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Properties

Label 2400.2.h.h
Level $2400$
Weight $2$
Character orbit 2400.h
Analytic conductor $19.164$
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] = [2400,2,Mod(1151,2400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2400.1151");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.h (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: \(19.1640964851\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 480)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q - 8 q^{21} - 40 q^{49} + 32 q^{61} + 56 q^{69} + 8 q^{81}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1151.1 0 −1.65670 − 0.505327i 0 0 0 2.36789i 0 2.48929 + 1.67435i 0
1151.2 0 −1.65670 − 0.505327i 0 0 0 2.36789i 0 2.48929 + 1.67435i 0
1151.3 0 −1.65670 + 0.505327i 0 0 0 − 2.36789i 0 2.48929 − 1.67435i 0
1151.4 0 −1.65670 + 0.505327i 0 0 0 − 2.36789i 0 2.48929 − 1.67435i 0
1151.5 0 −1.28241 − 1.16422i 0 0 0 − 4.22289i 0 0.289169 + 2.98603i 0
1151.6 0 −1.28241 − 1.16422i 0 0 0 − 4.22289i 0 0.289169 + 2.98603i 0
1151.7 0 −1.28241 + 1.16422i 0 0 0 4.22289i 0 0.289169 − 2.98603i 0
1151.8 0 −1.28241 + 1.16422i 0 0 0 4.22289i 0 0.289169 − 2.98603i 0
1151.9 0 −0.332823 − 1.69977i 0 0 0 1.60011i 0 −2.77846 + 1.13145i 0
1151.10 0 −0.332823 − 1.69977i 0 0 0 1.60011i 0 −2.77846 + 1.13145i 0
1151.11 0 −0.332823 + 1.69977i 0 0 0 − 1.60011i 0 −2.77846 − 1.13145i 0
1151.12 0 −0.332823 + 1.69977i 0 0 0 − 1.60011i 0 −2.77846 − 1.13145i 0
1151.13 0 0.332823 − 1.69977i 0 0 0 1.60011i 0 −2.77846 − 1.13145i 0
1151.14 0 0.332823 − 1.69977i 0 0 0 1.60011i 0 −2.77846 − 1.13145i 0
1151.15 0 0.332823 + 1.69977i 0 0 0 − 1.60011i 0 −2.77846 + 1.13145i 0
1151.16 0 0.332823 + 1.69977i 0 0 0 − 1.60011i 0 −2.77846 + 1.13145i 0
1151.17 0 1.28241 − 1.16422i 0 0 0 − 4.22289i 0 0.289169 − 2.98603i 0
1151.18 0 1.28241 − 1.16422i 0 0 0 − 4.22289i 0 0.289169 − 2.98603i 0
1151.19 0 1.28241 + 1.16422i 0 0 0 4.22289i 0 0.289169 + 2.98603i 0
1151.20 0 1.28241 + 1.16422i 0 0 0 4.22289i 0 0.289169 + 2.98603i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1151.24
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 2400.2.h.h 24
3.b odd 2 1 inner 2400.2.h.h 24
4.b odd 2 1 inner 2400.2.h.h 24
5.b even 2 1 inner 2400.2.h.h 24
5.c odd 4 2 480.2.o.a ✓ 24
12.b even 2 1 inner 2400.2.h.h 24
15.d odd 2 1 inner 2400.2.h.h 24
15.e even 4 2 480.2.o.a ✓ 24
20.d odd 2 1 inner 2400.2.h.h 24
20.e even 4 2 480.2.o.a ✓ 24
40.i odd 4 2 960.2.o.e 24
40.k even 4 2 960.2.o.e 24
60.h even 2 1 inner 2400.2.h.h 24
60.l odd 4 2 480.2.o.a ✓ 24
120.q odd 4 2 960.2.o.e 24
120.w even 4 2 960.2.o.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.o.a ✓ 24 5.c odd 4 2
480.2.o.a ✓ 24 15.e even 4 2
480.2.o.a ✓ 24 20.e even 4 2
480.2.o.a ✓ 24 60.l odd 4 2
960.2.o.e 24 40.i odd 4 2
960.2.o.e 24 40.k even 4 2
960.2.o.e 24 120.q odd 4 2
960.2.o.e 24 120.w even 4 2
2400.2.h.h 24 1.a even 1 1 trivial
2400.2.h.h 24 3.b odd 2 1 inner
2400.2.h.h 24 4.b odd 2 1 inner
2400.2.h.h 24 5.b even 2 1 inner
2400.2.h.h 24 12.b even 2 1 inner
2400.2.h.h 24 15.d odd 2 1 inner
2400.2.h.h 24 20.d odd 2 1 inner
2400.2.h.h 24 60.h even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2400, [\chi])\):

\( T_{7}^{6} + 26T_{7}^{4} + 160T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{6} - 40T_{11}^{4} + 288T_{11}^{2} - 512 \) Copy content Toggle raw display
\( T_{13}^{6} - 56T_{13}^{4} + 928T_{13}^{2} - 4096 \) Copy content Toggle raw display
\( T_{23}^{6} - 34T_{23}^{4} + 144T_{23}^{2} - 32 \) Copy content Toggle raw display