Properties

Label 2592.2.p.b
Level $2592$
Weight $2$
Character orbit 2592.p
Analytic conductor $20.697$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2592,2,Mod(431,2592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2592, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2592.431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.p (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: \(20.6972242039\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{11} - 2 \beta_{3} q^{17} - 2 q^{19} + ( - 5 \beta_{2} + 5) q^{25} + ( - 4 \beta_{3} + 4 \beta_1) q^{41} + (10 \beta_{2} - 10) q^{43} - 7 \beta_{2} q^{49} + ( - 5 \beta_{3} + 5 \beta_1) q^{59} + 14 \beta_{2} q^{67} + 2 q^{73} + \beta_1 q^{83} - 2 \beta_{3} q^{89} + ( - 10 \beta_{2} + 10) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{19} + 10 q^{25} - 20 q^{43} - 14 q^{49} + 28 q^{67} + 8 q^{73} + 20 q^{97}+O(q^{100}) \) 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

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2592\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
431.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
0 0 0 0 0 0 0 0 0
431.2 0 0 0 0 0 0 0 0 0
2159.1 0 0 0 0 0 0 0 0 0
2159.2 0 0 0 0 0 0 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
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
24.f even 2 1 inner
72.l even 6 1 inner
72.p odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.p.b 4
3.b odd 2 1 inner 2592.2.p.b 4
4.b odd 2 1 648.2.l.b 4
8.b even 2 1 648.2.l.b 4
8.d odd 2 1 CM 2592.2.p.b 4
9.c even 3 1 96.2.f.a 2
9.c even 3 1 inner 2592.2.p.b 4
9.d odd 6 1 96.2.f.a 2
9.d odd 6 1 inner 2592.2.p.b 4
12.b even 2 1 648.2.l.b 4
24.f even 2 1 inner 2592.2.p.b 4
24.h odd 2 1 648.2.l.b 4
36.f odd 6 1 24.2.f.a 2
36.f odd 6 1 648.2.l.b 4
36.h even 6 1 24.2.f.a 2
36.h even 6 1 648.2.l.b 4
45.h odd 6 1 2400.2.b.a 2
45.j even 6 1 2400.2.b.a 2
45.k odd 12 2 2400.2.m.a 4
45.l even 12 2 2400.2.m.a 4
72.j odd 6 1 24.2.f.a 2
72.j odd 6 1 648.2.l.b 4
72.l even 6 1 96.2.f.a 2
72.l even 6 1 inner 2592.2.p.b 4
72.n even 6 1 24.2.f.a 2
72.n even 6 1 648.2.l.b 4
72.p odd 6 1 96.2.f.a 2
72.p odd 6 1 inner 2592.2.p.b 4
144.u even 12 2 768.2.c.h 4
144.v odd 12 2 768.2.c.h 4
144.w odd 12 2 768.2.c.h 4
144.x even 12 2 768.2.c.h 4
180.n even 6 1 600.2.b.a 2
180.p odd 6 1 600.2.b.a 2
180.v odd 12 2 600.2.m.a 4
180.x even 12 2 600.2.m.a 4
360.z odd 6 1 2400.2.b.a 2
360.bd even 6 1 2400.2.b.a 2
360.bh odd 6 1 600.2.b.a 2
360.bk even 6 1 600.2.b.a 2
360.bo even 12 2 2400.2.m.a 4
360.br even 12 2 600.2.m.a 4
360.bt odd 12 2 2400.2.m.a 4
360.bu odd 12 2 600.2.m.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.2.f.a 2 36.f odd 6 1
24.2.f.a 2 36.h even 6 1
24.2.f.a 2 72.j odd 6 1
24.2.f.a 2 72.n even 6 1
96.2.f.a 2 9.c even 3 1
96.2.f.a 2 9.d odd 6 1
96.2.f.a 2 72.l even 6 1
96.2.f.a 2 72.p odd 6 1
600.2.b.a 2 180.n even 6 1
600.2.b.a 2 180.p odd 6 1
600.2.b.a 2 360.bh odd 6 1
600.2.b.a 2 360.bk even 6 1
600.2.m.a 4 180.v odd 12 2
600.2.m.a 4 180.x even 12 2
600.2.m.a 4 360.br even 12 2
600.2.m.a 4 360.bu odd 12 2
648.2.l.b 4 4.b odd 2 1
648.2.l.b 4 8.b even 2 1
648.2.l.b 4 12.b even 2 1
648.2.l.b 4 24.h odd 2 1
648.2.l.b 4 36.f odd 6 1
648.2.l.b 4 36.h even 6 1
648.2.l.b 4 72.j odd 6 1
648.2.l.b 4 72.n even 6 1
768.2.c.h 4 144.u even 12 2
768.2.c.h 4 144.v odd 12 2
768.2.c.h 4 144.w odd 12 2
768.2.c.h 4 144.x even 12 2
2400.2.b.a 2 45.h odd 6 1
2400.2.b.a 2 45.j even 6 1
2400.2.b.a 2 360.z odd 6 1
2400.2.b.a 2 360.bd even 6 1
2400.2.m.a 4 45.k odd 12 2
2400.2.m.a 4 45.l even 12 2
2400.2.m.a 4 360.bo even 12 2
2400.2.m.a 4 360.bt odd 12 2
2592.2.p.b 4 1.a even 1 1 trivial
2592.2.p.b 4 3.b odd 2 1 inner
2592.2.p.b 4 8.d odd 2 1 CM
2592.2.p.b 4 9.c even 3 1 inner
2592.2.p.b 4 9.d odd 6 1 inner
2592.2.p.b 4 24.f even 2 1 inner
2592.2.p.b 4 72.l even 6 1 inner
2592.2.p.b 4 72.p odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2592, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display
\( T_{41}^{4} - 128T_{41}^{2} + 16384 \) 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
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} - 8T^{2} + 64 \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$19$ \( (T + 2)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( T^{4} - 128 T^{2} + 16384 \) Copy content Toggle raw display
$43$ \( (T^{2} + 10 T + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} - 200 T^{2} + 40000 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T - 2)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 8T^{2} + 64 \) Copy content Toggle raw display
$89$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 10 T + 100)^{2} \) Copy content Toggle raw display
show more
show less