Properties

Label 3960.2.a.x
Level $3960$
Weight $2$
Character orbit 3960.a
Self dual yes
Analytic conductor $31.621$
Analytic rank $1$
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] = [3960,2,Mod(1,3960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3960, 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("3960.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3960.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: \(31.6207592004\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1320)
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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{5} + \beta q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{5} + \beta q^{7} - q^{11} + ( - \beta + 2) q^{13} + (\beta - 2) q^{17} - 2 \beta q^{23} + q^{25} + ( - 2 \beta - 2) q^{29} - 2 \beta q^{31} - \beta q^{35} - 2 q^{37} + (2 \beta - 6) q^{41} + ( - 3 \beta + 4) q^{43} + 2 \beta q^{47} + q^{49} + (2 \beta - 6) q^{53} + q^{55} + ( - 2 \beta + 4) q^{59} + (2 \beta - 2) q^{61} + (\beta - 2) q^{65} + (2 \beta - 4) q^{67} + 2 \beta q^{71} + (\beta + 6) q^{73} - \beta q^{77} - 4 q^{79} + ( - \beta - 8) q^{83} + ( - \beta + 2) q^{85} - 2 q^{89} + (2 \beta - 8) q^{91} + (2 \beta - 6) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} - 2 q^{11} + 4 q^{13} - 4 q^{17} + 2 q^{25} - 4 q^{29} - 4 q^{37} - 12 q^{41} + 8 q^{43} + 2 q^{49} - 12 q^{53} + 2 q^{55} + 8 q^{59} - 4 q^{61} - 4 q^{65} - 8 q^{67} + 12 q^{73} - 8 q^{79} - 16 q^{83} + 4 q^{85} - 4 q^{89} - 16 q^{91} - 12 q^{97}+O(q^{100}) \) 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
−1.41421
1.41421
0 0 0 −1.00000 0 −2.82843 0 0 0
1.2 0 0 0 −1.00000 0 2.82843 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)
\(5\) \( +1 \)
\(11\) \( +1 \)

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 3960.2.a.x 2
3.b odd 2 1 1320.2.a.q 2
4.b odd 2 1 7920.2.a.bt 2
12.b even 2 1 2640.2.a.z 2
15.d odd 2 1 6600.2.a.bh 2
15.e even 4 2 6600.2.d.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.a.q 2 3.b odd 2 1
2640.2.a.z 2 12.b even 2 1
3960.2.a.x 2 1.a even 1 1 trivial
6600.2.a.bh 2 15.d odd 2 1
6600.2.d.bb 4 15.e even 4 2
7920.2.a.bt 2 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(3960))\):

\( T_{7}^{2} - 8 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} - 4 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} - 4 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display
\( T_{23}^{2} - 32 \) Copy content Toggle raw display
\( T_{29}^{2} + 4T_{29} - 28 \) 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
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 8 \) Copy content Toggle raw display
$11$ \( (T + 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} + 4T - 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 32 \) Copy content Toggle raw display
$29$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$31$ \( T^{2} - 32 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T - 56 \) Copy content Toggle raw display
$47$ \( T^{2} - 32 \) Copy content Toggle raw display
$53$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$59$ \( T^{2} - 8T - 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 4T - 28 \) Copy content Toggle raw display
$67$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$71$ \( T^{2} - 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 12T + 28 \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16T + 56 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
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