Properties

Label 4032.2.a.bh
Level $4032$
Weight $2$
Character orbit 4032.a
Self dual yes
Analytic conductor $32.196$
Analytic rank $1$
Dimension $1$
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] = [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: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
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)
 
\(f(q)\) \(=\) \( q + 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{5} + q^{7} - 6 q^{13} + 2 q^{17} - 4 q^{19} + 4 q^{23} - q^{25} - 10 q^{29} - 8 q^{31} + 2 q^{35} - 6 q^{37} + 2 q^{41} + 4 q^{43} - 8 q^{47} + q^{49} - 10 q^{53} + 12 q^{59} + 2 q^{61} - 12 q^{65} - 12 q^{67} + 12 q^{71} - 14 q^{73} - 8 q^{79} + 12 q^{83} + 4 q^{85} + 2 q^{89} - 6 q^{91} - 8 q^{95} + 10 q^{97}+O(q^{100}) \) 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
0
0 0 0 2.00000 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

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 4032.2.a.bh 1
3.b odd 2 1 1344.2.a.m 1
4.b odd 2 1 4032.2.a.bc 1
8.b even 2 1 504.2.a.e 1
8.d odd 2 1 1008.2.a.b 1
12.b even 2 1 1344.2.a.b 1
21.c even 2 1 9408.2.a.be 1
24.f even 2 1 336.2.a.e 1
24.h odd 2 1 168.2.a.a 1
48.i odd 4 2 5376.2.c.d 2
48.k even 4 2 5376.2.c.bb 2
56.e even 2 1 7056.2.a.bq 1
56.h odd 2 1 3528.2.a.v 1
56.j odd 6 2 3528.2.s.g 2
56.p even 6 2 3528.2.s.w 2
84.h odd 2 1 9408.2.a.da 1
120.i odd 2 1 4200.2.a.t 1
120.m even 2 1 8400.2.a.y 1
120.w even 4 2 4200.2.t.j 2
168.e odd 2 1 2352.2.a.c 1
168.i even 2 1 1176.2.a.f 1
168.s odd 6 2 1176.2.q.f 2
168.v even 6 2 2352.2.q.d 2
168.ba even 6 2 1176.2.q.d 2
168.be odd 6 2 2352.2.q.w 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.2.a.a 1 24.h odd 2 1
336.2.a.e 1 24.f even 2 1
504.2.a.e 1 8.b even 2 1
1008.2.a.b 1 8.d odd 2 1
1176.2.a.f 1 168.i even 2 1
1176.2.q.d 2 168.ba even 6 2
1176.2.q.f 2 168.s odd 6 2
1344.2.a.b 1 12.b even 2 1
1344.2.a.m 1 3.b odd 2 1
2352.2.a.c 1 168.e odd 2 1
2352.2.q.d 2 168.v even 6 2
2352.2.q.w 2 168.be odd 6 2
3528.2.a.v 1 56.h odd 2 1
3528.2.s.g 2 56.j odd 6 2
3528.2.s.w 2 56.p even 6 2
4032.2.a.bc 1 4.b odd 2 1
4032.2.a.bh 1 1.a even 1 1 trivial
4200.2.a.t 1 120.i odd 2 1
4200.2.t.j 2 120.w even 4 2
5376.2.c.d 2 48.i odd 4 2
5376.2.c.bb 2 48.k even 4 2
7056.2.a.bq 1 56.e even 2 1
8400.2.a.y 1 120.m even 2 1
9408.2.a.be 1 21.c even 2 1
9408.2.a.da 1 84.h odd 2 1

Hecke kernels

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

\( T_{5} - 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{13} + 6 \) Copy content Toggle raw display
\( T_{17} - 2 \) Copy content Toggle raw display
\( T_{19} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 2 \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T + 6 \) Copy content Toggle raw display
$17$ \( T - 2 \) Copy content Toggle raw display
$19$ \( T + 4 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T + 10 \) Copy content Toggle raw display
$31$ \( T + 8 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T - 2 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T - 12 \) Copy content Toggle raw display
$61$ \( T - 2 \) Copy content Toggle raw display
$67$ \( T + 12 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T + 8 \) Copy content Toggle raw display
$83$ \( T - 12 \) Copy content Toggle raw display
$89$ \( T - 2 \) Copy content Toggle raw display
$97$ \( T - 10 \) Copy content Toggle raw display
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