# Properties

 Label 720.4.a.c 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 60) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 5 q^{5} - 32 q^{7}+O(q^{10})$$ q - 5 * q^5 - 32 * q^7 $$q - 5 q^{5} - 32 q^{7} + 36 q^{11} - 10 q^{13} + 78 q^{17} - 140 q^{19} - 192 q^{23} + 25 q^{25} - 6 q^{29} + 16 q^{31} + 160 q^{35} - 34 q^{37} + 390 q^{41} + 52 q^{43} + 408 q^{47} + 681 q^{49} + 114 q^{53} - 180 q^{55} + 516 q^{59} - 58 q^{61} + 50 q^{65} + 892 q^{67} - 120 q^{71} - 646 q^{73} - 1152 q^{77} + 1168 q^{79} - 732 q^{83} - 390 q^{85} + 1590 q^{89} + 320 q^{91} + 700 q^{95} + 194 q^{97}+O(q^{100})$$ q - 5 * q^5 - 32 * q^7 + 36 * q^11 - 10 * q^13 + 78 * q^17 - 140 * q^19 - 192 * q^23 + 25 * q^25 - 6 * q^29 + 16 * q^31 + 160 * q^35 - 34 * q^37 + 390 * q^41 + 52 * q^43 + 408 * q^47 + 681 * q^49 + 114 * q^53 - 180 * q^55 + 516 * q^59 - 58 * q^61 + 50 * q^65 + 892 * q^67 - 120 * q^71 - 646 * q^73 - 1152 * q^77 + 1168 * q^79 - 732 * q^83 - 390 * q^85 + 1590 * q^89 + 320 * q^91 + 700 * q^95 + 194 * 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 −32.0000 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.c 1
3.b odd 2 1 240.4.a.j 1
4.b odd 2 1 180.4.a.c 1
12.b even 2 1 60.4.a.b 1
15.d odd 2 1 1200.4.a.s 1
15.e even 4 2 1200.4.f.e 2
20.d odd 2 1 900.4.a.b 1
20.e even 4 2 900.4.d.b 2
24.f even 2 1 960.4.a.bb 1
24.h odd 2 1 960.4.a.a 1
36.f odd 6 2 1620.4.i.g 2
36.h even 6 2 1620.4.i.a 2
60.h even 2 1 300.4.a.e 1
60.l odd 4 2 300.4.d.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.b 1 12.b even 2 1
180.4.a.c 1 4.b odd 2 1
240.4.a.j 1 3.b odd 2 1
300.4.a.e 1 60.h even 2 1
300.4.d.d 2 60.l odd 4 2
720.4.a.c 1 1.a even 1 1 trivial
900.4.a.b 1 20.d odd 2 1
900.4.d.b 2 20.e even 4 2
960.4.a.a 1 24.h odd 2 1
960.4.a.bb 1 24.f even 2 1
1200.4.a.s 1 15.d odd 2 1
1200.4.f.e 2 15.e even 4 2
1620.4.i.a 2 36.h even 6 2
1620.4.i.g 2 36.f odd 6 2

## Hecke kernels

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

 $$T_{7} + 32$$ T7 + 32 $$T_{11} - 36$$ T11 - 36

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T + 32$$
$11$ $$T - 36$$
$13$ $$T + 10$$
$17$ $$T - 78$$
$19$ $$T + 140$$
$23$ $$T + 192$$
$29$ $$T + 6$$
$31$ $$T - 16$$
$37$ $$T + 34$$
$41$ $$T - 390$$
$43$ $$T - 52$$
$47$ $$T - 408$$
$53$ $$T - 114$$
$59$ $$T - 516$$
$61$ $$T + 58$$
$67$ $$T - 892$$
$71$ $$T + 120$$
$73$ $$T + 646$$
$79$ $$T - 1168$$
$83$ $$T + 732$$
$89$ $$T - 1590$$
$97$ $$T - 194$$