Properties

Label 48.22.a.d.1.1
Level $48$
Weight $22$
Character 48.1
Self dual yes
Analytic conductor $134.149$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [48,22,Mod(1,48)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("48.1"); S:= CuspForms(chi, 22); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(48, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 22, names="a")
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,59049,0,-41512770] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(134.149125258\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 48.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+59049.0 q^{3} -4.15128e7 q^{5} -5.38430e8 q^{7} +3.48678e9 q^{9} +6.41130e10 q^{11} -1.30980e11 q^{13} -2.45129e12 q^{15} +8.24203e12 q^{17} -1.34921e13 q^{19} -3.17937e13 q^{21} +2.33185e14 q^{23} +1.24647e15 q^{25} +2.05891e14 q^{27} -2.02456e15 q^{29} +6.86919e15 q^{31} +3.78581e15 q^{33} +2.23517e16 q^{35} +3.44400e15 q^{37} -7.73424e15 q^{39} -2.18424e16 q^{41} +7.17928e16 q^{43} -1.44746e17 q^{45} -2.83545e17 q^{47} -2.68639e17 q^{49} +4.86684e17 q^{51} -2.17229e18 q^{53} -2.66151e18 q^{55} -7.96695e17 q^{57} -1.53483e18 q^{59} +4.31159e18 q^{61} -1.87739e18 q^{63} +5.43735e18 q^{65} -9.24391e18 q^{67} +1.37693e19 q^{69} +2.03874e19 q^{71} +1.66178e19 q^{73} +7.36030e19 q^{75} -3.45204e19 q^{77} -6.79403e19 q^{79} +1.21577e19 q^{81} -3.95037e19 q^{83} -3.42149e20 q^{85} -1.19548e20 q^{87} +4.16117e19 q^{89} +7.05236e19 q^{91} +4.05619e20 q^{93} +5.60095e20 q^{95} +5.71815e19 q^{97} +2.23548e20 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display 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 0
\(3\) 59049.0 0.577350
\(4\) 0 0
\(5\) −4.15128e7 −1.90106 −0.950532 − 0.310627i \(-0.899461\pi\)
−0.950532 + 0.310627i \(0.899461\pi\)
\(6\) 0 0
\(7\) −5.38430e8 −0.720443 −0.360222 − 0.932867i \(-0.617299\pi\)
−0.360222 + 0.932867i \(0.617299\pi\)
\(8\) 0 0
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) 6.41130e10 0.745286 0.372643 − 0.927975i \(-0.378452\pi\)
0.372643 + 0.927975i \(0.378452\pi\)
\(12\) 0 0
\(13\) −1.30980e11 −0.263512 −0.131756 − 0.991282i \(-0.542062\pi\)
−0.131756 + 0.991282i \(0.542062\pi\)
\(14\) 0 0
\(15\) −2.45129e12 −1.09758
\(16\) 0 0
\(17\) 8.24203e12 0.991563 0.495782 − 0.868447i \(-0.334882\pi\)
0.495782 + 0.868447i \(0.334882\pi\)
\(18\) 0 0
\(19\) −1.34921e13 −0.504855 −0.252428 − 0.967616i \(-0.581229\pi\)
−0.252428 + 0.967616i \(0.581229\pi\)
\(20\) 0 0
\(21\) −3.17937e13 −0.415948
\(22\) 0 0
\(23\) 2.33185e14 1.17370 0.586851 − 0.809695i \(-0.300367\pi\)
0.586851 + 0.809695i \(0.300367\pi\)
\(24\) 0 0
\(25\) 1.24647e15 2.61404
\(26\) 0 0
\(27\) 2.05891e14 0.192450
\(28\) 0 0
\(29\) −2.02456e15 −0.893618 −0.446809 − 0.894629i \(-0.647440\pi\)
−0.446809 + 0.894629i \(0.647440\pi\)
\(30\) 0 0
\(31\) 6.86919e15 1.50525 0.752624 − 0.658451i \(-0.228788\pi\)
