 0 0
\(25\) −2.85086 −0.570171
\(26\) 0 0
\(27\) 0 0
\(28\) − 4.55496i − 0.860806i
\(29\) 7.89977 1.46695 0.733475 0.679716i \(-0.237897\pi\)
0.733475 + 0.679716i \(0.237897\pi\)
\(30\) 0 0
\(31\) − 5.89977i − 1.05963i −0.848113 0.529815i \(-0.822261\pi\)
0.848113 0.529815i \(-0.177739\pi\)
\(32\) − 5.34481i − 0.944839i
\(33\) 0 0
\(34\) − 0.631023i − 0.108219i
\(35\) −7.54288 −1.27498
\(36\) 0 0
\(37\) 0.951083i 0.156357i 0.996939 + 0.0781785i \(0.0249104\pi\)
−0.996939 + 0.0781785i \(0.975090\pi\)
\(38\) −1.07606 −0.174561
\(39\) 0 0
\(40\) −5.74094 −0.907722
\(41\) 3.31767i 0.518133i 0.965860 + 0.259066i \(0.0834148\pi\)
−0.965860 + 0.259066i \(0.916585\pi\)
\(42\) 0 0
\(43\) −7.15883 −1.09171 −0.545856 0.837879i \(-0.683795\pi\)
−0.545856 + 0.837879i \(0.683795\pi\)
\(44\) 2.02715i 0.305604i
\(45\) 0 0
\(46\) 2.55496i 0.376708i
\(47\) 7.69202i 1.12200i 0.827817 + 0.560998i \(0.189583\pi\)
−0.827817 + 0.560998i \(0.810417\pi\)
\(48\) 0 0
\(49\) −0.246980 −0.0352828
\(50\) 1.58211i 0.223743i
\(51\) 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
\(58\) − 4.38404i − 0.575653i
\(59\) 0.0120816i 0.00157289i 1.00000 0.000786444i \(0.000250333\pi\)
−1.00000 0.000786444i \(0.999750\pi\)
\(60\) 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
\(67\) 9.25667i 1.13088i 0.824789 + 0.565441i \(0.191294\pi\)
−0.824789 + 0.565441i \(0.808706\pi\)
\(68\) 1.92394 0.233311
\(69\) 0 0
\(70\) 4.18598i 0.500320i
\(71\) − 13.7409i − 1.63075i −0.578934 0.815375i \(-0.696531\pi\)
0.578934 0.815375i \(-0.303469\pi\)
\(72\) 0 0
\(73\) 12.8170i 1.50012i 0.661372 + 0.750058i \(0.269975\pi\)
−0.661372 + 0.750058i \(0.730025\pi\)
\(74\) 0.527811 0.0613568
\(75\) 0 0
\(76\) − 3.28083i − 0.376337i
\(77\) 3.22521 0.367547
\(78\) 0 0
\(79\) 0.807315 0.0908300 0.0454150 0.998968i \(-0.485539\pi\)
0.0454150 + 0.998968i \(0.485539\pi\)
\(80\) − 6.29590i − 0.703903i
\(81\) 0 0
\(82\) 1.84117 0.203323
\(83\) − 16.3327i − 1.79275i −0.443296 0.896375i \(-0.646191\pi\)
0.443296 0.896375i \(-0.353809\pi\)
\(84\) 0 0
\(85\) − 3.18598i − 0.345568i
\(86\) 3.97285i 0.428404i
\(87\) 0 0
\(88\) 2.45473 0.261675
\(89\) 14.7289i 1.56126i 0.624996 + 0.780628i \(0.285101\pi\)
−0.624996 + 0.780628i \(0.714899\pi\)
\(90\) 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
\(97\) − 3.13169i − 0.317975i −0.987281 0.158987i \(-0.949177\pi\)
0.987281 0.158987i \(-0.0508229\pi\)
\(98\) 0.137063i 0.0138455i
\(99\) 0 0
\(100\) −4.82371 −0.482371
\(101\) 5.29052 0.526426 0.263213 0.964738i \(-0.415218\pi\)
0.263213 + 0.964738i \(0.415218\pi\)
\(102\) 0 0
\(103\) 13.5308 1.33323 0.666614 0.745403i \(-0.267743\pi\)
0.666614 + 0.745403i \(0.267743\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 3.25906i 0.316548i
\(107\) −5.63102 −0.544371 −0.272186 0.962245i \(-0.587747\pi\)
−0.272186 + 0.962245i \(0.587747\pi\)
\(108\) 0 0
\(109\) − 4.17629i − 0.400016i −0.979794 0.200008i \(-0.935903\pi\)
0.979794 0.200008i \(-0.0640969\pi\)
\(110\) − 1.86294i − 0.177624i
\(111\) 0 0
\(112\) − 6.04892i − 0.571569i
\(113\) −7.64310 −0.719003 −0.359501 0.933145i \(-0.617053\pi\)
−0.359501 + 0.933145i \(0.617053\pi\)
\(114\) 0 0
\(115\) 12.8998i 1.20291i
\(116\) 13.3666 1.24106
\(117\) 0 0
\(118\) 0.00670477 0.000617224 0
\(119\) − 3.06100i − 0.280601i
\(120\) 0 0
\(121\) 9.56465 0.869513
\(122\) 4.46011i 0.403799i
\(123\) 0 0
\(124\) − 9.98254i − 0.896459i
\(125\) − 6.02177i − 0.538604i
\(126\) 0 0
\(127\) −6.77777 −0.601430 −0.300715 0.953714i \(-0.597225\pi\)
−0.300715 + 0.953714i \(0.597225\pi\)
\(128\) − 11.5375i − 1.01978i
\(129\) 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
\(136\) − 2.32975i − 0.199774i
\(137\) − 12.9879i − 1.10963i −0.831973 0.554816i \(-0.812789\pi\)
0.831973 0.554816i \(-0.187211\pi\)
\(138\) 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
