Properties

Label 54.8.a.g
Level $54$
Weight $8$
Character orbit 54.a
Self dual yes
Analytic conductor $16.869$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [54,8,Mod(1,54)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(54, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("54.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 54 = 2 \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 54.a (trivial)

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: yes
Analytic conductor: \(16.8687913761\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{329}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 82 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{3} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 27\sqrt{329}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 8 q^{2} + 64 q^{4} + (\beta + 24) q^{5} + (\beta + 440) q^{7} - 512 q^{8} + ( - 8 \beta - 192) q^{10} + ( - 8 \beta - 3615) q^{11} + (20 \beta + 4280) q^{13} + ( - 8 \beta - 3520) q^{14} + 4096 q^{16}+ \cdots + ( - 7040 \beta + 3120816) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{2} + 128 q^{4} + 48 q^{5} + 880 q^{7} - 1024 q^{8} - 384 q^{10} - 7230 q^{11} + 8560 q^{13} - 7040 q^{14} + 8192 q^{16} - 25704 q^{17} + 37048 q^{19} + 3072 q^{20} + 57840 q^{22} + 59628 q^{23}+ \cdots + 6241632 q^{98}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−8.56918
9.56918
−8.00000 0 64.0000 −465.736 0 −49.7356 −512.000 0 3725.89
1.2 −8.00000 0 64.0000 513.736 0 929.736 −512.000 0 −4109.89
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 54.8.a.g 2
3.b odd 2 1 54.8.a.h yes 2
4.b odd 2 1 432.8.a.p 2
9.c even 3 2 162.8.c.p 4
9.d odd 6 2 162.8.c.m 4
12.b even 2 1 432.8.a.k 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
54.8.a.g 2 1.a even 1 1 trivial
54.8.a.h yes 2 3.b odd 2 1
162.8.c.m 4 9.d odd 6 2
162.8.c.p 4 9.c even 3 2
432.8.a.k 2 12.b even 2 1
432.8.a.p 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 48T_{5} - 239265 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(54))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 48T - 239265 \) Copy content Toggle raw display
$7$ \( T^{2} - 880T - 46241 \) Copy content Toggle raw display
$11$ \( T^{2} + 7230 T - 2281599 \) Copy content Toggle raw display
$13$ \( T^{2} - 8560 T - 77618000 \) Copy content Toggle raw display
$17$ \( T^{2} + 25704 T + 69237504 \) Copy content Toggle raw display
$19$ \( T^{2} - 37048 T - 832082324 \) Copy content Toggle raw display
$23$ \( T^{2} - 59628 T + 792938196 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 6257180700 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 4403209961 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 179439584684 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 61090498764 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 146018088316 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 45928723716 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 463161939111 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots + 383589380304 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 2269257806656 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 71859687676 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 5355488275200 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 5381037627791 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 2609413236800 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 19347454919025 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 20597975704836 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 51902270530991 \) Copy content Toggle raw display
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