# Properties

 Label 720.4.a.ba 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 40) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 5 q^{5} + 18 q^{7}+O(q^{10})$$ q + 5 * q^5 + 18 * q^7 $$q + 5 q^{5} + 18 q^{7} - 16 q^{11} - 6 q^{13} + 6 q^{17} + 124 q^{19} + 42 q^{23} + 25 q^{25} - 142 q^{29} + 188 q^{31} + 90 q^{35} + 202 q^{37} - 54 q^{41} - 66 q^{43} + 38 q^{47} - 19 q^{49} - 738 q^{53} - 80 q^{55} + 564 q^{59} - 262 q^{61} - 30 q^{65} + 554 q^{67} + 140 q^{71} + 882 q^{73} - 288 q^{77} + 1160 q^{79} + 642 q^{83} + 30 q^{85} + 854 q^{89} - 108 q^{91} + 620 q^{95} - 478 q^{97}+O(q^{100})$$ q + 5 * q^5 + 18 * q^7 - 16 * q^11 - 6 * q^13 + 6 * q^17 + 124 * q^19 + 42 * q^23 + 25 * q^25 - 142 * q^29 + 188 * q^31 + 90 * q^35 + 202 * q^37 - 54 * q^41 - 66 * q^43 + 38 * q^47 - 19 * q^49 - 738 * q^53 - 80 * q^55 + 564 * q^59 - 262 * q^61 - 30 * q^65 + 554 * q^67 + 140 * q^71 + 882 * q^73 - 288 * q^77 + 1160 * q^79 + 642 * q^83 + 30 * q^85 + 854 * q^89 - 108 * q^91 + 620 * q^95 - 478 * 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 18.0000 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.ba 1
3.b odd 2 1 80.4.a.a 1
4.b odd 2 1 360.4.a.i 1
12.b even 2 1 40.4.a.c 1
15.d odd 2 1 400.4.a.u 1
15.e even 4 2 400.4.c.a 2
20.d odd 2 1 1800.4.a.bd 1
20.e even 4 2 1800.4.f.n 2
24.f even 2 1 320.4.a.a 1
24.h odd 2 1 320.4.a.n 1
48.i odd 4 2 1280.4.d.b 2
48.k even 4 2 1280.4.d.o 2
60.h even 2 1 200.4.a.a 1
60.l odd 4 2 200.4.c.a 2
84.h odd 2 1 1960.4.a.a 1
120.i odd 2 1 1600.4.a.a 1
120.m even 2 1 1600.4.a.ca 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.c 1 12.b even 2 1
80.4.a.a 1 3.b odd 2 1
200.4.a.a 1 60.h even 2 1
200.4.c.a 2 60.l odd 4 2
320.4.a.a 1 24.f even 2 1
320.4.a.n 1 24.h odd 2 1
360.4.a.i 1 4.b odd 2 1
400.4.a.u 1 15.d odd 2 1
400.4.c.a 2 15.e even 4 2
720.4.a.ba 1 1.a even 1 1 trivial
1280.4.d.b 2 48.i odd 4 2
1280.4.d.o 2 48.k even 4 2
1600.4.a.a 1 120.i odd 2 1
1600.4.a.ca 1 120.m even 2 1
1800.4.a.bd 1 20.d odd 2 1
1800.4.f.n 2 20.e even 4 2
1960.4.a.a 1 84.h odd 2 1

## Hecke kernels

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

 $$T_{7} - 18$$ T7 - 18 $$T_{11} + 16$$ T11 + 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T - 18$$
$11$ $$T + 16$$
$13$ $$T + 6$$
$17$ $$T - 6$$
$19$ $$T - 124$$
$23$ $$T - 42$$
$29$ $$T + 142$$
$31$ $$T - 188$$
$37$ $$T - 202$$
$41$ $$T + 54$$
$43$ $$T + 66$$
$47$ $$T - 38$$
$53$ $$T + 738$$
$59$ $$T - 564$$
$61$ $$T + 262$$
$67$ $$T - 554$$
$71$ $$T - 140$$
$73$ $$T - 882$$
$79$ $$T - 1160$$
$83$ $$T - 642$$
$89$ $$T - 854$$
$97$ $$T + 478$$