# Properties

 Label 1944.1.e.a Level $1944$ Weight $1$ Character orbit 1944.e Analytic conductor $0.970$ Analytic rank $0$ Dimension $2$ Projective image $S_{4}$ CM/RM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.970182384559$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$S_{4}$$ Projective field: Galois closure of 4.2.3888.1 Artin image: $\GL(2,3)$ Artin field: Galois closure of 8.2.181398528.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{5} +O(q^{10})$$ $$q -\beta q^{5} -\beta q^{11} + q^{13} + \beta q^{17} - q^{19} -\beta q^{23} - q^{25} - q^{31} + q^{43} - q^{49} -\beta q^{53} -2 q^{55} -\beta q^{59} + q^{61} -\beta q^{65} - q^{67} + \beta q^{71} + q^{73} + q^{79} + \beta q^{83} + 2 q^{85} + \beta q^{95} - q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + O(q^{10})$$ $$2 q + 2 q^{13} - 2 q^{19} - 2 q^{25} - 2 q^{31} + 2 q^{43} - 2 q^{49} - 4 q^{55} + 2 q^{61} - 2 q^{67} + 2 q^{73} + 2 q^{79} + 4 q^{85} - 2 q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1457.1
 1.41421i − 1.41421i
0 0 0 1.41421i 0 0 0 0 0
1457.2 0 0 0 1.41421i 0 0 0 0 0
 $$n$$: 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1944.1.e.a 2
3.b odd 2 1 inner 1944.1.e.a 2
4.b odd 2 1 3888.1.e.d 2
9.c even 3 2 1944.1.m.a 4
9.d odd 6 2 1944.1.m.a 4
12.b even 2 1 3888.1.e.d 2
36.f odd 6 2 3888.1.q.d 4
36.h even 6 2 3888.1.q.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1944.1.e.a 2 1.a even 1 1 trivial
1944.1.e.a 2 3.b odd 2 1 inner
1944.1.m.a 4 9.c even 3 2
1944.1.m.a 4 9.d odd 6 2
3888.1.e.d 2 4.b odd 2 1
3888.1.e.d 2 12.b even 2 1
3888.1.q.d 4 36.f odd 6 2
3888.1.q.d 4 36.h even 6 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1944, [\chi])$$.

## Hecke characteristic polynomials

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