Properties

Label 864.2.d.a
Level $864$
Weight $2$
Character orbit 864.d
Analytic conductor $6.899$
Analytic rank $0$
Dimension $4$
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(433,864)]
 
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, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("864.433");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 864 = 2^{5} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 864.d (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: \(6.89907473464\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 3x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 216)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{5} - q^{7} + 3 \beta_1 q^{11} - \beta_{3} q^{13} + \beta_{2} q^{17} - \beta_{3} q^{19} - \beta_{2} q^{23} + 4 q^{25} - 6 \beta_1 q^{29} + 7 q^{31} + \beta_1 q^{35} - \beta_{3} q^{37} - \beta_{2} q^{41} - 2 \beta_{3} q^{43} - 6 q^{49} + 9 \beta_1 q^{53} + 3 q^{55} - 4 \beta_1 q^{59} - \beta_{2} q^{65} + 3 \beta_{2} q^{71} + 3 q^{73} - 3 \beta_1 q^{77} - 4 q^{79} - 7 \beta_1 q^{83} - \beta_{3} q^{85} + 2 \beta_{2} q^{89} + \beta_{3} q^{91} - \beta_{2} q^{95} + 7 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} + 16 q^{25} + 28 q^{31} - 24 q^{49} + 12 q^{55} + 12 q^{73} - 16 q^{79} + 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 3x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 4\nu^{2} - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 6 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{2} + 10\beta_1 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
433.1
1.32288 + 0.500000i
−1.32288 + 0.500000i
−1.32288 0.500000i
1.32288 0.500000i
0 0 0 1.00000i 0 −1.00000 0 0 0
433.2 0 0 0 1.00000i 0 −1.00000 0 0 0
433.3 0 0 0 1.00000i 0 −1.00000 0 0 0
433.4 0 0 0 1.00000i 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.b even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 864.2.d.a 4
3.b odd 2 1 inner 864.2.d.a 4
4.b odd 2 1 216.2.d.b 4
8.b even 2 1 inner 864.2.d.a 4
8.d odd 2 1 216.2.d.b 4
9.c even 3 2 2592.2.r.p 8
9.d odd 6 2 2592.2.r.p 8
12.b even 2 1 216.2.d.b 4
16.e even 4 1 6912.2.a.bd 2
16.e even 4 1 6912.2.a.bv 2
16.f odd 4 1 6912.2.a.bc 2
16.f odd 4 1 6912.2.a.bu 2
24.f even 2 1 216.2.d.b 4
24.h odd 2 1 inner 864.2.d.a 4
36.f odd 6 2 648.2.n.n 8
36.h even 6 2 648.2.n.n 8
48.i odd 4 1 6912.2.a.bd 2
48.i odd 4 1 6912.2.a.bv 2
48.k even 4 1 6912.2.a.bc 2
48.k even 4 1 6912.2.a.bu 2
72.j odd 6 2 2592.2.r.p 8
72.l even 6 2 648.2.n.n 8
72.n even 6 2 2592.2.r.p 8
72.p odd 6 2 648.2.n.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.2.d.b 4 4.b odd 2 1
216.2.d.b 4 8.d odd 2 1
216.2.d.b 4 12.b even 2 1
216.2.d.b 4 24.f even 2 1
648.2.n.n 8 36.f odd 6 2
648.2.n.n 8 36.h even 6 2
648.2.n.n 8 72.l even 6 2
648.2.n.n 8 72.p odd 6 2
864.2.d.a 4 1.a even 1 1 trivial
864.2.d.a 4 3.b odd 2 1 inner
864.2.d.a 4 8.b even 2 1 inner
864.2.d.a 4 24.h odd 2 1 inner
2592.2.r.p 8 9.c even 3 2
2592.2.r.p 8 9.d odd 6 2
2592.2.r.p 8 72.j odd 6 2
2592.2.r.p 8 72.n even 6 2
6912.2.a.bc 2 16.f odd 4 1
6912.2.a.bc 2 48.k even 4 1
6912.2.a.bd 2 16.e even 4 1
6912.2.a.bd 2 48.i odd 4 1
6912.2.a.bu 2 16.f odd 4 1
6912.2.a.bu 2 48.k even 4 1
6912.2.a.bv 2 16.e even 4 1
6912.2.a.bv 2 48.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(864, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T - 7)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 28)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 28)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 112)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} - 252)^{2} \) Copy content Toggle raw display
$73$ \( (T - 3)^{4} \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 49)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 112)^{2} \) Copy content Toggle raw display
$97$ \( (T - 7)^{4} \) Copy content Toggle raw display
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