Properties

Label 2940.2.a.f
Level $2940$
Weight $2$
Character orbit 2940.a
Self dual yes
Analytic conductor $23.476$
Analytic rank $0$
Dimension $1$
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] = [2940,2,Mod(1,2940)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2940, 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("2940.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2940.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: \(23.4760181943\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 420)
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)
 
\(f(q)\) \(=\) \( q - q^{3} + q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + q^{5} + q^{9} + 6 q^{11} + 4 q^{13} - q^{15} - 6 q^{17} - 2 q^{19} + q^{25} - q^{27} + 6 q^{29} + 10 q^{31} - 6 q^{33} + 2 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} + q^{45} + 6 q^{51} - 12 q^{53} + 6 q^{55} + 2 q^{57} - 14 q^{61} + 4 q^{65} - 4 q^{67} + 6 q^{71} + 4 q^{73} - q^{75} - 16 q^{79} + q^{81} + 12 q^{83} - 6 q^{85} - 6 q^{87} - 6 q^{89} - 10 q^{93} - 2 q^{95} + 16 q^{97} + 6 q^{99}+O(q^{100}) \) 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
0
0 −1.00000 0 1.00000 0 0 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(-1\)
\(7\) \(-1\)

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 2940.2.a.f 1
3.b odd 2 1 8820.2.a.b 1
7.b odd 2 1 420.2.a.c 1
7.c even 3 2 2940.2.q.i 2
7.d odd 6 2 2940.2.q.e 2
21.c even 2 1 1260.2.a.i 1
28.d even 2 1 1680.2.a.a 1
35.c odd 2 1 2100.2.a.d 1
35.f even 4 2 2100.2.k.j 2
56.e even 2 1 6720.2.a.ch 1
56.h odd 2 1 6720.2.a.x 1
84.h odd 2 1 5040.2.a.bc 1
105.g even 2 1 6300.2.a.a 1
105.k odd 4 2 6300.2.k.a 2
140.c even 2 1 8400.2.a.cj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.a.c 1 7.b odd 2 1
1260.2.a.i 1 21.c even 2 1
1680.2.a.a 1 28.d even 2 1
2100.2.a.d 1 35.c odd 2 1
2100.2.k.j 2 35.f even 4 2
2940.2.a.f 1 1.a even 1 1 trivial
2940.2.q.e 2 7.d odd 6 2
2940.2.q.i 2 7.c even 3 2
5040.2.a.bc 1 84.h odd 2 1
6300.2.a.a 1 105.g even 2 1
6300.2.k.a 2 105.k odd 4 2
6720.2.a.x 1 56.h odd 2 1
6720.2.a.ch 1 56.e even 2 1
8400.2.a.cj 1 140.c even 2 1
8820.2.a.b 1 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2940))\):

\( T_{11} - 6 \) Copy content Toggle raw display
\( T_{13} - 4 \) Copy content Toggle raw display
\( T_{17} + 6 \) Copy content Toggle raw display
\( T_{31} - 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T - 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 6 \) Copy content Toggle raw display
$13$ \( T - 4 \) Copy content Toggle raw display
$17$ \( T + 6 \) Copy content Toggle raw display
$19$ \( T + 2 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 6 \) Copy content Toggle raw display
$31$ \( T - 10 \) Copy content Toggle raw display
$37$ \( T - 2 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 4 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T + 12 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 14 \) Copy content Toggle raw display
$67$ \( T + 4 \) Copy content Toggle raw display
$71$ \( T - 6 \) Copy content Toggle raw display
$73$ \( T - 4 \) Copy content Toggle raw display
$79$ \( T + 16 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T + 6 \) Copy content Toggle raw display
$97$ \( T - 16 \) Copy content Toggle raw display
show more
show less