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Properties

Label 448.4.q.b
Level $448$
Weight $4$
Character orbit 448.q
Analytic conductor $26.433$
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] = [448,4,Mod(31,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.31");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 448.q (of order \(6\), degree \(2\), 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: \(26.4328556826\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32 q + 80 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 32 q + 80 q^{9} - 288 q^{17} - 48 q^{25} - 1920 q^{33} + 784 q^{49} - 96 q^{57} + 768 q^{65} - 4752 q^{73} - 3616 q^{81} - 7920 q^{89}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1 0 −7.85823 − 4.53695i 0 −1.35499 − 2.34692i 0 17.1204 + 7.06332i 0 27.6679 + 47.9221i 0
31.2 0 −7.85823 − 4.53695i 0 1.35499 + 2.34692i 0 −17.1204 − 7.06332i 0 27.6679 + 47.9221i 0
31.3 0 −5.42062 − 3.12960i 0 −2.54036 − 4.40003i 0 18.4235 − 1.89073i 0 6.08873 + 10.5460i 0
31.4 0 −5.42062 − 3.12960i 0 2.54036 + 4.40003i 0 −18.4235 + 1.89073i 0 6.08873 + 10.5460i 0
31.5 0 −2.20410 − 1.27254i 0 5.37515 + 9.31003i 0 −4.23200 + 18.0303i 0 −10.2613 − 17.7731i 0
31.6 0 −2.20410 − 1.27254i 0 −5.37515 − 9.31003i 0 4.23200 − 18.0303i 0 −10.2613 − 17.7731i 0
31.7 0 −0.0840913 − 0.0485502i 0 9.52987 + 16.5062i 0 −9.19541 − 16.0762i 0 −13.4953 − 23.3745i 0
31.8 0 −0.0840913 − 0.0485502i 0 −9.52987 − 16.5062i 0 9.19541 + 16.0762i 0 −13.4953 − 23.3745i 0
31.9 0 0.0840913 + 0.0485502i 0 9.52987 + 16.5062i 0 9.19541 + 16.0762i 0 −13.4953 − 23.3745i 0
31.10 0 0.0840913 + 0.0485502i 0 −9.52987 − 16.5062i 0 −9.19541 − 16.0762i 0 −13.4953 − 23.3745i 0
31.11 0 2.20410 + 1.27254i 0 5.37515 + 9.31003i 0 4.23200 − 18.0303i 0 −10.2613 − 17.7731i 0
31.12 0 2.20410 + 1.27254i 0 −5.37515 − 9.31003i 0 −4.23200 + 18.0303i 0 −10.2613 − 17.7731i 0
31.13 0 5.42062 + 3.12960i 0 −2.54036 − 4.40003i 0 −18.4235 + 1.89073i 0 6.08873 + 10.5460i 0
31.14 0 5.42062 + 3.12960i 0 2.54036 + 4.40003i 0 18.4235 − 1.89073i 0 6.08873 + 10.5460i 0
31.15 0 7.85823 + 4.53695i 0 −1.35499 − 2.34692i 0 −17.1204 − 7.06332i 0 27.6679 + 47.9221i 0
31.16 0 7.85823 + 4.53695i 0 1.35499 + 2.34692i 0 17.1204 + 7.06332i 0 27.6679 + 47.9221i 0
159.1 0 −7.85823 + 4.53695i 0 −1.35499 + 2.34692i 0 17.1204 − 7.06332i 0 27.6679 − 47.9221i 0
159.2 0 −7.85823 + 4.53695i 0 1.35499 − 2.34692i 0 −17.1204 + 7.06332i 0 27.6679 − 47.9221i 0
159.3 0 −5.42062 + 3.12960i 0 −2.54036 + 4.40003i 0 18.4235 + 1.89073i 0 6.08873 − 10.5460i 0
159.4 0 −5.42062 + 3.12960i 0 2.54036 − 4.40003i 0 −18.4235 − 1.89073i 0 6.08873 − 10.5460i 0
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
28.f even 6 1 inner
56.j odd 6 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.4.q.b ✓ 32
4.b odd 2 1 inner 448.4.q.b ✓ 32
7.d odd 6 1 inner 448.4.q.b ✓ 32
8.b even 2 1 inner 448.4.q.b ✓ 32
8.d odd 2 1 inner 448.4.q.b ✓ 32
28.f even 6 1 inner 448.4.q.b ✓ 32
56.j odd 6 1 inner 448.4.q.b ✓ 32
56.m even 6 1 inner 448.4.q.b ✓ 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
448.4.q.b ✓ 32 1.a even 1 1 trivial
448.4.q.b ✓ 32 4.b odd 2 1 inner
448.4.q.b ✓ 32 7.d odd 6 1 inner
448.4.q.b ✓ 32 8.b even 2 1 inner
448.4.q.b ✓ 32 8.d odd 2 1 inner
448.4.q.b ✓ 32 28.f even 6 1 inner
448.4.q.b ✓ 32 56.j odd 6 1 inner
448.4.q.b ✓ 32 56.m even 6 1 inner