# Properties

 Label 720.4.a.a 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 360) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 5 q^{5} - 34 q^{7}+O(q^{10})$$ q - 5 * q^5 - 34 * q^7 $$q - 5 q^{5} - 34 q^{7} - 18 q^{11} + 12 q^{13} - 106 q^{17} + 44 q^{19} - 56 q^{23} + 25 q^{25} + 270 q^{29} - 204 q^{31} + 170 q^{35} + 120 q^{37} + 80 q^{41} - 536 q^{43} + 536 q^{47} + 813 q^{49} + 542 q^{53} + 90 q^{55} + 174 q^{59} + 186 q^{61} - 60 q^{65} - 332 q^{67} + 132 q^{71} - 602 q^{73} + 612 q^{77} + 548 q^{79} + 492 q^{83} + 530 q^{85} - 1052 q^{89} - 408 q^{91} - 220 q^{95} + 482 q^{97}+O(q^{100})$$ q - 5 * q^5 - 34 * q^7 - 18 * q^11 + 12 * q^13 - 106 * q^17 + 44 * q^19 - 56 * q^23 + 25 * q^25 + 270 * q^29 - 204 * q^31 + 170 * q^35 + 120 * q^37 + 80 * q^41 - 536 * q^43 + 536 * q^47 + 813 * q^49 + 542 * q^53 + 90 * q^55 + 174 * q^59 + 186 * q^61 - 60 * q^65 - 332 * q^67 + 132 * q^71 - 602 * q^73 + 612 * q^77 + 548 * q^79 + 492 * q^83 + 530 * q^85 - 1052 * q^89 - 408 * q^91 - 220 * q^95 + 482 * 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 −34.0000 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.a 1
3.b odd 2 1 720.4.a.p 1
4.b odd 2 1 360.4.a.g 1
12.b even 2 1 360.4.a.o yes 1
20.d odd 2 1 1800.4.a.b 1
20.e even 4 2 1800.4.f.o 2
60.h even 2 1 1800.4.a.a 1
60.l odd 4 2 1800.4.f.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.4.a.g 1 4.b odd 2 1
360.4.a.o yes 1 12.b even 2 1
720.4.a.a 1 1.a even 1 1 trivial
720.4.a.p 1 3.b odd 2 1
1800.4.a.a 1 60.h even 2 1
1800.4.a.b 1 20.d odd 2 1
1800.4.f.i 2 60.l odd 4 2
1800.4.f.o 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(720))$$:

 $$T_{7} + 34$$ T7 + 34 $$T_{11} + 18$$ T11 + 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T + 34$$
$11$ $$T + 18$$
$13$ $$T - 12$$
$17$ $$T + 106$$
$19$ $$T - 44$$
$23$ $$T + 56$$
$29$ $$T - 270$$
$31$ $$T + 204$$
$37$ $$T - 120$$
$41$ $$T - 80$$
$43$ $$T + 536$$
$47$ $$T - 536$$
$53$ $$T - 542$$
$59$ $$T - 174$$
$61$ $$T - 186$$
$67$ $$T + 332$$
$71$ $$T - 132$$
$73$ $$T + 602$$
$79$ $$T - 548$$
$83$ $$T - 492$$
$89$ $$T + 1052$$
$97$ $$T - 482$$