Properties

Label 4732.2.a.n
Level $4732$
Weight $2$
Character orbit 4732.a
Self dual yes
Analytic conductor $37.785$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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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: \(3\)
Coefficient field: \(\Q(\zeta_{14})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{5} - q^{7} - 3 q^{9} + ( - 4 \beta_{2} + \beta_1 - 2) q^{11} - 2 \beta_{2} q^{17} + (2 \beta_{2} + 4) q^{19} + (\beta_{2} + 2 \beta_1 + 2) q^{23} + (4 \beta_{2} + 3) q^{25} + ( - 3 \beta_{2} - \beta_1 - 3) q^{29}+ \cdots + (12 \beta_{2} - 3 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{5} - 3 q^{7} - 9 q^{9} - q^{11} + 2 q^{17} + 10 q^{19} + 7 q^{23} + 5 q^{25} - 7 q^{29} - 6 q^{31} - 2 q^{35} - 5 q^{37} + 4 q^{41} - q^{43} - 6 q^{45} + 3 q^{49} + 19 q^{53} - 10 q^{55} - 14 q^{59}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of \(\nu = \zeta_{14} + \zeta_{14}^{-1}\):

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 2 \) 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
−1.24698
0.445042
1.80194
0 0 0 −2.49396 0 −1.00000 0 −3.00000 0
1.2 0 0 0 0.890084 0 −1.00000 0 −3.00000 0
1.3 0 0 0 3.60388 0 −1.00000 0 −3.00000 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.n yes 3
13.b even 2 1 4732.2.a.m 3
13.d odd 4 2 4732.2.g.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4732.2.a.m 3 13.b even 2 1
4732.2.a.n yes 3 1.a even 1 1 trivial
4732.2.g.i 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4732))\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{3} - 2T_{5}^{2} - 8T_{5} + 8 \) Copy content Toggle raw display
\( T_{11}^{3} + T_{11}^{2} - 30T_{11} - 43 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 2 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + T^{2} + \cdots - 43 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 2 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$19$ \( T^{3} - 10 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$23$ \( T^{3} - 7T^{2} + 7 \) Copy content Toggle raw display
$29$ \( T^{3} + 7 T^{2} + \cdots - 7 \) Copy content Toggle raw display
$31$ \( T^{3} + 6 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$37$ \( T^{3} + 5 T^{2} + \cdots - 13 \) Copy content Toggle raw display
$41$ \( T^{3} - 4 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$43$ \( T^{3} + T^{2} - 2T - 1 \) Copy content Toggle raw display
$47$ \( T^{3} - 84T - 56 \) Copy content Toggle raw display
$53$ \( T^{3} - 19 T^{2} + \cdots + 83 \) Copy content Toggle raw display
$59$ \( T^{3} + 14T^{2} - 392 \) Copy content Toggle raw display
$61$ \( T^{3} - 22 T^{2} + \cdots + 232 \) Copy content Toggle raw display
$67$ \( T^{3} + 5 T^{2} + \cdots - 41 \) Copy content Toggle raw display
$71$ \( T^{3} - 21 T^{2} + \cdots - 287 \) Copy content Toggle raw display
$73$ \( T^{3} - 28T + 56 \) Copy content Toggle raw display
$79$ \( T^{3} - 3 T^{2} + \cdots + 433 \) Copy content Toggle raw display
$83$ \( T^{3} + 2 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$89$ \( T^{3} + 32 T^{2} + \cdots + 1112 \) Copy content Toggle raw display
$97$ \( T^{3} - 18 T^{2} + \cdots + 904 \) Copy content Toggle raw display
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