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Properties

Label 4704.2.p.b
Level $4704$
Weight $2$
Character orbit 4704.p
Analytic conductor $37.562$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4704,2,Mod(3919,4704)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4704, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4704.3919");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4704 = 2^{5} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4704.p (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: \(37.5616291108\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 1176)
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.

\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 48 q - 48 q^{9} - 32 q^{11} + 48 q^{25} - 32 q^{43} + 32 q^{67} + 48 q^{81} + 32 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3919.1 0 − 1.00000i 0 1.92559 0 0 0 −1.00000 0
3919.2 0 1.00000i 0 1.92559 0 0 0 −1.00000 0
3919.3 0 − 1.00000i 0 −2.66054 0 0 0 −1.00000 0
3919.4 0 1.00000i 0 −2.66054 0 0 0 −1.00000 0
3919.5 0 − 1.00000i 0 1.77992 0 0 0 −1.00000 0
3919.6 0 1.00000i 0 1.77992 0 0 0 −1.00000 0
3919.7 0 − 1.00000i 0 2.57843 0 0 0 −1.00000 0
3919.8 0 1.00000i 0 2.57843 0 0 0 −1.00000 0
3919.9 0 − 1.00000i 0 −2.73143 0 0 0 −1.00000 0
3919.10 0 1.00000i 0 −2.73143 0 0 0 −1.00000 0
3919.11 0 − 1.00000i 0 −4.01506 0 0 0 −1.00000 0
3919.12 0 1.00000i 0 −4.01506 0 0 0 −1.00000 0
3919.13 0 − 1.00000i 0 2.73143 0 0 0 −1.00000 0
3919.14 0 1.00000i 0 2.73143 0 0 0 −1.00000 0
3919.15 0 − 1.00000i 0 1.42504 0 0 0 −1.00000 0
3919.16 0 1.00000i 0 1.42504 0 0 0 −1.00000 0
3919.17 0 − 1.00000i 0 2.66054 0 0 0 −1.00000 0
3919.18 0 1.00000i 0 2.66054 0 0 0 −1.00000 0
3919.19 0 − 1.00000i 0 2.13212 0 0 0 −1.00000 0
3919.20 0 1.00000i 0 2.13212 0 0 0 −1.00000 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3919.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
8.d odd 2 1 inner
56.e even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4704.2.p.b 48
4.b odd 2 1 1176.2.p.b ✓ 48
7.b odd 2 1 inner 4704.2.p.b 48
8.b even 2 1 1176.2.p.b ✓ 48
8.d odd 2 1 inner 4704.2.p.b 48
28.d even 2 1 1176.2.p.b ✓ 48
56.e even 2 1 inner 4704.2.p.b 48
56.h odd 2 1 1176.2.p.b ✓ 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.p.b ✓ 48 4.b odd 2 1
1176.2.p.b ✓ 48 8.b even 2 1
1176.2.p.b ✓ 48 28.d even 2 1
1176.2.p.b ✓ 48 56.h odd 2 1
4704.2.p.b 48 1.a even 1 1 trivial
4704.2.p.b 48 7.b odd 2 1 inner
4704.2.p.b 48 8.d odd 2 1 inner
4704.2.p.b 48 56.e even 2 1 inner