\(145\) − 22.1347i − 1.83818i
\(146\) 7.11290 0.588668
\(147\) 0 0
\(148\) 1.60925i 0.132280i
\(149\) − 0.740939i − 0.0607001i −0.999539 0.0303500i \(-0.990338\pi\)
0.999539 0.0303500i \(-0.00966220\pi\)
\(150\) 0 0
\(151\) − 19.0737i − 1.55219i −0.630614 0.776097i \(-0.717197\pi\)
0.630614 0.776097i \(-0.282803\pi\)
\(152\) −3.97285 −0.322241
\(153\) 0 0
\(154\) − 1.78986i − 0.144231i
\(155\) −16.5308 −1.32779
\(156\) 0 0
\(157\) −4.02177 −0.320972 −0.160486 0.987038i \(-0.551306\pi\)
−0.160486 + 0.987038i \(0.551306\pi\)
\(158\) − 0.448026i − 0.0356430i
\(159\) 0 0
\(160\) −14.9758 −1.18394
\(161\) 12.3937i 0.976763i
\(162\) 0 0
\(163\) 15.1371i 1.18563i 0.805340 + 0.592813i \(0.201983\pi\)
−0.805340 + 0.592813i \(0.798017\pi\)
\(164\) 5.61356i 0.438346i
\(165\) 0 0
\(166\) −9.06398 −0.703502
\(167\) 6.26337i 0.484674i 0.970192 + 0.242337i \(0.0779140\pi\)
−0.970192 + 0.242337i \(0.922086\pi\)
\(168\) 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 \(-0.285871\pi\)
0.623105 + 0.782138i \(0.285871\pi\)
\(174\) 0 0
\(175\) 7.67456i 0.580142i
\(176\) 2.69202i 0.202919i
\(177\) 0 0
\(178\) 8.17390 0.612660
\(179\) −2.45473 −0.183475 −0.0917376 0.995783i \(-0.529242\pi\)
−0.0917376 + 0.995783i \(0.529242\pi\)
\(180\) 0 0
\(181\) −11.8073 −0.877631 −0.438815 0.898577i \(-0.644602\pi\)
−0.438815 + 0.898577i \(0.644602\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 9.43296i 0.695407i
\(185\) 2.66487 0.195925
\(186\) 0 0
\(187\) 1.36227i 0.0996192i
\(188\) 13.0151i 0.949221i
\(189\) 0 0
\(190\) 3.01507i 0.218736i
\(191\) 8.99330 0.650732 0.325366 0.945588i \(-0.394512\pi\)
0.325366 + 0.945588i \(0.394512\pi\)
\(192\) 0 0
\(193\) 13.5254i 0.973581i 0.873519 + 0.486790i \(0.161832\pi\)
−0.873519 + 0.486790i \(0.838168\pi\)
\(194\) −1.73795 −0.124778
\(195\) 0 0
\(196\) −0.417895 −0.0298496
\(197\) 12.9758i 0.924490i 0.886752 + 0.462245i \(0.152956\pi\)
−0.886752 + 0.462245i \(0.847044\pi\)
\(198\) 0 0
\(199\) 13.5864 0.963116 0.481558 0.876414i \(-0.340071\pi\)
0.481558 + 0.876414i \(0.340071\pi\)
\(200\) 5.84117i 0.413033i
\(201\) 0 0
\(202\) − 2.93602i − 0.206577i
\(203\) − 21.2664i − 1.49261i
\(204\) 0 0
\(205\) 9.29590 0.649254
\(206\) − 7.50902i − 0.523179i
\(207\) 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
\(214\) 3.12498i 0.213619i
\(215\) 20.0586i 1.36799i
\(216\) 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
\(223\) 11.4058i 0.763790i 0.924206 + 0.381895i \(0.124728\pi\)
−0.924206 + 0.381895i \(0.875272\pi\)
\(224\) −14.3884 −0.961362
\(225\) 0 0
\(226\) 4.24160i 0.282147i
\(227\) 10.6407i 0.706249i 0.935576 + 0.353124i \(0.114881\pi\)
−0.935576 + 0.353124i \(0.885119\pi\)
\(228\) 0 0
\(229\) − 1.13946i − 0.0752974i −0.999291 0.0376487i \(-0.988013\pi\)
0.999291 0.0376487i \(-0.0119868\pi\)
\(230\) 7.15883 0.472040
\(231\) 0 0
\(232\) − 16.1860i − 1.06266i
\(233\) −10.8509 −0.710863 −0.355432 0.934702i \(-0.615666\pi\)
−0.355432 + 0.934702i \(0.615666\pi\)
\(234\) 0 0
\(235\) 21.5526 1.40593
\(236\) 0.0204423i 0.00133068i
\(237\) 0 0
\(238\) −1.69873 −0.110112
\(239\) − 11.9293i − 0.771643i −0.922573 0.385822i \(-0.873918\pi\)
0.922573 0.385822i \(-0.126082\pi\)
\(240\) 0 0
\(241\) − 3.64848i − 0.235019i −0.993072 0.117510i \(-0.962509\pi\)
0.993072 0.117510i \(-0.0374911\pi\)
\(242\) − 5.30798i − 0.341210i
\(243\) 0 0
\(244\) −13.5985 −0.870555
\(245\) 0.692021i 0.0442116i
\(246\) 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 \(-0.486200\pi\)
0.0433406 + 0.999060i \(0.486200\pi\)
\(252\) 0 0
\(253\) − 5.51573i − 0.346771i
\(254\) 3.76138i 0.236010i
\(255\) 0 0
\(256\) −3.34721 −0.209200
\(257\) 29.4359 1.83616 0.918082 0.396391i \(-0.129737\pi\)
0.918082 + 0.396391i \(0.129737\pi\)
\(258\) 0 0
\(259\) 2.56033 0.159091
\(260\) 0 0
\(261\) 0 0
\(262\) − 7.59179i − 0.469023i
\(263\) −10.6963 −0.659564 −0.329782 0.944057i \(-0.606975\pi\)
