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Properties

Label 138.4.f.a
Level $138$
Weight $4$
Character orbit 138.f
Analytic conductor $8.142$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [138,4,Mod(5,138)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(138, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([11, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("138.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 138.f (of order \(22\), degree \(10\), 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: \(8.14226358079\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(24\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 240 q - 8 q^{3} + 96 q^{4} - 8 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 240 q - 8 q^{3} + 96 q^{4} - 8 q^{6} - 36 q^{9} + 32 q^{12} - 96 q^{13} - 748 q^{15} - 384 q^{16} - 40 q^{18} + 1320 q^{21} + 32 q^{24} - 144 q^{25} - 56 q^{27} - 1496 q^{30} - 72 q^{31} - 1936 q^{33} + 144 q^{36} - 1188 q^{37} + 660 q^{39} + 1716 q^{43} + 96 q^{46} - 128 q^{48} + 4860 q^{49} + 384 q^{52} + 3036 q^{54} + 3708 q^{55} + 4224 q^{57} + 1032 q^{58} + 88 q^{60} - 1056 q^{61} - 4180 q^{63} + 1536 q^{64} - 5456 q^{66} - 2904 q^{67} - 7832 q^{69} - 6432 q^{70} - 1952 q^{72} + 576 q^{73} - 6914 q^{75} + 1372 q^{78} + 3696 q^{79} + 9264 q^{81} + 3672 q^{82} + 3784 q^{84} + 4104 q^{85} + 938 q^{87} + 2924 q^{93} + 1296 q^{94} - 128 q^{96} - 25344 q^{97} + 2310 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1 −0.563465 + 1.91899i −4.94437 − 1.59789i −3.36501 − 2.16256i 0.0111281 − 0.0773979i 5.85230 − 8.58781i −31.2205 + 14.2579i 6.04600 − 5.23889i 21.8935 + 15.8011i 0.142255 + 0.0649658i
5.2 −0.563465 + 1.91899i −4.74206 + 2.12435i −3.36501 − 2.16256i 0.739504 − 5.14336i −1.40462 − 10.2969i 12.2116 − 5.57687i 6.04600 − 5.23889i 17.9742 − 20.1476i 9.45336 + 4.31720i
5.3 −0.563465 + 1.91899i −3.97934 − 3.34138i −3.36501 − 2.16256i −0.171172 + 1.19053i 8.65429 − 5.75355i 15.2724 − 6.97468i 6.04600 − 5.23889i 4.67033 + 26.5930i −2.18816 − 0.999300i
5.4 −0.563465 + 1.91899i −2.96132 + 4.26973i −3.36501 − 2.16256i 0.260866 − 1.81436i −6.52496 − 8.08857i −4.44632 + 2.03056i 6.04600 − 5.23889i −9.46122 − 25.2880i 3.33474 + 1.52293i
5.5 −0.563465 + 1.91899i −1.79919 − 4.87472i −3.36501 − 2.16256i 2.74427 − 19.0868i 10.3683 − 0.705890i −11.8366 + 5.40561i 6.04600 − 5.23889i −20.5258 + 17.5411i 35.0810 + 16.0210i
5.6 −0.563465 + 1.91899i −0.214595 − 5.19172i −3.36501 − 2.16256i −1.46724 + 10.2049i 10.0838 + 2.51355i 12.0935 − 5.52291i 6.04600 − 5.23889i −26.9079 + 2.22824i −18.7563 − 8.56571i
5.7 −0.563465 + 1.91899i 0.590201 + 5.16252i −3.36501 − 2.16256i −2.48225 + 17.2644i −10.2394 − 1.77632i −6.69475 + 3.05739i 6.04600 − 5.23889i −26.3033 + 6.09385i −31.7316 − 14.4913i
