Properties

Label 864.2.a.n
Level $864$
Weight $2$
Character orbit 864.a
Self dual yes
Analytic conductor $6.899$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [864,2,Mod(1,864)]
 
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([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.a (trivial)

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: yes
Analytic conductor: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = \sqrt{13}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta q^{5} - \beta q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - \beta q^{5} - \beta q^{7} + q^{11} + 4 q^{13} + 2 \beta q^{19} - 6 q^{23} + 8 q^{25} + 2 \beta q^{29} + \beta q^{31} + 13 q^{35} + 10 q^{37} + 2 \beta q^{41} - 2 \beta q^{43} - 10 q^{47} + 6 q^{49} + \beta q^{53} - \beta q^{55} - 4 q^{59} - 4 \beta q^{65} - 2 \beta q^{67} + 8 q^{71} - 3 q^{73} - \beta q^{77} + 4 \beta q^{79} + 9 q^{83} - 2 \beta q^{89} - 4 \beta q^{91} - 26 q^{95} + 7 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{11} + 8 q^{13} - 12 q^{23} + 16 q^{25} + 26 q^{35} + 20 q^{37} - 20 q^{47} + 12 q^{49} - 8 q^{59} + 16 q^{71} - 6 q^{73} + 18 q^{83} - 52 q^{95} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.30278
−1.30278
0 0 0 −3.60555 0 −3.60555 0 0 0
1.2 0 0 0 3.60555 0 3.60555 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.a.n yes 2
3.b odd 2 1 864.2.a.m 2
4.b odd 2 1 864.2.a.m 2
8.b even 2 1 1728.2.a.bc 2
8.d odd 2 1 1728.2.a.bd 2
9.c even 3 2 2592.2.i.bb 4
9.d odd 6 2 2592.2.i.bc 4
12.b even 2 1 inner 864.2.a.n yes 2
24.f even 2 1 1728.2.a.bc 2
24.h odd 2 1 1728.2.a.bd 2
36.f odd 6 2 2592.2.i.bc 4
36.h even 6 2 2592.2.i.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
864.2.a.m 2 3.b odd 2 1
864.2.a.m 2 4.b odd 2 1
864.2.a.n yes 2 1.a even 1 1 trivial
864.2.a.n yes 2 12.b even 2 1 inner
1728.2.a.bc 2 8.b even 2 1
1728.2.a.bc 2 24.f even 2 1
1728.2.a.bd 2 8.d odd 2 1
1728.2.a.bd 2 24.h odd 2 1
2592.2.i.bb 4 9.c even 3 2
2592.2.i.bb 4 36.h even 6 2
2592.2.i.bc 4 9.d odd 6 2
2592.2.i.bc 4 36.f odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(864))\):

\( T_{5}^{2} - 13 \) Copy content Toggle raw display
\( T_{7}^{2} - 13 \) Copy content Toggle raw display
\( T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 13 \) Copy content Toggle raw display
$7$ \( T^{2} - 13 \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( (T - 4)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - 52 \) Copy content Toggle raw display
$23$ \( (T + 6)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 52 \) Copy content Toggle raw display
$31$ \( T^{2} - 13 \) Copy content Toggle raw display
$37$ \( (T - 10)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} - 52 \) Copy content Toggle raw display
$43$ \( T^{2} - 52 \) Copy content Toggle raw display
$47$ \( (T + 10)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 13 \) Copy content Toggle raw display
$59$ \( (T + 4)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 52 \) Copy content Toggle raw display
$71$ \( (T - 8)^{2} \) Copy content Toggle raw display
$73$ \( (T + 3)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 208 \) Copy content Toggle raw display
$83$ \( (T - 9)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 52 \) Copy content Toggle raw display
$97$ \( (T - 7)^{2} \) Copy content Toggle raw display
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