−0.329782 + 0.944057i \(0.606975\pi\)
\(264\) 0 0
\(265\) 16.4547i 1.01081i
\(266\) 2.89679i 0.177613i
\(267\) 0 0
\(268\) 15.6625i 0.956738i
\(269\) 10.1860 0.621050 0.310525 0.950565i \(-0.399495\pi\)
0.310525 + 0.950565i \(0.399495\pi\)
\(270\) 0 0
\(271\) − 29.4523i − 1.78910i −0.446966 0.894551i \(-0.647495\pi\)
0.446966 0.894551i \(-0.352505\pi\)
\(272\) 2.55496 0.154917
\(273\) 0 0
\(274\) −7.20775 −0.435436
\(275\) − 3.41550i − 0.205963i
\(276\) 0 0
\(277\) 10.2446 0.615538 0.307769 0.951461i \(-0.400418\pi\)
0.307769 + 0.951461i \(0.400418\pi\)
\(278\) − 6.68532i − 0.400959i
\(279\) 0 0
\(280\) 15.4547i 0.923597i
\(281\) 11.5646i 0.689889i 0.938623 + 0.344944i \(0.112102\pi\)
−0.938623 + 0.344944i \(0.887898\pi\)
\(282\) 0 0
\(283\) 30.7090 1.82546 0.912730 0.408562i \(-0.133970\pi\)
0.912730 + 0.408562i \(0.133970\pi\)
\(284\) − 23.2500i − 1.37963i
\(285\) 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
\(292\) 21.6866i 1.26911i
\(293\) 18.6082i 1.08710i 0.839376 + 0.543551i \(0.182921\pi\)
−0.839376 + 0.543551i \(0.817079\pi\)
\(294\) 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
\(301\) 19.2717i 1.11080i
\(302\) −10.5851 −0.609103
\(303\) 0 0
\(304\) − 4.35690i − 0.249885i
\(305\) 22.5187i 1.28942i
\(306\) 0 0
\(307\) 8.94438i 0.510483i 0.966877 + 0.255241i \(0.0821549\pi\)
−0.966877 + 0.255241i \(0.917845\pi\)
\(308\) 5.45712 0.310948
\(309\) 0 0
\(310\) 9.17390i 0.521042i
\(311\) 21.0398 1.19306 0.596529 0.802591i \(-0.296546\pi\)
0.596529 + 0.802591i \(0.296546\pi\)
\(312\) 0 0
\(313\) −7.12737 −0.402863 −0.201432 0.979503i \(-0.564559\pi\)
−0.201432 + 0.979503i \(0.564559\pi\)
\(314\) 2.23191i 0.125954i
\(315\) 0 0
\(316\) 1.36599 0.0768431
\(317\) 23.9651i 1.34601i 0.739636 + 0.673007i \(0.234997\pi\)
−0.739636 + 0.673007i \(0.765003\pi\)
\(318\) 0 0
\(319\) 9.46442i 0.529906i
\(320\) − 4.28083i − 0.239306i
\(321\) 0 0
\(322\) 6.87800 0.383296
\(323\) − 2.20477i − 0.122677i
\(324\) 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
\(331\) − 2.89546i − 0.159149i −0.996829 0.0795745i \(-0.974644\pi\)
0.996829 0.0795745i \(-0.0253561\pi\)
\(332\) − 27.6353i − 1.51669i
\(333\) 0 0
\(334\) 3.47591 0.190193
\(335\) 25.9366 1.41707
\(336\) 0 0
\(337\) 3.10560 0.169173 0.0845865 0.996416i \(-0.473043\pi\)
0.0845865 + 0.996416i \(0.473043\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) − 5.39075i − 0.292354i
\(341\) 7.06829 0.382770
\(342\) 0 0
\(343\) − 18.1793i − 0.981589i
\(344\) 14.6679i 0.790838i
\(345\) 0 0
\(346\) − 9.09651i − 0.489031i
\(347\) 11.3787 0.610839 0.305419 0.952218i \(-0.401203\pi\)
0.305419 + 0.952218i \(0.401203\pi\)
\(348\) 0 0
\(349\) − 3.34721i − 0.179172i −0.995979 0.0895859i \(-0.971446\pi\)
0.995979 0.0895859i \(-0.0285544\pi\)
\(350\) 4.25906 0.227656
\(351\) 0 0
\(352\) 6.40342 0.341303
\(353\) 0.637727i 0.0339428i 0.999856 + 0.0169714i \(0.00540242\pi\)
−0.999856 + 0.0169714i \(0.994598\pi\)
\(354\) 0 0
\(355\) −38.5013 −2.04343
\(356\) 24.9215i 1.32084i
\(357\) 0 0
\(358\) 1.36227i 0.0719983i
\(359\) 21.4590i 1.13256i 0.824211 + 0.566282i \(0.191619\pi\)
−0.824211 + 0.566282i \(0.808381\pi\)
\(360\) 0 0
\(361\) 15.2403 0.802120
\(362\) 6.55257i 0.344395i
\(363\) 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
\(370\) − 1.47889i − 0.0768840i
\(371\) 15.8092i 0.820775i
\(372\) 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
\(379\) − 35.8702i − 1.84253i −0.388935 0.921265i \(-0.627157\pi\)
0.388935 0.921265i \(-0.372843\pi\)
\(380\) −9.19269 −0.471575
\(381\) 0 0
\(382\) − 4.99090i − 0.255357i
\(383\) 4.85517i 0.248087i 0.992277 + 0.124044i \(0.0395863\pi\)
−0.992277 + 0.124044i \(0.960414\pi\)
\(384\) 0 0
\(385\) − 9.03684i − 0.460560i
\(386\) 7.50604 0.382047
\(387\) 0 0
\(388\) − 5.29888i − 0.269010i
