Properties

Label 5040.2.a.by
Level $5040$
Weight $2$
Character orbit 5040.a
Self dual yes
Analytic conductor $40.245$
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] = [5040,2,Mod(1,5040)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5040, 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("5040.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5040 = 2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5040.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: \(40.2446026187\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 280)
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 \(\beta = \frac{1}{2}(1 + \sqrt{33})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{5} + q^{7} + ( - \beta + 4) q^{11} + ( - \beta + 2) q^{13} + ( - \beta - 2) q^{17} - 2 \beta q^{19} + 2 \beta q^{23} + q^{25} + ( - \beta + 2) q^{29} + 8 q^{31} + q^{35} - 2 q^{37} + (2 \beta - 2) q^{41} + ( - 2 \beta + 4) q^{43} + 3 \beta q^{47} + q^{49} + (2 \beta - 6) q^{53} + ( - \beta + 4) q^{55} + 8 q^{59} + (2 \beta + 2) q^{61} + ( - \beta + 2) q^{65} + 4 q^{67} + 8 q^{71} - 6 q^{73} + ( - \beta + 4) q^{77} + (3 \beta - 8) q^{79} + 4 \beta q^{83} + ( - \beta - 2) q^{85} + (2 \beta - 10) q^{89} + ( - \beta + 2) q^{91} - 2 \beta q^{95} + (5 \beta + 2) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{5} + 2 q^{7} + 7 q^{11} + 3 q^{13} - 5 q^{17} - 2 q^{19} + 2 q^{23} + 2 q^{25} + 3 q^{29} + 16 q^{31} + 2 q^{35} - 4 q^{37} - 2 q^{41} + 6 q^{43} + 3 q^{47} + 2 q^{49} - 10 q^{53} + 7 q^{55} + 16 q^{59} + 6 q^{61} + 3 q^{65} + 8 q^{67} + 16 q^{71} - 12 q^{73} + 7 q^{77} - 13 q^{79} + 4 q^{83} - 5 q^{85} - 18 q^{89} + 3 q^{91} - 2 q^{95} + 9 q^{97}+O(q^{100}) \) 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
3.37228
−2.37228
0 0 0 1.00000 0 1.00000 0 0 0
1.2 0 0 0 1.00000 0 1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 5040.2.a.by 2
3.b odd 2 1 560.2.a.h 2
4.b odd 2 1 2520.2.a.x 2
12.b even 2 1 280.2.a.c 2
15.d odd 2 1 2800.2.a.bk 2
15.e even 4 2 2800.2.g.r 4
21.c even 2 1 3920.2.a.bt 2
24.f even 2 1 2240.2.a.bk 2
24.h odd 2 1 2240.2.a.bg 2
60.h even 2 1 1400.2.a.r 2
60.l odd 4 2 1400.2.g.i 4
84.h odd 2 1 1960.2.a.s 2
84.j odd 6 2 1960.2.q.r 4
84.n even 6 2 1960.2.q.t 4
420.o odd 2 1 9800.2.a.bu 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.a.c 2 12.b even 2 1
560.2.a.h 2 3.b odd 2 1
1400.2.a.r 2 60.h even 2 1
1400.2.g.i 4 60.l odd 4 2
1960.2.a.s 2 84.h odd 2 1
1960.2.q.r 4 84.j odd 6 2
1960.2.q.t 4 84.n even 6 2
2240.2.a.bg 2 24.h odd 2 1
2240.2.a.bk 2 24.f even 2 1
2520.2.a.x 2 4.b odd 2 1
2800.2.a.bk 2 15.d odd 2 1
2800.2.g.r 4 15.e even 4 2
3920.2.a.bt 2 21.c even 2 1
5040.2.a.by 2 1.a even 1 1 trivial
9800.2.a.bu 2 420.o odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5040))\):

\( T_{11}^{2} - 7T_{11} + 4 \) Copy content Toggle raw display
\( T_{13}^{2} - 3T_{13} - 6 \) Copy content Toggle raw display
\( T_{17}^{2} + 5T_{17} - 2 \) Copy content Toggle raw display
\( T_{19}^{2} + 2T_{19} - 32 \) Copy content Toggle raw display
\( T_{29}^{2} - 3T_{29} - 6 \) 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
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 7T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$17$ \( T^{2} + 5T - 2 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 32 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 6 \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 2T - 32 \) Copy content Toggle raw display
$43$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T - 72 \) Copy content Toggle raw display
$53$ \( T^{2} + 10T - 8 \) Copy content Toggle raw display
$59$ \( (T - 8)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 6T - 24 \) Copy content Toggle raw display
$67$ \( (T - 4)^{2} \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( (T + 6)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 13T - 32 \) Copy content Toggle raw display
$83$ \( T^{2} - 4T - 128 \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 48 \) Copy content Toggle raw display
$97$ \( T^{2} - 9T - 186 \) Copy content Toggle raw display
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