# Properties

 Label 1444.1.b.a Level $1444$ Weight $1$ Character orbit 1444.b Self dual yes Analytic conductor $0.721$ Analytic rank $0$ Dimension $2$ Projective image $D_{5}$ CM discriminant -4 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1444 = 2^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1444.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.720649878242$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{5}$$ Projective field: Galois closure of 5.1.2085136.1

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + ( -1 + \beta ) q^{5} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + q^{4} + ( -1 + \beta ) q^{5} - q^{8} + q^{9} + ( 1 - \beta ) q^{10} + \beta q^{13} + q^{16} -\beta q^{17} - q^{18} + ( -1 + \beta ) q^{20} + ( 1 - \beta ) q^{25} -\beta q^{26} + \beta q^{29} - q^{32} + \beta q^{34} + q^{36} + ( 1 - \beta ) q^{37} + ( 1 - \beta ) q^{40} + ( 1 - \beta ) q^{41} + ( -1 + \beta ) q^{45} + q^{49} + ( -1 + \beta ) q^{50} + \beta q^{52} + ( 1 - \beta ) q^{53} -\beta q^{58} -\beta q^{61} + q^{64} + q^{65} -\beta q^{68} - q^{72} + ( -1 + \beta ) q^{73} + ( -1 + \beta ) q^{74} + ( -1 + \beta ) q^{80} + q^{81} + ( -1 + \beta ) q^{82} - q^{85} + ( 1 - \beta ) q^{89} + ( 1 - \beta ) q^{90} + \beta q^{97} - q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - q^{5} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - q^{5} - 2q^{8} + 2q^{9} + q^{10} + q^{13} + 2q^{16} - q^{17} - 2q^{18} - q^{20} + q^{25} - q^{26} + q^{29} - 2q^{32} + q^{34} + 2q^{36} + q^{37} + q^{40} + q^{41} - q^{45} + 2q^{49} - q^{50} + q^{52} + q^{53} - q^{58} - q^{61} + 2q^{64} + 2q^{65} - q^{68} - 2q^{72} - q^{73} - q^{74} - q^{80} + 2q^{81} - q^{82} - 2q^{85} + q^{89} + q^{90} + q^{97} - 2q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1444\mathbb{Z}\right)^\times$$.

 $$n$$ $$723$$ $$1085$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
723.1
 −0.618034 1.61803
−1.00000 0 1.00000 −1.61803 0 0 −1.00000 1.00000 1.61803
723.2 −1.00000 0 1.00000 0.618034 0 0 −1.00000 1.00000 −0.618034
 $$n$$: 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
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.1.b.a 2
4.b odd 2 1 CM 1444.1.b.a 2
19.b odd 2 1 1444.1.b.b yes 2
19.c even 3 2 1444.1.g.b 4
19.d odd 6 2 1444.1.g.a 4
19.e even 9 6 1444.1.l.b 12
19.f odd 18 6 1444.1.l.a 12
76.d even 2 1 1444.1.b.b yes 2
76.f even 6 2 1444.1.g.a 4
76.g odd 6 2 1444.1.g.b 4
76.k even 18 6 1444.1.l.a 12
76.l odd 18 6 1444.1.l.b 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1444.1.b.a 2 1.a even 1 1 trivial
1444.1.b.a 2 4.b odd 2 1 CM
1444.1.b.b yes 2 19.b odd 2 1
1444.1.b.b yes 2 76.d even 2 1
1444.1.g.a 4 19.d odd 6 2
1444.1.g.a 4 76.f even 6 2
1444.1.g.b 4 19.c even 3 2
1444.1.g.b 4 76.g odd 6 2
1444.1.l.a 12 19.f odd 18 6
1444.1.l.a 12 76.k even 18 6
1444.1.l.b 12 19.e even 9 6
1444.1.l.b 12 76.l odd 18 6

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{13}^{2} - T_{13} - 1$$ acting on $$S_{1}^{\mathrm{new}}(1444, [\chi])$$.

## Hecke characteristic polynomials

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