\(389\) 2.38537 0.120943 0.0604716 0.998170i \(-0.480740\pi\)
0.0604716 + 0.998170i \(0.480740\pi\)
\(390\) 0 0
\(391\) −5.23490 −0.264740
\(392\) 0.506041i 0.0255589i
\(393\) 0 0
\(394\) 7.20105 0.362783
\(395\) − 2.26205i − 0.113816i
\(396\) 0 0
\(397\) − 15.2664i − 0.766196i −0.923708 0.383098i \(-0.874857\pi\)
0.923708 0.383098i \(-0.125143\pi\)
\(398\) − 7.53989i − 0.377941i
\(399\) 0 0
\(400\) −6.40581 −0.320291
\(401\) − 12.7584i − 0.637124i −0.947902 0.318562i \(-0.896800\pi\)
0.947902 0.318562i \(-0.103200\pi\)
\(402\) 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
\(409\) 25.3588i 1.25391i 0.779054 + 0.626956i \(0.215700\pi\)
−0.779054 + 0.626956i \(0.784300\pi\)
\(410\) − 5.15883i − 0.254777i
\(411\) 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
\(418\) − 1.28919i − 0.0630565i
\(419\) 11.6673 0.569983 0.284992 0.958530i \(-0.408009\pi\)
0.284992 + 0.958530i \(0.408009\pi\)
\(420\) 0 0
\(421\) 8.29291i 0.404172i 0.979368 + 0.202086i \(0.0647720\pi\)
−0.979368 + 0.202086i \(0.935228\pi\)
\(422\) − 5.80492i − 0.282579i
\(423\) 0 0
\(424\) 12.0325i 0.584351i
\(425\) −3.24160 −0.157241
\(426\) 0 0
\(427\) 21.6353i 1.04701i
\(428\) −9.52781 −0.460544
\(429\) 0 0
\(430\) 11.1317 0.536818
\(431\) 0.932296i 0.0449071i 0.999748 + 0.0224536i \(0.00714779\pi\)
−0.999748 + 0.0224536i \(0.992852\pi\)
\(432\) 0 0
\(433\) 13.3502 0.641569 0.320785 0.947152i \(-0.396053\pi\)
0.320785 + 0.947152i \(0.396053\pi\)
\(434\) 8.81402i 0.423086i
\(435\) 0 0
\(436\) − 7.06638i − 0.338418i
\(437\) 8.92692i 0.427032i
\(438\) 0 0
\(439\) −13.9922 −0.667813 −0.333906 0.942606i \(-0.608367\pi\)
−0.333906 + 0.942606i \(0.608367\pi\)
\(440\) − 6.87800i − 0.327896i
\(441\) 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
\(448\) − 4.11290i − 0.194316i
\(449\) 12.5864i 0.593990i 0.954879 + 0.296995i \(0.0959844\pi\)
−0.954879 + 0.296995i \(0.904016\pi\)
\(450\) 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
\(457\) 33.6383i 1.57353i 0.617250 + 0.786767i \(0.288247\pi\)
−0.617250 + 0.786767i \(0.711753\pi\)
\(458\) −0.632351 −0.0295478
\(459\) 0 0
\(460\) 21.8267i 1.01767i
\(461\) − 1.40283i − 0.0653363i −0.999466 0.0326681i \(-0.989600\pi\)
0.999466 0.0326681i \(-0.0104004\pi\)
\(462\) 0 0
\(463\) − 15.2010i − 0.706453i −0.935538 0.353226i \(-0.885085\pi\)
0.935538 0.353226i \(-0.114915\pi\)
\(464\) 17.7506 0.824052
\(465\) 0 0
\(466\) 6.02177i 0.278953i
\(467\) −39.3414 −1.82050 −0.910250 0.414058i \(-0.864111\pi\)
−0.910250 + 0.414058i \(0.864111\pi\)
\(468\) 0 0
\(469\) 24.9191 1.15066
\(470\) − 11.9608i − 0.551709i
\(471\) 0 0
\(472\) 0.0247542 0.00113940
\(473\) − 8.57673i − 0.394358i
\(474\) 0 0
\(475\) 5.52781i 0.253633i
\(476\) − 5.17928i − 0.237392i
\(477\) 0 0
\(478\) −6.62027 −0.302804
\(479\) 22.3690i 1.02206i 0.859561 + 0.511032i \(0.170737\pi\)
−0.859561 + 0.511032i \(0.829263\pi\)
\(480\) 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
\(487\) − 22.9205i − 1.03863i −0.854584 0.519313i \(-0.826188\pi\)
0.854584 0.519313i \(-0.173812\pi\)
\(488\) 16.4668i 0.745418i
\(489\) 0 0
\(490\) 0.384043 0.0173493
\(491\) 1.84356 0.0831987 0.0415993 0.999134i \(-0.486755\pi\)
0.0415993 + 0.999134i \(0.486755\pi\)
\(492\) 0 0
\(493\) 8.98254 0.404553
\(494\) 0 0
\(495\) 0 0
\(496\) − 13.2567i − 0.595242i
\(497\) −36.9909 −1.65927
\(498\) 0 0
\(499\) 12.0344i 0.538736i 0.963037 + 0.269368i \(0.0868147\pi\)
−0.963037 + 0.269368i \(0.913185\pi\)
\(500\) − 10.1890i − 0.455664i
\(501\) 0 0
\(502\) − 0.762118i − 0.0340150i
\(503\) 30.5056 1.36018 0.680088 0.733130i \(-0.261942\pi\)
0.680088 + 0.733130i \(0.261942\pi\)
\(504\) 0 0
\(505\) − 14.8237i − 0.659646i
\(506\) −3.06100 −0.136078
\(507\) 0 0
\(508\) −11.4681 −0.508816
\(509\) − 1.51142i − 0.0669924i −0.999439 0.0334962i \(-0.989336\pi\)
0.999439 0.0334962i \(-0.0106642\pi\)
\(510\) 0 0
\(511\) 34.5036 1.52635
