Properties

Label 9576.2.a.bd
Level $9576$
Weight $2$
Character orbit 9576.a
Self dual yes
Analytic conductor $76.465$
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] = [9576,2,Mod(1,9576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9576, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9576.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9576 = 2^{3} \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9576.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: \(76.4647449756\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
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 \(\beta = \sqrt{17}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 q^{5} - q^{7} + 2 q^{11} + (\beta + 1) q^{13} + (\beta + 1) q^{17} - q^{19} + (\beta - 3) q^{23} - q^{25} + ( - \beta + 5) q^{29} + (\beta + 3) q^{31} + 2 q^{35} + ( - \beta - 5) q^{37} + 2 \beta q^{41} + 4 q^{43} + q^{49} + ( - \beta + 1) q^{53} - 4 q^{55} + ( - 2 \beta - 2) q^{59} - 2 q^{61} + ( - 2 \beta - 2) q^{65} + (2 \beta - 6) q^{67} + (3 \beta + 3) q^{71} - 10 q^{73} - 2 q^{77} + ( - \beta + 9) q^{79} + (\beta - 5) q^{83} + ( - 2 \beta - 2) q^{85} + (2 \beta + 8) q^{89} + ( - \beta - 1) q^{91} + 2 q^{95} - 2 \beta q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{5} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{5} - 2 q^{7} + 4 q^{11} + 2 q^{13} + 2 q^{17} - 2 q^{19} - 6 q^{23} - 2 q^{25} + 10 q^{29} + 6 q^{31} + 4 q^{35} - 10 q^{37} + 8 q^{43} + 2 q^{49} + 2 q^{53} - 8 q^{55} - 4 q^{59} - 4 q^{61} - 4 q^{65} - 12 q^{67} + 6 q^{71} - 20 q^{73} - 4 q^{77} + 18 q^{79} - 10 q^{83} - 4 q^{85} + 16 q^{89} - 2 q^{91} + 4 q^{95}+O(q^{100}) \) 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
−1.56155
2.56155
0 0 0 −2.00000 0 −1.00000 0 0 0
1.2 0 0 0 −2.00000 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(7\) \( +1 \)
\(19\) \( +1 \)

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 9576.2.a.bd 2
3.b odd 2 1 9576.2.a.bx yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9576.2.a.bd 2 1.a even 1 1 trivial
9576.2.a.bx yes 2 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9576))\):

\( T_{5} + 2 \) Copy content Toggle raw display
\( T_{11} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 2T_{13} - 16 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T + 2)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 6T - 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 10T + 8 \) Copy content Toggle raw display
$31$ \( T^{2} - 6T - 8 \) Copy content Toggle raw display
$37$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$41$ \( T^{2} - 68 \) Copy content Toggle raw display
$43$ \( (T - 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 2T - 16 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 64 \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 12T - 32 \) Copy content Toggle raw display
$71$ \( T^{2} - 6T - 144 \) Copy content Toggle raw display
$73$ \( (T + 10)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 18T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} + 10T + 8 \) Copy content Toggle raw display
$89$ \( T^{2} - 16T - 4 \) Copy content Toggle raw display
$97$ \( T^{2} - 68 \) Copy content Toggle raw display
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