5.8 −0.563465 + 1.91899i 1.00414 + 5.09821i −3.36501 − 2.16256i 1.76162 − 12.2524i −10.3492 − 0.945735i 28.0507 − 12.8103i 6.04600 − 5.23889i −24.9834 + 10.2386i 22.5195 + 10.2843i
5.9 −0.563465 + 1.91899i 3.18891 + 4.10254i −3.36501 − 2.16256i 0.894547 − 6.22171i −9.66956 + 3.80784i −12.2766 + 5.60652i 6.04600 − 5.23889i −6.66169 + 26.1653i 11.4353 + 5.22234i
5.10 −0.563465 + 1.91899i 3.83621 − 3.50479i −3.36501 − 2.16256i −1.71402 + 11.9213i 4.56407 + 9.33645i −26.6981 + 12.1926i 6.04600 − 5.23889i 2.43295 − 26.8902i −21.9110 − 10.0064i
5.11 −0.563465 + 1.91899i 5.18749 + 0.299872i −3.36501 − 2.16256i −2.08799 + 14.5223i −3.49842 + 9.78576i 33.4115 − 15.2585i 6.04600 − 5.23889i 26.8202 + 3.11117i −26.6915 − 12.1896i
5.12 −0.563465 + 1.91899i 5.19604 − 0.0342665i −3.36501 − 2.16256i 1.51074 − 10.5074i −2.86203 + 9.99043i −7.86690 + 3.59269i 6.04600 − 5.23889i 26.9977 − 0.356101i 19.3123 + 8.81965i
5.13 0.563465 − 1.91899i −5.19397 + 0.150510i −3.36501 − 2.16256i 2.48225 − 17.2644i −2.63779 + 10.0520i −6.69475 + 3.05739i −6.04600 + 5.23889i 26.9547 − 1.56349i −31.7316 − 14.4913i
5.14 0.563465 − 1.91899i −5.18922 − 0.268367i −3.36501 − 2.16256i −1.76162 + 12.2524i −3.43894 + 9.80682i 28.0507 − 12.8103i −6.04600 + 5.23889i 26.8560 + 2.78523i 22.5195 + 10.2843i
5.15 0.563465 − 1.91899i −4.51461 − 2.57260i −3.36501 − 2.16256i −0.894547 + 6.22171i −7.48061 + 7.21391i −12.2766 + 5.60652i −6.04600 + 5.23889i 13.7635 + 23.2286i 11.4353 + 5.22234i
5.16 0.563465 − 1.91899i −3.80483 + 3.53882i −3.36501 − 2.16256i −0.260866 + 1.81436i 4.64705 + 9.29542i −4.44632 + 2.03056i −6.04600 + 5.23889i 1.95351 − 26.9292i 3.33474 + 1.52293i
5.17 0.563465 − 1.91899i −1.42787 + 4.99612i −3.36501 − 2.16256i −0.739504 + 5.14336i 8.78293 + 5.55519i 12.2116 − 5.57687i −6.04600 + 5.23889i −22.9224 − 14.2676i 9.45336 + 4.31720i
5.18 0.563465 − 1.91899i −1.03508 − 5.09201i −3.36501 − 2.16256i 2.08799 − 14.5223i −10.3547 − 0.882874i 33.4115 − 15.2585i −6.04600 + 5.23889i −24.8572 + 10.5413i −26.6915 − 12.1896i
5.19 0.563465 − 1.91899i −0.705556 − 5.14803i −3.36501 − 2.16256i −1.51074 + 10.5074i −10.2765 − 1.54678i −7.86690 + 3.59269i −6.04600 + 5.23889i −26.0044 + 7.26444i 19.3123 + 8.81965i
5.20 0.563465 − 1.91899i 2.28528 + 4.66664i −3.36501 − 2.16256i −0.0111281 + 0.0773979i 10.2429 − 1.75593i −31.2205 + 14.2579i −6.04600 + 5.23889i −16.5550 + 21.3291i 0.142255 + 0.0649658i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
23.d odd 22 1 inner
69.g even 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.4.f.a ✓ 240
3.b odd 2 1 inner 138.4.f.a ✓ 240
23.d odd 22 1 inner 138.4.f.a ✓ 240
69.g even 22 1 inner 138.4.f.a ✓ 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.4.f.a ✓ 240 1.a even 1 1 trivial
138.4.f.a ✓ 240 3.b odd 2 1 inner
138.4.f.a ✓ 240 23.d odd 22 1 inner
138.4.f.a ✓ 240 69.g even 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(138, [\chi])\).