Properties

Label 4032.2.p.a
Level $4032$
Weight $2$
Character orbit 4032.p
Analytic conductor $32.196$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(1567,4032)]
 
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([1, 1, 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("4032.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.p (of order \(2\), degree \(1\), 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: \(32.1956820950\)
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: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1344)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{7} + ( - 2 \beta_{2} - \beta_1) q^{11} - 4 q^{13} - \beta_{3} q^{17} + 2 \beta_1 q^{19} + 3 \beta_1 q^{23} - 5 q^{25} - \beta_{3} q^{29} + (2 \beta_{2} + \beta_1) q^{31} + 2 \beta_{3} q^{37} - \beta_{3} q^{41} + (4 \beta_{2} + 2 \beta_1) q^{43} + (4 \beta_{2} + 2 \beta_1) q^{47} + ( - \beta_{3} + 5) q^{49} - \beta_{3} q^{53} - 6 \beta_1 q^{59} + 4 q^{61} - 3 \beta_1 q^{71} + ( - \beta_{3} + 12) q^{77} + \beta_1 q^{79} + 6 \beta_1 q^{83} - \beta_{3} q^{89} + 4 \beta_{2} q^{91} - 2 \beta_{3} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{13} - 20 q^{25} + 20 q^{49} + 16 q^{61} + 48 q^{77}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( 2\nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} - \nu^{2} + 3\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2\nu^{3} + 6\nu ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + 2\beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{3} - 6\beta_{2} - 3\beta_1 ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
−1.22474 1.22474i
0 0 0 0 0 −2.44949 1.00000i 0 0 0
1567.2 0 0 0 0 0 −2.44949 + 1.00000i 0 0 0
1567.3 0 0 0 0 0 2.44949 1.00000i 0 0 0
1567.4 0 0 0 0 0 2.44949 + 1.00000i 0 0 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
4.b odd 2 1 inner
56.e even 2 1 inner
56.h 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.p.a 4
3.b odd 2 1 1344.2.p.a 4
4.b odd 2 1 inner 4032.2.p.a 4
7.b odd 2 1 4032.2.p.d 4
8.b even 2 1 4032.2.p.d 4
8.d odd 2 1 4032.2.p.d 4
12.b even 2 1 1344.2.p.a 4
21.c even 2 1 1344.2.p.b yes 4
24.f even 2 1 1344.2.p.b yes 4
24.h odd 2 1 1344.2.p.b yes 4
28.d even 2 1 4032.2.p.d 4
56.e even 2 1 inner 4032.2.p.a 4
56.h odd 2 1 inner 4032.2.p.a 4
84.h odd 2 1 1344.2.p.b yes 4
168.e odd 2 1 1344.2.p.a 4
168.i even 2 1 1344.2.p.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.2.p.a 4 3.b odd 2 1
1344.2.p.a 4 12.b even 2 1
1344.2.p.a 4 168.e odd 2 1
1344.2.p.a 4 168.i even 2 1
1344.2.p.b yes 4 21.c even 2 1
1344.2.p.b yes 4 24.f even 2 1
1344.2.p.b yes 4 24.h odd 2 1
1344.2.p.b yes 4 84.h odd 2 1
4032.2.p.a 4 1.a even 1 1 trivial
4032.2.p.a 4 4.b odd 2 1 inner
4032.2.p.a 4 56.e even 2 1 inner
4032.2.p.a 4 56.h odd 2 1 inner
4032.2.p.d 4 7.b odd 2 1
4032.2.p.d 4 8.b even 2 1
4032.2.p.d 4 8.d odd 2 1
4032.2.p.d 4 28.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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{2} - 24 \) Copy content Toggle raw display
\( T_{13} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 10T^{2} + 49 \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$13$ \( (T + 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 24)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 96)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T - 4)^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 96)^{2} \) Copy content Toggle raw display
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