# Properties

 Label 720.4.a.x Level $720$ Weight $4$ Character orbit 720.a Self dual yes Analytic conductor $42.481$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 360) 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 $$q + 5 q^{5} - 2 q^{7} + 34 q^{11} - 68 q^{13} - 38 q^{17} - 4 q^{19} - 152 q^{23} + 25 q^{25} - 46 q^{29} + 260 q^{31} - 10 q^{35} - 312 q^{37} + 48 q^{41} + 200 q^{43} - 104 q^{47} - 339 q^{49} - 414 q^{53} + 170 q^{55} + 2 q^{59} - 38 q^{61} - 340 q^{65} + 244 q^{67} - 708 q^{71} - 378 q^{73} - 68 q^{77} + 852 q^{79} - 844 q^{83} - 190 q^{85} - 1380 q^{89} + 136 q^{91} - 20 q^{95} + 514 q^{97}+O(q^{100})$$ q + 5 * q^5 - 2 * q^7 + 34 * q^11 - 68 * q^13 - 38 * q^17 - 4 * q^19 - 152 * q^23 + 25 * q^25 - 46 * q^29 + 260 * q^31 - 10 * q^35 - 312 * q^37 + 48 * q^41 + 200 * q^43 - 104 * q^47 - 339 * q^49 - 414 * q^53 + 170 * q^55 + 2 * q^59 - 38 * q^61 - 340 * q^65 + 244 * q^67 - 708 * q^71 - 378 * q^73 - 68 * q^77 + 852 * q^79 - 844 * q^83 - 190 * q^85 - 1380 * q^89 + 136 * q^91 - 20 * q^95 + 514 * 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.x 1
3.b odd 2 1 720.4.a.g 1
4.b odd 2 1 360.4.a.k yes 1
12.b even 2 1 360.4.a.d 1
20.d odd 2 1 1800.4.a.q 1
20.e even 4 2 1800.4.f.f 2
60.h even 2 1 1800.4.a.r 1
60.l odd 4 2 1800.4.f.t 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.a.d 1 12.b even 2 1
360.4.a.k yes 1 4.b odd 2 1
720.4.a.g 1 3.b odd 2 1
720.4.a.x 1 1.a even 1 1 trivial
1800.4.a.q 1 20.d odd 2 1
1800.4.a.r 1 60.h even 2 1
1800.4.f.f 2 20.e even 4 2
1800.4.f.t 2 60.l odd 4 2

## Hecke kernels

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

 $$T_{7} + 2$$ T7 + 2 $$T_{11} - 34$$ T11 - 34

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T + 2$$
$11$ $$T - 34$$
$13$ $$T + 68$$
$17$ $$T + 38$$
$19$ $$T + 4$$
$23$ $$T + 152$$
$29$ $$T + 46$$
$31$ $$T - 260$$
$37$ $$T + 312$$
$41$ $$T - 48$$
$43$ $$T - 200$$
$47$ $$T + 104$$
$53$ $$T + 414$$
$59$ $$T - 2$$
$61$ $$T + 38$$
$67$ $$T - 244$$
$71$ $$T + 708$$
$73$ $$T + 378$$
$79$ $$T - 852$$
$83$ $$T + 844$$
$89$ $$T + 1380$$
$97$ $$T - 514$$