Properties

Label 448.2.z.a
Level $448$
Weight $2$
Character orbit 448.z
Analytic conductor $3.577$
Analytic rank $0$
Dimension $56$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [448,2,Mod(47,448)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(448, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 9, 10]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("448.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 448 = 2^{6} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 448.z (of order \(12\), degree \(4\), 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: \(3.57729801055\)
Analytic rank: \(0\)
Dimension: \(56\)
Relative dimension: \(14\) over \(\Q(\zeta_{12})\)
Twist minimal: no (minimal twist has level 112)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 56 q + 6 q^{3} - 6 q^{5} + 8 q^{7} - 2 q^{11} - 12 q^{17} + 6 q^{19} - 10 q^{21} + 12 q^{23} - 24 q^{29} - 12 q^{33} + 2 q^{35} + 6 q^{37} + 4 q^{39} + 12 q^{45} - 8 q^{49} + 34 q^{51} + 6 q^{53} - 42 q^{59}+ \cdots + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1 0 −0.743580 + 2.77508i 0 −3.38680 + 0.907491i 0 −0.0791552 2.64457i 0 −4.55008 2.62699i 0
47.2 0 −0.615336 + 2.29647i 0 −0.835825 + 0.223959i 0 1.25761 + 2.32775i 0 −2.29704 1.32620i 0
47.3 0 −0.524583 + 1.95777i 0 2.48890 0.666898i 0 2.38027 + 1.15512i 0 −0.959602 0.554026i 0
47.4 0 −0.449868 + 1.67893i 0 0.731029 0.195879i 0 −2.52163 0.800849i 0 −0.0183525 0.0105958i 0
47.5 0 −0.350301 + 1.30734i 0 1.09872 0.294400i 0 1.56831 2.13082i 0 1.01165 + 0.584076i 0
47.6 0 −0.0530773 + 0.198087i 0 −1.82029 + 0.487744i 0 −1.84933 + 1.89208i 0 2.56165 + 1.47897i 0
47.7 0 0.0661591 0.246909i 0 −0.499339 + 0.133797i 0 −2.45011 0.998472i 0 2.54149 + 1.46733i 0
47.8 0 0.204601 0.763582i 0 −3.81370 + 1.02188i 0 2.64575 + 0.00379639i 0 2.05688 + 1.18754i 0
47.9 0 0.237312 0.885661i 0 3.05579 0.818796i 0 0.640837 + 2.56697i 0 1.87000 + 1.07964i 0
47.10 0 0.282507 1.05433i 0 −1.39922 + 0.374919i 0 0.298614 + 2.62885i 0 1.56627 + 0.904287i 0
47.11 0 0.388227 1.44888i 0 2.81809 0.755106i 0 1.47726 2.19493i 0 0.649537 + 0.375010i 0
47.12 0 0.665801 2.48480i 0 −3.12401 + 0.837076i 0 −1.56215 2.13534i 0 −3.13287 1.80877i 0
47.13 0 0.700587 2.61463i 0 0.450647 0.120751i 0 2.37326 1.16946i 0 −3.74737 2.16355i 0
47.14 0 0.825526 3.08091i 0 1.86998 0.501060i 0 −2.17953 + 1.49988i 0 −6.21241 3.58674i 0
143.1 0 −0.743580 2.77508i 0 −3.38680 0.907491i 0 −0.0791552 + 2.64457i 0 −4.55008 + 2.62699i 0
143.2 0 −0.615336 2.29647i 0 −0.835825 0.223959i 0 1.25761 2.32775i 0 −2.29704 + 1.32620i 0
143.3 0 −0.524583 1.95777i 0 2.48890 + 0.666898i 0 2.38027 1.15512i 0 −0.959602 + 0.554026i 0
143.4 0 −0.449868 1.67893i 0 0.731029 + 0.195879i 0 −2.52163 + 0.800849i 0 −0.0183525 + 0.0105958i 0
143.5 0 −0.350301 1.30734i 0 1.09872 + 0.294400i 0 1.56831 + 2.13082i 0 1.01165 0.584076i 0
143.6 0 −0.0530773 0.198087i 0 −1.82029 0.487744i 0 −1.84933 1.89208i 0 2.56165 1.47897i 0
See all 56 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner
16.f odd 4 1 inner
112.v even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 448.2.z.a 56
4.b odd 2 1 112.2.v.a 56
7.d odd 6 1 inner 448.2.z.a 56
8.b even 2 1 896.2.z.a 56
8.d odd 2 1 896.2.z.b 56
16.e even 4 1 112.2.v.a 56
16.e even 4 1 896.2.z.b 56
16.f odd 4 1 inner 448.2.z.a 56
16.f odd 4 1 896.2.z.a 56
28.d even 2 1 784.2.w.f 56
28.f even 6 1 112.2.v.a 56
28.f even 6 1 784.2.j.a 56
28.g odd 6 1 784.2.j.a 56
28.g odd 6 1 784.2.w.f 56
56.j odd 6 1 896.2.z.a 56
56.m even 6 1 896.2.z.b 56
112.l odd 4 1 784.2.w.f 56
112.v even 12 1 inner 448.2.z.a 56
112.v even 12 1 896.2.z.a 56
112.w even 12 1 784.2.j.a 56
112.w even 12 1 784.2.w.f 56
112.x odd 12 1 112.2.v.a 56
112.x odd 12 1 784.2.j.a 56
112.x odd 12 1 896.2.z.b 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.v.a 56 4.b odd 2 1
112.2.v.a 56 16.e even 4 1
112.2.v.a 56 28.f even 6 1
112.2.v.a 56 112.x odd 12 1
448.2.z.a 56 1.a even 1 1 trivial
448.2.z.a 56 7.d odd 6 1 inner
448.2.z.a 56 16.f odd 4 1 inner
448.2.z.a 56 112.v even 12 1 inner
784.2.j.a 56 28.f even 6 1
784.2.j.a 56 28.g odd 6 1
784.2.j.a 56 112.w even 12 1
784.2.j.a 56 112.x odd 12 1
784.2.w.f 56 28.d even 2 1
784.2.w.f 56 28.g odd 6 1
784.2.w.f 56 112.l odd 4 1
784.2.w.f 56 112.w even 12 1
896.2.z.a 56 8.b even 2 1
896.2.z.a 56 16.f odd 4 1
896.2.z.a 56 56.j odd 6 1
896.2.z.a 56 112.v even 12 1
896.2.z.b 56 8.d odd 2 1
896.2.z.b 56 16.e even 4 1
896.2.z.b 56 56.m even 6 1
896.2.z.b 56 112.x odd 12 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(448, [\chi])\).