\(512\) − 21.2174i − 0.937687i
\(513\) 0 0
\(514\) − 16.3357i − 0.720538i
\(515\) − 37.9124i − 1.67062i
\(516\) 0 0
\(517\) −9.21552 −0.405298
\(518\) − 1.42088i − 0.0624298i
\(519\) 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
\(526\) 5.93602i 0.258823i
\(527\) − 6.70841i − 0.292223i
\(528\) 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
\(535\) 15.7778i 0.682133i
\(536\) 18.9661 0.819213
\(537\) 0 0
\(538\) − 5.65279i − 0.243709i
\(539\) − 0.295897i − 0.0127452i
\(540\) 0 0
\(541\) 24.3297i 1.04602i 0.852327 + 0.523009i \(0.175191\pi\)
−0.852327 + 0.523009i \(0.824809\pi\)
\(542\) −16.3448 −0.702070
\(543\) 0 0
\(544\) − 6.07739i − 0.260566i
\(545\) −11.7017 −0.501246
\(546\) 0 0
\(547\) −8.18896 −0.350135 −0.175067 0.984556i \(-0.556014\pi\)
−0.175067 + 0.984556i \(0.556014\pi\)
\(548\) − 21.9758i − 0.938761i
\(549\) 0 0
\(550\) −1.89546 −0.0808227
\(551\) − 15.3177i − 0.652555i
\(552\) 0 0
\(553\) − 2.17331i − 0.0924185i
\(554\) − 5.68532i − 0.241546i
\(555\) 0 0
\(556\) 20.3830 0.864431
\(557\) − 25.3327i − 1.07338i −0.843779 0.536691i \(-0.819674\pi\)
0.843779 0.536691i \(-0.180326\pi\)
\(558\) 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 \(-0.679730\pi\)
−0.535109 + 0.844783i \(0.679730\pi\)
\(564\) 0 0
\(565\) 21.4155i 0.900957i
\(566\) − 17.0422i − 0.716338i
\(567\) 0 0
\(568\) −28.1540 −1.18132
\(569\) −31.1347 −1.30523 −0.652617 0.757688i \(-0.726329\pi\)
−0.652617 + 0.757688i \(0.726329\pi\)
\(570\) 0 0
\(571\) 20.5090 0.858276 0.429138 0.903239i \(-0.358817\pi\)
0.429138 + 0.903239i \(0.358817\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) − 4.95646i − 0.206879i
\(575\) 13.1250 0.547350
\(576\) 0 0
\(577\) − 15.6890i − 0.653143i −0.945172 0.326572i \(-0.894107\pi\)
0.945172 0.326572i \(-0.105893\pi\)
\(578\) 8.71678i 0.362570i
\(579\) 0 0
\(580\) − 37.4523i − 1.55512i
\(581\) −43.9681 −1.82410
\(582\) 0 0
\(583\) − 7.03577i − 0.291392i
\(584\) 26.2610 1.08669
\(585\) 0 0
\(586\) 10.3268 0.426595
\(587\) 30.5687i 1.26171i 0.775903 + 0.630853i \(0.217295\pi\)
−0.775903 + 0.630853i \(0.782705\pi\)
\(588\) 0 0
\(589\) −11.4397 −0.471363
\(590\) − 0.0187864i 0 0.000773422i
\(591\) 0 0
\(592\) 2.13706i 0.0878328i
\(593\) 29.6883i 1.21915i 0.792727 + 0.609576i \(0.208660\pi\)
−0.792727 + 0.609576i \(0.791340\pi\)
\(594\) 0 0
\(595\) −8.57673 −0.351612
\(596\) − 1.25368i − 0.0513529i
\(597\) 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
\(604\) − 32.2731i − 1.31317i
\(605\) − 26.7995i − 1.08956i
\(606\) 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
\(613\) − 3.84846i − 0.155438i −0.996975 0.0777190i \(-0.975236\pi\)
0.996975 0.0777190i \(-0.0247637\pi\)
\(614\) 4.96376 0.200321
\(615\) 0 0
\(616\) − 6.60819i − 0.266251i
\(617\) − 15.0388i − 0.605437i −0.953080 0.302719i \(-0.902106\pi\)
0.953080 0.302719i \(-0.0978943\pi\)
\(618\) 0 0
\(619\) 12.8170i 0.515159i 0.966257 + 0.257579i \(0.0829249\pi\)
−0.966257 + 0.257579i \(0.917075\pi\)
\(620\) −27.9705 −1.12332
\(621\) 0 0
\(622\) − 11.6762i − 0.468174i
\(623\) 39.6504 1.58856
\(624\) 0 0
\(625\) −31.1269 −1.24508
\(626\) 3.95539i 0.158089i
\(627\) 0 0
\(628\) −6.80492 −0.271546
\(629\) 1.08144i 0.0431199i
\(630\) 0 0
\(631\) 25.7517i 1.02516i 0.858640 + 0.512579i \(0.171310\pi\)
−0.858640 + 0.512579i \(0.828690\pi\)
\(632\) − 1.65412i − 0.0657974i
\(633\) 0 0
\(634\) 13.2996 0.528195
\(635\) 18.9909i 0.753631i
\(636\) 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 \(-0.339547\pi\)
0.482999 + 0.875621i \(0.339547\pi\)
\(642\) 0 0
\(643\) 9.97344i 0.393314i 0.980472 + 0.196657i \(0.0630086\pi\)
−0.980472 + 0.196657i \(0.936991\pi\)
\(644\) 20.9705i 0.826352i
\(645\) 0 0
\(646\) −1.22355 −0.0481401
\(647\) 11.8431 0.465600 0.232800 0.972525i \(-0.425211\pi\)
0.232800 + 0.972525i \(0.425211\pi\)
\(648\) 0 0
