# 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 $2$

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## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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