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Properties

Label 240.4.w.b
Level $240$
Weight $4$
Character orbit 240.w
Analytic conductor $14.160$
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] = [240,4,Mod(127,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.127");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 240.w (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: \(14.1604584014\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 32 q^{5}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 32 q^{5} + 280 q^{13} + 152 q^{17} - 192 q^{21} - 360 q^{25} + 336 q^{33} + 936 q^{37} + 1056 q^{41} + 72 q^{45} + 1192 q^{53} + 768 q^{57} - 2368 q^{61} - 4440 q^{65} + 3992 q^{73} + 1952 q^{77} - 1944 q^{81} - 8 q^{85} - 2016 q^{93} - 1192 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
127.1 0 −2.12132 − 2.12132i 0 −10.9075 + 2.45489i 0 −15.9725 + 15.9725i 0 9.00000i 0
127.2 0 −2.12132 − 2.12132i 0 −6.27665 + 9.25222i 0 9.71832 − 9.71832i 0 9.00000i 0
127.3 0 −2.12132 − 2.12132i 0 −5.97150 − 9.45204i 0 1.79978 − 1.79978i 0 9.00000i 0
127.4 0 −2.12132 − 2.12132i 0 −0.178759 − 11.1789i 0 18.2648 − 18.2648i 0 9.00000i 0
127.5 0 −2.12132 − 2.12132i 0 4.63677 + 10.1735i 0 6.82867 − 6.82867i 0 9.00000i 0
127.6 0 −2.12132 − 2.12132i 0 10.6976 − 3.24967i 0 −9.32534 + 9.32534i 0 9.00000i 0
127.7 0 2.12132 + 2.12132i 0 −10.9075 + 2.45489i 0 15.9725 − 15.9725i 0 9.00000i 0
127.8 0 2.12132 + 2.12132i 0 −6.27665 + 9.25222i 0 −9.71832 + 9.71832i 0 9.00000i 0
127.9 0 2.12132 + 2.12132i 0 −5.97150 − 9.45204i 0 −1.79978 + 1.79978i 0 9.00000i 0
127.10 0 2.12132 + 2.12132i 0 −0.178759 − 11.1789i 0 −18.2648 + 18.2648i 0 9.00000i 0
127.11 0 2.12132 + 2.12132i 0 4.63677 + 10.1735i 0 −6.82867 + 6.82867i 0 9.00000i 0
127.12 0 2.12132 + 2.12132i 0 10.6976 − 3.24967i 0 9.32534 − 9.32534i 0 9.00000i 0
223.1 0 −2.12132 + 2.12132i 0 −10.9075 − 2.45489i 0 −15.9725 − 15.9725i 0 − 9.00000i 0
223.2 0 −2.12132 + 2.12132i 0 −6.27665 − 9.25222i 0 9.71832 + 9.71832i 0 − 9.00000i 0
223.3 0 −2.12132 + 2.12132i 0 −5.97150 + 9.45204i 0 1.79978 + 1.79978i 0 − 9.00000i 0
223.4 0 −2.12132 + 2.12132i 0 −0.178759 + 11.1789i 0 18.2648 + 18.2648i 0 − 9.00000i 0
223.5 0 −2.12132 + 2.12132i 0 4.63677 − 10.1735i 0 6.82867 + 6.82867i 0 − 9.00000i 0
223.6 0 −2.12132 + 2.12132i 0 10.6976 + 3.24967i 0 −9.32534 − 9.32534i 0 − 9.00000i 0
223.7 0 2.12132 − 2.12132i 0 −10.9075 − 2.45489i 0 15.9725 + 15.9725i 0 − 9.00000i 0
223.8 0 2.12132 − 2.12132i 0 −6.27665 − 9.25222i 0 −9.71832 − 9.71832i 0 − 9.00000i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 127.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.4.w.b ✓ 24
3.b odd 2 1 720.4.x.i 24
4.b odd 2 1 inner 240.4.w.b ✓ 24
5.c odd 4 1 inner 240.4.w.b ✓ 24
12.b even 2 1 720.4.x.i 24
15.e even 4 1 720.4.x.i 24
20.e even 4 1 inner 240.4.w.b ✓ 24
60.l odd 4 1 720.4.x.i 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.4.w.b ✓ 24 1.a even 1 1 trivial
240.4.w.b ✓ 24 4.b odd 2 1 inner
240.4.w.b ✓ 24 5.c odd 4 1 inner
240.4.w.b ✓ 24 20.e even 4 1 inner
720.4.x.i 24 3.b odd 2 1
720.4.x.i 24 12.b even 2 1
720.4.x.i 24 15.e even 4 1
720.4.x.i 24 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{24} + 780176 T_{7}^{20} + 170231248480 T_{7}^{16} + \cdots + 45\!\cdots\!36 \) acting on \(S_{4}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display