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Properties

Label 648.1.j.b
Level $648$
Weight $1$
Character orbit 648.j
Analytic conductor $0.323$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -24
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,1,Mod(53,648)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(648, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.53");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 648.j (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: \(0.323394128186\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 216)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.216.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of 12.0.101559956668416.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q - \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} - \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} - q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{6}^{2} q^{2} - \zeta_{6} q^{4} - \zeta_{6} q^{5} - \zeta_{6}^{2} q^{7} - q^{8} - q^{10} + \zeta_{6}^{2} q^{11} - \zeta_{6} q^{14} + \zeta_{6}^{2} q^{16} + \zeta_{6}^{2} q^{20} + \zeta_{6} q^{22} - q^{28} - \zeta_{6}^{2} q^{29} + \zeta_{6} q^{31} + \zeta_{6} q^{32} - q^{35} + \zeta_{6} q^{40} + q^{44} + q^{53} + q^{55} + \zeta_{6}^{2} q^{56} - 2 \zeta_{6} q^{58} + \zeta_{6} q^{59} + q^{62} + q^{64} + \zeta_{6}^{2} q^{70} - q^{73} + \zeta_{6} q^{77} + \zeta_{6}^{2} q^{79} + q^{80} + \zeta_{6}^{2} q^{83} - \zeta_{6}^{2} q^{88} - \zeta_{6}^{2} q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - q^{4} - q^{5} + q^{7} - 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - q^{4} - q^{5} + q^{7} - 2 q^{8} - 2 q^{10} - q^{11} - q^{14} - q^{16} - q^{20} + q^{22} - 2 q^{28} + 2 q^{29} + q^{31} + q^{32} - 2 q^{35} + q^{40} + 2 q^{44} + 2 q^{53} + 2 q^{55} - q^{56} - 2 q^{58} + 2 q^{59} + 2 q^{62} + 2 q^{64} - q^{70} - 2 q^{73} + q^{77} - 2 q^{79} + 2 q^{80} - q^{83} + q^{88} + q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/648\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
0.500000 − 0.866025i
0.500000 + 0.866025i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 + 0.866025i −1.00000 0 −1.00000
269.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.500000 − 0.866025i 0 0.500000 − 0.866025i −1.00000 0 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
9.c even 3 1 inner
72.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.1.j.b 2
3.b odd 2 1 648.1.j.a 2
4.b odd 2 1 2592.1.n.a 2
8.b even 2 1 648.1.j.a 2
8.d odd 2 1 2592.1.n.b 2
9.c even 3 1 216.1.h.a ✓ 1
9.c even 3 1 inner 648.1.j.b 2
9.d odd 6 1 216.1.h.b yes 1
9.d odd 6 1 648.1.j.a 2
12.b even 2 1 2592.1.n.b 2
24.f even 2 1 2592.1.n.a 2
24.h odd 2 1 CM 648.1.j.b 2
36.f odd 6 1 864.1.h.b 1
36.f odd 6 1 2592.1.n.a 2
36.h even 6 1 864.1.h.a 1
36.h even 6 1 2592.1.n.b 2
72.j odd 6 1 216.1.h.a ✓ 1
72.j odd 6 1 inner 648.1.j.b 2
72.l even 6 1 864.1.h.b 1
72.l even 6 1 2592.1.n.a 2
72.n even 6 1 216.1.h.b yes 1
72.n even 6 1 648.1.j.a 2
72.p odd 6 1 864.1.h.a 1
72.p odd 6 1 2592.1.n.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
216.1.h.a ✓ 1 9.c even 3 1
216.1.h.a ✓ 1 72.j odd 6 1
216.1.h.b yes 1 9.d odd 6 1
216.1.h.b yes 1 72.n even 6 1
648.1.j.a 2 3.b odd 2 1
648.1.j.a 2 8.b even 2 1
648.1.j.a 2 9.d odd 6 1
648.1.j.a 2 72.n even 6 1
648.1.j.b 2 1.a even 1 1 trivial
648.1.j.b 2 9.c even 3 1 inner
648.1.j.b 2 24.h odd 2 1 CM
648.1.j.b 2 72.j odd 6 1 inner
864.1.h.a 1 36.h even 6 1
864.1.h.a 1 72.p odd 6 1
864.1.h.b 1 36.f odd 6 1
864.1.h.b 1 72.l even 6 1
2592.1.n.a 2 4.b odd 2 1
2592.1.n.a 2 24.f even 2 1
2592.1.n.a 2 36.f odd 6 1
2592.1.n.a 2 72.l even 6 1
2592.1.n.b 2 8.d odd 2 1
2592.1.n.b 2 12.b even 2 1
2592.1.n.b 2 36.h even 6 1
2592.1.n.b 2 72.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(648, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

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