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Properties

Label 240.4.o.c
Level $240$
Weight $4$
Character orbit 240.o
Analytic conductor $14.160$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,4,Mod(239,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.239");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.o (of order \(2\), degree \(1\), 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: \(14.1604584014\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 32 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 32 q^{9} - 72 q^{21} - 24 q^{25} + 520 q^{45} + 936 q^{49} - 864 q^{61} - 1656 q^{69} + 1288 q^{81} - 1056 q^{85}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
239.1 0 −4.99387 − 1.43571i 0 −6.60176 − 9.02312i 0 20.9188 0 22.8775 + 14.3395i 0
239.2 0 −4.99387 − 1.43571i 0 6.60176 − 9.02312i 0 20.9188 0 22.8775 + 14.3395i 0
239.3 0 −4.99387 + 1.43571i 0 −6.60176 + 9.02312i 0 20.9188 0 22.8775 − 14.3395i 0
239.4 0 −4.99387 + 1.43571i 0 6.60176 + 9.02312i 0 20.9188 0 22.8775 − 14.3395i 0
239.5 0 −4.05776 − 3.24571i 0 −11.0511 − 1.69525i 0 −25.4968 0 5.93076 + 26.3406i 0
239.6 0 −4.05776 − 3.24571i 0 11.0511 − 1.69525i 0 −25.4968 0 5.93076 + 26.3406i 0
239.7 0 −4.05776 + 3.24571i 0 −11.0511 + 1.69525i 0 −25.4968 0 5.93076 − 26.3406i 0
239.8 0 −4.05776 + 3.24571i 0 11.0511 + 1.69525i 0 −25.4968 0 5.93076 − 26.3406i 0
239.9 0 −1.04685 − 5.08961i 0 −4.50451 + 10.2328i 0 7.63631 0 −24.8082 + 10.6561i 0
239.10 0 −1.04685 − 5.08961i 0 4.50451 + 10.2328i 0 7.63631 0 −24.8082 + 10.6561i 0
239.11 0 −1.04685 + 5.08961i 0 −4.50451 − 10.2328i 0 7.63631 0 −24.8082 − 10.6561i 0
239.12 0 −1.04685 + 5.08961i 0 4.50451 − 10.2328i 0 7.63631 0 −24.8082 − 10.6561i 0
239.13 0 1.04685 − 5.08961i 0 −4.50451 − 10.2328i 0 −7.63631 0 −24.8082 − 10.6561i 0
239.14 0 1.04685 − 5.08961i 0 4.50451 − 10.2328i 0 −7.63631 0 −24.8082 − 10.6561i 0
239.15 0 1.04685 + 5.08961i 0 −4.50451 + 10.2328i 0 −7.63631 0 −24.8082 + 10.6561i 0
239.16 0 1.04685 + 5.08961i 0 4.50451 + 10.2328i 0 −7.63631 0 −24.8082 + 10.6561i 0
239.17 0 4.05776 − 3.24571i 0 −11.0511 + 1.69525i 0 25.4968 0 5.93076 − 26.3406i 0
239.18 0 4.05776 − 3.24571i 0 11.0511 + 1.69525i 0 25.4968 0 5.93076 − 26.3406i 0
239.19 0 4.05776 + 3.24571i 0 −11.0511 − 1.69525i 0 25.4968 0 5.93076 + 26.3406i 0
239.20 0 4.05776 + 3.24571i 0 11.0511 − 1.69525i 0 25.4968 0 5.93076 + 26.3406i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 239.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.4.o.c ✓ 24
3.b odd 2 1 inner 240.4.o.c ✓ 24
4.b odd 2 1 inner 240.4.o.c ✓ 24
5.b even 2 1 inner 240.4.o.c ✓ 24
12.b even 2 1 inner 240.4.o.c ✓ 24
15.d odd 2 1 inner 240.4.o.c ✓ 24
20.d odd 2 1 inner 240.4.o.c ✓ 24
60.h even 2 1 inner 240.4.o.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.o.c ✓ 24 1.a even 1 1 trivial
240.4.o.c ✓ 24 3.b odd 2 1 inner
240.4.o.c ✓ 24 4.b odd 2 1 inner
240.4.o.c ✓ 24 5.b even 2 1 inner
240.4.o.c ✓ 24 12.b even 2 1 inner
240.4.o.c ✓ 24 15.d odd 2 1 inner
240.4.o.c ✓ 24 20.d odd 2 1 inner
240.4.o.c ✓ 24 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{6} - 1146T_{7}^{4} + 347904T_{7}^{2} - 16588800 \) acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display