Properties

Label 648.2.n.h
Level $648$
Weight $2$
Character orbit 648.n
Analytic conductor $5.174$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [648,2,Mod(109,648)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("648.109");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 648 = 2^{3} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 648.n (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: \(5.17430605098\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + (2 \beta_{3} - 2 \beta_1) q^{5} + (2 \beta_{2} - 2) q^{7} + 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + 2 \beta_{2} q^{4} + (2 \beta_{3} - 2 \beta_1) q^{5} + (2 \beta_{2} - 2) q^{7} + 2 \beta_{3} q^{8} - 4 q^{10} - 4 \beta_1 q^{11} + (2 \beta_{3} - 2 \beta_1) q^{14} + (4 \beta_{2} - 4) q^{16} - 4 \beta_1 q^{20} - 8 \beta_{2} q^{22} + ( - 3 \beta_{2} + 3) q^{25} - 4 q^{28} + 2 \beta_1 q^{29} + 10 \beta_{2} q^{31} + (4 \beta_{3} - 4 \beta_1) q^{32} - 4 \beta_{3} q^{35} - 8 \beta_{2} q^{40} - 8 \beta_{3} q^{44} + 3 \beta_{2} q^{49} + ( - 3 \beta_{3} + 3 \beta_1) q^{50} + 10 \beta_{3} q^{53} + 16 q^{55} - 4 \beta_1 q^{56} + 4 \beta_{2} q^{58} + (8 \beta_{3} - 8 \beta_1) q^{59} + 10 \beta_{3} q^{62} - 8 q^{64} + ( - 8 \beta_{2} + 8) q^{70} + 14 q^{73} + ( - 8 \beta_{3} + 8 \beta_1) q^{77} + ( - 10 \beta_{2} + 10) q^{79} - 8 \beta_{3} q^{80} - 4 \beta_1 q^{83} + ( - 16 \beta_{2} + 16) q^{88} + (2 \beta_{2} - 2) q^{97} + 3 \beta_{3} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{4} - 4 q^{7} - 16 q^{10} - 8 q^{16} - 16 q^{22} + 6 q^{25} - 16 q^{28} + 20 q^{31} - 16 q^{40} + 6 q^{49} + 64 q^{55} + 8 q^{58} - 32 q^{64} + 16 q^{70} + 56 q^{73} + 20 q^{79} + 32 q^{88} - 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(487\) \(569\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
−1.22474 + 0.707107i
1.22474 0.707107i
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i 0 1.00000 1.73205i 2.44949 + 1.41421i 0 −1.00000 1.73205i 2.82843i 0 −4.00000
109.2 1.22474 0.707107i 0 1.00000 1.73205i −2.44949 1.41421i 0 −1.00000 1.73205i 2.82843i 0 −4.00000
541.1 −1.22474 0.707107i 0 1.00000 + 1.73205i 2.44949 1.41421i 0 −1.00000 + 1.73205i 2.82843i 0 −4.00000
541.2 1.22474 + 0.707107i 0 1.00000 + 1.73205i −2.44949 + 1.41421i 0 −1.00000 + 1.73205i 2.82843i 0 −4.00000
\(n\): 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 \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
8.b even 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
72.j odd 6 1 inner
72.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 648.2.n.h 4
3.b odd 2 1 inner 648.2.n.h 4
4.b odd 2 1 2592.2.r.i 4
8.b even 2 1 inner 648.2.n.h 4
8.d odd 2 1 2592.2.r.i 4
9.c even 3 1 72.2.d.a 2
9.c even 3 1 inner 648.2.n.h 4
9.d odd 6 1 72.2.d.a 2
9.d odd 6 1 inner 648.2.n.h 4
12.b even 2 1 2592.2.r.i 4
24.f even 2 1 2592.2.r.i 4
24.h odd 2 1 CM 648.2.n.h 4
36.f odd 6 1 288.2.d.a 2
