Properties

Label 1728.1.g.a
Level $1728$
Weight $1$
Character orbit 1728.g
Analytic conductor $0.862$
Analytic rank $0$
Dimension $2$
Projective image $D_{6}$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1728.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.862384341830\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 432)
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.186624.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( \zeta_{6} + \zeta_{6}^{2} ) q^{7} +O(q^{10})\) \( q + ( \zeta_{6} + \zeta_{6}^{2} ) q^{7} - q^{13} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{19} - q^{25} + q^{37} + ( -1 - \zeta_{6} + \zeta_{6}^{2} ) q^{49} + q^{61} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{67} + q^{73} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{79} + ( -\zeta_{6} - \zeta_{6}^{2} ) q^{91} + q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + O(q^{10}) \) \( 2 q - 2 q^{13} - 2 q^{25} + 2 q^{37} - 4 q^{49} + 2 q^{61} + 2 q^{73} + 2 q^{97} + O(q^{100}) \)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
703.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 0 0 1.73205i 0 0 0
703.2 0 0 0 0 0 1.73205i 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
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1728.1.g.a 2
3.b odd 2 1 CM 1728.1.g.a 2
4.b odd 2 1 inner 1728.1.g.a 2
8.b even 2 1 432.1.g.a 2
8.d odd 2 1 432.1.g.a 2
12.b even 2 1 inner 1728.1.g.a 2
24.f even 2 1 432.1.g.a 2
24.h odd 2 1 432.1.g.a 2
72.j odd 6 1 1296.1.o.a 2
72.j odd 6 1 1296.1.o.c 2
72.l even 6 1 1296.1.o.a 2
72.l even 6 1 1296.1.o.c 2
72.n even 6 1 1296.1.o.a 2
72.n even 6 1 1296.1.o.c 2
72.p odd 6 1 1296.1.o.a 2
72.p odd 6 1 1296.1.o.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
432.1.g.a 2 8.b even 2 1
432.1.g.a 2 8.d odd 2 1
432.1.g.a 2 24.f even 2 1
432.1.g.a 2 24.h odd 2 1
1296.1.o.a 2 72.j odd 6 1
1296.1.o.a 2 72.l even 6 1
1296.1.o.a 2 72.n even 6 1
1296.1.o.a 2 72.p odd 6 1
1296.1.o.c 2 72.j odd 6 1
1296.1.o.c 2 72.l even 6 1
1296.1.o.c 2 72.n even 6 1
1296.1.o.c 2 72.p odd 6 1
1728.1.g.a 2 1.a even 1 1 trivial
1728.1.g.a 2 3.b odd 2 1 CM
1728.1.g.a 2 4.b odd 2 1 inner
1728.1.g.a 2 12.b even 2 1 inner

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 3 + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 3 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( ( -1 + T )^{2} \)
$67$ \( 3 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( ( -1 + T )^{2} \)
$79$ \( 3 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -1 + T )^{2} \)
show more
show less