Properties

Label 4032.2.p.i
Level $4032$
Weight $2$
Character orbit 4032.p
Analytic conductor $32.196$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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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: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{5} - \beta_{5} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{5} - \beta_{5} q^{7} - \beta_{3} q^{11} + 6 q^{13} - \beta_{6} q^{17} + 3 \beta_{2} q^{23} + 3 q^{25} + ( - \beta_{5} - \beta_1) q^{31} + (\beta_{3} + 2 \beta_{2}) q^{35} + \beta_{7} q^{37} - \beta_{6} q^{41} + ( - \beta_{5} - \beta_1) q^{43} + 2 \beta_{3} q^{47} + ( - \beta_{7} - 1) q^{49} + 2 \beta_{6} q^{53} + (4 \beta_{5} + 4 \beta_1) q^{55} - 2 \beta_{2} q^{59} + 6 q^{61} - 6 \beta_{4} q^{65} + ( - 3 \beta_{5} - 3 \beta_1) q^{67} - 3 \beta_{2} q^{71} - 2 \beta_{7} q^{73} + ( - 2 \beta_{6} + 3 \beta_{4}) q^{77} + ( - \beta_{5} + \beta_1) q^{79} + 2 \beta_{2} q^{83} + 2 \beta_{7} q^{85} + 3 \beta_{6} q^{89} - 6 \beta_{5} q^{91} + 2 \beta_{7} q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 48 q^{13} + 24 q^{25} - 8 q^{49} + 48 q^{61}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -4\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -3\zeta_{24}^{6} + 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( 4\zeta_{24}^{7} + 2\zeta_{24}^{5} - 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 8\zeta_{24}^{4} - 4 \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{6} + \beta_{4} + \beta_{3} + \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{5} + 3\beta_1 ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{4} - \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{7} + 4 ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{5} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} ) / 8 \) 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
0.965926 0.258819i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.258819 0.965926i
−0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 0.965926i
0.258819 + 0.965926i
0 0 0 −2.82843 0 −1.73205 2.00000i 0 0 0
1567.2 0 0 0 −2.82843 0 −1.73205 + 2.00000i 0 0 0
1567.3 0 0 0 −2.82843 0 1.73205 2.00000i 0 0 0
1567.4 0 0 0 −2.82843 0 1.73205 + 2.00000i 0 0 0
1567.5 0 0 0 2.82843 0 −1.73205 2.00000i 0 0 0
1567.6 0 0 0 2.82843 0 −1.73205 + 2.00000i 0 0 0
1567.7 0 0 0 2.82843 0 1.73205 2.00000i 0 0 0
1567.8 0 0 0 2.82843 0 1.73205 + 2.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1567.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner
56.e even 2 1 inner
56.h odd 2 1 inner
168.e odd 2 1 inner
168.i even 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.i yes 8
3.b odd 2 1 inner 4032.2.p.i yes 8
4.b odd 2 1 inner 4032.2.p.i yes 8
7.b odd 2 1 4032.2.p.e 8
8.b even 2 1 4032.2.p.e 8
8.d odd 2 1 4032.2.p.e 8
12.b even 2 1 inner 4032.2.p.i yes 8
21.c even 2 1 4032.2.p.e 8
24.f even 2 1 4032.2.p.e 8
24.h odd 2 1 4032.2.p.e 8
28.d even 2 1 4032.2.p.e 8
56.e even 2 1 inner 4032.2.p.i yes 8
56.h odd 2 1 inner 4032.2.p.i yes 8
84.h odd 2 1 4032.2.p.e 8
168.e odd 2 1 inner 4032.2.p.i yes 8
168.i even 2 1 inner 4032.2.p.i yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4032.2.p.e 8 7.b odd 2 1
4032.2.p.e 8 8.b even 2 1
4032.2.p.e 8 8.d odd 2 1
4032.2.p.e 8 21.c even 2 1
4032.2.p.e 8 24.f even 2 1
4032.2.p.e 8 24.h odd 2 1
4032.2.p.e 8 28.d even 2 1
4032.2.p.e 8 84.h odd 2 1
4032.2.p.i yes 8 1.a even 1 1 trivial
4032.2.p.i yes 8 3.b odd 2 1 inner
4032.2.p.i yes 8 4.b odd 2 1 inner
4032.2.p.i yes 8 12.b even 2 1 inner
4032.2.p.i yes 8 56.e even 2 1 inner
4032.2.p.i yes 8 56.h odd 2 1 inner
4032.2.p.i yes 8 168.e odd 2 1 inner
4032.2.p.i yes 8 168.i even 2 1 inner

Hecke kernels

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

\( T_{5}^{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} - 24 \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{2} + 49)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 24)^{4} \) Copy content Toggle raw display
$13$ \( (T - 6)^{8} \) Copy content Toggle raw display
$17$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 24)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 96)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 96)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$61$ \( (T - 6)^{8} \) Copy content Toggle raw display
$67$ \( (T^{2} - 108)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 192)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 16)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 32)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 216)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 192)^{4} \) Copy content Toggle raw display
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