36.f odd 6 1 2592.2.r.i 4
36.h even 6 1 288.2.d.a 2
36.h even 6 1 2592.2.r.i 4
45.h odd 6 1 1800.2.k.e 2
45.j even 6 1 1800.2.k.e 2
45.k odd 12 2 1800.2.d.n 4
45.l even 12 2 1800.2.d.n 4
72.j odd 6 1 72.2.d.a 2
72.j odd 6 1 inner 648.2.n.h 4
72.l even 6 1 288.2.d.a 2
72.l even 6 1 2592.2.r.i 4
72.n even 6 1 72.2.d.a 2
72.n even 6 1 inner 648.2.n.h 4
72.p odd 6 1 288.2.d.a 2
72.p odd 6 1 2592.2.r.i 4
144.u even 12 2 2304.2.a.y 2
144.v odd 12 2 2304.2.a.y 2
144.w odd 12 2 2304.2.a.q 2
144.x even 12 2 2304.2.a.q 2
180.n even 6 1 7200.2.k.h 2
180.p odd 6 1 7200.2.k.h 2
180.v odd 12 2 7200.2.d.p 4
180.x even 12 2 7200.2.d.p 4
360.z odd 6 1 7200.2.k.h 2
360.bd even 6 1 7200.2.k.h 2
360.bh odd 6 1 1800.2.k.e 2
360.bk even 6 1 1800.2.k.e 2
360.bo even 12 2 7200.2.d.p 4
360.br even 12 2 1800.2.d.n 4
360.bt odd 12 2 7200.2.d.p 4
360.bu odd 12 2 1800.2.d.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.2.d.a 2 9.c even 3 1
72.2.d.a 2 9.d odd 6 1
72.2.d.a 2 72.j odd 6 1
72.2.d.a 2 72.n even 6 1
288.2.d.a 2 36.f odd 6 1
288.2.d.a 2 36.h even 6 1
288.2.d.a 2 72.l even 6 1
288.2.d.a 2 72.p odd 6 1
648.2.n.h 4 1.a even 1 1 trivial
648.2.n.h 4 3.b odd 2 1 inner
648.2.n.h 4 8.b even 2 1 inner
648.2.n.h 4 9.c even 3 1 inner
648.2.n.h 4 9.d odd 6 1 inner
648.2.n.h 4 24.h odd 2 1 CM
648.2.n.h 4 72.j odd 6 1 inner
648.2.n.h 4 72.n even 6 1 inner
1800.2.d.n 4 45.k odd 12 2
1800.2.d.n 4 45.l even 12 2
1800.2.d.n 4 360.br even 12 2
1800.2.d.n 4 360.bu odd 12 2
1800.2.k.e 2 45.h odd 6 1
1800.2.k.e 2 45.j even 6 1
1800.2.k.e 2 360.bh odd 6 1
1800.2.k.e 2 360.bk even 6 1
2304.2.a.q 2 144.w odd 12 2
2304.2.a.q 2 144.x even 12 2
2304.2.a.y 2 144.u even 12 2
2304.2.a.y 2 144.v odd 12 2
2592.2.r.i 4 4.b odd 2 1
2592.2.r.i 4 8.d odd 2 1
2592.2.r.i 4 12.b even 2 1
2592.2.r.i 4 24.f even 2 1
2592.2.r.i 4 36.f odd 6 1
2592.2.r.i 4 36.h even 6 1
2592.2.r.i 4 72.l even 6 1
2592.2.r.i 4 72.p odd 6 1
7200.2.d.p 4 180.v odd 12 2
7200.2.d.p 4 180.x even 12 2
7200.2.d.p 4 360.bo even 12 2
7200.2.d.p 4 360.bt odd 12 2
7200.2.k.h 2 180.n even 6 1
7200.2.k.h 2 180.p odd 6 1
7200.2.k.h 2 360.z odd 6 1
7200.2.k.h 2 360.bd even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(648, [\chi])\):

\( T_{5}^{4} - 8T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{13} \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( T^{4} - 8T^{2} + 64 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} - 32T^{2} + 1024 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} - 8T^{2} + 64 \) Copy content Toggle raw display
$31$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( (T^{2} + 200)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 128 T^{2} + 16384 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T - 14)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 32T^{2} + 1024 \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
show more
show less