Properties

Label 9.48.a.c
Level $9$
Weight $48$
Character orbit 9.a
Self dual yes
Analytic conductor $125.917$
Analytic rank $1$
Dimension $4$
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] = [9,48,Mod(1,9)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 48, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 48);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 48 \)
Character orbit: \([\chi]\) \(=\) 9.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: \(125.916896390\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 832803191366x^{2} + 3710135215485780x + 13175318942671469337000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{20}\cdot 3^{11}\cdot 5^{3}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 1)
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 a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1446390) q^{2} + (\beta_{3} + 6 \beta_{2} + 2732420 \beta_1 + 101201874790288) q^{4} + (96 \beta_{3} - 1796 \beta_{2} - 214856224 \beta_1 + 77\!\cdots\!50) q^{5}+ \cdots + ( - 2897880 \beta_{3} - 105238288 \beta_{2} + \cdots - 59\!\cdots\!80) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1446390) q^{2} + (\beta_{3} + 6 \beta_{2} + 2732420 \beta_1 + 101201874790288) q^{4} + (96 \beta_{3} - 1796 \beta_{2} - 214856224 \beta_1 + 77\!\cdots\!50) q^{5}+ \cdots + (39\!\cdots\!60 \beta_{3} + \cdots + 74\!\cdots\!30) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 5785560 q^{2} + 404807499161152 q^{4} + 31\!\cdots\!00 q^{5}+ \cdots - 23\!\cdots\!20 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 5785560 q^{2} + 404807499161152 q^{4} + 31\!\cdots\!00 q^{5}+ \cdots + 29\!\cdots\!20 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 832803191366x^{2} + 3710135215485780x + 13175318942671469337000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 24\nu - 6 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 27\nu^{3} + 1621431\nu^{2} - 21620549846568\nu - 600047676703864212 ) / 171584 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -81\nu^{3} + 44551899\nu^{2} + 65191807354488\nu - 18776838253817714244 ) / 85792 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 6 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6\beta_{2} - 160348\beta _1 + 239847319113552 ) / 576 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -180159\beta_{3} + 9900422\beta_{2} + 57683687726180\beta _1 - 4807255926333560112 ) / 1728 \) 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
901372.
129356.
−124721.
−906006.
−2.30793e7 0 3.91917e14 1.15086e16 0 1.54211e19 −5.79706e21 0 −2.65611e23
1.2 −4.55092e6 0 −1.20027e14 3.08341e16 0 1.11073e20 1.18672e21 0 −1.40324e23
1.3 1.54692e6 0 −1.38345e14 −4.23962e16 0 −3.90714e19 −4.31719e20 0 −6.55837e22
1.4 2.02978e7 0 2.71261e14 3.11682e16 0 −1.26592e20 2.64934e21 0 6.32644e23
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 9.48.a.c 4
3.b odd 2 1 1.48.a.a 4
12.b even 2 1 16.48.a.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.48.a.a 4 3.b odd 2 1
9.48.a.c 4 1.a even 1 1 trivial
16.48.a.d 4 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 5785560 T_{2}^{3} - 467142374034432 T_{2}^{2} + \cdots + 32\!\cdots\!56 \) acting on \(S_{48}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 5785560 T^{3} + \cdots + 32\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} + \cdots - 46\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 84\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 18\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots - 14\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 56\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots - 69\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 15\!\cdots\!64 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 89\!\cdots\!76 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 57\!\cdots\!04 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 84\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 10\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 14\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots - 47\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots - 23\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 15\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 60\!\cdots\!84 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots - 10\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots - 18\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots - 75\!\cdots\!84 \) Copy content Toggle raw display
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