# Properties

 Label 720.4.a.q 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 120) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 5 q^{5} - 20 q^{7}+O(q^{10})$$ q + 5 * q^5 - 20 * q^7 $$q + 5 q^{5} - 20 q^{7} - 56 q^{11} - 86 q^{13} + 106 q^{17} - 4 q^{19} + 136 q^{23} + 25 q^{25} + 206 q^{29} + 152 q^{31} - 100 q^{35} + 282 q^{37} + 246 q^{41} - 412 q^{43} + 40 q^{47} + 57 q^{49} + 126 q^{53} - 280 q^{55} + 56 q^{59} - 2 q^{61} - 430 q^{65} + 388 q^{67} - 672 q^{71} + 1170 q^{73} + 1120 q^{77} - 408 q^{79} + 668 q^{83} + 530 q^{85} - 66 q^{89} + 1720 q^{91} - 20 q^{95} - 926 q^{97}+O(q^{100})$$ q + 5 * q^5 - 20 * q^7 - 56 * q^11 - 86 * q^13 + 106 * q^17 - 4 * q^19 + 136 * q^23 + 25 * q^25 + 206 * q^29 + 152 * q^31 - 100 * q^35 + 282 * q^37 + 246 * q^41 - 412 * q^43 + 40 * q^47 + 57 * q^49 + 126 * q^53 - 280 * q^55 + 56 * q^59 - 2 * q^61 - 430 * q^65 + 388 * q^67 - 672 * q^71 + 1170 * q^73 + 1120 * q^77 - 408 * q^79 + 668 * q^83 + 530 * q^85 - 66 * q^89 + 1720 * q^91 - 20 * q^95 - 926 * 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 −20.0000 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.q 1
3.b odd 2 1 240.4.a.g 1
4.b odd 2 1 360.4.a.n 1
12.b even 2 1 120.4.a.b 1
15.d odd 2 1 1200.4.a.p 1
15.e even 4 2 1200.4.f.t 2
20.d odd 2 1 1800.4.a.f 1
20.e even 4 2 1800.4.f.v 2
24.f even 2 1 960.4.a.bj 1
24.h odd 2 1 960.4.a.k 1
60.h even 2 1 600.4.a.i 1
60.l odd 4 2 600.4.f.a 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.b 1 12.b even 2 1
240.4.a.g 1 3.b odd 2 1
360.4.a.n 1 4.b odd 2 1
600.4.a.i 1 60.h even 2 1
600.4.f.a 2 60.l odd 4 2
720.4.a.q 1 1.a even 1 1 trivial
960.4.a.k 1 24.h odd 2 1
960.4.a.bj 1 24.f even 2 1
1200.4.a.p 1 15.d odd 2 1
1200.4.f.t 2 15.e even 4 2
1800.4.a.f 1 20.d odd 2 1
1800.4.f.v 2 20.e even 4 2

## Hecke kernels

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

 $$T_{7} + 20$$ T7 + 20 $$T_{11} + 56$$ T11 + 56

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T + 20$$
$11$ $$T + 56$$
$13$ $$T + 86$$
$17$ $$T - 106$$
$19$ $$T + 4$$
$23$ $$T - 136$$
$29$ $$T - 206$$
$31$ $$T - 152$$
$37$ $$T - 282$$
$41$ $$T - 246$$
$43$ $$T + 412$$
$47$ $$T - 40$$
$53$ $$T - 126$$
$59$ $$T - 56$$
$61$ $$T + 2$$
$67$ $$T - 388$$
$71$ $$T + 672$$
$73$ $$T - 1170$$
$79$ $$T + 408$$
$83$ $$T - 668$$
$89$ $$T + 66$$
$97$ $$T + 926$$