\(649\) −0.0144745 −0.000568173 0
\(650\) 0 0
\(651\) 0 0
\(652\) 25.6122i 1.00305i
\(653\) 7.47411 0.292484 0.146242 0.989249i \(-0.453282\pi\)
0.146242 + 0.989249i \(0.453282\pi\)
\(654\) 0 0
\(655\) − 38.3303i − 1.49769i
\(656\) 7.45473i 0.291058i
\(657\) 0 0
\(658\) − 11.4916i − 0.447988i
\(659\) −34.1739 −1.33123 −0.665613 0.746297i \(-0.731830\pi\)
−0.665613 + 0.746297i \(0.731830\pi\)
\(660\) 0 0
\(661\) 33.6088i 1.30723i 0.756827 + 0.653615i \(0.226748\pi\)
−0.756827 + 0.653615i \(0.773252\pi\)
\(662\) −1.60686 −0.0624524
\(663\) 0 0
\(664\) −33.4644 −1.29867
\(665\) 14.6256i 0.567158i
\(666\) 0 0
\(667\) −36.3696 −1.40824
\(668\) 10.5978i 0.410040i
\(669\) 0 0
\(670\) − 14.3937i − 0.556078i
\(671\) − 9.62863i − 0.371709i
\(672\) 0 0
\(673\) −48.0320 −1.85150 −0.925750 0.378137i \(-0.876565\pi\)
−0.925750 + 0.378137i \(0.876565\pi\)
\(674\) − 1.72348i − 0.0663860i
\(675\) 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
\(682\) − 3.92261i − 0.150204i
\(683\) − 15.9041i − 0.608553i −0.952584 0.304276i \(-0.901585\pi\)
0.952584 0.304276i \(-0.0984147\pi\)
\(684\) 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
\(691\) − 33.1903i − 1.26262i −0.775531 0.631309i \(-0.782518\pi\)
0.775531 0.631309i \(-0.217482\pi\)
\(692\) 27.7345 1.05431
\(693\) 0 0
\(694\) − 6.31468i − 0.239702i
\(695\) − 33.7536i − 1.28035i
\(696\) 0 0
\(697\) 3.77240i 0.142890i
\(698\) −1.85756 −0.0703097
\(699\) 0 0
\(700\) 12.9855i 0.490807i
\(701\) −14.9129 −0.563253 −0.281627 0.959524i \(-0.590874\pi\)
−0.281627 + 0.959524i \(0.590874\pi\)
\(702\) 0 0
\(703\) 1.84415 0.0695534
\(704\) 1.83041i 0.0689863i
\(705\) 0 0
\(706\) 0.353912 0.0133197
\(707\) − 14.2422i − 0.535633i
\(708\) 0 0
\(709\) 38.4312i 1.44331i 0.692252 + 0.721656i \(0.256619\pi\)
−0.692252 + 0.721656i \(0.743381\pi\)
\(710\) 21.3666i 0.801874i
\(711\) 0 0
\(712\) 30.1782 1.13098
\(713\) 27.1618i 1.01722i
\(714\) 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 \(-0.568411\pi\)
−0.213269 + 0.976993i \(0.568411\pi\)
\(720\) 0 0
\(721\) − 36.4252i − 1.35654i
\(722\) − 8.45771i − 0.314764i
\(723\) 0 0
\(724\) −19.9782 −0.742485
\(725\) −22.5211 −0.836413
\(726\) 0 0
\(727\) −3.63640 −0.134867 −0.0674333 0.997724i \(-0.521481\pi\)
−0.0674333 + 0.997724i \(0.521481\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) − 19.9299i − 0.737639i
\(731\) −8.14005 −0.301071
\(732\) 0 0
\(733\) 3.52217i 0.130094i 0.997882 + 0.0650472i \(0.0207198\pi\)
−0.997882 + 0.0650472i \(0.979280\pi\)
\(734\) 5.20941i 0.192283i
\(735\) 0 0
\(736\) 24.6069i 0.907021i
\(737\) −11.0901 −0.408508
\(738\) 0 0
\(739\) 0.420288i 0.0154605i 0.999970 + 0.00773027i \(0.00246064\pi\)
−0.999970 + 0.00773027i \(0.997539\pi\)
\(740\) 4.50902 0.165755
\(741\) 0 0
\(742\) 8.77346 0.322084
\(743\) − 25.3623i − 0.930452i −0.885192 0.465226i \(-0.845973\pi\)
0.885192 0.465226i \(-0.154027\pi\)
\(744\) 0 0
\(745\) −2.07606 −0.0760611
\(746\) − 15.3870i − 0.563359i
\(747\) 0 0
\(748\) 2.30499i 0.0842790i
\(749\) 15.1588i 0.553892i
\(750\) 0 0
\(751\) −0.650874 −0.0237507 −0.0118754 0.999929i \(-0.503780\pi\)
−0.0118754 + 0.999929i \(0.503780\pi\)
\(752\) 17.2838i 0.630276i
\(753\) 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
\(760\) 11.1317i 0.403789i
\(761\) − 30.9221i − 1.12093i −0.828179 0.560463i \(-0.810623\pi\)
0.828179 0.560463i \(-0.189377\pi\)
\(762\) 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
\(769\) 43.7689i 1.57835i 0.614169 + 0.789174i \(0.289491\pi\)
−0.614169 + 0.789174i \(0.710509\pi\)
\(770\) −5.01507 −0.180730
\(771\) 0 0
\(772\) 22.8853i 0.823660i
\(773\) − 42.4209i − 1.52577i −0.646532 0.762886i \(-0.723782\pi\)
0.646532 0.762886i \(-0.276218\pi\)
\(774\) 0 0
\(775\) 16.8194i 0.604171i
\(776\) −6.41657 −0.230341
\(777\) 0 0
\(778\) − 1.32378i − 0.0474598i
\(779\) 6.43296 0.230485
\(780\) 0 0
