# Properties

 Label 1440.4.a.bb Level $1440$ Weight $4$ Character orbit 1440.a Self dual yes Analytic conductor $84.963$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1440.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$84.9627504083$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 160) 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.

 $$f(q)$$ $$=$$ $$q + 5 q^{5} - 7 \beta q^{7} +O(q^{10})$$ q + 5 * q^5 - 7*b * q^7 $$q + 5 q^{5} - 7 \beta q^{7} + 2 \beta q^{11} - 62 q^{13} + 46 q^{17} + 24 \beta q^{19} - 43 \beta q^{23} + 25 q^{25} + 90 q^{29} - 34 \beta q^{31} - 35 \beta q^{35} - 214 q^{37} + 10 q^{41} - 15 \beta q^{43} - 89 \beta q^{47} + 637 q^{49} + 678 q^{53} + 10 \beta q^{55} + 92 \beta q^{59} + 250 q^{61} - 310 q^{65} + 11 \beta q^{67} + 82 \beta q^{71} + 522 q^{73} - 280 q^{77} + 196 \beta q^{79} - 85 \beta q^{83} + 230 q^{85} - 970 q^{89} + 434 \beta q^{91} + 120 \beta q^{95} - 934 q^{97} +O(q^{100})$$ q + 5 * q^5 - 7*b * q^7 + 2*b * q^11 - 62 * q^13 + 46 * q^17 + 24*b * q^19 - 43*b * q^23 + 25 * q^25 + 90 * q^29 - 34*b * q^31 - 35*b * q^35 - 214 * q^37 + 10 * q^41 - 15*b * q^43 - 89*b * q^47 + 637 * q^49 + 678 * q^53 + 10*b * q^55 + 92*b * q^59 + 250 * q^61 - 310 * q^65 + 11*b * q^67 + 82*b * q^71 + 522 * q^73 - 280 * q^77 + 196*b * q^79 - 85*b * q^83 + 230 * q^85 - 970 * q^89 + 434*b * q^91 + 120*b * q^95 - 934 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 10 q^{5}+O(q^{10})$$ 2 * q + 10 * q^5 $$2 q + 10 q^{5} - 124 q^{13} + 92 q^{17} + 50 q^{25} + 180 q^{29} - 428 q^{37} + 20 q^{41} + 1274 q^{49} + 1356 q^{53} + 500 q^{61} - 620 q^{65} + 1044 q^{73} - 560 q^{77} + 460 q^{85} - 1940 q^{89} - 1868 q^{97}+O(q^{100})$$ 2 * q + 10 * q^5 - 124 * q^13 + 92 * q^17 + 50 * q^25 + 180 * q^29 - 428 * q^37 + 20 * q^41 + 1274 * q^49 + 1356 * q^53 + 500 * q^61 - 620 * q^65 + 1044 * q^73 - 560 * q^77 + 460 * q^85 - 1940 * q^89 - 1868 * q^97

## 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 5.00000 0 −31.3050 0 0 0
1.2 0 0 0 5.00000 0 31.3050 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 1440.4.a.bb 2
3.b odd 2 1 160.4.a.d 2
4.b odd 2 1 inner 1440.4.a.bb 2
12.b even 2 1 160.4.a.d 2
15.d odd 2 1 800.4.a.o 2
15.e even 4 2 800.4.c.l 4
24.f even 2 1 320.4.a.q 2
24.h odd 2 1 320.4.a.q 2
48.i odd 4 2 1280.4.d.v 4
48.k even 4 2 1280.4.d.v 4
60.h even 2 1 800.4.a.o 2
60.l odd 4 2 800.4.c.l 4
120.i odd 2 1 1600.4.a.cg 2
120.m even 2 1 1600.4.a.cg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.4.a.d 2 3.b odd 2 1
160.4.a.d 2 12.b even 2 1
320.4.a.q 2 24.f even 2 1
320.4.a.q 2 24.h odd 2 1
800.4.a.o 2 15.d odd 2 1
800.4.a.o 2 60.h even 2 1
800.4.c.l 4 15.e even 4 2
800.4.c.l 4 60.l odd 4 2
1280.4.d.v 4 48.i odd 4 2
1280.4.d.v 4 48.k even 4 2
1440.4.a.bb 2 1.a even 1 1 trivial
1440.4.a.bb 2 4.b odd 2 1 inner
1600.4.a.cg 2 120.i odd 2 1
1600.4.a.cg 2 120.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1440))$$:

 $$T_{7}^{2} - 980$$ T7^2 - 980 $$T_{11}^{2} - 80$$ T11^2 - 80 $$T_{17} - 46$$ T17 - 46

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$T^{2} - 980$$
$11$ $$T^{2} - 80$$
$13$ $$(T + 62)^{2}$$
$17$ $$(T - 46)^{2}$$
$19$ $$T^{2} - 11520$$
$23$ $$T^{2} - 36980$$
$29$ $$(T - 90)^{2}$$
$31$ $$T^{2} - 23120$$
$37$ $$(T + 214)^{2}$$
$41$ $$(T - 10)^{2}$$
$43$ $$T^{2} - 4500$$
$47$ $$T^{2} - 158420$$
$53$ $$(T - 678)^{2}$$
$59$ $$T^{2} - 169280$$
$61$ $$(T - 250)^{2}$$
$67$ $$T^{2} - 2420$$
$71$ $$T^{2} - 134480$$
$73$ $$(T - 522)^{2}$$
$79$ $$T^{2} - 768320$$
$83$ $$T^{2} - 144500$$
$89$ $$(T + 970)^{2}$$
$97$ $$(T + 934)^{2}$$