Properties

Label 4032.2.a.bq
Level $4032$
Weight $2$
Character orbit 4032.a
Self dual yes
Analytic conductor $32.196$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(1,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("4032.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 63)
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 \(\beta = 2\sqrt{3}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{5} - q^{7} + \beta q^{11} - 2 q^{13} + \beta q^{17} - 4 q^{19} + \beta q^{23} + 7 q^{25} + 4 q^{31} - \beta q^{35} - 2 q^{37} + 3 \beta q^{41} - 4 q^{43} - 2 \beta q^{47} + q^{49} + 2 \beta q^{53} + 12 q^{55} - 2 \beta q^{59} + 10 q^{61} - 2 \beta q^{65} - 4 q^{67} + 3 \beta q^{71} + 14 q^{73} - \beta q^{77} - 8 q^{79} + 12 q^{85} - \beta q^{89} + 2 q^{91} - 4 \beta q^{95} + 14 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{7} - 4 q^{13} - 8 q^{19} + 14 q^{25} + 8 q^{31} - 4 q^{37} - 8 q^{43} + 2 q^{49} + 24 q^{55} + 20 q^{61} - 8 q^{67} + 28 q^{73} - 16 q^{79} + 24 q^{85} + 4 q^{91} + 28 q^{97}+O(q^{100}) \) 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
−1.73205
1.73205
0 0 0 −3.46410 0 −1.00000 0 0 0
1.2 0 0 0 3.46410 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(7\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4032.2.a.bq 2
3.b odd 2 1 inner 4032.2.a.bq 2
4.b odd 2 1 4032.2.a.bt 2
8.b even 2 1 1008.2.a.n 2
8.d odd 2 1 63.2.a.b 2
12.b even 2 1 4032.2.a.bt 2
24.f even 2 1 63.2.a.b 2
24.h odd 2 1 1008.2.a.n 2
40.e odd 2 1 1575.2.a.q 2
40.k even 4 2 1575.2.d.i 4
56.e even 2 1 441.2.a.g 2
56.h odd 2 1 7056.2.a.cm 2
56.k odd 6 2 441.2.e.j 4
56.m even 6 2 441.2.e.i 4
72.l even 6 2 567.2.f.j 4
72.p odd 6 2 567.2.f.j 4
88.g even 2 1 7623.2.a.bi 2
120.m even 2 1 1575.2.a.q 2
120.q odd 4 2 1575.2.d.i 4
168.e odd 2 1 441.2.a.g 2
168.i even 2 1 7056.2.a.cm 2
168.v even 6 2 441.2.e.j 4
168.be odd 6 2 441.2.e.i 4
264.p odd 2 1 7623.2.a.bi 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.a.b 2 8.d odd 2 1
63.2.a.b 2 24.f even 2 1
441.2.a.g 2 56.e even 2 1
441.2.a.g 2 168.e odd 2 1
441.2.e.i 4 56.m even 6 2
441.2.e.i 4 168.be odd 6 2
441.2.e.j 4 56.k odd 6 2
441.2.e.j 4 168.v even 6 2
567.2.f.j 4 72.l even 6 2
567.2.f.j 4 72.p odd 6 2
1008.2.a.n 2 8.b even 2 1
1008.2.a.n 2 24.h odd 2 1
1575.2.a.q 2 40.e odd 2 1
1575.2.a.q 2 120.m even 2 1
1575.2.d.i 4 40.k even 4 2
1575.2.d.i 4 120.q odd 4 2
4032.2.a.bq 2 1.a even 1 1 trivial
4032.2.a.bq 2 3.b odd 2 1 inner
4032.2.a.bt 2 4.b odd 2 1
4032.2.a.bt 2 12.b even 2 1
7056.2.a.cm 2 56.h odd 2 1
7056.2.a.cm 2 168.i even 2 1
7623.2.a.bi 2 88.g even 2 1
7623.2.a.bi 2 264.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4032))\):

\( T_{5}^{2} - 12 \) Copy content Toggle raw display
\( T_{11}^{2} - 12 \) Copy content Toggle raw display
\( T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 12 \) Copy content Toggle raw display
\( T_{19} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 12 \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 12 \) Copy content Toggle raw display
$13$ \( (T + 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 12 \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 12 \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 4)^{2} \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 108 \) Copy content Toggle raw display
$43$ \( (T + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 48 \) Copy content Toggle raw display
$53$ \( T^{2} - 48 \) Copy content Toggle raw display
$59$ \( T^{2} - 48 \) Copy content Toggle raw display
$61$ \( (T - 10)^{2} \) Copy content Toggle raw display
$67$ \( (T + 4)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 108 \) Copy content Toggle raw display
$73$ \( (T - 14)^{2} \) Copy content Toggle raw display
$79$ \( (T + 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 12 \) Copy content Toggle raw display
$97$ \( (T - 14)^{2} \) Copy content Toggle raw display
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