Properties

Label 9.32.a.a
Level $9$
Weight $32$
Character orbit 9.a
Self dual yes
Analytic conductor $54.789$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9,32,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, 32, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9.1");
 
S:= CuspForms(chi, 32);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 32 \)
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: \(54.7894195371\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4573872 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3}\cdot 3 \)
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 \(\beta = 12\sqrt{18295489}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 19980) q^{2} + (39960 \beta + 886267168) q^{4} + (1418560 \beta + 9695609010) q^{5} + ( - 71928864 \beta + 15128763788600) q^{7} + (462815680 \beta - 80077529352960) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 19980) q^{2} + (39960 \beta + 886267168) q^{4} + (1418560 \beta + 9695609010) q^{5} + ( - 71928864 \beta + 15128763788600) q^{7} + (462815680 \beta - 80077529352960) q^{8} + ( - 38038437810 \beta - 39\!\cdots\!60) q^{10}+ \cdots + ( - 41\!\cdots\!93 \beta + 40\!\cdots\!60) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 39960 q^{2} + 1772534336 q^{4} + 19391218020 q^{5} + 30257527577200 q^{7} - 160155058705920 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 39960 q^{2} + 1772534336 q^{4} + 19391218020 q^{5} + 30257527577200 q^{7} - 160155058705920 q^{8} - 78\!\cdots\!20 q^{10}+ \cdots + 80\!\cdots\!20 q^{98}+O(q^{100}) \) 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
2139.16
−2138.16
−71307.9 0 2.93733e9 8.25073e10 0 1.14368e13 −5.63222e13 0 −5.88342e15
1.2 31347.9 0 −1.16479e9 −6.31161e10 0 1.88207e13 −1.03833e14 0 −1.97855e15
\(n\): 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 9.32.a.a 2
3.b odd 2 1 1.32.a.a 2
12.b even 2 1 16.32.a.b 2
15.d odd 2 1 25.32.a.a 2
15.e even 4 2 25.32.b.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.32.a.a 2 3.b odd 2 1
9.32.a.a 2 1.a even 1 1 trivial
16.32.a.b 2 12.b even 2 1
25.32.a.a 2 15.d odd 2 1
25.32.b.a 4 15.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 39960T_{2} - 2235350016 \) acting on \(S_{32}^{\mathrm{new}}(\Gamma_0(9))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 39960 T - 2235350016 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 19391218020 T - 52\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{2} - 30257527577200 T + 21\!\cdots\!64 \) Copy content Toggle raw display
$11$ \( T^{2} + \cdots + 12\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{2} + \cdots - 55\!\cdots\!04 \) Copy content Toggle raw display
$17$ \( T^{2} + \cdots + 68\!\cdots\!24 \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 73\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 11\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 25\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 70\!\cdots\!36 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots - 22\!\cdots\!64 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 56\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 51\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 26\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 16\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 90\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 53\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 86\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 62\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
show more
show less