Properties

Label 4732.2.a.k
Level $4732$
Weight $2$
Character orbit 4732.a
Self dual yes
Analytic conductor $37.785$
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] = [4732,2,Mod(1,4732)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4732, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4732.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4732.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: \(37.7852102365\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 364)
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 \(\beta = \frac{1}{2}(1 + \sqrt{13})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( - \beta + 2) q^{5} - q^{7} + \beta q^{9} + (\beta + 4) q^{11} + (\beta - 3) q^{15} + ( - 2 \beta + 3) q^{17} + ( - \beta + 6) q^{19} - \beta q^{21} + 2 q^{23} + ( - 3 \beta + 2) q^{25} + \cdots + (5 \beta + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{3} + 3 q^{5} - 2 q^{7} + q^{9} + 9 q^{11} - 5 q^{15} + 4 q^{17} + 11 q^{19} - q^{21} + 4 q^{23} + q^{25} + 4 q^{27} + 3 q^{29} - 8 q^{31} + 11 q^{33} - 3 q^{35} + 4 q^{37} - 10 q^{41} - q^{43}+ \cdots + 11 q^{99}+O(q^{100}) \) 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
−1.30278
2.30278
0 −1.30278 0 3.30278 0 −1.00000 0 −1.30278 0
1.2 0 2.30278 0 −0.302776 0 −1.00000 0 2.30278 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(7\) \( +1 \)
\(13\) \( +1 \)

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 4732.2.a.k 2
13.b even 2 1 4732.2.a.j 2
13.c even 3 2 364.2.k.c 4
13.d odd 4 2 4732.2.g.g 4
39.i odd 6 2 3276.2.z.d 4
52.j odd 6 2 1456.2.s.m 4
91.g even 3 2 2548.2.l.k 4
91.h even 3 2 2548.2.i.j 4
91.m odd 6 2 2548.2.l.i 4
91.n odd 6 2 2548.2.k.f 4
91.v odd 6 2 2548.2.i.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.k.c 4 13.c even 3 2
1456.2.s.m 4 52.j odd 6 2
2548.2.i.j 4 91.h even 3 2
2548.2.i.l 4 91.v odd 6 2
2548.2.k.f 4 91.n odd 6 2
2548.2.l.i 4 91.m odd 6 2
2548.2.l.k 4 91.g even 3 2
3276.2.z.d 4 39.i odd 6 2
4732.2.a.j 2 13.b even 2 1
4732.2.a.k 2 1.a even 1 1 trivial
4732.2.g.g 4 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4732))\):

\( T_{3}^{2} - T_{3} - 3 \) Copy content Toggle raw display
\( T_{5}^{2} - 3T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 9T_{11} + 17 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - T - 3 \) Copy content Toggle raw display
$5$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 9T + 17 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 4T - 9 \) Copy content Toggle raw display
$19$ \( T^{2} - 11T + 27 \) Copy content Toggle raw display
$23$ \( (T - 2)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 3T - 1 \) Copy content Toggle raw display
$31$ \( T^{2} + 8T + 3 \) Copy content Toggle raw display
$37$ \( T^{2} - 4T - 48 \) Copy content Toggle raw display
$41$ \( T^{2} + 10T + 12 \) Copy content Toggle raw display
$43$ \( T^{2} + T - 29 \) Copy content Toggle raw display
$47$ \( T^{2} - 16T + 51 \) Copy content Toggle raw display
$53$ \( T^{2} - 117 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 9 \) Copy content Toggle raw display
$61$ \( T^{2} - 52 \) Copy content Toggle raw display
$67$ \( (T - 13)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 2T - 116 \) Copy content Toggle raw display
$73$ \( T^{2} - 8T - 36 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 16T + 51 \) Copy content Toggle raw display
$89$ \( T^{2} - 21T + 81 \) Copy content Toggle raw display
$97$ \( T^{2} + 23T + 103 \) Copy content Toggle raw display
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