Properties

Label 8028.2.h.a
Level $8028$
Weight $2$
Character orbit 8028.h
Analytic conductor $64.104$
Analytic rank $0$
Dimension $76$
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] = [8028,2,Mod(4013,8028)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8028, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("8028.4013");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8028 = 2^{2} \cdot 3^{2} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8028.h (of order \(2\), degree \(1\), 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: \(64.1039027427\)
Analytic rank: \(0\)
Dimension: \(76\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 76 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 76 q + 8 q^{7} + 16 q^{19} + 100 q^{25} - 8 q^{31} + 32 q^{37} - 24 q^{43} + 68 q^{49} + 24 q^{55} + 8 q^{73}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
4013.1 0 0 0 −4.23105 0 2.43977 0 0 0
4013.2 0 0 0 −4.23105 0 2.43977 0 0 0
4013.3 0 0 0 −3.84036 0 1.75909 0 0 0
4013.4 0 0 0 −3.84036 0 1.75909 0 0 0
4013.5 0 0 0 −3.83180 0 −1.12103 0 0 0
4013.6 0 0 0 −3.83180 0 −1.12103 0 0 0
4013.7 0 0 0 −3.80256 0 0.208948 0 0 0
4013.8 0 0 0 −3.80256 0 0.208948 0 0 0
4013.9 0 0 0 −3.65965 0 −4.62592 0 0 0
4013.10 0 0 0 −3.65965 0 −4.62592 0 0 0
4013.11 0 0 0 −2.81288 0 3.64674 0 0 0
4013.12 0 0 0 −2.81288 0 3.64674 0 0 0
4013.13 0 0 0 −2.68681 0 −3.11614 0 0 0
4013.14 0 0 0 −2.68681 0 −3.11614 0 0 0
4013.15 0 0 0 −2.24499 0 −1.57279 0 0 0
4013.16 0 0 0 −2.24499 0 −1.57279 0 0 0
4013.17 0 0 0 −2.12158 0 2.81365 0 0 0
4013.18 0 0 0 −2.12158 0 2.81365 0 0 0
4013.19 0 0 0 −1.82157 0 −3.59530 0 0 0
4013.20 0 0 0 −1.82157 0 −3.59530 0 0 0
See all 76 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 4013.76
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
223.b odd 2 1 inner
669.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8028.2.h.a 76
3.b odd 2 1 inner 8028.2.h.a 76
223.b odd 2 1 inner 8028.2.h.a 76
669.c even 2 1 inner 8028.2.h.a 76
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8028.2.h.a 76 1.a even 1 1 trivial
8028.2.h.a 76 3.b odd 2 1 inner
8028.2.h.a 76 223.b odd 2 1 inner
8028.2.h.a 76 669.c even 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(8028, [\chi])\).