\(781\) 16.4625 0.589075
\(782\) 2.90515i 0.103888i
\(783\) 0 0
\(784\) −0.554958 −0.0198199
\(785\) 11.2687i 0.402199i
\(786\) 0 0
\(787\) − 36.0116i − 1.28368i −0.766841 0.641838i \(-0.778172\pi\)
0.766841 0.641838i \(-0.221828\pi\)
\(788\) 21.9554i 0.782129i
\(789\) 0 0
\(790\) −1.25534 −0.0446630
\(791\) 20.5754i 0.731577i
\(792\) 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 \(-0.689819\pi\)
−0.561614 + 0.827399i \(0.689819\pi\)
\(798\) 0 0
\(799\) 8.74632i 0.309422i
\(800\) 15.2373i 0.538720i
\(801\) 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
\(808\) − 10.8398i − 0.381344i
\(809\) 45.2814 1.59201 0.796005 0.605290i \(-0.206943\pi\)
0.796005 + 0.605290i \(0.206943\pi\)
\(810\) 0 0
\(811\) 42.8635i 1.50514i 0.658511 + 0.752571i \(0.271187\pi\)
−0.658511 + 0.752571i \(0.728813\pi\)
\(812\) − 35.9831i − 1.26276i
\(813\) 0 0
\(814\) 0.632351i 0.0221639i
\(815\) 42.4131 1.48567
\(816\) 0 0
\(817\) 13.8810i 0.485634i
\(818\) 14.0731 0.492054
\(819\) 0 0
\(820\) 15.7289 0.549276
\(821\) − 7.82776i − 0.273191i −0.990627 0.136595i \(-0.956384\pi\)
0.990627 0.136595i \(-0.0436160\pi\)
\(822\) 0 0
\(823\) 36.7754 1.28191 0.640955 0.767579i \(-0.278539\pi\)
0.640955 + 0.767579i \(0.278539\pi\)
\(824\) − 27.7235i − 0.965793i
\(825\) 0 0
\(826\) − 0.0180494i 0 0.000628019i
\(827\) − 47.3293i − 1.64580i −0.568186 0.822900i \(-0.692355\pi\)
0.568186 0.822900i \(-0.307645\pi\)
\(828\) 0 0
\(829\) −25.2687 −0.877620 −0.438810 0.898580i \(-0.644600\pi\)
−0.438810 + 0.898580i \(0.644600\pi\)
\(830\) 25.3967i 0.881533i
\(831\) 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
\(838\) − 6.47484i − 0.223670i
\(839\) 37.6883i 1.30114i 0.759444 + 0.650572i \(0.225471\pi\)
−0.759444 + 0.650572i \(0.774529\pi\)
\(840\) 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
\(847\) − 25.7482i − 0.884720i
\(848\) −13.1957 −0.453141
\(849\) 0 0
\(850\) 1.79895i 0.0617036i
\(851\) − 4.37867i − 0.150099i
\(852\) 0 0
\(853\) − 31.0121i − 1.06183i −0.847424 0.530917i \(-0.821848\pi\)
0.847424 0.530917i \(-0.178152\pi\)
\(854\) 12.0067 0.410861
\(855\) 0 0
\(856\) 11.5375i 0.394344i
\(857\) 12.4692 0.425940 0.212970 0.977059i \(-0.431686\pi\)
0.212970 + 0.977059i \(0.431686\pi\)
\(858\) 0 0
\(859\) −17.3163 −0.590826 −0.295413 0.955370i \(-0.595457\pi\)
−0.295413 + 0.955370i \(0.595457\pi\)
\(860\) 33.9396i 1.15733i
\(861\) 0 0
\(862\) 0.517385 0.0176222
\(863\) − 3.46383i − 0.117910i −0.998261 0.0589550i \(-0.981223\pi\)
0.998261 0.0589550i \(-0.0187769\pi\)
\(864\) 0 0
\(865\) − 45.9275i − 1.56158i
\(866\) − 7.40880i − 0.251761i
\(867\) 0 0
\(868\) −26.8732 −0.912136
\(869\) 0.967213i 0.0328105i
\(870\) 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
\(877\) − 57.2549i − 1.93336i −0.255989 0.966680i \(-0.582401\pi\)
0.255989 0.966680i \(-0.417599\pi\)
\(878\) 7.76510i 0.262059i
\(879\) 0 0
\(880\) 7.54288 0.254270
\(881\) −43.1782 −1.45471 −0.727355 0.686261i \(-0.759251\pi\)
−0.727355 + 0.686261i \(0.759251\pi\)
\(882\) 0 0
\(883\) −49.9560 −1.68115 −0.840576 0.541693i \(-0.817784\pi\)
−0.840576 + 0.541693i \(0.817784\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 13.1535i 0.441899i
\(887\) −17.6746 −0.593454 −0.296727 0.954962i \(-0.595895\pi\)
−0.296727 + 0.954962i \(0.595895\pi\)
\(888\) 0 0
\(889\) 18.2459i 0.611948i
\(890\) − 22.9028i − 0.767702i
\(891\) 0 0
\(892\) 19.2989i 0.646174i
\(893\) 14.9148 0.499106
\(894\) 0 0
\(895\) 6.87800i 0.229906i
\(896\) −31.0592 −1.03761
\(897\) 0 0
\(898\) 6.98493 0.233090
\(899\) − 46.6069i − 1.55443i
\(900\) 0 0
\(901\) −6.67755 −0.222461
\(902\) 2.20583i 0.0734462i
\(903\) 0 0
\(904\) 15.6601i 0.520847i
\(905\) 33.0834i 1.09973i
\(906\) 0 0
\(907\) −7.73423 −0.256811 −0.128406 0.991722i \(-0.540986\pi\)
−0.128406 + 0.991722i \(0.540986\pi\)
\(908\) 18.0043i 0.597494i
