Properties

Label 4732.2.a.q
Level $4732$
Weight $2$
Character orbit 4732.a
Self dual yes
Analytic conductor $37.785$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.2854789.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 6x^{4} + 17x^{3} + 11x^{2} - 20x - 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} - \beta_{3} - \beta_{2}) q^{3} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{5} - q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 3) q^{9} + ( - \beta_{5} + \beta_{3} + \cdots + \beta_1) q^{11}+ \cdots + ( - \beta_{5} - \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} - 6 q^{7} + 20 q^{9} - 4 q^{11} - 3 q^{15} - 3 q^{17} - 5 q^{19} - 6 q^{21} + 18 q^{23} + 2 q^{25} + 24 q^{27} + 17 q^{29} + 17 q^{31} - 21 q^{33} - q^{37} - 14 q^{41} + 14 q^{43} + 29 q^{45}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 6x^{4} + 17x^{3} + 11x^{2} - 20x - 13 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{4} - 2\nu^{3} - 6\nu^{2} + 7\nu + 7 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 2\nu^{3} - 7\nu^{2} + 8\nu + 10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 8\nu^{2} + 11\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{5} - 4\nu^{4} - 13\nu^{3} + 15\nu^{2} + 18\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} - 4\beta_{4} - 8\beta_{3} + 9\beta_{2} + 9\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 11\beta_{5} - 21\beta_{4} - 15\beta_{3} + 17\beta_{2} + 34\beta _1 + 31 \) 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
−0.748190
3.00976
−0.875901
−2.00976
1.74819
1.87590
0 −2.48311 0 1.84698 0 −1.00000 0 3.16581 0
1.2 0 −1.85869 0 −0.131645 0 −1.00000 0 0.454712 0
1.3 0 1.76089 0 −2.41317 0 −1.00000 0 0.100737 0
1.4 0 2.16666 0 −2.91727 0 −1.00000 0 1.69443 0
1.5 0 3.12621 0 −0.154960 0 −1.00000 0 6.77319 0
1.6 0 3.28803 0 3.77007 0 −1.00000 0 7.81112 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.q 6
13.b even 2 1 4732.2.a.r yes 6
13.d odd 4 2 4732.2.g.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4732.2.a.q 6 1.a even 1 1 trivial
4732.2.a.r yes 6 13.b even 2 1
4732.2.g.j 12 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4732))\):

\( T_{3}^{6} - 6T_{3}^{5} - T_{3}^{4} + 58T_{3}^{3} - 61T_{3}^{2} - 129T_{3} + 181 \) Copy content Toggle raw display
\( T_{5}^{6} - 16T_{5}^{4} - 7T_{5}^{3} + 48T_{5}^{2} + 14T_{5} + 1 \) Copy content Toggle raw display
\( T_{11}^{6} + 4T_{11}^{5} - 14T_{11}^{4} - 9T_{11}^{3} + 14T_{11}^{2} + 4T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 6 T^{5} + \cdots + 181 \) Copy content Toggle raw display
$5$ \( T^{6} - 16 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( (T + 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 4 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 3 T^{5} + \cdots + 727 \) Copy content Toggle raw display
$19$ \( T^{6} + 5 T^{5} + \cdots + 13 \) Copy content Toggle raw display
$23$ \( T^{6} - 18 T^{5} + \cdots - 97 \) Copy content Toggle raw display
$29$ \( T^{6} - 17 T^{5} + \cdots + 29 \) Copy content Toggle raw display
$31$ \( T^{6} - 17 T^{5} + \cdots + 937 \) Copy content Toggle raw display
$37$ \( T^{6} + T^{5} + \cdots + 281 \) Copy content Toggle raw display
$41$ \( T^{6} + 14 T^{5} + \cdots + 36541 \) Copy content Toggle raw display
$43$ \( T^{6} - 14 T^{5} + \cdots + 4067 \) Copy content Toggle raw display
$47$ \( T^{6} - 6 T^{5} + \cdots - 9143 \) Copy content Toggle raw display
$53$ \( T^{6} - 3 T^{5} + \cdots + 3653 \) Copy content Toggle raw display
$59$ \( T^{6} - 14 T^{5} + \cdots - 103249 \) Copy content Toggle raw display
$61$ \( T^{6} + 4 T^{5} + \cdots + 3263 \) Copy content Toggle raw display
$67$ \( T^{6} + 4 T^{5} + \cdots - 7267 \) Copy content Toggle raw display
$71$ \( T^{6} + 3 T^{5} + \cdots + 21629 \) Copy content Toggle raw display
$73$ \( T^{6} - 31 T^{5} + \cdots - 74593 \) Copy content Toggle raw display
$79$ \( T^{6} - 22 T^{5} + \cdots - 22919 \) Copy content Toggle raw display
$83$ \( T^{6} - 15 T^{5} + \cdots - 607711 \) Copy content Toggle raw display
$89$ \( T^{6} - 20 T^{5} + \cdots - 119671 \) Copy content Toggle raw display
$97$ \( T^{6} - 29 T^{5} + \cdots + 1424683 \) Copy content Toggle raw display
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