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Properties

Label 135.4.f.b
Level $135$
Weight $4$
Character orbit 135.f
Analytic conductor $7.965$
Analytic rank $0$
Dimension $24$
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] = [135,4,Mod(53,135)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(135, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("135.53");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.f (of order \(4\), degree \(2\), 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: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 12 q^{7} - 120 q^{10} - 72 q^{13} - 612 q^{16} - 552 q^{22} + 144 q^{25} + 1008 q^{28} + 456 q^{31} + 360 q^{37} - 1092 q^{40} - 1848 q^{43} + 732 q^{46} + 2868 q^{52} - 2052 q^{55} + 2424 q^{58} - 2544 q^{61} - 1776 q^{67} + 7332 q^{70} + 4236 q^{73} + 5844 q^{76} + 1188 q^{82} - 48 q^{85} - 8472 q^{88} - 7056 q^{91} + 2916 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1 −3.78617 − 3.78617i 0 20.6702i −5.68236 − 9.62864i 0 20.4442 − 20.4442i 47.9717 − 47.9717i 0 −14.9413 + 57.9701i
53.2 −3.43319 − 3.43319i 0 15.5735i 10.5926 + 3.57723i 0 −15.7386 + 15.7386i 26.0014 − 26.0014i 0 −24.0851 − 48.6477i
53.3 −2.22348 − 2.22348i 0 1.88775i −10.9208 + 2.39508i 0 5.29400 − 5.29400i −13.5905 + 13.5905i 0 29.6076 + 18.9568i
53.4 −2.11625 − 2.11625i 0 0.957003i 8.31186 + 7.47750i 0 9.53687 − 9.53687i −14.9047 + 14.9047i 0 −1.76571 − 33.4142i
53.5 −1.34872 − 1.34872i 0 − 4.36192i −1.46492 − 11.0840i 0 −21.5088 + 21.5088i −16.6727 + 16.6727i 0 −12.9734 + 16.9249i
53.6 −0.369727 − 0.369727i 0 − 7.72660i 7.61648 − 8.18470i 0 4.97241 − 4.97241i −5.81454 + 5.81454i 0 −5.84211 + 0.210086i
53.7 0.369727 + 0.369727i 0 − 7.72660i −7.61648 + 8.18470i 0 4.97241 − 4.97241i 5.81454 − 5.81454i 0 −5.84211 + 0.210086i
53.8 1.34872 + 1.34872i 0 − 4.36192i 1.46492 + 11.0840i 0 −21.5088 + 21.5088i 16.6727 − 16.6727i 0 −12.9734 + 16.9249i
53.9 2.11625 + 2.11625i 0 0.957003i −8.31186 − 7.47750i 0 9.53687 − 9.53687i 14.9047 − 14.9047i 0 −1.76571 − 33.4142i
53.10 2.22348 + 2.22348i 0 1.88775i 10.9208 − 2.39508i 0 5.29400 − 5.29400i 13.5905 − 13.5905i 0 29.6076 + 18.9568i
53.11 3.43319 + 3.43319i 0 15.5735i −10.5926 − 3.57723i 0 −15.7386 + 15.7386i −26.0014 + 26.0014i 0 −24.0851 − 48.6477i
53.12 3.78617 + 3.78617i 0 20.6702i 5.68236 + 9.62864i 0 20.4442 − 20.4442i −47.9717 + 47.9717i 0 −14.9413 + 57.9701i
107.1 −3.78617 + 3.78617i 0 − 20.6702i −5.68236 + 9.62864i 0 20.4442 + 20.4442i 47.9717 + 47.9717i 0 −14.9413 − 57.9701i
107.2 −3.43319 + 3.43319i 0 − 15.5735i 10.5926 − 3.57723i 0 −15.7386 − 15.7386i 26.0014 + 26.0014i 0 −24.0851 + 48.6477i
107.3 −2.22348 + 2.22348i 0 − 1.88775i −10.9208 − 2.39508i 0 5.29400 + 5.29400i −13.5905 − 13.5905i 0 29.6076 − 18.9568i
107.4 −2.11625 + 2.11625i 0 − 0.957003i 8.31186 − 7.47750i 0 9.53687 + 9.53687i −14.9047 − 14.9047i 0 −1.76571 + 33.4142i
107.5 −1.34872 + 1.34872i 0 4.36192i −1.46492 + 11.0840i 0 −21.5088 − 21.5088i −16.6727 − 16.6727i 0 −12.9734 − 16.9249i
107.6 −0.369727 + 0.369727i 0 7.72660i 7.61648 + 8.18470i 0 4.97241 + 4.97241i −5.81454 − 5.81454i 0 −5.84211 − 0.210086i
107.7 0.369727 − 0.369727i 0 7.72660i −7.61648 − 8.18470i 0 4.97241 + 4.97241i 5.81454 + 5.81454i 0 −5.84211 − 0.210086i
107.8 1.34872 − 1.34872i 0 4.36192i 1.46492 − 11.0840i 0 −21.5088 − 21.5088i 16.6727 + 16.6727i 0 −12.9734 − 16.9249i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 135.4.f.b ✓ 24
3.b odd 2 1 inner 135.4.f.b ✓ 24
5.c odd 4 1 inner 135.4.f.b ✓ 24
15.e even 4 1 inner 135.4.f.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.4.f.b ✓ 24 1.a even 1 1 trivial
135.4.f.b ✓ 24 3.b odd 2 1 inner
135.4.f.b ✓ 24 5.c odd 4 1 inner
135.4.f.b ✓ 24 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + 1569 T_{2}^{20} + 730560 T_{2}^{16} + 101561825 T_{2}^{12} + 4809625185 T_{2}^{8} + \cdots + 3544535296 \) acting on \(S_{4}^{\mathrm{new}}(135, [\chi])\). Copy content Toggle raw display