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Properties

Label 4032.2.b.p
Level $4032$
Weight $2$
Character orbit 4032.b
Analytic conductor $32.196$
Analytic rank $0$
Dimension $8$
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(3583,4032)]
 
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, 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("4032.3583");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.b (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: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 224)
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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} + \beta_{4} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{5} + \beta_{4} q^{7} + \beta_1 q^{11} + ( - \beta_{5} - \beta_{3}) q^{13} + (2 \beta_{5} - \beta_{3}) q^{17} + (\beta_{6} - \beta_{4} - \beta_{2}) q^{19} + (2 \beta_{6} + 2 \beta_{4} - 3 \beta_1) q^{23} + (\beta_{7} + 1) q^{25} + ( - 2 \beta_{7} - 2) q^{29} + (\beta_{6} - \beta_{4} - 2 \beta_{2}) q^{31} + (\beta_{6} + \beta_{4} + \cdots - 3 \beta_1) q^{35}+ \cdots + (6 \beta_{5} - \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{25} - 16 q^{29} + 16 q^{37} + 8 q^{49} - 16 q^{53} + 32 q^{65} - 16 q^{77} - 64 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{16}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{16}^{7} - \zeta_{16}^{5} + \zeta_{16}^{3} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{16}^{7} + 2\zeta_{16}^{5} + 2\zeta_{16}^{3} + 2\zeta_{16} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\zeta_{16}^{7} + \zeta_{16}^{6} + \zeta_{16}^{5} + \zeta_{16}^{4} - \zeta_{16}^{3} + \zeta_{16}^{2} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \zeta_{16}^{7} - \zeta_{16}^{5} - \zeta_{16}^{3} + \zeta_{16} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{16}^{7} + \zeta_{16}^{6} - \zeta_{16}^{5} + \zeta_{16}^{4} + \zeta_{16}^{3} + \zeta_{16}^{2} - \zeta_{16} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{16}^{6} + 2\zeta_{16}^{2} \) Copy content Toggle raw display

Display \(\zeta_{16}^j\) in terms of \(\beta_i\)

\(\zeta_{16}\)\(=\) \( ( -\beta_{6} + 2\beta_{5} + \beta_{4} + \beta_{3} + 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{16}^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{4} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{16}^{3}\)\(=\) \( ( \beta_{6} - 2\beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{16}^{4}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{16}^{5}\)\(=\) \( ( -\beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{16}^{6}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{16}^{7}\)\(=\) \( ( \beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} ) / 8 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{16}^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3583.1
−0.382683 − 0.923880i
0.382683 − 0.923880i
−0.923880 + 0.382683i
0.923880 + 0.382683i
−0.923880 − 0.382683i
0.923880 − 0.382683i
−0.382683 + 0.923880i
0.382683 + 0.923880i
0 0 0 − 2.61313i 0 −2.61313 + 0.414214i 0 0 0
3583.2 0 0 0 − 2.61313i 0 2.61313 − 0.414214i 0 0 0
3583.3 0 0 0 − 1.08239i 0 −1.08239 − 2.41421i 0 0 0
3583.4 0 0 0 − 1.08239i 0 1.08239 + 2.41421i 0 0 0
3583.5 0 0 0 1.08239i 0 −1.08239 + 2.41421i 0 0 0
3583.6 0 0 0 1.08239i 0 1.08239 − 2.41421i 0 0 0
3583.7 0 0 0 2.61313i 0 −2.61313 − 0.414214i 0 0 0
3583.8 0 0 0 2.61313i 0 2.61313 + 0.414214i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3583.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d 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.b.p 8
3.b odd 2 1 448.2.f.d 8
4.b odd 2 1 inner 4032.2.b.p 8
7.b odd 2 1 inner 4032.2.b.p 8
8.b even 2 1 2016.2.b.b 8
8.d odd 2 1 2016.2.b.b 8
12.b even 2 1 448.2.f.d 8
21.c even 2 1 448.2.f.d 8
24.f even 2 1 224.2.f.a ✓ 8
24.h odd 2 1 224.2.f.a ✓ 8
28.d even 2 1 inner 4032.2.b.p 8
48.i odd 4 1 1792.2.e.f 8
48.i odd 4 1 1792.2.e.g 8
48.k even 4 1 1792.2.e.f 8
48.k even 4 1 1792.2.e.g 8
56.e even 2 1 2016.2.b.b 8
56.h odd 2 1 2016.2.b.b 8
84.h odd 2 1 448.2.f.d 8
168.e odd 2 1 224.2.f.a ✓ 8
168.i even 2 1 224.2.f.a ✓ 8
168.s odd 6 2 1568.2.p.a 16
168.v even 6 2 1568.2.p.a 16
168.ba even 6 2 1568.2.p.a 16
168.be odd 6 2 1568.2.p.a 16
336.v odd 4 1 1792.2.e.f 8
336.v odd 4 1 1792.2.e.g 8
336.y even 4 1 1792.2.e.f 8
336.y even 4 1 1792.2.e.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.f.a ✓ 8 24.f even 2 1
224.2.f.a ✓ 8 24.h odd 2 1
224.2.f.a ✓ 8 168.e odd 2 1
224.2.f.a ✓ 8 168.i even 2 1
448.2.f.d 8 3.b odd 2 1
448.2.f.d 8 12.b even 2 1
448.2.f.d 8 21.c even 2 1
448.2.f.d 8 84.h odd 2 1
1568.2.p.a 16 168.s odd 6 2
1568.2.p.a 16 168.v even 6 2
1568.2.p.a 16 168.ba even 6 2
1568.2.p.a 16 168.be odd 6 2
1792.2.e.f 8 48.i odd 4 1
1792.2.e.f 8 48.k even 4 1
1792.2.e.f 8 336.v odd 4 1
1792.2.e.f 8 336.y even 4 1
1792.2.e.g 8 48.i odd 4 1
1792.2.e.g 8 48.k even 4 1
1792.2.e.g 8 336.v odd 4 1
1792.2.e.g 8 336.y even 4 1
2016.2.b.b 8 8.b even 2 1
2016.2.b.b 8 8.d odd 2 1
2016.2.b.b 8 56.e even 2 1
2016.2.b.b 8 56.h odd 2 1
4032.2.b.p 8 1.a even 1 1 trivial
4032.2.b.p 8 4.b odd 2 1 inner
4032.2.b.p 8 7.b odd 2 1 inner
4032.2.b.p 8 28.d 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 \(S_{2}^{\mathrm{new}}(4032, [\chi])\):

\( T_{5}^{4} + 8T_{5}^{2} + 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 4 \) Copy content Toggle raw display
\( T_{19}^{4} - 40T_{19}^{2} + 392 \) 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
$5$ \( (T^{4} + 8 T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 40 T^{2} + 392)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 40 T^{2} + 392)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 72 T^{2} + 784)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 28)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 4 T - 28)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 160 T^{2} + 6272)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 72 T^{2} + 784)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 64 T^{2} + 512)^{2} \) Copy content Toggle raw display
$53$ \( (T + 2)^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 104 T^{2} + 2312)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 8 T^{2} + 8)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 264 T^{2} + 4624)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 88 T^{2} + 784)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 160 T^{2} + 128)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 216 T^{2} + 8464)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 136 T^{2} + 392)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 32 T^{2} + 128)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 320 T^{2} + 25088)^{2} \) Copy content Toggle raw display
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