Properties

Label 2592.2.s.e
Level $2592$
Weight $2$
Character orbit 2592.s
Analytic conductor $20.697$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2592,2,Mod(863,2592)]
 
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, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2592.863");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2592 = 2^{5} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2592.s (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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 96)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{5} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{5} q^{5} - \beta_1 q^{7} + ( - \beta_{7} + \beta_{4}) q^{11} + 2 \beta_{2} q^{13} - 3 \beta_{3} q^{19} - 2 \beta_{4} q^{23} + ( - 3 \beta_{2} + 3) q^{25} + \beta_{6} q^{29} + ( - \beta_{3} + \beta_1) q^{31} - 2 \beta_{7} q^{35} + 6 q^{37} + 2 \beta_{5} q^{41} - \beta_1 q^{43} + (4 \beta_{7} - 4 \beta_{4}) q^{47} - 3 \beta_{2} q^{49} + ( - 3 \beta_{6} + 3 \beta_{5}) q^{53} + 4 \beta_{3} q^{55} - \beta_{4} q^{59} + ( - 2 \beta_{2} + 2) q^{61} + 2 \beta_{6} q^{65} + ( - \beta_{3} + \beta_1) q^{67} - 2 \beta_{7} q^{71} - 6 q^{73} + 2 \beta_{5} q^{77} + 7 \beta_1 q^{79} + (\beta_{7} - \beta_{4}) q^{83} + ( - 6 \beta_{6} + 6 \beta_{5}) q^{89} - 2 \beta_{3} q^{91} - 6 \beta_{4} q^{95} + (10 \beta_{2} - 10) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{13} + 12 q^{25} + 48 q^{37} - 12 q^{49} + 8 q^{61} - 48 q^{73} - 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( 2\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\zeta_{24}^{6} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -2\zeta_{24}^{7} + 2\zeta_{24}^{5} + 2\zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -2\zeta_{24}^{5} + 2\zeta_{24}^{3} + 2\zeta_{24} \) Copy content Toggle raw display
\(\zeta_{24}\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( -\beta_{7} + \beta_{6} + \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( -\beta_{5} + \beta_{4} ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
863.1
−0.965926 + 0.258819i
−0.258819 0.965926i
0.965926 0.258819i
0.258819 + 0.965926i
−0.965926 0.258819i
−0.258819 + 0.965926i
0.965926 + 0.258819i
0.258819 0.965926i
0 0 0 −2.44949 1.41421i 0 −1.73205 + 1.00000i 0 0 0
863.2 0 0 0 −2.44949 1.41421i 0 1.73205 1.00000i 0 0 0
863.3 0 0 0 2.44949 + 1.41421i 0 −1.73205 + 1.00000i 0 0 0
863.4 0 0 0 2.44949 + 1.41421i 0 1.73205 1.00000i 0 0 0
1727.1 0 0 0 −2.44949 + 1.41421i 0 −1.73205 1.00000i 0 0 0
1727.2 0 0 0 −2.44949 + 1.41421i 0 1.73205 + 1.00000i 0 0 0
1727.3 0 0 0 2.44949 1.41421i 0 −1.73205 1.00000i 0 0 0
1727.4 0 0 0 2.44949 1.41421i 0 1.73205 + 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 863.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner
12.b even 2 1 inner
36.f odd 6 1 inner
36.h even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2592.2.s.e 8
3.b odd 2 1 inner 2592.2.s.e 8
4.b odd 2 1 inner 2592.2.s.e 8
9.c even 3 1 96.2.c.a 4
9.c even 3 1 inner 2592.2.s.e 8
9.d odd 6 1 96.2.c.a 4
9.d odd 6 1 inner 2592.2.s.e 8
12.b even 2 1 inner 2592.2.s.e 8
36.f odd 6 1 96.2.c.a 4
36.f odd 6 1 inner 2592.2.s.e 8
36.h even 6 1 96.2.c.a 4
36.h even 6 1 inner 2592.2.s.e 8
45.h odd 6 1 2400.2.h.c 4
45.j even 6 1 2400.2.h.c 4
45.k odd 12 1 2400.2.o.a 4
45.k odd 12 1 2400.2.o.h 4
45.l even 12 1 2400.2.o.a 4
45.l even 12 1 2400.2.o.h 4
72.j odd 6 1 192.2.c.b 4
72.l even 6 1 192.2.c.b 4
72.n even 6 1 192.2.c.b 4
72.p odd 6 1 192.2.c.b 4
144.u even 12 1 768.2.f.a 4
144.u even 12 1 768.2.f.g 4
144.v odd 12 1 768.2.f.a 4
144.v odd 12 1 768.2.f.g 4
144.w odd 12 1 768.2.f.a 4
144.w odd 12 1 768.2.f.g 4
144.x even 12 1 768.2.f.a 4
144.x even 12 1 768.2.f.g 4
180.n even 6 1 2400.2.h.c 4
180.p odd 6 1 2400.2.h.c 4
180.v odd 12 1 2400.2.o.a 4
180.v odd 12 1 2400.2.o.h 4
180.x even 12 1 2400.2.o.a 4
180.x even 12 1 2400.2.o.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
96.2.c.a 4 9.c even 3 1
96.2.c.a 4 9.d odd 6 1
96.2.c.a 4 36.f odd 6 1
96.2.c.a 4 36.h even 6 1
192.2.c.b 4 72.j odd 6 1
192.2.c.b 4 72.l even 6 1
192.2.c.b 4 72.n even 6 1
192.2.c.b 4 72.p odd 6 1
768.2.f.a 4 144.u even 12 1
768.2.f.a 4 144.v odd 12 1
768.2.f.a 4 144.w odd 12 1
768.2.f.a 4 144.x even 12 1
768.2.f.g 4 144.u even 12 1
768.2.f.g 4 144.v odd 12 1
768.2.f.g 4 144.w odd 12 1
768.2.f.g 4 144.x even 12 1
2400.2.h.c 4 45.h odd 6 1
2400.2.h.c 4 45.j even 6 1
2400.2.h.c 4 180.n even 6 1
2400.2.h.c 4 180.p odd 6 1
2400.2.o.a 4 45.k odd 12 1
2400.2.o.a 4 45.l even 12 1
2400.2.o.a 4 180.v odd 12 1
2400.2.o.a 4 180.x even 12 1
2400.2.o.h 4 45.k odd 12 1
2400.2.o.h 4 45.l even 12 1
2400.2.o.h 4 180.v odd 12 1
2400.2.o.h 4 180.x even 12 1
2592.2.s.e 8 1.a even 1 1 trivial
2592.2.s.e 8 3.b odd 2 1 inner
2592.2.s.e 8 4.b odd 2 1 inner
2592.2.s.e 8 9.c even 3 1 inner
2592.2.s.e 8 9.d odd 6 1 inner
2592.2.s.e 8 12.b even 2 1 inner
2592.2.s.e 8 36.f odd 6 1 inner
2592.2.s.e 8 36.h even 6 1 inner

Hecke kernels

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

\( T_{5}^{4} - 8T_{5}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{2} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{2} + 36)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$37$ \( (T - 6)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 128 T^{2} + 16384)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 72)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 32)^{4} \) Copy content Toggle raw display
$73$ \( (T + 6)^{8} \) Copy content Toggle raw display
$79$ \( (T^{4} - 196 T^{2} + 38416)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 8 T^{2} + 64)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 288)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} + 10 T + 100)^{4} \) Copy content Toggle raw display
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