Properties

Label 4732.2.a.i
Level $4732$
Weight $2$
Character orbit 4732.a
Self dual yes
Analytic conductor $37.785$
Analytic rank $1$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) 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 \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{3} + ( - \beta + 1) q^{5} + q^{7} + 3 q^{9} + (\beta - 4) q^{11} + (\beta - 6) q^{15} - \beta q^{17} + ( - \beta - 3) q^{19} + \beta q^{21} + ( - 2 \beta - 1) q^{23} + ( - 2 \beta + 2) q^{25} + \cdots + (3 \beta - 12) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{5} + 2 q^{7} + 6 q^{9} - 8 q^{11} - 12 q^{15} - 6 q^{19} - 2 q^{23} + 4 q^{25} - 2 q^{29} - 2 q^{31} + 12 q^{33} + 2 q^{35} + 12 q^{37} - 12 q^{41} + 2 q^{43} + 6 q^{45} - 2 q^{47} + 2 q^{49}+ \cdots - 24 q^{99}+O(q^{100}) \) 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
−2.44949
2.44949
0 −2.44949 0 3.44949 0 1.00000 0 3.00000 0
1.2 0 2.44949 0 −1.44949 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.i 2
13.b even 2 1 364.2.a.c 2
13.d odd 4 2 4732.2.g.f 4
39.d odd 2 1 3276.2.a.q 2
52.b odd 2 1 1456.2.a.p 2
65.d even 2 1 9100.2.a.v 2
91.b odd 2 1 2548.2.a.m 2
91.r even 6 2 2548.2.j.m 4
91.s odd 6 2 2548.2.j.l 4
104.e even 2 1 5824.2.a.bm 2
104.h odd 2 1 5824.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.a.c 2 13.b even 2 1
1456.2.a.p 2 52.b odd 2 1
2548.2.a.m 2 91.b odd 2 1
2548.2.j.l 4 91.s odd 6 2
2548.2.j.m 4 91.r even 6 2
3276.2.a.q 2 39.d odd 2 1
4732.2.a.i 2 1.a even 1 1 trivial
4732.2.g.f 4 13.d odd 4 2
5824.2.a.bm 2 104.e even 2 1
5824.2.a.bn 2 104.h odd 2 1
9100.2.a.v 2 65.d even 2 1

Hecke kernels

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

\( T_{3}^{2} - 6 \) Copy content Toggle raw display
\( T_{5}^{2} - 2T_{5} - 5 \) Copy content Toggle raw display
\( T_{11}^{2} + 8T_{11} + 10 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 6 \) Copy content Toggle raw display
$5$ \( T^{2} - 2T - 5 \) Copy content Toggle raw display
$7$ \( (T - 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 8T + 10 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 6 \) Copy content Toggle raw display
$19$ \( T^{2} + 6T + 3 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 23 \) Copy content Toggle raw display
$29$ \( T^{2} + 2T - 23 \) Copy content Toggle raw display
$31$ \( T^{2} + 2T - 5 \) Copy content Toggle raw display
$37$ \( T^{2} - 12T + 30 \) Copy content Toggle raw display
$41$ \( T^{2} + 12T + 12 \) Copy content Toggle raw display
$43$ \( (T - 1)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 2T - 5 \) Copy content Toggle raw display
$53$ \( T^{2} + 6T - 15 \) Copy content Toggle raw display
$59$ \( (T + 14)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 20 \) Copy content Toggle raw display
$71$ \( T^{2} + 20T + 94 \) Copy content Toggle raw display
$73$ \( T^{2} - 2T - 149 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 25 \) Copy content Toggle raw display
$83$ \( T^{2} + 6T - 45 \) Copy content Toggle raw display
$89$ \( T^{2} - 18T + 75 \) Copy content Toggle raw display
$97$ \( T^{2} - 26T + 163 \) Copy content Toggle raw display
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