Properties

Label 1728.2.i.l
Level $1728$
Weight $2$
Character orbit 1728.i
Analytic conductor $13.798$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,2,Mod(577,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.577");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.i (of order \(3\), 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: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
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: \( 3 \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{5} - \beta_{2} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{5} - \beta_{2} q^{7} + \beta_{2} q^{11} + (3 \beta_1 - 3) q^{13} - 4 q^{17} + 4 \beta_{3} q^{19} + (5 \beta_{3} - 5 \beta_{2}) q^{23} + 4 \beta_1 q^{25} - \beta_1 q^{29} + (3 \beta_{3} - 3 \beta_{2}) q^{31} - \beta_{3} q^{35} + 8 q^{37} + ( - 5 \beta_1 + 5) q^{41} + 5 \beta_{2} q^{43} - 7 \beta_{2} q^{47} + ( - 4 \beta_1 + 4) q^{49} - 8 q^{53} + \beta_{3} q^{55} + (\beta_{3} - \beta_{2}) q^{59} - 7 \beta_1 q^{61} + 3 \beta_1 q^{65} + (5 \beta_{3} - 5 \beta_{2}) q^{67} + 2 \beta_{3} q^{71} - 12 q^{73} + ( - 3 \beta_1 + 3) q^{77} - 3 \beta_{2} q^{79} - 5 \beta_{2} q^{83} + (4 \beta_1 - 4) q^{85} + 4 q^{89} + 3 \beta_{3} q^{91} + (4 \beta_{3} - 4 \beta_{2}) q^{95} + 3 \beta_1 q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{5} - 6 q^{13} - 16 q^{17} + 8 q^{25} - 2 q^{29} + 32 q^{37} + 10 q^{41} + 8 q^{49} - 32 q^{53} - 14 q^{61} + 6 q^{65} - 48 q^{73} + 6 q^{77} - 8 q^{85} + 16 q^{89} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
577.1
0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i
0 0 0 0.500000 0.866025i 0 −0.866025 1.50000i 0 0 0
577.2 0 0 0 0.500000 0.866025i 0 0.866025 + 1.50000i 0 0 0
1153.1 0 0 0 0.500000 + 0.866025i 0 −0.866025 + 1.50000i 0 0 0
1153.2 0 0 0 0.500000 + 0.866025i 0 0.866025 1.50000i 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
4.b odd 2 1 inner
9.c even 3 1 inner
36.f odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.2.i.l 4
3.b odd 2 1 576.2.i.k 4
4.b odd 2 1 inner 1728.2.i.l 4
8.b even 2 1 864.2.i.d 4
8.d odd 2 1 864.2.i.d 4
9.c even 3 1 inner 1728.2.i.l 4
9.c even 3 1 5184.2.a.bl 2
9.d odd 6 1 576.2.i.k 4
9.d odd 6 1 5184.2.a.bx 2
12.b even 2 1 576.2.i.k 4
24.f even 2 1 288.2.i.d 4
24.h odd 2 1 288.2.i.d 4
36.f odd 6 1 inner 1728.2.i.l 4
36.f odd 6 1 5184.2.a.bl 2
36.h even 6 1 576.2.i.k 4
36.h even 6 1 5184.2.a.bx 2
72.j odd 6 1 288.2.i.d 4
72.j odd 6 1 2592.2.a.l 2
72.l even 6 1 288.2.i.d 4
72.l even 6 1 2592.2.a.l 2
72.n even 6 1 864.2.i.d 4
72.n even 6 1 2592.2.a.p 2
72.p odd 6 1 864.2.i.d 4
72.p odd 6 1 2592.2.a.p 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
288.2.i.d 4 24.f even 2 1
288.2.i.d 4 24.h odd 2 1
288.2.i.d 4 72.j odd 6 1
288.2.i.d 4 72.l even 6 1
576.2.i.k 4 3.b odd 2 1
576.2.i.k 4 9.d odd 6 1
576.2.i.k 4 12.b even 2 1
576.2.i.k 4 36.h even 6 1
864.2.i.d 4 8.b even 2 1
864.2.i.d 4 8.d odd 2 1
864.2.i.d 4 72.n even 6 1
864.2.i.d 4 72.p odd 6 1
1728.2.i.l 4 1.a even 1 1 trivial
1728.2.i.l 4 4.b odd 2 1 inner
1728.2.i.l 4 9.c even 3 1 inner
1728.2.i.l 4 36.f odd 6 1 inner
2592.2.a.l 2 72.j odd 6 1
2592.2.a.l 2 72.l even 6 1
2592.2.a.p 2 72.n even 6 1
2592.2.a.p 2 72.p odd 6 1
5184.2.a.bl 2 9.c even 3 1
5184.2.a.bl 2 36.f odd 6 1
5184.2.a.bx 2 9.d odd 6 1
5184.2.a.bx 2 36.h even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1728, [\chi])\):

\( T_{5}^{2} - T_{5} + 1 \) Copy content Toggle raw display
\( T_{7}^{4} + 3T_{7}^{2} + 9 \) 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
$5$ \( (T^{2} - T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$11$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$13$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$17$ \( (T + 4)^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} - 48)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$29$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$37$ \( (T - 8)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 5 T + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$47$ \( T^{4} + 147 T^{2} + 21609 \) Copy content Toggle raw display
$53$ \( (T + 8)^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$61$ \( (T^{2} + 7 T + 49)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$71$ \( (T^{2} - 12)^{2} \) Copy content Toggle raw display
$73$ \( (T + 12)^{4} \) Copy content Toggle raw display
$79$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$83$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$89$ \( (T - 4)^{4} \) Copy content Toggle raw display
$97$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
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