↑

Properties

Label 2400.2.m.e
Level $2400$
Weight $2$
Character orbit 2400.m
Analytic conductor $19.164$
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] = [2400,2,Mod(1199,2400)]
 
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, 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("2400.1199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2400 = 2^{5} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2400.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: \(19.1640964851\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: no (minimal twist has level 600)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 4 q^{9} - 8 q^{19} + 72 q^{49} + 60 q^{51} - 20 q^{81} + 48 q^{91} + 116 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1199.1 0 −1.71500 − 0.242431i 0 0 0 −3.08957 0 2.88245 + 0.831539i 0
1199.2 0 −1.71500 − 0.242431i 0 0 0 3.08957 0 2.88245 + 0.831539i 0
1199.3 0 −1.71500 + 0.242431i 0 0 0 −3.08957 0 2.88245 − 0.831539i 0
1199.4 0 −1.71500 + 0.242431i 0 0 0 3.08957 0 2.88245 − 0.831539i 0
1199.5 0 −1.12950 − 1.31310i 0 0 0 −4.34495 0 −0.448458 + 2.96629i 0
1199.6 0 −1.12950 − 1.31310i 0 0 0 4.34495 0 −0.448458 + 2.96629i 0
1199.7 0 −1.12950 + 1.31310i 0 0 0 −4.34495 0 −0.448458 − 2.96629i 0
1199.8 0 −1.12950 + 1.31310i 0 0 0 4.34495 0 −0.448458 − 2.96629i 0
1199.9 0 −0.730070 − 1.57067i 0 0 0 −1.25539 0 −1.93400 + 2.29339i 0
1199.10 0 −0.730070 − 1.57067i 0 0 0 1.25539 0 −1.93400 + 2.29339i 0
1199.11 0 −0.730070 + 1.57067i 0 0 0 −1.25539 0 −1.93400 − 2.29339i 0
1199.12 0 −0.730070 + 1.57067i 0 0 0 1.25539 0 −1.93400 − 2.29339i 0
1199.13 0 0.730070 − 1.57067i 0 0 0 −1.25539 0 −1.93400 − 2.29339i 0
1199.14 0 0.730070 − 1.57067i 0 0 0 1.25539 0 −1.93400 − 2.29339i 0
1199.15 0 0.730070 + 1.57067i 0 0 0 −1.25539 0 −1.93400 + 2.29339i 0
1199.16 0 0.730070 + 1.57067i 0 0 0 1.25539 0 −1.93400 + 2.29339i 0
1199.17 0 1.12950 − 1.31310i 0 0 0 −4.34495 0 −0.448458 − 2.96629i 0
1199.18 0 1.12950 − 1.31310i 0 0 0 4.34495 0 −0.448458 − 2.96629i 0
1199.19 0 1.12950 + 1.31310i 0 0 0 −4.34495 0 −0.448458 + 2.96629i 0
1199.20 0 1.12950 + 1.31310i 0 0 0 4.34495 0 −0.448458 + 2.96629i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1199.24
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 2400.2.m.e 24
3.b odd 2 1 inner 2400.2.m.e 24
4.b odd 2 1 600.2.m.e 24
5.b even 2 1 inner 2400.2.m.e 24
5.c odd 4 1 2400.2.b.g 12
5.c odd 4 1 2400.2.b.h 12
8.b even 2 1 600.2.m.e 24
8.d odd 2 1 inner 2400.2.m.e 24
12.b even 2 1 600.2.m.e 24
15.d odd 2 1 inner 2400.2.m.e 24
15.e even 4 1 2400.2.b.g 12
15.e even 4 1 2400.2.b.h 12
20.d odd 2 1 600.2.m.e 24
20.e even 4 1 600.2.b.g ✓ 12
20.e even 4 1 600.2.b.h yes 12
24.f even 2 1 inner 2400.2.m.e 24
24.h odd 2 1 600.2.m.e 24
40.e odd 2 1 inner 2400.2.m.e 24
40.f even 2 1 600.2.m.e 24
40.i odd 4 1 600.2.b.g ✓ 12
40.i odd 4 1 600.2.b.h yes 12
40.k even 4 1 2400.2.b.g 12
40.k even 4 1 2400.2.b.h 12
60.h even 2 1 600.2.m.e 24
60.l odd 4 1 600.2.b.g ✓ 12
60.l odd 4 1 600.2.b.h yes 12
120.i odd 2 1 600.2.m.e 24
120.m even 2 1 inner 2400.2.m.e 24
120.q odd 4 1 2400.2.b.g 12
120.q odd 4 1 2400.2.b.h 12
120.w even 4 1 600.2.b.g ✓ 12
120.w even 4 1 600.2.b.h yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.b.g ✓ 12 20.e even 4 1
600.2.b.g ✓ 12 40.i odd 4 1
600.2.b.g ✓ 12 60.l odd 4 1
600.2.b.g ✓ 12 120.w even 4 1
600.2.b.h yes 12 20.e even 4 1
600.2.b.h yes 12 40.i odd 4 1
600.2.b.h yes 12 60.l odd 4 1
600.2.b.h yes 12 120.w even 4 1
600.2.m.e 24 4.b odd 2 1
600.2.m.e 24 8.b even 2 1
600.2.m.e 24 12.b even 2 1
600.2.m.e 24 20.d odd 2 1
600.2.m.e 24 24.h odd 2 1
600.2.m.e 24 40.f even 2 1
600.2.m.e 24 60.h even 2 1
600.2.m.e 24 120.i odd 2 1
2400.2.b.g 12 5.c odd 4 1
2400.2.b.g 12 15.e even 4 1
2400.2.b.g 12 40.k even 4 1
2400.2.b.g 12 120.q odd 4 1
2400.2.b.h 12 5.c odd 4 1
2400.2.b.h 12 15.e even 4 1
2400.2.b.h 12 40.k even 4 1
2400.2.b.h 12 120.q odd 4 1
2400.2.m.e 24 1.a even 1 1 trivial
2400.2.m.e 24 3.b odd 2 1 inner
2400.2.m.e 24 5.b even 2 1 inner
2400.2.m.e 24 8.d odd 2 1 inner
2400.2.m.e 24 15.d odd 2 1 inner
2400.2.m.e 24 24.f even 2 1 inner
2400.2.m.e 24 40.e odd 2 1 inner
2400.2.m.e 24 120.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 \(S_{2}^{\mathrm{new}}(2400, [\chi])\):

\( T_{7}^{6} - 30T_{7}^{4} + 225T_{7}^{2} - 284 \) Copy content Toggle raw display
\( T_{11}^{6} + 19T_{11}^{4} + 112T_{11}^{2} + 200 \) Copy content Toggle raw display
\( T_{29}^{6} - 140T_{29}^{4} + 5752T_{29}^{2} - 56800 \) Copy content Toggle raw display