# Properties

 Label 1440.4.a.o Level $1440$ Weight $4$ Character orbit 1440.a Self dual yes Analytic conductor $84.963$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1440 = 2^{5} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1440.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$84.9627504083$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 160) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 5 q^{5} + 6 q^{7} + O(q^{10})$$ $$q + 5 q^{5} + 6 q^{7} - 60 q^{11} + 50 q^{13} + 30 q^{17} + 40 q^{19} - 178 q^{23} + 25 q^{25} - 166 q^{29} + 20 q^{31} + 30 q^{35} + 10 q^{37} + 250 q^{41} + 142 q^{43} - 214 q^{47} - 307 q^{49} - 490 q^{53} - 300 q^{55} + 800 q^{59} + 250 q^{61} + 250 q^{65} - 774 q^{67} - 100 q^{71} - 230 q^{73} - 360 q^{77} - 1320 q^{79} - 982 q^{83} + 150 q^{85} - 874 q^{89} + 300 q^{91} + 200 q^{95} - 310 q^{97} + O(q^{100})$$

## 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 5.00000 0 6.00000 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 1440.4.a.o 1
3.b odd 2 1 160.4.a.a 1
4.b odd 2 1 1440.4.a.n 1
12.b even 2 1 160.4.a.b yes 1
15.d odd 2 1 800.4.a.h 1
15.e even 4 2 800.4.c.f 2
24.f even 2 1 320.4.a.f 1
24.h odd 2 1 320.4.a.i 1
48.i odd 4 2 1280.4.d.f 2
48.k even 4 2 1280.4.d.k 2
60.h even 2 1 800.4.a.d 1
60.l odd 4 2 800.4.c.e 2
120.i odd 2 1 1600.4.a.r 1
120.m even 2 1 1600.4.a.bj 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.a 1 3.b odd 2 1
160.4.a.b yes 1 12.b even 2 1
320.4.a.f 1 24.f even 2 1
320.4.a.i 1 24.h odd 2 1
800.4.a.d 1 60.h even 2 1
800.4.a.h 1 15.d odd 2 1
800.4.c.e 2 60.l odd 4 2
800.4.c.f 2 15.e even 4 2
1280.4.d.f 2 48.i odd 4 2
1280.4.d.k 2 48.k even 4 2
1440.4.a.n 1 4.b odd 2 1
1440.4.a.o 1 1.a even 1 1 trivial
1600.4.a.r 1 120.i odd 2 1
1600.4.a.bj 1 120.m even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1440))$$:

 $$T_{7} - 6$$ $$T_{11} + 60$$ $$T_{17} - 30$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-5 + T$$
$7$ $$-6 + T$$
$11$ $$60 + T$$
$13$ $$-50 + T$$
$17$ $$-30 + T$$
$19$ $$-40 + T$$
$23$ $$178 + T$$
$29$ $$166 + T$$
$31$ $$-20 + T$$
$37$ $$-10 + T$$
$41$ $$-250 + T$$
$43$ $$-142 + T$$
$47$ $$214 + T$$
$53$ $$490 + T$$
$59$ $$-800 + T$$
$61$ $$-250 + T$$
$67$ $$774 + T$$
$71$ $$100 + T$$
$73$ $$230 + T$$
$79$ $$1320 + T$$
$83$ $$982 + T$$
$89$ $$874 + T$$
$97$ $$310 + T$$