Properties

Label 864.2.c.b
Level $864$
Weight $2$
Character orbit 864.c
Analytic conductor $6.899$
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] = [864,2,Mod(863,864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(864, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("864.863");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.c (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: \(6.89907473464\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{5} + ( - \beta_{3} - \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{5} + ( - \beta_{3} - \beta_1) q^{7} + \beta_{6} q^{11} + (\beta_{4} + 1) q^{13} + \beta_{7} q^{17} + \beta_{3} q^{19} + (2 \beta_{6} + \beta_{5}) q^{23} + ( - 2 \beta_{4} - 3) q^{25} + 2 \beta_{2} q^{29} + (2 \beta_{3} + 2 \beta_1) q^{31} + (\beta_{6} + 2 \beta_{5}) q^{35} + (\beta_{4} - 3) q^{37} + 2 \beta_{2} q^{41} + ( - 2 \beta_{3} + 2 \beta_1) q^{43} + (2 \beta_{6} - \beta_{5}) q^{47} - 2 \beta_{4} q^{49} + 2 \beta_{7} q^{53} + 2 \beta_1 q^{55} + (\beta_{6} - 2 \beta_{5}) q^{59} + (3 \beta_{4} + 1) q^{61} + ( - \beta_{7} + 4 \beta_{2}) q^{65} + (\beta_{3} - 4 \beta_1) q^{67} + ( - 2 \beta_{6} + 2 \beta_{5}) q^{71} + ( - 2 \beta_{4} + 3) q^{73} - \beta_{2} q^{77} + (3 \beta_{3} - 5 \beta_1) q^{79} + 2 \beta_{6} q^{83} + 2 \beta_{4} q^{85} + \beta_{7} q^{89} + ( - 5 \beta_{3} - 4 \beta_1) q^{91} - \beta_{5} q^{95} + 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{13} - 24 q^{25} - 24 q^{37} + 8 q^{61} + 24 q^{73} + 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{24}^{7} + 2\zeta_{24}^{5} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{24}^{4} - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\zeta_{24}^{6} + 4\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} - 2\zeta_{24}^{5} + 4\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24}^{5} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 2\zeta_{24}^{7} - 2\zeta_{24}^{5} - 4\zeta_{24}^{3} + 4\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} ) / 8 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( ( \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( \beta_{6} + \beta_{2} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{6} + \beta_{2} ) / 4 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/864\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(353\) \(703\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
863.1
−0.965926 0.258819i
0.965926 0.258819i
−0.258819 0.965926i
0.258819 0.965926i
0.258819 + 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0 0 0 3.86370i 0 3.73205i 0 0 0
863.2 0 0 0 3.86370i 0 3.73205i 0 0 0
863.3 0 0 0 1.03528i 0 0.267949i 0 0 0
863.4 0 0 0 1.03528i 0 0.267949i 0 0 0
863.5 0 0 0 1.03528i 0 0.267949i 0 0 0
863.6 0 0 0 1.03528i 0 0.267949i 0 0 0
863.7 0 0 0 3.86370i 0 3.73205i 0 0 0
863.8 0 0 0 3.86370i 0 3.73205i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.8
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.c.b 8
3.b odd 2 1 inner 864.2.c.b 8
4.b odd 2 1 inner 864.2.c.b 8
8.b even 2 1 1728.2.c.f 8
8.d odd 2 1 1728.2.c.f 8
9.c even 3 1 2592.2.s.c 8
9.c even 3 1 2592.2.s.g 8
9.d odd 6 1 2592.2.s.c 8
9.d odd 6 1 2592.2.s.g 8
12.b even 2 1 inner 864.2.c.b 8
24.f even 2 1 1728.2.c.f 8
24.h odd 2 1 1728.2.c.f 8
36.f odd 6 1 2592.2.s.c 8
36.f odd 6 1 2592.2.s.g 8
36.h even 6 1 2592.2.s.c 8
36.h even 6 1 2592.2.s.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.c.b 8 1.a even 1 1 trivial
864.2.c.b 8 3.b odd 2 1 inner
864.2.c.b 8 4.b odd 2 1 inner
864.2.c.b 8 12.b even 2 1 inner
1728.2.c.f 8 8.b even 2 1
1728.2.c.f 8 8.d odd 2 1
1728.2.c.f 8 24.f even 2 1
1728.2.c.f 8 24.h odd 2 1
2592.2.s.c 8 9.c even 3 1
2592.2.s.c 8 9.d odd 6 1
2592.2.s.c 8 36.f odd 6 1
2592.2.s.c 8 36.h even 6 1
2592.2.s.g 8 9.c even 3 1
2592.2.s.g 8 9.d odd 6 1
2592.2.s.g 8 36.f odd 6 1
2592.2.s.g 8 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 16T_{5}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\). 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
$5$ \( (T^{4} + 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 14 T^{2} + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 16 T^{2} + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T - 11)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 112 T^{2} + 2704)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 64 T^{2} + 256)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 56 T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 6 T - 3)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} + 64 T^{2} + 256)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 56 T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 112 T^{2} + 1936)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 192 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 208 T^{2} + 8464)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T - 107)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 134 T^{2} + 3721)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 128)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} - 6 T - 39)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 254 T^{2} + 5329)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} - 64 T^{2} + 256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 48 T^{2} + 144)^{2} \) Copy content Toggle raw display
$97$ \( (T - 7)^{8} \) Copy content Toggle raw display
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