Properties

Label 9648.2.a.bn
Level $9648$
Weight $2$
Character orbit 9648.a
Self dual yes
Analytic conductor $77.040$
Analytic rank $1$
Dimension $3$
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] = [9648,2,Mod(1,9648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9648, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9648.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9648 = 2^{4} \cdot 3^{2} \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9648.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: \(77.0396678701\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 201)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + ( - \beta_{2} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} + ( - \beta_{2} - \beta_1) q^{7} + ( - \beta_{2} + \beta_1 + 3) q^{11} + ( - \beta_{2} + \beta_1 - 3) q^{13} + (\beta_{2} + 3 \beta_1 - 1) q^{17} + (3 \beta_{2} - \beta_1 + 1) q^{19} + ( - 2 \beta_{2} - 3 \beta_1 + 2) q^{23} + (\beta_{2} + \beta_1 - 3) q^{25} - 4 \beta_1 q^{29} + (2 \beta_{2} + 4 \beta_1 - 5) q^{31} + (\beta_{2} + 2 \beta_1 + 1) q^{35} + (\beta_{2} - 3 \beta_1 - 2) q^{37} + ( - 3 \beta_{2} + 2 \beta_1 - 1) q^{41} + q^{43} + (3 \beta_{2} + 3 \beta_1 + 5) q^{47} + (2 \beta_1 - 4) q^{49} + (5 \beta_{2} + 2 \beta_1 - 3) q^{53} + ( - \beta_{2} - 3 \beta_1 - 3) q^{55} + (5 \beta_{2} + 5) q^{59} + ( - \beta_{2} - 5 \beta_1 + 1) q^{61} + ( - \beta_{2} + 3 \beta_1 - 3) q^{65} - q^{67} + ( - \beta_{2} + 3 \beta_1 + 5) q^{71} + ( - 2 \beta_{2} + 2 \beta_1 - 7) q^{73} + ( - 5 \beta_{2} - 5 \beta_1 + 1) q^{77} + (2 \beta_1 - 10) q^{79} + (3 \beta_{2} + 2 \beta_1 - 3) q^{83} + ( - 3 \beta_{2} - 3 \beta_1 - 5) q^{85} + (5 \beta_{2} - 3 \beta_1 + 3) q^{89} + (\beta_{2} + \beta_1 + 1) q^{91} + (\beta_{2} - 3 \beta_1 + 5) q^{95} + (7 \beta_{2} + 7 \beta_1 - 5) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{5} - q^{7} + 10 q^{11} - 8 q^{13} + 2 q^{19} + 3 q^{23} - 8 q^{25} - 4 q^{29} - 11 q^{31} + 5 q^{35} - 9 q^{37} - q^{41} + 3 q^{43} + 18 q^{47} - 10 q^{49} - 7 q^{53} - 12 q^{55} + 15 q^{59} - 2 q^{61} - 6 q^{65} - 3 q^{67} + 18 q^{71} - 19 q^{73} - 2 q^{77} - 28 q^{79} - 7 q^{83} - 18 q^{85} + 6 q^{89} + 4 q^{91} + 12 q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
2.17009
0.311108
−1.48119
0 0 0 −2.17009 0 −2.70928 0 0 0
1.2 0 0 0 −0.311108 0 1.90321 0 0 0
1.3 0 0 0 1.48119 0 −0.193937 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(67\) \(1\)

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 9648.2.a.bn 3
3.b odd 2 1 3216.2.a.u 3
4.b odd 2 1 603.2.a.i 3
12.b even 2 1 201.2.a.d 3
60.h even 2 1 5025.2.a.m 3
84.h odd 2 1 9849.2.a.ba 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
201.2.a.d 3 12.b even 2 1
603.2.a.i 3 4.b odd 2 1
3216.2.a.u 3 3.b odd 2 1
5025.2.a.m 3 60.h even 2 1
9648.2.a.bn 3 1.a even 1 1 trivial
9849.2.a.ba 3 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9648))\):

\( T_{5}^{3} + T_{5}^{2} - 3T_{5} - 1 \) Copy content Toggle raw display
\( T_{7}^{3} + T_{7}^{2} - 5T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{3} - 10T_{11}^{2} + 24T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{3} + 8T_{13}^{2} + 12T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + T^{2} - 3T - 1 \) Copy content Toggle raw display
$7$ \( T^{3} + T^{2} - 5T - 1 \) Copy content Toggle raw display
$11$ \( T^{3} - 10 T^{2} + 24 T + 4 \) Copy content Toggle raw display
$13$ \( T^{3} + 8 T^{2} + 12 T + 4 \) Copy content Toggle raw display
$17$ \( T^{3} - 28T - 52 \) Copy content Toggle raw display
$19$ \( T^{3} - 2 T^{2} - 44 T + 20 \) Copy content Toggle raw display
$23$ \( T^{3} - 3 T^{2} - 31 T + 95 \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} - 48 T - 64 \) Copy content Toggle raw display
$31$ \( T^{3} + 11 T^{2} - 13 T - 295 \) Copy content Toggle raw display
$37$ \( T^{3} + 9 T^{2} - 13 T - 169 \) Copy content Toggle raw display
$41$ \( T^{3} + T^{2} - 61 T + 97 \) Copy content Toggle raw display
$43$ \( (T - 1)^{3} \) Copy content Toggle raw display
$47$ \( T^{3} - 18 T^{2} + 60 T + 52 \) Copy content Toggle raw display
$53$ \( T^{3} + 7 T^{2} - 77 T + 131 \) Copy content Toggle raw display
$59$ \( T^{3} - 15 T^{2} - 25 T + 625 \) Copy content Toggle raw display
$61$ \( T^{3} + 2 T^{2} - 76 T + 116 \) Copy content Toggle raw display
$67$ \( (T + 1)^{3} \) Copy content Toggle raw display
$71$ \( T^{3} - 18 T^{2} + 68 T + 100 \) Copy content Toggle raw display
$73$ \( T^{3} + 19 T^{2} + 83 T + 97 \) Copy content Toggle raw display
$79$ \( T^{3} + 28 T^{2} + 248 T + 688 \) Copy content Toggle raw display
$83$ \( T^{3} + 7 T^{2} - 21 T - 25 \) Copy content Toggle raw display
$89$ \( T^{3} - 6 T^{2} - 148 T - 116 \) Copy content Toggle raw display
$97$ \( T^{3} + 8 T^{2} - 240 T - 932 \) Copy content Toggle raw display
show more
show less