# Properties

 Label 720.2.a.j Level $720$ Weight $2$ Character orbit 720.a Self dual yes Analytic conductor $5.749$ 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: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{5} + 4q^{7} + O(q^{10})$$ $$q + q^{5} + 4q^{7} + 2q^{13} - 6q^{17} + 4q^{19} + q^{25} + 6q^{29} - 8q^{31} + 4q^{35} + 2q^{37} + 6q^{41} + 4q^{43} + 9q^{49} + 6q^{53} - 10q^{61} + 2q^{65} + 4q^{67} + 2q^{73} - 8q^{79} + 12q^{83} - 6q^{85} - 18q^{89} + 8q^{91} + 4q^{95} + 2q^{97} + O(q^{100})$$

## 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 1.00000 0 4.00000 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 720.2.a.j 1
3.b odd 2 1 240.2.a.b 1
4.b odd 2 1 90.2.a.c 1
5.b even 2 1 3600.2.a.f 1
5.c odd 4 2 3600.2.f.i 2
8.b even 2 1 2880.2.a.q 1
8.d odd 2 1 2880.2.a.a 1
12.b even 2 1 30.2.a.a 1
15.d odd 2 1 1200.2.a.k 1
15.e even 4 2 1200.2.f.e 2
20.d odd 2 1 450.2.a.d 1
20.e even 4 2 450.2.c.b 2
24.f even 2 1 960.2.a.e 1
24.h odd 2 1 960.2.a.p 1
28.d even 2 1 4410.2.a.z 1
36.f odd 6 2 810.2.e.b 2
36.h even 6 2 810.2.e.l 2
48.i odd 4 2 3840.2.k.f 2
48.k even 4 2 3840.2.k.y 2
60.h even 2 1 150.2.a.b 1
60.l odd 4 2 150.2.c.a 2
84.h odd 2 1 1470.2.a.d 1
84.j odd 6 2 1470.2.i.q 2
84.n even 6 2 1470.2.i.o 2
120.i odd 2 1 4800.2.a.d 1
120.m even 2 1 4800.2.a.cq 1
120.q odd 4 2 4800.2.f.p 2
120.w even 4 2 4800.2.f.w 2
132.d odd 2 1 3630.2.a.w 1
156.h even 2 1 5070.2.a.w 1
156.l odd 4 2 5070.2.b.k 2
204.h even 2 1 8670.2.a.g 1
420.o odd 2 1 7350.2.a.ct 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 12.b even 2 1
90.2.a.c 1 4.b odd 2 1
150.2.a.b 1 60.h even 2 1
150.2.c.a 2 60.l odd 4 2
240.2.a.b 1 3.b odd 2 1
450.2.a.d 1 20.d odd 2 1
450.2.c.b 2 20.e even 4 2
720.2.a.j 1 1.a even 1 1 trivial
810.2.e.b 2 36.f odd 6 2
810.2.e.l 2 36.h even 6 2
960.2.a.e 1 24.f even 2 1
960.2.a.p 1 24.h odd 2 1
1200.2.a.k 1 15.d odd 2 1
1200.2.f.e 2 15.e even 4 2
1470.2.a.d 1 84.h odd 2 1
1470.2.i.o 2 84.n even 6 2
1470.2.i.q 2 84.j odd 6 2
2880.2.a.a 1 8.d odd 2 1
2880.2.a.q 1 8.b even 2 1
3600.2.a.f 1 5.b even 2 1
3600.2.f.i 2 5.c odd 4 2
3630.2.a.w 1 132.d odd 2 1
3840.2.k.f 2 48.i odd 4 2
3840.2.k.y 2 48.k even 4 2
4410.2.a.z 1 28.d even 2 1
4800.2.a.d 1 120.i odd 2 1
4800.2.a.cq 1 120.m even 2 1
4800.2.f.p 2 120.q odd 4 2
4800.2.f.w 2 120.w even 4 2
5070.2.a.w 1 156.h even 2 1
5070.2.b.k 2 156.l odd 4 2
7350.2.a.ct 1 420.o odd 2 1
8670.2.a.g 1 204.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(720))$$:

 $$T_{7} - 4$$ $$T_{11}$$ $$T_{13} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$-1 + T$$
$7$ $$-4 + T$$
$11$ $$T$$
$13$ $$-2 + T$$
$17$ $$6 + T$$
$19$ $$-4 + T$$
$23$ $$T$$
$29$ $$-6 + T$$
$31$ $$8 + T$$
$37$ $$-2 + T$$
$41$ $$-6 + T$$
$43$ $$-4 + T$$
$47$ $$T$$
$53$ $$-6 + T$$
$59$ $$T$$
$61$ $$10 + T$$
$67$ $$-4 + T$$
$71$ $$T$$
$73$ $$-2 + T$$
$79$ $$8 + T$$
$83$ $$-12 + T$$
$89$ $$18 + T$$
$97$ $$-2 + T$$