0.752624 + 0.658451i \(0.228788\pi\)
\(32\) 0 0
\(33\) 3.78581e15 0.430291
\(34\) 0 0
\(35\) 2.23517e16 1.36961
\(36\) 0 0
\(37\) 3.44400e15 0.117746 0.0588728 − 0.998265i \(-0.481249\pi\)
0.0588728 + 0.998265i \(0.481249\pi\)
\(38\) 0 0
\(39\) −7.73424e15 −0.152139
\(40\) 0 0
\(41\) −2.18424e16 −0.254138 −0.127069 − 0.991894i \(-0.540557\pi\)
−0.127069 + 0.991894i \(0.540557\pi\)
\(42\) 0 0
\(43\) 7.17928e16 0.506597 0.253298 − 0.967388i \(-0.418485\pi\)
0.253298 + 0.967388i \(0.418485\pi\)
\(44\) 0 0
\(45\) −1.44746e17 −0.633688
\(46\) 0 0
\(47\) −2.83545e17 −0.786310 −0.393155 − 0.919472i \(-0.628617\pi\)
−0.393155 + 0.919472i \(0.628617\pi\)
\(48\) 0 0
\(49\) −2.68639e17 −0.480962
\(50\) 0 0
\(51\) 4.86684e17 0.572479
\(52\) 0 0
\(53\) −2.17229e18 −1.70616 −0.853081 − 0.521779i \(-0.825269\pi\)
−0.853081 + 0.521779i \(0.825269\pi\)
\(54\) 0 0
\(55\) −2.66151e18 −1.41684
\(56\) 0 0
\(57\) −7.96695e17 −0.291478
\(58\) 0 0
\(59\) −1.53483e18 −0.390944 −0.195472 − 0.980709i \(-0.562624\pi\)
−0.195472 + 0.980709i \(0.562624\pi\)
\(60\) 0 0
\(61\) 4.31159e18 0.773881 0.386940 − 0.922105i \(-0.373532\pi\)
0.386940 + 0.922105i \(0.373532\pi\)
\(62\) 0 0
\(63\) −1.87739e18 −0.240148
\(64\) 0 0
\(65\) 5.43735e18 0.500953
\(66\) 0 0
\(67\) −9.24391e18 −0.619541 −0.309771 − 0.950811i \(-0.600252\pi\)
−0.309771 + 0.950811i \(0.600252\pi\)
\(68\) 0 0
\(69\) 1.37693e19 0.677637
\(70\) 0 0
\(71\) 2.03874e19 0.743273 0.371636 − 0.928378i \(-0.378797\pi\)
0.371636 + 0.928378i \(0.378797\pi\)
\(72\) 0 0
\(73\) 1.66178e19 0.452566 0.226283 − 0.974062i \(-0.427343\pi\)
0.226283 + 0.974062i \(0.427343\pi\)
\(74\) 0 0
\(75\) 7.36030e19 1.50922
\(76\) 0 0
\(77\) −3.45204e19 −0.536936
\(78\) 0 0
\(79\) −6.79403e19 −0.807315 −0.403658 − 0.914910i \(-0.632261\pi\)
−0.403658 + 0.914910i \(0.632261\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 0 0
\(83\) −3.95037e19 −0.279459 −0.139730 − 0.990190i \(-0.544623\pi\)
−0.139730 + 0.990190i \(0.544623\pi\)
\(84\) 0 0
\(85\) −3.42149e20 −1.88503
\(86\) 0 0
\(87\) −1.19548e20 −0.515931
\(88\) 0 0
\(89\) 4.16117e19 0.141456 0.0707278 − 0.997496i \(-0.477468\pi\)
0.0707278 + 0.997496i \(0.477468\pi\)
\(90\) 0 0
\(91\) 7.05236e19 0.189845
\(92\) 0 0
\(93\) 4.05619e20 0.869055
\(94\) 0 0
\(95\) 5.60095e20 0.959762
\(96\) 0 0
\(97\) 5.71815e19 0.0787322 0.0393661 − 0.999225i \(-0.487466\pi\)
0.0393661 + 0.999225i \(0.487466\pi\)
\(98\) 0 0
\(99\) 2.23548e20 0.248429
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 48.22.a.d.1.1 1
4.3 odd 2 3.22.a.b.1.1 ✓ 1
12.11 even 2 9.22.a.a.1.1 1
20.3 even 4 75.22.b.b.49.1 2
20.7 even 4 75.22.b.b.49.2 2
20.19 odd 2 75.22.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.b.1.1 ✓ 1 4.3 odd 2
9.22.a.a.1.1 1 12.11 even 2
48.22.a.d.1.1 1 1.1 even 1 trivial
75.22.a.a.1.1 1 20.19 odd 2
75.22.b.b.49.1 2 20.3 even 4
75.22.b.b.49.2 2 20.7 even 4