# Properties

 Label 1440.2.a.f 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: no (minimal twist has level 480) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} + 4 q^{7} + O(q^{10})$$ $$q - q^{5} + 4 q^{7} - 2 q^{13} + 6 q^{17} - 4 q^{23} + q^{25} + 2 q^{29} + 8 q^{31} - 4 q^{35} + 6 q^{37} + 6 q^{41} - 12 q^{43} - 12 q^{47} + 9 q^{49} + 10 q^{53} + 8 q^{59} - 10 q^{61} + 2 q^{65} + 12 q^{67} + 8 q^{71} + 10 q^{73} - 16 q^{79} + 12 q^{83} - 6 q^{85} + 6 q^{89} - 8 q^{91} + 18 q^{97} + O(q^{100})$$

## 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 4.00000 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.f 1
3.b odd 2 1 480.2.a.h yes 1
4.b odd 2 1 1440.2.a.a 1
5.b even 2 1 7200.2.a.b 1
5.c odd 4 2 7200.2.f.p 2
8.b even 2 1 2880.2.a.bh 1
8.d odd 2 1 2880.2.a.s 1
12.b even 2 1 480.2.a.c 1
15.d odd 2 1 2400.2.a.b 1
15.e even 4 2 2400.2.f.j 2
20.d odd 2 1 7200.2.a.bz 1
20.e even 4 2 7200.2.f.o 2
24.f even 2 1 960.2.a.i 1
24.h odd 2 1 960.2.a.d 1
48.i odd 4 2 3840.2.k.d 2
48.k even 4 2 3840.2.k.w 2
60.h even 2 1 2400.2.a.bg 1
60.l odd 4 2 2400.2.f.i 2
120.i odd 2 1 4800.2.a.bm 1
120.m even 2 1 4800.2.a.bi 1
120.q odd 4 2 4800.2.f.r 2
120.w even 4 2 4800.2.f.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
480.2.a.c 1 12.b even 2 1
480.2.a.h yes 1 3.b odd 2 1
960.2.a.d 1 24.h odd 2 1
960.2.a.i 1 24.f even 2 1
1440.2.a.a 1 4.b odd 2 1
1440.2.a.f 1 1.a even 1 1 trivial
2400.2.a.b 1 15.d odd 2 1
2400.2.a.bg 1 60.h even 2 1
2400.2.f.i 2 60.l odd 4 2
2400.2.f.j 2 15.e even 4 2
2880.2.a.s 1 8.d odd 2 1
2880.2.a.bh 1 8.b even 2 1
3840.2.k.d 2 48.i odd 4 2
3840.2.k.w 2 48.k even 4 2
4800.2.a.bi 1 120.m even 2 1
4800.2.a.bm 1 120.i odd 2 1
4800.2.f.r 2 120.q odd 4 2
4800.2.f.s 2 120.w even 4 2
7200.2.a.b 1 5.b even 2 1
7200.2.a.bz 1 20.d odd 2 1
7200.2.f.o 2 20.e even 4 2
7200.2.f.p 2 5.c odd 4 2

## Hecke kernels

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

 $$T_{7} - 4$$ $$T_{11}$$ $$T_{17} - 6$$ $$T_{19}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$1 + T$$
$7$ $$-4 + T$$
$11$ $$T$$
$13$ $$2 + T$$
$17$ $$-6 + T$$
$19$ $$T$$
$23$ $$4 + T$$
$29$ $$-2 + T$$
$31$ $$-8 + T$$
$37$ $$-6 + T$$
$41$ $$-6 + T$$
$43$ $$12 + T$$
$47$ $$12 + T$$
$53$ $$-10 + T$$
$59$ $$-8 + T$$
$61$ $$10 + T$$
$67$ $$-12 + T$$
$71$ $$-8 + T$$
$73$ $$-10 + T$$
$79$ $$16 + T$$
$83$ $$-12 + T$$
$89$ $$-6 + T$$
$97$ $$-18 + T$$