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Properties

Label 6336.2.m.a
Level $6336$
Weight $2$
Character orbit 6336.m
Analytic conductor $50.593$
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] = [6336,2,Mod(5345,6336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6336.5345");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6336 = 2^{6} \cdot 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6336.m (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: \(50.5932147207\)
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: \(\Z[a_1, \ldots, a_{67}]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{6} + \beta_{2}) q^{11} + 3 \beta_{6} q^{13} - 2 \beta_{4} q^{17} + \beta_{5} q^{19} - \beta_{7} q^{23} - 5 q^{25} - 3 \beta_1 q^{29} + 2 \beta_{4} q^{31} + 4 \beta_{2} q^{37} + 2 \beta_{4} q^{41} + \beta_{5} q^{43} + \beta_{7} q^{47} + 7 q^{49} - 4 \beta_{5} q^{53} + 8 \beta_{6} q^{59} - 9 \beta_{6} q^{61} - 4 \beta_1 q^{67} + 5 \beta_{7} q^{71} + 4 \beta_{7} q^{73} - 12 \beta_{3} q^{79} - 6 \beta_{2} q^{83} - 7 \beta_{3} q^{89} + 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{25} + 56 q^{49} + 64 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6336\mathbb{Z}\right)^\times\).

\(n\) \(1729\) \(3521\) \(4159\) \(4357\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5345.1
0.965926 − 0.258819i
−0.258819 − 0.965926i
0.965926 + 0.258819i
−0.258819 + 0.965926i
−0.965926 + 0.258819i
0.258819 + 0.965926i
−0.965926 − 0.258819i
0.258819 − 0.965926i
0 0 0 0 0 0 0 0 0
5345.2 0 0 0 0 0 0 0 0 0
5345.3 0 0 0 0 0 0 0 0 0
5345.4 0 0 0 0 0 0 0 0 0
5345.5 0 0 0 0 0 0 0 0 0
5345.6 0 0 0 0 0 0 0 0 0
5345.7 0 0 0 0 0 0 0 0 0
5345.8 0 0 0 0 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5345.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
33.d even 2 1 inner
44.c even 2 1 inner
88.g even 2 1 inner
264.m even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6336.2.m.a ✓ 8
3.b odd 2 1 6336.2.m.b yes 8
4.b odd 2 1 6336.2.m.b yes 8
8.b even 2 1 inner 6336.2.m.a ✓ 8
8.d odd 2 1 6336.2.m.b yes 8
11.b odd 2 1 6336.2.m.b yes 8
12.b even 2 1 inner 6336.2.m.a ✓ 8
24.f even 2 1 inner 6336.2.m.a ✓ 8
24.h odd 2 1 6336.2.m.b yes 8
33.d even 2 1 inner 6336.2.m.a ✓ 8
44.c even 2 1 inner 6336.2.m.a ✓ 8
88.b odd 2 1 6336.2.m.b yes 8
88.g even 2 1 inner 6336.2.m.a ✓ 8
132.d odd 2 1 6336.2.m.b yes 8
264.m even 2 1 inner 6336.2.m.a ✓ 8
264.p odd 2 1 6336.2.m.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6336.2.m.a ✓ 8 1.a even 1 1 trivial
6336.2.m.a ✓ 8 8.b even 2 1 inner
6336.2.m.a ✓ 8 12.b even 2 1 inner
6336.2.m.a ✓ 8 24.f even 2 1 inner
6336.2.m.a ✓ 8 33.d even 2 1 inner
6336.2.m.a ✓ 8 44.c even 2 1 inner
6336.2.m.a ✓ 8 88.g even 2 1 inner
6336.2.m.a ✓ 8 264.m even 2 1 inner
6336.2.m.b yes 8 3.b odd 2 1
6336.2.m.b yes 8 4.b odd 2 1
6336.2.m.b yes 8 8.d odd 2 1
6336.2.m.b yes 8 11.b odd 2 1
6336.2.m.b yes 8 24.h odd 2 1
6336.2.m.b yes 8 88.b odd 2 1
6336.2.m.b yes 8 132.d odd 2 1
6336.2.m.b yes 8 264.p odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(6336, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{167} + 24 \) 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
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} - 10 T^{2} + 121)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 18)^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 48)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 48)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 6)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 6)^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} - 96)^{4} \) Copy content Toggle raw display
$59$ \( (T^{2} - 128)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} - 162)^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 64)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 150)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 96)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 288)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 108)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 98)^{4} \) Copy content Toggle raw display
$97$ \( (T - 8)^{8} \) Copy content Toggle raw display
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