Properties

 Label 720.4.a.k 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 20) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - 5 q^{5} + 16 q^{7}+O(q^{10})$$ q - 5 * q^5 + 16 * q^7 $$q - 5 q^{5} + 16 q^{7} - 60 q^{11} + 86 q^{13} - 18 q^{17} - 44 q^{19} + 48 q^{23} + 25 q^{25} + 186 q^{29} - 176 q^{31} - 80 q^{35} + 254 q^{37} - 186 q^{41} + 100 q^{43} + 168 q^{47} - 87 q^{49} + 498 q^{53} + 300 q^{55} - 252 q^{59} - 58 q^{61} - 430 q^{65} + 1036 q^{67} + 168 q^{71} + 506 q^{73} - 960 q^{77} - 272 q^{79} + 948 q^{83} + 90 q^{85} + 1014 q^{89} + 1376 q^{91} + 220 q^{95} - 766 q^{97}+O(q^{100})$$ q - 5 * q^5 + 16 * q^7 - 60 * q^11 + 86 * q^13 - 18 * q^17 - 44 * q^19 + 48 * q^23 + 25 * q^25 + 186 * q^29 - 176 * q^31 - 80 * q^35 + 254 * q^37 - 186 * q^41 + 100 * q^43 + 168 * q^47 - 87 * q^49 + 498 * q^53 + 300 * q^55 - 252 * q^59 - 58 * q^61 - 430 * q^65 + 1036 * q^67 + 168 * q^71 + 506 * q^73 - 960 * q^77 - 272 * q^79 + 948 * q^83 + 90 * q^85 + 1014 * q^89 + 1376 * q^91 + 220 * q^95 - 766 * 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 16.0000 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.k 1
3.b odd 2 1 80.4.a.c 1
4.b odd 2 1 180.4.a.a 1
12.b even 2 1 20.4.a.a 1
15.d odd 2 1 400.4.a.o 1
15.e even 4 2 400.4.c.j 2
20.d odd 2 1 900.4.a.m 1
20.e even 4 2 900.4.d.k 2
24.f even 2 1 320.4.a.d 1
24.h odd 2 1 320.4.a.k 1
36.f odd 6 2 1620.4.i.j 2
36.h even 6 2 1620.4.i.d 2
48.i odd 4 2 1280.4.d.c 2
48.k even 4 2 1280.4.d.n 2
60.h even 2 1 100.4.a.a 1
60.l odd 4 2 100.4.c.a 2
84.h odd 2 1 980.4.a.c 1
84.j odd 6 2 980.4.i.n 2
84.n even 6 2 980.4.i.e 2
120.i odd 2 1 1600.4.a.p 1
120.m even 2 1 1600.4.a.bl 1
132.d odd 2 1 2420.4.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.4.a.a 1 12.b even 2 1
80.4.a.c 1 3.b odd 2 1
100.4.a.a 1 60.h even 2 1
100.4.c.a 2 60.l odd 4 2
180.4.a.a 1 4.b odd 2 1
320.4.a.d 1 24.f even 2 1
320.4.a.k 1 24.h odd 2 1
400.4.a.o 1 15.d odd 2 1
400.4.c.j 2 15.e even 4 2
720.4.a.k 1 1.a even 1 1 trivial
900.4.a.m 1 20.d odd 2 1
900.4.d.k 2 20.e even 4 2
980.4.a.c 1 84.h odd 2 1
980.4.i.e 2 84.n even 6 2
980.4.i.n 2 84.j odd 6 2
1280.4.d.c 2 48.i odd 4 2
1280.4.d.n 2 48.k even 4 2
1600.4.a.p 1 120.i odd 2 1
1600.4.a.bl 1 120.m even 2 1
1620.4.i.d 2 36.h even 6 2
1620.4.i.j 2 36.f odd 6 2
2420.4.a.d 1 132.d odd 2 1

Hecke kernels

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

 $$T_{7} - 16$$ T7 - 16 $$T_{11} + 60$$ T11 + 60

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 16$$
$11$ $$T + 60$$
$13$ $$T - 86$$
$17$ $$T + 18$$
$19$ $$T + 44$$
$23$ $$T - 48$$
$29$ $$T - 186$$
$31$ $$T + 176$$
$37$ $$T - 254$$
$41$ $$T + 186$$
$43$ $$T - 100$$
$47$ $$T - 168$$
$53$ $$T - 498$$
$59$ $$T + 252$$
$61$ $$T + 58$$
$67$ $$T - 1036$$
$71$ $$T - 168$$
$73$ $$T - 506$$
$79$ $$T + 272$$
$83$ $$T - 948$$
$89$ $$T - 1014$$
$97$ $$T + 766$$