# Properties

 Label 1440.2.a.q Level $1440$ Weight $2$ Character orbit 1440.a Self dual yes Analytic conductor $11.498$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1440.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$11.4984578911$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + q^{5} -\beta q^{7} +O(q^{10})$$ $$q + q^{5} -\beta q^{7} + \beta q^{11} + 4 q^{13} + 2 q^{17} -2 \beta q^{23} + q^{25} -6 q^{29} + 2 \beta q^{31} -\beta q^{35} + 8 q^{37} + 8 q^{41} + 2 \beta q^{47} + 13 q^{49} -6 q^{53} + \beta q^{55} + \beta q^{59} + 10 q^{61} + 4 q^{65} + 2 \beta q^{67} + 2 \beta q^{71} + 6 q^{73} -20 q^{77} -2 \beta q^{79} -2 \beta q^{83} + 2 q^{85} + 4 q^{89} -4 \beta q^{91} + 2 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} + O(q^{10})$$ $$2q + 2q^{5} + 8q^{13} + 4q^{17} + 2q^{25} - 12q^{29} + 16q^{37} + 16q^{41} + 26q^{49} - 12q^{53} + 20q^{61} + 8q^{65} + 12q^{73} - 40q^{77} + 4q^{85} + 8q^{89} + 4q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.61803 −0.618034
0 0 0 1.00000 0 −4.47214 0 0 0
1.2 0 0 0 1.00000 0 4.47214 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.a.q yes 2
3.b odd 2 1 1440.2.a.p 2
4.b odd 2 1 inner 1440.2.a.q yes 2
5.b even 2 1 7200.2.a.cg 2
5.c odd 4 2 7200.2.f.bj 4
8.b even 2 1 2880.2.a.bi 2
8.d odd 2 1 2880.2.a.bi 2
12.b even 2 1 1440.2.a.p 2
15.d odd 2 1 7200.2.a.ch 2
15.e even 4 2 7200.2.f.be 4
20.d odd 2 1 7200.2.a.cg 2
20.e even 4 2 7200.2.f.bj 4
24.f even 2 1 2880.2.a.bj 2
24.h odd 2 1 2880.2.a.bj 2
60.h even 2 1 7200.2.a.ch 2
60.l odd 4 2 7200.2.f.be 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.a.p 2 3.b odd 2 1
1440.2.a.p 2 12.b even 2 1
1440.2.a.q yes 2 1.a even 1 1 trivial
1440.2.a.q yes 2 4.b odd 2 1 inner
2880.2.a.bi 2 8.b even 2 1
2880.2.a.bi 2 8.d odd 2 1
2880.2.a.bj 2 24.f even 2 1
2880.2.a.bj 2 24.h odd 2 1
7200.2.a.cg 2 5.b even 2 1
7200.2.a.cg 2 20.d odd 2 1
7200.2.a.ch 2 15.d odd 2 1
7200.2.a.ch 2 60.h even 2 1
7200.2.f.be 4 15.e even 4 2
7200.2.f.be 4 60.l odd 4 2
7200.2.f.bj 4 5.c odd 4 2
7200.2.f.bj 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1440))$$:

 $$T_{7}^{2} - 20$$ $$T_{11}^{2} - 20$$ $$T_{17} - 2$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$-20 + T^{2}$$
$11$ $$-20 + T^{2}$$
$13$ $$( -4 + T )^{2}$$
$17$ $$( -2 + T )^{2}$$
$19$ $$T^{2}$$
$23$ $$-80 + T^{2}$$
$29$ $$( 6 + T )^{2}$$
$31$ $$-80 + T^{2}$$
$37$ $$( -8 + T )^{2}$$
$41$ $$( -8 + T )^{2}$$
$43$ $$T^{2}$$
$47$ $$-80 + T^{2}$$
$53$ $$( 6 + T )^{2}$$
$59$ $$-20 + T^{2}$$
$61$ $$( -10 + T )^{2}$$
$67$ $$-80 + T^{2}$$
$71$ $$-80 + T^{2}$$
$73$ $$( -6 + T )^{2}$$
$79$ $$-80 + T^{2}$$
$83$ $$-80 + T^{2}$$
$89$ $$( -4 + T )^{2}$$
$97$ $$( -2 + T )^{2}$$