Properties

Label 504.6.a.f
Level $504$
Weight $6$
Character orbit 504.a
Self dual yes
Analytic conductor $80.833$
Analytic rank $0$
Dimension $1$
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] = [504,6,Mod(1,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 504.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: \(80.8334451857\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
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)
 
\(f(q)\) \(=\) \( q + 34 q^{5} + 49 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 34 q^{5} + 49 q^{7} + 756 q^{11} + 678 q^{13} + 1838 q^{17} + 604 q^{19} - 2840 q^{23} - 1969 q^{25} - 6878 q^{29} + 3568 q^{31} + 1666 q^{35} + 14598 q^{37} - 5962 q^{41} - 676 q^{43} + 20800 q^{47} + 2401 q^{49} - 32390 q^{53} + 25704 q^{55} - 42948 q^{59} + 44806 q^{61} + 23052 q^{65} - 39708 q^{67} + 25800 q^{71} + 58954 q^{73} + 37044 q^{77} - 77648 q^{79} - 35964 q^{83} + 62492 q^{85} - 80842 q^{89} + 33222 q^{91} + 20536 q^{95} - 64334 q^{97}+O(q^{100}) \) 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
0
0 0 0 34.0000 0 49.0000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)

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 504.6.a.f 1
3.b odd 2 1 168.6.a.a 1
4.b odd 2 1 1008.6.a.s 1
12.b even 2 1 336.6.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.a.a 1 3.b odd 2 1
336.6.a.k 1 12.b even 2 1
504.6.a.f 1 1.a even 1 1 trivial
1008.6.a.s 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(504))\):

\( T_{5} - 34 \) Copy content Toggle raw display
\( T_{11} - 756 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 34 \) Copy content Toggle raw display
$7$ \( T - 49 \) Copy content Toggle raw display
$11$ \( T - 756 \) Copy content Toggle raw display
$13$ \( T - 678 \) Copy content Toggle raw display
$17$ \( T - 1838 \) Copy content Toggle raw display
$19$ \( T - 604 \) Copy content Toggle raw display
$23$ \( T + 2840 \) Copy content Toggle raw display
$29$ \( T + 6878 \) Copy content Toggle raw display
$31$ \( T - 3568 \) Copy content Toggle raw display
$37$ \( T - 14598 \) Copy content Toggle raw display
$41$ \( T + 5962 \) Copy content Toggle raw display
$43$ \( T + 676 \) Copy content Toggle raw display
$47$ \( T - 20800 \) Copy content Toggle raw display
$53$ \( T + 32390 \) Copy content Toggle raw display
$59$ \( T + 42948 \) Copy content Toggle raw display
$61$ \( T - 44806 \) Copy content Toggle raw display
$67$ \( T + 39708 \) Copy content Toggle raw display
$71$ \( T - 25800 \) Copy content Toggle raw display
$73$ \( T - 58954 \) Copy content Toggle raw display
$79$ \( T + 77648 \) Copy content Toggle raw display
$83$ \( T + 35964 \) Copy content Toggle raw display
$89$ \( T + 80842 \) Copy content Toggle raw display
$97$ \( T + 64334 \) Copy content Toggle raw display
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