# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1444 = 2^{2} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1444.g (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.720649878242$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta_{3} ) q^{2} + \beta_{3} q^{4} + ( 1 - \beta_{1} + \beta_{3} ) q^{5} - q^{8} + \beta_{3} q^{9} +O(q^{10})$$ $$q + ( 1 + \beta_{3} ) q^{2} + \beta_{3} q^{4} + ( 1 - \beta_{1} + \beta_{3} ) q^{5} - q^{8} + \beta_{3} q^{9} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{10} + ( \beta_{1} + \beta_{2} ) q^{13} + ( -1 - \beta_{3} ) q^{16} + \beta_{1} q^{17} - q^{18} + ( -1 - \beta_{2} ) q^{20} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + \beta_{2} q^{26} + ( \beta_{1} + \beta_{2} ) q^{29} -\beta_{3} q^{32} + ( \beta_{1} + \beta_{2} ) q^{34} + ( -1 - \beta_{3} ) q^{36} + ( 1 + \beta_{2} ) q^{37} + ( -1 + \beta_{1} - \beta_{3} ) q^{40} + ( -1 + \beta_{1} - \beta_{3} ) q^{41} + ( -1 - \beta_{2} ) q^{45} + q^{49} + ( -1 - \beta_{2} ) q^{50} -\beta_{1} q^{52} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{53} + \beta_{2} q^{58} + ( -\beta_{1} - \beta_{2} ) q^{61} + q^{64} + q^{65} + \beta_{2} q^{68} -\beta_{3} q^{72} + ( 1 - \beta_{1} + \beta_{3} ) q^{73} + ( 1 - \beta_{1} + \beta_{3} ) q^{74} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{80} + ( -1 - \beta_{3} ) q^{81} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{82} -\beta_{3} q^{85} + ( -\beta_{1} - \beta_{2} + \beta_{3} ) q^{89} + ( -1 + \beta_{1} - \beta_{3} ) q^{90} -\beta_{1} q^{97} + ( 1 + \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{2} - 2q^{4} + q^{5} - 4q^{8} - 2q^{9} + O(q^{10})$$ $$4q + 2q^{2} - 2q^{4} + q^{5} - 4q^{8} - 2q^{9} - q^{10} - q^{13} - 2q^{16} + q^{17} - 4q^{18} - 2q^{20} - q^{25} - 2q^{26} - q^{29} + 2q^{32} - q^{34} - 2q^{36} + 2q^{37} - q^{40} - q^{41} - 2q^{45} + 4q^{49} - 2q^{50} - q^{52} - q^{53} - 2q^{58} + q^{61} + 4q^{64} + 4q^{65} - 2q^{68} + 2q^{72} + q^{73} + q^{74} + q^{80} - 2q^{81} + q^{82} + 2q^{85} - q^{89} - q^{90} - q^{97} + 2q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 1$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} - 2 \nu - 1$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{2} - 1$$

## Character values

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

 $$n$$ $$723$$ $$1085$$ $$\chi(n)$$ $$-1$$ $$\beta_{3}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1151.1
 0.809017 − 1.40126i −0.309017 + 0.535233i 0.809017 + 1.40126i −0.309017 − 0.535233i
0.500000 0.866025i 0 −0.500000 0.866025i −0.309017 + 0.535233i 0 0 −1.00000 −0.500000 0.866025i 0.309017 + 0.535233i
1151.2 0.500000 0.866025i 0 −0.500000 0.866025i 0.809017 1.40126i 0 0 −1.00000 −0.500000 0.866025i −0.809017 1.40126i
1375.1 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.309017 0.535233i 0 0 −1.00000 −0.500000 + 0.866025i 0.309017 0.535233i
1375.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.809017 + 1.40126i 0 0 −1.00000 −0.500000 + 0.866025i −0.809017 + 1.40126i
 $$n$$: 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 $$\Q(\sqrt{-1})$$
19.c even 3 1 inner
76.g odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1444.1.g.b 4
4.b odd 2 1 CM 1444.1.g.b 4
19.b odd 2 1 1444.1.g.a 4
19.c even 3 1 1444.1.b.a 2
19.c even 3 1 inner 1444.1.g.b 4
19.d odd 6 1 1444.1.b.b yes 2
19.d odd 6 1 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.g.a 4
76.f even 6 1 1444.1.b.b yes 2
76.f even 6 1 1444.1.g.a 4
76.g odd 6 1 1444.1.b.a 2
76.g odd 6 1 inner 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 19.c even 3 1
1444.1.b.a 2 76.g odd 6 1
1444.1.b.b yes 2 19.d odd 6 1
1444.1.b.b yes 2 76.f even 6 1
1444.1.g.a 4 19.b odd 2 1
1444.1.g.a 4 19.d odd 6 1
1444.1.g.a 4 76.d even 2 1
1444.1.g.a 4 76.f even 6 1
1444.1.g.b 4 1.a even 1 1 trivial
1444.1.g.b 4 4.b odd 2 1 CM
1444.1.g.b 4 19.c even 3 1 inner
1444.1.g.b 4 76.g odd 6 1 inner
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 $$T_{13}^{4} + T_{13}^{3} + 2 T_{13}^{2} - T_{13} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1444, [\chi])$$.

## Hecke characteristic polynomials

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