# Properties

 Label 1440.2.a.m Level $1440$ Weight $2$ Character orbit 1440.a Self dual yes Analytic conductor $11.498$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} + 2 q^{7} + O(q^{10})$$ $$q + q^{5} + 2 q^{7} + 2 q^{11} - 2 q^{17} + 4 q^{19} + q^{25} + 2 q^{29} + 8 q^{31} + 2 q^{35} - 4 q^{37} - 8 q^{41} + 8 q^{43} - 8 q^{47} - 3 q^{49} + 10 q^{53} + 2 q^{55} - 6 q^{59} + 2 q^{61} + 12 q^{67} + 12 q^{71} - 2 q^{73} + 4 q^{77} + 8 q^{79} - 4 q^{83} - 2 q^{85} - 12 q^{89} + 4 q^{95} + 10 q^{97} + O(q^{100})$$

## 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
 0
0 0 0 1.00000 0 2.00000 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

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1440.2.a.m yes 1
3.b odd 2 1 1440.2.a.e yes 1
4.b odd 2 1 1440.2.a.h yes 1
5.b even 2 1 7200.2.a.n 1
5.c odd 4 2 7200.2.f.u 2
8.b even 2 1 2880.2.a.l 1
8.d odd 2 1 2880.2.a.g 1
12.b even 2 1 1440.2.a.b 1
15.d odd 2 1 7200.2.a.m 1
15.e even 4 2 7200.2.f.h 2
20.d odd 2 1 7200.2.a.bn 1
20.e even 4 2 7200.2.f.i 2
24.f even 2 1 2880.2.a.v 1
24.h odd 2 1 2880.2.a.be 1
60.h even 2 1 7200.2.a.bo 1
60.l odd 4 2 7200.2.f.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.a.b 1 12.b even 2 1
1440.2.a.e yes 1 3.b odd 2 1
1440.2.a.h yes 1 4.b odd 2 1
1440.2.a.m yes 1 1.a even 1 1 trivial
2880.2.a.g 1 8.d odd 2 1
2880.2.a.l 1 8.b even 2 1
2880.2.a.v 1 24.f even 2 1
2880.2.a.be 1 24.h odd 2 1
7200.2.a.m 1 15.d odd 2 1
7200.2.a.n 1 5.b even 2 1
7200.2.a.bn 1 20.d odd 2 1
7200.2.a.bo 1 60.h even 2 1
7200.2.f.h 2 15.e even 4 2
7200.2.f.i 2 20.e even 4 2
7200.2.f.u 2 5.c odd 4 2
7200.2.f.v 2 60.l odd 4 2

## Hecke kernels

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

 $$T_{7} - 2$$ $$T_{11} - 2$$ $$T_{17} + 2$$ $$T_{19} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$-2 + T$$
$11$ $$-2 + T$$
$13$ $$T$$
$17$ $$2 + T$$
$19$ $$-4 + T$$
$23$ $$T$$
$29$ $$-2 + T$$
$31$ $$-8 + T$$
$37$ $$4 + T$$
$41$ $$8 + T$$
$43$ $$-8 + T$$
$47$ $$8 + T$$
$53$ $$-10 + T$$
$59$ $$6 + T$$
$61$ $$-2 + T$$
$67$ $$-12 + T$$
$71$ $$-12 + T$$
$73$ $$2 + T$$
$79$ $$-8 + T$$
$83$ $$4 + T$$
$89$ $$12 + T$$
$97$ $$-10 + T$$