Properties

Label 1944.1.h.a
Level $1944$
Weight $1$
Character orbit 1944.h
Self dual yes
Analytic conductor $0.970$
Analytic rank $0$
Dimension $3$
Projective image $D_{9}$
CM discriminant -24
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1944 = 2^{3} \cdot 3^{5} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1944.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(0.970182384559\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
Defining polynomial: \(x^{3} - 3 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{9}\)
Projective field: Galois closure of 9.1.1586874322944.5

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{1} q^{5} + ( \beta_{1} - \beta_{2} ) q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{1} q^{5} + ( \beta_{1} - \beta_{2} ) q^{7} - q^{8} -\beta_{1} q^{10} -\beta_{2} q^{11} + ( -\beta_{1} + \beta_{2} ) q^{14} + q^{16} + \beta_{1} q^{20} + \beta_{2} q^{22} + ( 1 + \beta_{2} ) q^{25} + ( \beta_{1} - \beta_{2} ) q^{28} + q^{29} + \beta_{2} q^{31} - q^{32} + ( 1 - \beta_{1} + \beta_{2} ) q^{35} -\beta_{1} q^{40} -\beta_{2} q^{44} + ( 1 - \beta_{1} ) q^{49} + ( -1 - \beta_{2} ) q^{50} + ( -\beta_{1} + \beta_{2} ) q^{53} + ( -1 - \beta_{1} ) q^{55} + ( -\beta_{1} + \beta_{2} ) q^{56} - q^{58} + q^{59} -\beta_{2} q^{62} + q^{64} + ( -1 + \beta_{1} - \beta_{2} ) q^{70} + ( \beta_{1} - \beta_{2} ) q^{73} + ( 1 - \beta_{2} ) q^{77} - q^{79} + \beta_{1} q^{80} + ( -\beta_{1} + \beta_{2} ) q^{83} + \beta_{2} q^{88} -\beta_{1} q^{97} + ( -1 + \beta_{1} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + O(q^{10}) \) \( 3 q - 3 q^{2} + 3 q^{4} - 3 q^{8} + 3 q^{16} + 3 q^{25} + 3 q^{29} - 3 q^{32} + 3 q^{35} + 3 q^{49} - 3 q^{50} - 3 q^{55} - 3 q^{58} + 3 q^{59} + 3 q^{64} - 3 q^{70} + 3 q^{77} - 3 q^{79} - 3 q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1944\mathbb{Z}\right)^\times\).

\(n\) \(487\) \(973\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
485.1
−1.53209
−0.347296
1.87939
−1.00000 0 1.00000 −1.53209 0 −1.87939 −1.00000 0 1.53209
485.2 −1.00000 0 1.00000 −0.347296 0 1.53209 −1.00000 0 0.347296
485.3 −1.00000 0 1.00000 1.87939 0 0.347296 −1.00000 0 −1.87939
\(n\): 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 \(\Q(\sqrt{-6}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1944.1.h.a 3
3.b odd 2 1 1944.1.h.b yes 3
8.b even 2 1 1944.1.h.b yes 3
9.c even 3 2 1944.1.j.b 6
9.d odd 6 2 1944.1.j.a 6
24.h odd 2 1 CM 1944.1.h.a 3
72.j odd 6 2 1944.1.j.b 6
72.n even 6 2 1944.1.j.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1944.1.h.a 3 1.a even 1 1 trivial
1944.1.h.a 3 24.h odd 2 1 CM
1944.1.h.b yes 3 3.b odd 2 1
1944.1.h.b yes 3 8.b even 2 1
1944.1.j.a 6 9.d odd 6 2
1944.1.j.a 6 72.n even 6 2
1944.1.j.b 6 9.c even 3 2
1944.1.j.b 6 72.j odd 6 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} - 3 T_{5} - 1 \) acting on \(S_{1}^{\mathrm{new}}(1944, [\chi])\).

Hecke characteristic polynomials

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