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Properties

Label 2400.2.bh.c
Level $2400$
Weight $2$
Character orbit 2400.bh
Analytic conductor $19.164$
Analytic rank $0$
Dimension $32$
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] = [2400,2,Mod(943,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 0, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2400.943");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.bh (of order \(4\), degree \(2\), not 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: \(19.1640964851\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 600)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{11} - 32 q^{51} - 32 q^{81} + 224 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
943.1 0 −0.707107 − 0.707107i 0 0 0 0.337834 + 0.337834i 0 1.00000i 0
943.2 0 −0.707107 − 0.707107i 0 0 0 2.62409 + 2.62409i 0 1.00000i 0
943.3 0 −0.707107 − 0.707107i 0 0 0 −1.17057 − 1.17057i 0 1.00000i 0
943.4 0 −0.707107 − 0.707107i 0 0 0 −0.337834 − 0.337834i 0 1.00000i 0
943.5 0 −0.707107 − 0.707107i 0 0 0 −2.37271 − 2.37271i 0 1.00000i 0
943.6 0 −0.707107 − 0.707107i 0 0 0 1.17057 + 1.17057i 0 1.00000i 0
943.7 0 −0.707107 − 0.707107i 0 0 0 −2.62409 − 2.62409i 0 1.00000i 0
943.8 0 −0.707107 − 0.707107i 0 0 0 2.37271 + 2.37271i 0 1.00000i 0
943.9 0 0.707107 + 0.707107i 0 0 0 2.37271 + 2.37271i 0 1.00000i 0
943.10 0 0.707107 + 0.707107i 0 0 0 2.62409 + 2.62409i 0 1.00000i 0
943.11 0 0.707107 + 0.707107i 0 0 0 1.17057 + 1.17057i 0 1.00000i 0
943.12 0 0.707107 + 0.707107i 0 0 0 0.337834 + 0.337834i 0 1.00000i 0
943.13 0 0.707107 + 0.707107i 0 0 0 −1.17057 − 1.17057i 0 1.00000i 0
943.14 0 0.707107 + 0.707107i 0 0 0 −2.37271 − 2.37271i 0 1.00000i 0
943.15 0 0.707107 + 0.707107i 0 0 0 −2.62409 − 2.62409i 0 1.00000i 0
943.16 0 0.707107 + 0.707107i 0 0 0 −0.337834 − 0.337834i 0 1.00000i 0
1807.1 0 −0.707107 + 0.707107i 0 0 0 0.337834 − 0.337834i 0 − 1.00000i 0
1807.2 0 −0.707107 + 0.707107i 0 0 0 2.62409 − 2.62409i 0 − 1.00000i 0
1807.3 0 −0.707107 + 0.707107i 0 0 0 −1.17057 + 1.17057i 0 − 1.00000i 0
1807.4 0 −0.707107 + 0.707107i 0 0 0 −0.337834 + 0.337834i 0 − 1.00000i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 943.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
5.c odd 4 2 inner
8.d odd 2 1 inner
40.e odd 2 1 inner
40.k even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2400.2.bh.c 32
4.b odd 2 1 600.2.v.c ✓ 32
5.b even 2 1 inner 2400.2.bh.c 32
5.c odd 4 2 inner 2400.2.bh.c 32
8.b even 2 1 600.2.v.c ✓ 32
8.d odd 2 1 inner 2400.2.bh.c 32
20.d odd 2 1 600.2.v.c ✓ 32
20.e even 4 2 600.2.v.c ✓ 32
40.e odd 2 1 inner 2400.2.bh.c 32
40.f even 2 1 600.2.v.c ✓ 32
40.i odd 4 2 600.2.v.c ✓ 32
40.k even 4 2 inner 2400.2.bh.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.v.c ✓ 32 4.b odd 2 1
600.2.v.c ✓ 32 8.b even 2 1
600.2.v.c ✓ 32 20.d odd 2 1
600.2.v.c ✓ 32 20.e even 4 2
600.2.v.c ✓ 32 40.f even 2 1
600.2.v.c ✓ 32 40.i odd 4 2
2400.2.bh.c 32 1.a even 1 1 trivial
2400.2.bh.c 32 5.b even 2 1 inner
2400.2.bh.c 32 5.c odd 4 2 inner
2400.2.bh.c 32 8.d odd 2 1 inner
2400.2.bh.c 32 40.e odd 2 1 inner
2400.2.bh.c 32 40.k even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{16} + 324T_{7}^{12} + 26438T_{7}^{8} + 181956T_{7}^{4} + 9409 \) acting on \(S_{2}^{\mathrm{new}}(2400, [\chi])\). Copy content Toggle raw display