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Properties

Label 1296.2.s.k
Level $1296$
Weight $2$
Character orbit 1296.s
Analytic conductor $10.349$
Analytic rank $0$
Dimension $8$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1296,2,Mod(431,1296)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1296, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 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("1296.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1296.s (of order \(6\), degree \(2\), 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: \(10.3486121020\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{5}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{5} + ( - \beta_{2} + 2 \beta_1) q^{13} + ( - \beta_{7} - \beta_{6} - \beta_{5}) q^{17} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{25} + ( - \beta_{6} - 3 \beta_{5} + 3 \beta_{4}) q^{29} + (2 \beta_{3} - 1) q^{37} + ( - \beta_{7} + 4 \beta_{4}) q^{41} - 7 \beta_1 q^{49} + ( - \beta_{7} - \beta_{6} + 4 \beta_{5}) q^{53} + ( - 2 \beta_{3} + 2 \beta_{2} + \cdots - 5) q^{61}+ \cdots + ( - 8 \beta_1 + 8) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{13} + 4 q^{25} - 8 q^{37} - 28 q^{49} - 20 q^{61} - 64 q^{73} + 12 q^{85} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

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

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

\(\zeta_{24}\)\(=\) \( ( \beta_{6} + 2\beta_{5} + \beta_{4} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{6} - \beta_{5} - 2\beta_{4} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + 3\beta_{4} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{3} + 2\beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{5} - 3\beta_{4} ) / 9 \) 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 \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
−0.258819 + 0.965926i
−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 0 0 −2.89778 + 1.67303i 0 0 0 0 0
431.2 0 0 0 −0.776457 + 0.448288i 0 0 0 0 0
431.3 0 0 0 0.776457 − 0.448288i 0 0 0 0 0
431.4 0 0 0 2.89778 − 1.67303i 0 0 0 0 0
863.1 0 0 0 −2.89778 − 1.67303i 0 0 0 0 0
863.2 0 0 0 −0.776457 − 0.448288i 0 0 0 0 0
863.3 0 0 0 0.776457 + 0.448288i 0 0 0 0 0
863.4 0 0 0 2.89778 + 1.67303i 0 0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 431.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by \(\Q(\sqrt{-1}) \)
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
12.b even 2 1 inner
36.f odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1296.2.s.k 8
3.b odd 2 1 inner 1296.2.s.k 8
4.b odd 2 1 CM 1296.2.s.k 8
9.c even 3 1 1296.2.c.e ✓ 4
9.c even 3 1 inner 1296.2.s.k 8
9.d odd 6 1 1296.2.c.e ✓ 4
9.d odd 6 1 inner 1296.2.s.k 8
12.b even 2 1 inner 1296.2.s.k 8
36.f odd 6 1 1296.2.c.e ✓ 4
36.f odd 6 1 inner 1296.2.s.k 8
36.h even 6 1 1296.2.c.e ✓ 4
36.h even 6 1 inner 1296.2.s.k 8
72.j odd 6 1 5184.2.c.g 4
72.l even 6 1 5184.2.c.g 4
72.n even 6 1 5184.2.c.g 4
72.p odd 6 1 5184.2.c.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1296.2.c.e ✓ 4 9.c even 3 1
1296.2.c.e ✓ 4 9.d odd 6 1
1296.2.c.e ✓ 4 36.f odd 6 1
1296.2.c.e ✓ 4 36.h even 6 1
1296.2.s.k 8 1.a even 1 1 trivial
1296.2.s.k 8 3.b odd 2 1 inner
1296.2.s.k 8 4.b odd 2 1 CM
1296.2.s.k 8 9.c even 3 1 inner
1296.2.s.k 8 9.d odd 6 1 inner
1296.2.s.k 8 12.b even 2 1 inner
1296.2.s.k 8 36.f odd 6 1 inner
1296.2.s.k 8 36.h even 6 1 inner
5184.2.c.g 4 72.j odd 6 1
5184.2.c.g 4 72.l even 6 1
5184.2.c.g 4 72.n even 6 1
5184.2.c.g 4 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1296, [\chi])\):

\( T_{5}^{8} - 12T_{5}^{6} + 135T_{5}^{4} - 108T_{5}^{2} + 81 \) Copy content Toggle raw display
\( T_{7} \) 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
$5$ \( T^{8} - 12 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + \cdots + 529)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 84 T^{2} + 1089)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} - 156 T^{6} + \cdots + 22667121 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} + 2 T - 107)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 162 T^{2} + 26244)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{2} + 162)^{4} \) Copy content Toggle raw display
$59$ \( T^{8} \) Copy content Toggle raw display
$61$ \( (T^{4} + 10 T^{3} + \cdots + 6889)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( T^{8} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16 T + 37)^{4} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{4} + 516 T^{2} + 62001)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 8 T + 64)^{4} \) Copy content Toggle raw display
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