Properties

Label 252.8.a.b
Level $252$
Weight $8$
Character orbit 252.a
Self dual yes
Analytic conductor $78.721$
Analytic rank $1$
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] = [252,8,Mod(1,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.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: \(78.7210264220\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
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)
 
\(f(q)\) \(=\) \( q + 240 q^{5} + 343 q^{7} - 702 q^{11} - 3958 q^{13} + 3408 q^{17} - 49036 q^{19} + 11514 q^{23} - 20525 q^{25} - 49662 q^{29} - 113320 q^{31} + 82320 q^{35} - 66886 q^{37} + 360900 q^{41} - 765292 q^{43}+ \cdots + 773846 q^{97}+O(q^{100}) \) 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 0 240.000 0 343.000 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 252.8.a.b 1
3.b odd 2 1 84.8.a.b 1
12.b even 2 1 336.8.a.c 1
21.c even 2 1 588.8.a.b 1
21.g even 6 2 588.8.i.e 2
21.h odd 6 2 588.8.i.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.8.a.b 1 3.b odd 2 1
252.8.a.b 1 1.a even 1 1 trivial
336.8.a.c 1 12.b even 2 1
588.8.a.b 1 21.c even 2 1
588.8.i.d 2 21.h odd 6 2
588.8.i.e 2 21.g even 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 240 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(252))\). 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
$5$ \( T - 240 \) Copy content Toggle raw display
$7$ \( T - 343 \) Copy content Toggle raw display
$11$ \( T + 702 \) Copy content Toggle raw display
$13$ \( T + 3958 \) Copy content Toggle raw display
$17$ \( T - 3408 \) Copy content Toggle raw display
$19$ \( T + 49036 \) Copy content Toggle raw display
$23$ \( T - 11514 \) Copy content Toggle raw display
$29$ \( T + 49662 \) Copy content Toggle raw display
$31$ \( T + 113320 \) Copy content Toggle raw display
$37$ \( T + 66886 \) Copy content Toggle raw display
$41$ \( T - 360900 \) Copy content Toggle raw display
$43$ \( T + 765292 \) Copy content Toggle raw display
$47$ \( T - 1344876 \) Copy content Toggle raw display
$53$ \( T + 358962 \) Copy content Toggle raw display
$59$ \( T + 930528 \) Copy content Toggle raw display
$61$ \( T + 1318834 \) Copy content Toggle raw display
$67$ \( T - 1893464 \) Copy content Toggle raw display
$71$ \( T + 227994 \) Copy content Toggle raw display
$73$ \( T - 784934 \) Copy content Toggle raw display
$79$ \( T + 2100892 \) Copy content Toggle raw display
$83$ \( T + 8629308 \) Copy content Toggle raw display
$89$ \( T + 5903100 \) Copy content Toggle raw display
$97$ \( T - 773846 \) Copy content Toggle raw display
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