Properties

Label 2592.2.d.g
Level $2592$
Weight $2$
Character orbit 2592.d
Analytic conductor $20.697$
Analytic rank $0$
Dimension $4$
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] = [2592,2,Mod(1297,2592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2592, 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("2592.1297");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.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: \(20.6972242039\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 648)
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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 2 \beta_1) q^{5} + (\beta_{3} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 2 \beta_1) q^{5} + (\beta_{3} - 1) q^{7} + ( - \beta_{2} - 3 \beta_1) q^{11} + (2 \beta_{2} - \beta_1) q^{13} + ( - \beta_{3} - 2) q^{17} + ( - \beta_{2} + 5 \beta_1) q^{19} + ( - \beta_{3} - 7) q^{23} + ( - 4 \beta_{3} - 2) q^{25} - 3 \beta_{2} q^{29} - 2 q^{31} + (\beta_{2} + \beta_1) q^{35} + 5 \beta_1 q^{37} + ( - 2 \beta_{3} + 8) q^{41} + (5 \beta_{2} + \beta_1) q^{43} - 2 \beta_{3} q^{47} + ( - 2 \beta_{3} - 3) q^{49} + ( - 4 \beta_{2} + 6 \beta_1) q^{53} + (5 \beta_{3} + 9) q^{55} + ( - 6 \beta_{2} + 2 \beta_1) q^{59} - 9 \beta_1 q^{61} + ( - 3 \beta_{3} - 4) q^{65} + (\beta_{2} + 3 \beta_1) q^{67} + ( - 5 \beta_{3} + 3) q^{71} + 9 q^{73} - 2 \beta_{2} q^{77} + (\beta_{3} - 7) q^{79} + 14 \beta_1 q^{83} + ( - 4 \beta_{2} - 7 \beta_1) q^{85} + ( - \beta_{3} - 10) q^{89} + ( - 3 \beta_{2} + 7 \beta_1) q^{91} + ( - 3 \beta_{3} - 7) q^{95} - 8 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{7} - 8 q^{17} - 28 q^{23} - 8 q^{25} - 8 q^{31} + 32 q^{41} - 12 q^{49} + 36 q^{55} - 16 q^{65} + 12 q^{71} + 36 q^{73} - 28 q^{79} - 40 q^{89} - 28 q^{95} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{12}^{3} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 2\zeta_{12}^{2} - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{12}^{3} + 2\zeta_{12} \) Copy content Toggle raw display
\(\zeta_{12}\)\(=\) \( ( \beta_{3} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{2}\)\(=\) \( ( \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{12}^{3}\)\(=\) \( \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(1217\) \(2431\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1297.1
0.866025 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0 0 0 3.73205i 0 0.732051 0 0 0
1297.2 0 0 0 0.267949i 0 −2.73205 0 0 0
1297.3 0 0 0 0.267949i 0 −2.73205 0 0 0
1297.4 0 0 0 3.73205i 0 0.732051 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.d.g 4
3.b odd 2 1 2592.2.d.h 4
4.b odd 2 1 648.2.d.i yes 4
8.b even 2 1 inner 2592.2.d.g 4
8.d odd 2 1 648.2.d.i yes 4
9.c even 3 1 2592.2.r.a 4
9.c even 3 1 2592.2.r.k 4
9.d odd 6 1 2592.2.r.b 4
9.d odd 6 1 2592.2.r.j 4
12.b even 2 1 648.2.d.e 4
24.f even 2 1 648.2.d.e 4
24.h odd 2 1 2592.2.d.h 4
36.f odd 6 1 648.2.n.a 4
36.f odd 6 1 648.2.n.l 4
36.h even 6 1 648.2.n.b 4
36.h even 6 1 648.2.n.m 4
72.j odd 6 1 2592.2.r.b 4
72.j odd 6 1 2592.2.r.j 4
72.l even 6 1 648.2.n.b 4
72.l even 6 1 648.2.n.m 4
72.n even 6 1 2592.2.r.a 4
72.n even 6 1 2592.2.r.k 4
72.p odd 6 1 648.2.n.a 4
72.p odd 6 1 648.2.n.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
648.2.d.e 4 12.b even 2 1
648.2.d.e 4 24.f even 2 1
648.2.d.i yes 4 4.b odd 2 1
648.2.d.i yes 4 8.d odd 2 1
648.2.n.a 4 36.f odd 6 1
648.2.n.a 4 72.p odd 6 1
648.2.n.b 4 36.h even 6 1
648.2.n.b 4 72.l even 6 1
648.2.n.l 4 36.f odd 6 1
648.2.n.l 4 72.p odd 6 1
648.2.n.m 4 36.h even 6 1
648.2.n.m 4 72.l even 6 1
2592.2.d.g 4 1.a even 1 1 trivial
2592.2.d.g 4 8.b even 2 1 inner
2592.2.d.h 4 3.b odd 2 1
2592.2.d.h 4 24.h odd 2 1
2592.2.r.a 4 9.c even 3 1
2592.2.r.a 4 72.n even 6 1
2592.2.r.b 4 9.d odd 6 1
2592.2.r.b 4 72.j odd 6 1
2592.2.r.j 4 9.d odd 6 1
2592.2.r.j 4 72.j odd 6 1
2592.2.r.k 4 9.c even 3 1
2592.2.r.k 4 72.n even 6 1

Hecke kernels

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

\( T_{5}^{4} + 14T_{5}^{2} + 1 \) Copy content Toggle raw display
\( T_{17}^{2} + 4T_{17} + 1 \) 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
$5$ \( T^{4} + 14T^{2} + 1 \) Copy content Toggle raw display
$7$ \( (T^{2} + 2 T - 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$13$ \( T^{4} + 26T^{2} + 121 \) Copy content Toggle raw display
$17$ \( (T^{2} + 4 T + 1)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 56T^{2} + 484 \) Copy content Toggle raw display
$23$ \( (T^{2} + 14 T + 46)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$31$ \( (T + 2)^{4} \) Copy content Toggle raw display
$37$ \( (T^{2} + 25)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 16 T + 52)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 152T^{2} + 5476 \) Copy content Toggle raw display
$47$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 144 \) Copy content Toggle raw display
$59$ \( T^{4} + 224 T^{2} + 10816 \) Copy content Toggle raw display
$61$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 24T^{2} + 36 \) Copy content Toggle raw display
$71$ \( (T^{2} - 6 T - 66)^{2} \) Copy content Toggle raw display
$73$ \( (T - 9)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 14 T + 46)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 20 T + 97)^{2} \) Copy content Toggle raw display
$97$ \( (T + 8)^{4} \) Copy content Toggle raw display
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