↑

Properties

Label 2940.2.a.r
Level $2940$
Weight $2$
Character orbit 2940.a
Self dual yes
Analytic conductor $23.476$
Analytic rank $1$
Dimension $2$
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: \(1\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 3\sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} - q^{5} + q^{9} + \beta q^{11} + ( - \beta - 1) q^{13} - q^{15} - \beta q^{17} - 7 q^{19} + \beta q^{23} + q^{25} + q^{27} + ( - \beta - 6) q^{29} + (2 \beta - 1) q^{31} + \beta q^{33} + \cdots + \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} - 2 q^{13} - 2 q^{15} - 14 q^{19} + 2 q^{25} + 2 q^{27} - 12 q^{29} - 2 q^{31} - 2 q^{37} - 2 q^{39} - 2 q^{43} - 2 q^{45} - 12 q^{47} - 14 q^{57} - 12 q^{59} - 8 q^{61}+ \cdots + 16 q^{97}+O(q^{100}) \) 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 1.00000 0 −1.00000 0 0 0 1.00000 0
1.2 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.r 2
3.b odd 2 1 8820.2.a.bk 2
7.b odd 2 1 2940.2.a.p 2
7.c even 3 2 420.2.q.d ✓ 4
7.d odd 6 2 2940.2.q.q 4
21.c even 2 1 8820.2.a.bf 2
21.h odd 6 2 1260.2.s.e 4
28.g odd 6 2 1680.2.bg.t 4
35.j even 6 2 2100.2.q.k 4
35.l odd 12 4 2100.2.bc.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.q.d ✓ 4 7.c even 3 2
1260.2.s.e 4 21.h odd 6 2
1680.2.bg.t 4 28.g odd 6 2
2100.2.q.k 4 35.j even 6 2
2100.2.bc.f 8 35.l odd 12 4
2940.2.a.p 2 7.b odd 2 1
2940.2.a.r 2 1.a even 1 1 trivial
2940.2.q.q 4 7.d odd 6 2
8820.2.a.bf 2 21.c even 2 1
8820.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(2940))\):

\( T_{11}^{2} - 18 \) Copy content Toggle raw display
\( T_{13}^{2} + 2T_{13} - 17 \) Copy content Toggle raw display
\( T_{17}^{2} - 18 \) Copy content Toggle raw display
\( T_{31}^{2} + 2T_{31} - 71 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 18 \) Copy content Toggle raw display
$13$ \( T^{2} + 2T - 17 \) Copy content Toggle raw display
$17$ \( T^{2} - 18 \) Copy content Toggle raw display
$19$ \( (T + 7)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 18 \) Copy content Toggle raw display
$29$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 71 \) Copy content Toggle raw display
$37$ \( T^{2} + 2T - 17 \) Copy content Toggle raw display
$41$ \( T^{2} - 18 \) Copy content Toggle raw display
$43$ \( T^{2} + 2T - 17 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 72 \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$61$ \( T^{2} + 8T - 56 \) Copy content Toggle raw display
$67$ \( T^{2} + 2T - 17 \) Copy content Toggle raw display
$71$ \( T^{2} - 162 \) Copy content Toggle raw display
$73$ \( T^{2} - 10T + 7 \) Copy content Toggle raw display
$79$ \( (T - 11)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$89$ \( T^{2} + 12T + 18 \) Copy content Toggle raw display
$97$ \( T^{2} - 16T - 8 \) Copy content Toggle raw display
show more
show less