Properties

Label 3040.2.a.g
Level $3040$
Weight $2$
Character orbit 3040.a
Self dual yes
Analytic conductor $24.275$
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] = [3040,2,Mod(1,3040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3040, 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("3040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3040 = 2^{5} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3040.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: \(24.2745222145\)
Analytic rank: \(0\)
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: 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 \(\beta = 2\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + \beta q^{7} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + \beta q^{7} - 3 q^{9} + 4 q^{11} + (\beta + 2) q^{13} + (2 \beta + 2) q^{17} - q^{19} + \beta q^{23} + q^{25} + ( - 2 \beta + 2) q^{29} - 2 \beta q^{31} + \beta q^{35} + ( - 3 \beta + 2) q^{37} + ( - 2 \beta + 2) q^{41} + 3 \beta q^{43} - 3 q^{45} + (\beta - 8) q^{47} + q^{49} + ( - \beta + 10) q^{53} + 4 q^{55} + ( - 2 \beta - 4) q^{59} + 6 q^{61} - 3 \beta q^{63} + (\beta + 2) q^{65} - 4 \beta q^{67} - 2 \beta q^{71} + (2 \beta + 10) q^{73} + 4 \beta q^{77} + 2 \beta q^{79} + 9 q^{81} + (\beta - 8) q^{83} + (2 \beta + 2) q^{85} + ( - 2 \beta - 6) q^{89} + (2 \beta + 8) q^{91} - q^{95} + ( - \beta + 10) q^{97} - 12 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} - 6 q^{9} + 8 q^{11} + 4 q^{13} + 4 q^{17} - 2 q^{19} + 2 q^{25} + 4 q^{29} + 4 q^{37} + 4 q^{41} - 6 q^{45} - 16 q^{47} + 2 q^{49} + 20 q^{53} + 8 q^{55} - 8 q^{59} + 12 q^{61} + 4 q^{65} + 20 q^{73} + 18 q^{81} - 16 q^{83} + 4 q^{85} - 12 q^{89} + 16 q^{91} - 2 q^{95} + 20 q^{97} - 24 q^{99}+O(q^{100}) \) 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 −3.00000 0
1.2 0 0 0 1.00000 0 2.82843 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(19\) \(1\)

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 3040.2.a.g yes 2
4.b odd 2 1 3040.2.a.f 2
8.b even 2 1 6080.2.a.bd 2
8.d odd 2 1 6080.2.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3040.2.a.f 2 4.b odd 2 1
3040.2.a.g yes 2 1.a even 1 1 trivial
6080.2.a.bd 2 8.b even 2 1
6080.2.a.be 2 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3040))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{7}^{2} - 8 \) Copy content Toggle raw display
\( T_{11} - 4 \) 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
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4T - 4 \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$19$ \( (T + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 8 \) Copy content Toggle raw display
$29$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$31$ \( T^{2} - 32 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 68 \) Copy content Toggle raw display
$41$ \( T^{2} - 4T - 28 \) Copy content Toggle raw display
$43$ \( T^{2} - 72 \) Copy content Toggle raw display
$47$ \( T^{2} + 16T + 56 \) Copy content Toggle raw display
$53$ \( T^{2} - 20T + 92 \) Copy content Toggle raw display
$59$ \( T^{2} + 8T - 16 \) Copy content Toggle raw display
$61$ \( (T - 6)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 128 \) Copy content Toggle raw display
$71$ \( T^{2} - 32 \) Copy content Toggle raw display
$73$ \( T^{2} - 20T + 68 \) Copy content Toggle raw display
$79$ \( T^{2} - 32 \) Copy content Toggle raw display
$83$ \( T^{2} + 16T + 56 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 4 \) Copy content Toggle raw display
$97$ \( T^{2} - 20T + 92 \) Copy content Toggle raw display
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