↑

Properties

Label 240.3.bj.c
Level $240$
Weight $3$
Character orbit 240.bj
Analytic conductor $6.540$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [240,3,Mod(47,240)]
 
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, 2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.47");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 240.bj (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: \(6.53952634465\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 40 q^{13} - 152 q^{21} + 24 q^{25} + 232 q^{33} - 72 q^{37} + 88 q^{45} - 136 q^{57} + 16 q^{61} - 296 q^{73} + 88 q^{81} + 632 q^{85} + 240 q^{93} + 472 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −2.99979 + 0.0354758i 0 −0.855169 + 4.92633i 0 6.42842 − 6.42842i 0 8.99748 − 0.212840i 0
47.2 0 −2.74268 − 1.21562i 0 4.97111 + 0.536719i 0 0.494192 − 0.494192i 0 6.04455 + 6.66809i 0
47.3 0 −2.45801 + 1.71994i 0 −3.68195 + 3.38279i 0 −2.72603 + 2.72603i 0 3.08362 − 8.45525i 0
47.4 0 −1.71994 + 2.45801i 0 3.68195 − 3.38279i 0 2.72603 − 2.72603i 0 −3.08362 − 8.45525i 0
47.5 0 −1.21562 − 2.74268i 0 −4.97111 − 0.536719i 0 0.494192 − 0.494192i 0 −6.04455 + 6.66809i 0
47.6 0 −0.0354758 + 2.99979i 0 0.855169 − 4.92633i 0 −6.42842 + 6.42842i 0 −8.99748 − 0.212840i 0
47.7 0 0.0354758 − 2.99979i 0 0.855169 − 4.92633i 0 6.42842 − 6.42842i 0 −8.99748 − 0.212840i 0
47.8 0 1.21562 + 2.74268i 0 −4.97111 − 0.536719i 0 −0.494192 + 0.494192i 0 −6.04455 + 6.66809i 0
47.9 0 1.71994 − 2.45801i 0 3.68195 − 3.38279i 0 −2.72603 + 2.72603i 0 −3.08362 − 8.45525i 0
47.10 0 2.45801 − 1.71994i 0 −3.68195 + 3.38279i 0 2.72603 − 2.72603i 0 3.08362 − 8.45525i 0
47.11 0 2.74268 + 1.21562i 0 4.97111 + 0.536719i 0 −0.494192 + 0.494192i 0 6.04455 + 6.66809i 0
47.12 0 2.99979 − 0.0354758i 0 −0.855169 + 4.92633i 0 −6.42842 + 6.42842i 0 8.99748 − 0.212840i 0
143.1 0 −2.99979 − 0.0354758i 0 −0.855169 − 4.92633i 0 6.42842 + 6.42842i 0 8.99748 + 0.212840i 0
143.2 0 −2.74268 + 1.21562i 0 4.97111 − 0.536719i 0 0.494192 + 0.494192i 0 6.04455 − 6.66809i 0
143.3 0 −2.45801 − 1.71994i 0 −3.68195 − 3.38279i 0 −2.72603 − 2.72603i 0 3.08362 + 8.45525i 0
143.4 0 −1.71994 − 2.45801i 0 3.68195 + 3.38279i 0 2.72603 + 2.72603i 0 −3.08362 + 8.45525i 0
143.5 0 −1.21562 + 2.74268i 0 −4.97111 + 0.536719i 0 0.494192 + 0.494192i 0 −6.04455 − 6.66809i 0
143.6 0 −0.0354758 − 2.99979i 0 0.855169 + 4.92633i 0 −6.42842 − 6.42842i 0 −8.99748 + 0.212840i 0
143.7 0 0.0354758 + 2.99979i 0 0.855169 + 4.92633i 0 6.42842 + 6.42842i 0 −8.99748 + 0.212840i 0
143.8 0 1.21562 − 2.74268i 0 −4.97111 + 0.536719i 0 −0.494192 − 0.494192i 0 −6.04455 − 6.66809i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.12
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.c odd 4 1 inner
12.b even 2 1 inner
15.e even 4 1 inner
20.e even 4 1 inner
60.l odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.3.bj.c ✓ 24
3.b odd 2 1 inner 240.3.bj.c ✓ 24
4.b odd 2 1 inner 240.3.bj.c ✓ 24
5.c odd 4 1 inner 240.3.bj.c ✓ 24
12.b even 2 1 inner 240.3.bj.c ✓ 24
15.e even 4 1 inner 240.3.bj.c ✓ 24
20.e even 4 1 inner 240.3.bj.c ✓ 24
60.l odd 4 1 inner 240.3.bj.c ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.bj.c ✓ 24 1.a even 1 1 trivial
240.3.bj.c ✓ 24 3.b odd 2 1 inner
240.3.bj.c ✓ 24 4.b odd 2 1 inner
240.3.bj.c ✓ 24 5.c odd 4 1 inner
240.3.bj.c ✓ 24 12.b even 2 1 inner
240.3.bj.c ✓ 24 15.e even 4 1 inner
240.3.bj.c ✓ 24 20.e even 4 1 inner
240.3.bj.c ✓ 24 60.l odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{12} + 7052T_{7}^{8} + 1510576T_{7}^{4} + 360000 \) acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\). Copy content Toggle raw display