Properties

Label 1728.1.h.a
Level $1728$
Weight $1$
Character orbit 1728.h
Analytic conductor $0.862$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -3
Inner twists $8$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,1,Mod(161,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.161");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1728.h (of order \(2\), degree \(1\), 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: \(0.862384341830\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.1492992.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{5} + \zeta_{12}) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{5} + \zeta_{12}) q^{7} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{13} + \zeta_{12}^{3} q^{19} - q^{25} + ( - \zeta_{12}^{4} - \zeta_{12}^{2}) q^{37} + \zeta_{12}^{3} q^{43} + ( - \zeta_{12}^{4} + \zeta_{12}^{2} + 1) q^{49} + (\zeta_{12}^{4} + \zeta_{12}^{2}) q^{61} - \zeta_{12}^{3} q^{67} - q^{73} + ( - \zeta_{12}^{5} + \zeta_{12}) q^{79} + ( - \zeta_{12}^{5} - \zeta_{12}^{3} - \zeta_{12}) q^{91} + q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{25} + 8 q^{49} - 4 q^{73} + 4 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
0.866025 0.500000i
0 0 0 0 0 −1.73205 0 0 0
161.2 0 0 0 0 0 −1.73205 0 0 0
161.3 0 0 0 0 0 1.73205 0 0 0
161.4 0 0 0 0 0 1.73205 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
4.b odd 2 1 inner
8.b even 2 1 inner
8.d odd 2 1 inner
12.b even 2 1 inner
24.f even 2 1 inner
24.h odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.1.h.a 4
3.b odd 2 1 CM 1728.1.h.a 4
4.b odd 2 1 inner 1728.1.h.a 4
8.b even 2 1 inner 1728.1.h.a 4
8.d odd 2 1 inner 1728.1.h.a 4
12.b even 2 1 inner 1728.1.h.a 4
24.f even 2 1 inner 1728.1.h.a 4
24.h odd 2 1 inner 1728.1.h.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1728.1.h.a 4 1.a even 1 1 trivial
1728.1.h.a 4 3.b odd 2 1 CM
1728.1.h.a 4 4.b odd 2 1 inner
1728.1.h.a 4 8.b even 2 1 inner
1728.1.h.a 4 8.d odd 2 1 inner
1728.1.h.a 4 12.b even 2 1 inner
1728.1.h.a 4 24.f even 2 1 inner
1728.1.h.a 4 24.h odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(1728, [\chi])\).

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
$7$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( (T^{2} + 1)^{2} \) 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
$37$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 3)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( (T + 1)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( T^{4} \) Copy content Toggle raw display
$97$ \( (T - 1)^{4} \) Copy content Toggle raw display
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