# Properties

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

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## 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

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

## 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.e yes 1
3.b odd 2 1 1440.2.a.m yes 1
4.b odd 2 1 1440.2.a.b 1
5.b even 2 1 7200.2.a.m 1
5.c odd 4 2 7200.2.f.h 2
8.b even 2 1 2880.2.a.be 1
8.d odd 2 1 2880.2.a.v 1
12.b even 2 1 1440.2.a.h yes 1
15.d odd 2 1 7200.2.a.n 1
15.e even 4 2 7200.2.f.u 2
20.d odd 2 1 7200.2.a.bo 1
20.e even 4 2 7200.2.f.v 2
24.f even 2 1 2880.2.a.g 1
24.h odd 2 1 2880.2.a.l 1
60.h even 2 1 7200.2.a.bn 1
60.l odd 4 2 7200.2.f.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1440.2.a.b 1 4.b odd 2 1
1440.2.a.e yes 1 1.a even 1 1 trivial
1440.2.a.h yes 1 12.b even 2 1
1440.2.a.m yes 1 3.b odd 2 1
2880.2.a.g 1 24.f even 2 1
2880.2.a.l 1 24.h odd 2 1
2880.2.a.v 1 8.d odd 2 1
2880.2.a.be 1 8.b even 2 1
7200.2.a.m 1 5.b even 2 1
7200.2.a.n 1 15.d odd 2 1
7200.2.a.bn 1 60.h even 2 1
7200.2.a.bo 1 20.d odd 2 1
7200.2.f.h 2 5.c odd 4 2
7200.2.f.i 2 60.l odd 4 2
7200.2.f.u 2 15.e even 4 2
7200.2.f.v 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1440))$$:

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

## Hecke characteristic polynomials

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