# Properties

 Label 720.4.a.w Level $720$ Weight $4$ Character orbit 720.a Self dual yes Analytic conductor $42.481$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 720.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$42.4813752041$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 180) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 5 q^{5} - 2 q^{7}+O(q^{10})$$ q + 5 * q^5 - 2 * q^7 $$q + 5 q^{5} - 2 q^{7} - 30 q^{11} - 4 q^{13} + 90 q^{17} + 28 q^{19} - 120 q^{23} + 25 q^{25} + 210 q^{29} + 4 q^{31} - 10 q^{35} + 200 q^{37} + 240 q^{41} + 136 q^{43} + 120 q^{47} - 339 q^{49} - 30 q^{53} - 150 q^{55} + 450 q^{59} - 166 q^{61} - 20 q^{65} - 908 q^{67} + 1020 q^{71} - 250 q^{73} + 60 q^{77} + 916 q^{79} + 1140 q^{83} + 450 q^{85} - 420 q^{89} + 8 q^{91} + 140 q^{95} + 1538 q^{97}+O(q^{100})$$ q + 5 * q^5 - 2 * q^7 - 30 * q^11 - 4 * q^13 + 90 * q^17 + 28 * q^19 - 120 * q^23 + 25 * q^25 + 210 * q^29 + 4 * q^31 - 10 * q^35 + 200 * q^37 + 240 * q^41 + 136 * q^43 + 120 * q^47 - 339 * q^49 - 30 * q^53 - 150 * q^55 + 450 * q^59 - 166 * q^61 - 20 * q^65 - 908 * q^67 + 1020 * q^71 - 250 * q^73 + 60 * q^77 + 916 * q^79 + 1140 * q^83 + 450 * q^85 - 420 * q^89 + 8 * q^91 + 140 * q^95 + 1538 * 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
 0
0 0 0 5.00000 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 720.4.a.w 1
3.b odd 2 1 720.4.a.h 1
4.b odd 2 1 180.4.a.e yes 1
12.b even 2 1 180.4.a.b 1
20.d odd 2 1 900.4.a.j 1
20.e even 4 2 900.4.d.i 2
36.f odd 6 2 1620.4.i.c 2
36.h even 6 2 1620.4.i.i 2
60.h even 2 1 900.4.a.i 1
60.l odd 4 2 900.4.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.4.a.b 1 12.b even 2 1
180.4.a.e yes 1 4.b odd 2 1
720.4.a.h 1 3.b odd 2 1
720.4.a.w 1 1.a even 1 1 trivial
900.4.a.i 1 60.h even 2 1
900.4.a.j 1 20.d odd 2 1
900.4.d.d 2 60.l odd 4 2
900.4.d.i 2 20.e even 4 2
1620.4.i.c 2 36.f odd 6 2
1620.4.i.i 2 36.h even 6 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(720))$$:

 $$T_{7} + 2$$ T7 + 2 $$T_{11} + 30$$ T11 + 30

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T + 2$$
$11$ $$T + 30$$
$13$ $$T + 4$$
$17$ $$T - 90$$
$19$ $$T - 28$$
$23$ $$T + 120$$
$29$ $$T - 210$$
$31$ $$T - 4$$
$37$ $$T - 200$$
$41$ $$T - 240$$
$43$ $$T - 136$$
$47$ $$T - 120$$
$53$ $$T + 30$$
$59$ $$T - 450$$
$61$ $$T + 166$$
$67$ $$T + 908$$
$71$ $$T - 1020$$
$73$ $$T + 250$$
$79$ $$T - 916$$
$83$ $$T - 1140$$
$89$ $$T + 420$$
$97$ $$T - 1538$$