\(909\) 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
\(916\) − 1.92798i − 0.0637024i
\(917\) − 36.8267i − 1.21612i
\(918\) 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
\(925\) − 2.71140i − 0.0891502i
\(926\) −8.43594 −0.277222
\(927\) 0 0
\(928\) − 42.2228i − 1.38603i
\(929\) 3.55735i 0.116713i 0.998296 + 0.0583565i \(0.0185860\pi\)
−0.998296 + 0.0583565i \(0.981414\pi\)
\(930\) 0 0
\(931\) 0.478894i 0.0156951i
\(932\) −18.3599 −0.601398
\(933\) 0 0
\(934\) 21.8328i 0.714391i
\(935\) 3.81700 0.124829
\(936\) 0 0
\(937\) 34.5526 1.12878 0.564392 0.825507i \(-0.309111\pi\)
0.564392 + 0.825507i \(0.309111\pi\)
\(938\) − 13.8291i − 0.451536i
\(939\) 0 0
\(940\) 36.4674 1.18944
\(941\) 20.6233i 0.672299i 0.941809 + 0.336149i \(0.109125\pi\)
−0.941809 + 0.336149i \(0.890875\pi\)
\(942\) 0 0
\(943\) − 15.2741i − 0.497394i
\(944\) 0.0271471i 0 0.000883562i
\(945\) 0 0
\(946\) −4.75973 −0.154752
\(947\) − 29.4999i − 0.958619i −0.877646 0.479309i \(-0.840887\pi\)
0.877646 0.479309i \(-0.159113\pi\)
\(948\) 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 \(-0.360281\pi\)
0.424981 + 0.905202i \(0.360281\pi\)
\(954\) 0 0
\(955\) − 25.1987i − 0.815409i
\(956\) − 20.1847i − 0.652818i
\(957\) 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
\(964\) − 6.17331i − 0.198829i
\(965\) 37.8974 1.21996
\(966\) 0 0
\(967\) − 17.5176i − 0.563330i −0.959513 0.281665i \(-0.909113\pi\)
0.959513 0.281665i \(-0.0908866\pi\)
\(968\) − 19.5972i − 0.629877i
\(969\) 0 0
\(970\) 4.86964i 0.156355i
\(971\) −20.5120 −0.658262 −0.329131 0.944284i \(-0.606756\pi\)
−0.329131 + 0.944284i \(0.606756\pi\)
\(972\) 0 0
\(973\) − 32.4295i − 1.03964i
\(974\) −12.7199 −0.407572
\(975\) 0 0
\(976\) −18.0586 −0.578042
\(977\) 25.4450i 0.814059i 0.913415 + 0.407030i \(0.133435\pi\)
−0.913415 + 0.407030i \(0.866565\pi\)
\(978\) 0 0
\(979\) −17.6461 −0.563971
\(980\) 1.17092i 0.0374035i
\(981\) 0 0
\(982\) − 1.02310i − 0.0326484i
\(983\) 39.5244i 1.26063i 0.776339 + 0.630316i \(0.217074\pi\)
−0.776339 + 0.630316i \(0.782926\pi\)
\(984\) 0 0
\(985\) 36.3575 1.15845
\(986\) − 4.98493i − 0.158753i
\(987\) 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
\(994\) 20.5284i 0.651121i
\(995\) − 38.0683i − 1.20685i
\(996\) 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.b.l.1351.3 6
3.2 odd 2 169.2.b.b.168.4 6
12.11 even 2 2704.2.f.o.337.2 6
13.5 odd 4 1521.2.a.o.1.2 3
13.8 odd 4 1521.2.a.r.1.2 3
13.12 even 2 inner 1521.2.b.l.1351.4 6
39.2 even 12 169.2.c.b.22.2 6
39.5 even 4 169.2.a.c.1.2 yes 3
39.8 even 4 169.2.a.b.1.2 3
39.11 even 12 169.2.c.c.22.2 6
39.17 odd 6 169.2.e.b.23.3 12
39.20 even 12 169.2.c.c.146.2 6
39.23 odd 6 169.2.e.b.147.4 12
39.29 odd 6 169.2.e.b.147.3 12
39.32 even 12 169.2.c.b.146.2 6
39.35 odd 6 169.2.e.b.23.4 12
39.38 odd 2 169.2.b.b.168.3 6
156.47 odd 4 2704.2.a.z.1.1 3
156.83 odd 4 2704.2.a.ba.1.1 3
156.155 even 2 2704.2.f.o.337.1 6
195.44 even 4 4225.2.a.bb.1.2 3
195.164 even 4 4225.2.a.bg.1.2 3
273.83 odd 4 8281.2.a.bj.1.2 3
273.125 odd 4 8281.2.a.bf.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
169.2.a.b.1.2 3 39.8 even 4
169.2.a.c.1.2 yes 3 39.5 even 4
169.2.b.b.168.3 6 39.38 odd 2
169.2.b.b.168.4 6 3.2 odd 2
169.2.c.b.22.2 6 39.2 even 12
169.2.c.b.146.2 6 39.32 even 12
169.2.c.c.22.2 6 39.11 even 12
169.2.c.c.146.2 6 39.20 even 12
169.2.e.b.23.3 12 39.17 odd 6
169.2.e.b.23.4 12 39.35 odd 6
169.2.e.b.147.3 12 39.29 odd 6
169.2.e.b.147.4 12 39.23 odd 6
1521.2.a.o.1.2 3 13.5 odd 4
1521.2.a.r.1.2 3 13.8 odd 4
1521.2.b.l.1351.3 6 1.1 even 1 trivial
1521.2.b.l.1351.4 6 13.12 even 2 inner
2704.2.a.z.1.1 3 156.47 odd 4
2704.2.a.ba.1.1 3 156.83 odd 4
2704.2.f.o.337.1 6 156.155 even 2
2704.2.f.o.337.2 6 12.11 even 2
4225.2.a.bb.1.2 3 195.44 even 4
4225.2.a.bg.1.2 3 195.164 even 4
8281.2.a.bf.1.2 3 273.125 odd 4
8281.2.a.bj.1.2 3 273.83 odd 4