Properties

Label 1728.1.b.a
Level $1728$
Weight $1$
Character orbit 1728.b
Analytic conductor $0.862$
Analytic rank $0$
Dimension $4$
Projective image $D_{6}$
CM discriminant -24
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1728.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.862384341830\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.2.373248.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{5} -\zeta_{12}^{3} q^{7} +O(q^{10})\) \( q + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{5} -\zeta_{12}^{3} q^{7} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{11} + ( -1 - \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{25} + \zeta_{12}^{3} q^{31} + ( -\zeta_{12} + \zeta_{12}^{5} ) q^{35} + ( -\zeta_{12}^{2} - \zeta_{12}^{4} ) q^{53} + ( \zeta_{12} + 2 \zeta_{12}^{3} + \zeta_{12}^{5} ) q^{55} + q^{73} + ( \zeta_{12}^{2} + \zeta_{12}^{4} ) q^{77} -2 \zeta_{12}^{3} q^{79} + ( \zeta_{12} - \zeta_{12}^{5} ) q^{83} - q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + O(q^{10}) \) \( 4q - 8q^{25} + 4q^{73} - 4q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1567.1
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0 0 0 1.73205i 0 1.00000i 0 0 0
1567.2 0 0 0 1.73205i 0 1.00000i 0 0 0
1567.3 0 0 0 1.73205i 0 1.00000i 0 0 0
1567.4 0 0 0 1.73205i 0 1.00000i 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
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
3.b odd 2 1 inner
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

Twists

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3 \) acting on \(S_{1}^{\mathrm{new}}(1728, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \)
$3$ \( T^{4} \)
$5$ \( ( 3 + T^{2} )^{2} \)
$7$ \( ( 1 + T^{2} )^{2} \)
$11$ \( ( -3 + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( T^{4} \)
$19$ \( T^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( ( 1 + T^{2} )^{2} \)
$37$ \( T^{4} \)
$41$ \( T^{4} \)
$43$ \( T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 3 + T^{2} )^{2} \)
$59$ \( T^{4} \)
$61$ \( T^{4} \)
$67$ \( T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( -1 + T )^{4} \)
$79$ \( ( 4 + T^{2} )^{2} \)
$83$ \( ( -3 + T^{2} )^{2} \)
$89$ \( T^{4} \)
$97$ \( ( 1 + T )^{4} \)
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