Properties

Label 4732.2.g.f
Level $4732$
Weight $2$
Character orbit 4732.g
Analytic conductor $37.785$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4732,2,Mod(337,4732)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4732.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4732.g (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(37.7852102365\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + (\beta_{2} - \beta_1) q^{5} + \beta_1 q^{7} + 3 q^{9} + (\beta_{2} - 4 \beta_1) q^{11} + ( - \beta_{2} + 6 \beta_1) q^{15} + \beta_{3} q^{17} + (\beta_{2} + 3 \beta_1) q^{19} + \beta_{2} q^{21}+ \cdots + (3 \beta_{2} - 12 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 12 q^{9} + 4 q^{23} - 8 q^{25} - 4 q^{29} + 4 q^{35} - 4 q^{43} - 4 q^{49} + 24 q^{51} - 12 q^{53} - 40 q^{55} - 8 q^{61} + 48 q^{69} + 48 q^{75} + 16 q^{77} - 28 q^{79} - 36 q^{81} + 48 q^{87} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{3} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{3} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4732\mathbb{Z}\right)^\times\).

\(n\) \(2367\) \(2705\) \(4565\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
−1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
1.22474 + 1.22474i
0 −2.44949 0 3.44949i 0 1.00000i 0 3.00000 0
337.2 0 −2.44949 0 3.44949i 0 1.00000i 0 3.00000 0
337.3 0 2.44949 0 1.44949i 0 1.00000i 0 3.00000 0
337.4 0 2.44949 0 1.44949i 0 1.00000i 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4732.2.g.f 4
13.b even 2 1 inner 4732.2.g.f 4
13.d odd 4 1 364.2.a.c 2
13.d odd 4 1 4732.2.a.i 2
39.f even 4 1 3276.2.a.q 2
52.f even 4 1 1456.2.a.p 2
65.g odd 4 1 9100.2.a.v 2
91.i even 4 1 2548.2.a.m 2
91.z odd 12 2 2548.2.j.m 4
91.bb even 12 2 2548.2.j.l 4
104.j odd 4 1 5824.2.a.bm 2
104.m even 4 1 5824.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.a.c 2 13.d odd 4 1
1456.2.a.p 2 52.f even 4 1
2548.2.a.m 2 91.i even 4 1
2548.2.j.l 4 91.bb even 12 2
2548.2.j.m 4 91.z odd 12 2
3276.2.a.q 2 39.f even 4 1
4732.2.a.i 2 13.d odd 4 1
4732.2.g.f 4 1.a even 1 1 trivial
4732.2.g.f 4 13.b even 2 1 inner
5824.2.a.bm 2 104.j odd 4 1
5824.2.a.bn 2 104.m even 4 1
9100.2.a.v 2 65.g odd 4 1

Hecke kernels

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

\( T_{3}^{2} - 6 \) Copy content Toggle raw display
\( T_{5}^{4} + 14T_{5}^{2} + 25 \) Copy content Toggle raw display
\( T_{17}^{2} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 44T^{2} + 100 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 6)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 30T^{2} + 9 \) Copy content Toggle raw display
$23$ \( (T^{2} - 2 T - 23)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 2 T - 23)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$37$ \( T^{4} + 84T^{2} + 900 \) Copy content Toggle raw display
$41$ \( T^{4} + 120T^{2} + 144 \) Copy content Toggle raw display
$43$ \( (T + 1)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} + 14T^{2} + 25 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 15)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$61$ \( (T + 2)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 56T^{2} + 400 \) Copy content Toggle raw display
$71$ \( T^{4} + 212T^{2} + 8836 \) Copy content Toggle raw display
$73$ \( T^{4} + 302 T^{2} + 22201 \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T + 25)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 126T^{2} + 2025 \) Copy content Toggle raw display
$89$ \( T^{4} + 174T^{2} + 5625 \) Copy content Toggle raw display
$97$ \( T^{4} + 350 T^{2} + 26569 \) Copy content Toggle raw display
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