Properties

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

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(575,4032)]
 
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, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.h (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: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2016)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{7} + 3 \beta_{3} q^{11} + 2 q^{13} + 2 \beta_{2} q^{17} + 4 \beta_1 q^{19} + \beta_{3} q^{23} + 5 q^{25} + 5 \beta_{2} q^{29} - 8 \beta_1 q^{31} - 8 q^{37} - 4 \beta_{2} q^{41} + 4 \beta_1 q^{43} - 2 \beta_{3} q^{47} - q^{49} + 5 \beta_{2} q^{53} - 2 \beta_{3} q^{59} + 2 q^{61} - 2 \beta_1 q^{67} - 9 \beta_{3} q^{71} + 14 q^{73} + 3 \beta_{2} q^{77} - 2 \beta_1 q^{79} + 4 \beta_{3} q^{83} + 8 \beta_{2} q^{89} + 2 \beta_1 q^{91} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{13} + 20 q^{25} - 32 q^{37} - 4 q^{49} + 8 q^{61} + 56 q^{73} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
575.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
0 0 0 0 0 1.00000i 0 0 0
575.2 0 0 0 0 0 1.00000i 0 0 0
575.3 0 0 0 0 0 1.00000i 0 0 0
575.4 0 0 0 0 0 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
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b 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.h.d 4
3.b odd 2 1 inner 4032.2.h.d 4
4.b odd 2 1 inner 4032.2.h.d 4
8.b even 2 1 2016.2.h.b 4
8.d odd 2 1 2016.2.h.b 4
12.b even 2 1 inner 4032.2.h.d 4
24.f even 2 1 2016.2.h.b 4
24.h odd 2 1 2016.2.h.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2016.2.h.b 4 8.b even 2 1
2016.2.h.b 4 8.d odd 2 1
2016.2.h.b 4 24.f even 2 1
2016.2.h.b 4 24.h odd 2 1
4032.2.h.d 4 1.a even 1 1 trivial
4032.2.h.d 4 3.b odd 2 1 inner
4032.2.h.d 4 4.b odd 2 1 inner
4032.2.h.d 4 12.b 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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{11}^{2} - 18 \) Copy content Toggle raw display
\( T_{13} - 2 \) 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
$7$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$13$ \( (T - 2)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 2)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 64)^{2} \) Copy content Toggle raw display
$37$ \( (T + 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 50)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 8)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 162)^{2} \) Copy content Toggle raw display
$73$ \( (T - 14)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} - 32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 128)^{2} \) Copy content Toggle raw display
$97$ \( (T - 14)^{4} \) Copy content Toggle raw display
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