↑

Properties

Label 7200.2.m.f
Level $7200$
Weight $2$
Character orbit 7200.m
Analytic conductor $57.492$
Analytic rank $0$
Dimension $32$
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] = [7200,2,Mod(3599,7200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7200.3599");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7200 = 2^{5} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7200.m (of order \(2\), degree \(1\), 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: \(57.4922894553\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 1800)
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.

\(\operatorname{Tr}(f)(q) = \) \( 32 q+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 32 q^{19} + 64 q^{49} - 96 q^{91}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3599.1 0 0 0 0 0 −4.53014 0 0 0
3599.2 0 0 0 0 0 −4.53014 0 0 0
3599.3 0 0 0 0 0 −4.53014 0 0 0
3599.4 0 0 0 0 0 −4.53014 0 0 0
3599.5 0 0 0 0 0 −3.40004 0 0 0
3599.6 0 0 0 0 0 −3.40004 0 0 0
3599.7 0 0 0 0 0 −3.40004 0 0 0
3599.8 0 0 0 0 0 −3.40004 0 0 0
3599.9 0 0 0 0 0 −1.68313 0 0 0
3599.10 0 0 0 0 0 −1.68313 0 0 0
3599.11 0 0 0 0 0 −1.68313 0 0 0
3599.12 0 0 0 0 0 −1.68313 0 0 0
3599.13 0 0 0 0 0 −1.04148 0 0 0
3599.14 0 0 0 0 0 −1.04148 0 0 0
3599.15 0 0 0 0 0 −1.04148 0 0 0
3599.16 0 0 0 0 0 −1.04148 0 0 0
3599.17 0 0 0 0 0 1.04148 0 0 0
3599.18 0 0 0 0 0 1.04148 0 0 0
3599.19 0 0 0 0 0 1.04148 0 0 0
3599.20 0 0 0 0 0 1.04148 0 0 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3599.32
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
8.d odd 2 1 inner
15.d odd 2 1 inner
24.f even 2 1 inner
40.e odd 2 1 inner
120.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7200.2.m.f 32
3.b odd 2 1 inner 7200.2.m.f 32
4.b odd 2 1 1800.2.m.f 32
5.b even 2 1 inner 7200.2.m.f 32
5.c odd 4 1 7200.2.b.g 16
5.c odd 4 1 7200.2.b.h 16
8.b even 2 1 1800.2.m.f 32
8.d odd 2 1 inner 7200.2.m.f 32
12.b even 2 1 1800.2.m.f 32
15.d odd 2 1 inner 7200.2.m.f 32
15.e even 4 1 7200.2.b.g 16
15.e even 4 1 7200.2.b.h 16
20.d odd 2 1 1800.2.m.f 32
20.e even 4 1 1800.2.b.h ✓ 16
20.e even 4 1 1800.2.b.i yes 16
24.f even 2 1 inner 7200.2.m.f 32
24.h odd 2 1 1800.2.m.f 32
40.e odd 2 1 inner 7200.2.m.f 32
40.f even 2 1 1800.2.m.f 32
40.i odd 4 1 1800.2.b.h ✓ 16
40.i odd 4 1 1800.2.b.i yes 16
40.k even 4 1 7200.2.b.g 16
40.k even 4 1 7200.2.b.h 16
60.h even 2 1 1800.2.m.f 32
60.l odd 4 1 1800.2.b.h ✓ 16
60.l odd 4 1 1800.2.b.i yes 16
120.i odd 2 1 1800.2.m.f 32
120.m even 2 1 inner 7200.2.m.f 32
120.q odd 4 1 7200.2.b.g 16
120.q odd 4 1 7200.2.b.h 16
120.w even 4 1 1800.2.b.h ✓ 16
120.w even 4 1 1800.2.b.i yes 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.2.b.h ✓ 16 20.e even 4 1
1800.2.b.h ✓ 16 40.i odd 4 1
1800.2.b.h ✓ 16 60.l odd 4 1
1800.2.b.h ✓ 16 120.w even 4 1
1800.2.b.i yes 16 20.e even 4 1
1800.2.b.i yes 16 40.i odd 4 1
1800.2.b.i yes 16 60.l odd 4 1
1800.2.b.i yes 16 120.w even 4 1
1800.2.m.f 32 4.b odd 2 1
1800.2.m.f 32 8.b even 2 1
1800.2.m.f 32 12.b even 2 1
1800.2.m.f 32 20.d odd 2 1
1800.2.m.f 32 24.h odd 2 1
1800.2.m.f 32 40.f even 2 1
1800.2.m.f 32 60.h even 2 1
1800.2.m.f 32 120.i odd 2 1
7200.2.b.g 16 5.c odd 4 1
7200.2.b.g 16 15.e even 4 1
7200.2.b.g 16 40.k even 4 1
7200.2.b.g 16 120.q odd 4 1
7200.2.b.h 16 5.c odd 4 1
7200.2.b.h 16 15.e even 4 1
7200.2.b.h 16 40.k even 4 1
7200.2.b.h 16 120.q odd 4 1
7200.2.m.f 32 1.a even 1 1 trivial
7200.2.m.f 32 3.b odd 2 1 inner
7200.2.m.f 32 5.b even 2 1 inner
7200.2.m.f 32 8.d odd 2 1 inner
7200.2.m.f 32 15.d odd 2 1 inner
7200.2.m.f 32 24.f even 2 1 inner
7200.2.m.f 32 40.e odd 2 1 inner
7200.2.m.f 32 120.m 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 \(S_{2}^{\mathrm{new}}(7200, [\chi])\):

\( T_{7}^{8} - 36T_{7}^{6} + 366T_{7}^{4} - 1028T_{7}^{2} + 729 \) Copy content Toggle raw display
\( T_{29}^{8} - 104T_{29}^{6} + 2776T_{29}^{4} - 11008T_{29}^{2} + 3600 \) Copy content Toggle raw display