# Properties

 Label 720.4.a.bd Level $720$ Weight $4$ Character orbit 720.a Self dual yes Analytic conductor $42.481$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

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## 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

 $$f(q)$$ $$=$$ $$q + 5 q^{5} + 34 q^{7}+O(q^{10})$$ q + 5 * q^5 + 34 * q^7 $$q + 5 q^{5} + 34 q^{7} + 16 q^{11} + 58 q^{13} + 70 q^{17} - 4 q^{19} - 134 q^{23} + 25 q^{25} + 242 q^{29} - 100 q^{31} + 170 q^{35} - 438 q^{37} + 138 q^{41} - 178 q^{43} + 22 q^{47} + 813 q^{49} - 162 q^{53} + 80 q^{55} - 268 q^{59} + 250 q^{61} + 290 q^{65} - 422 q^{67} - 852 q^{71} + 306 q^{73} + 544 q^{77} + 456 q^{79} + 434 q^{83} + 350 q^{85} + 726 q^{89} + 1972 q^{91} - 20 q^{95} + 1378 q^{97}+O(q^{100})$$ q + 5 * q^5 + 34 * q^7 + 16 * q^11 + 58 * q^13 + 70 * q^17 - 4 * q^19 - 134 * q^23 + 25 * q^25 + 242 * q^29 - 100 * q^31 + 170 * q^35 - 438 * q^37 + 138 * q^41 - 178 * q^43 + 22 * q^47 + 813 * q^49 - 162 * q^53 + 80 * q^55 - 268 * q^59 + 250 * q^61 + 290 * q^65 - 422 * q^67 - 852 * q^71 + 306 * q^73 + 544 * q^77 + 456 * q^79 + 434 * q^83 + 350 * q^85 + 726 * q^89 + 1972 * q^91 - 20 * q^95 + 1378 * 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.bd 1
3.b odd 2 1 80.4.a.e 1
4.b odd 2 1 360.4.a.h 1
12.b even 2 1 40.4.a.a 1
15.d odd 2 1 400.4.a.e 1
15.e even 4 2 400.4.c.f 2
20.d odd 2 1 1800.4.a.bi 1
20.e even 4 2 1800.4.f.j 2
24.f even 2 1 320.4.a.l 1
24.h odd 2 1 320.4.a.c 1
48.i odd 4 2 1280.4.d.a 2
48.k even 4 2 1280.4.d.p 2
60.h even 2 1 200.4.a.i 1
60.l odd 4 2 200.4.c.c 2
84.h odd 2 1 1960.4.a.h 1
120.i odd 2 1 1600.4.a.br 1
120.m even 2 1 1600.4.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.4.a.a 1 12.b even 2 1
80.4.a.e 1 3.b odd 2 1
200.4.a.i 1 60.h even 2 1
200.4.c.c 2 60.l odd 4 2
320.4.a.c 1 24.h odd 2 1
320.4.a.l 1 24.f even 2 1
360.4.a.h 1 4.b odd 2 1
400.4.a.e 1 15.d odd 2 1
400.4.c.f 2 15.e even 4 2
720.4.a.bd 1 1.a even 1 1 trivial
1280.4.d.a 2 48.i odd 4 2
1280.4.d.p 2 48.k even 4 2
1600.4.a.j 1 120.m even 2 1
1600.4.a.br 1 120.i odd 2 1
1800.4.a.bi 1 20.d odd 2 1
1800.4.f.j 2 20.e even 4 2
1960.4.a.h 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(720))$$:

 $$T_{7} - 34$$ T7 - 34 $$T_{11} - 16$$ T11 - 16

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 5$$
$7$ $$T - 34$$
$11$ $$T - 16$$
$13$ $$T - 58$$
$17$ $$T - 70$$
$19$ $$T + 4$$
$23$ $$T + 134$$
$29$ $$T - 242$$
$31$ $$T + 100$$
$37$ $$T + 438$$
$41$ $$T - 138$$
$43$ $$T + 178$$
$47$ $$T - 22$$
$53$ $$T + 162$$
$59$ $$T + 268$$
$61$ $$T - 250$$
$67$ $$T + 422$$
$71$ $$T + 852$$
$73$ $$T - 306$$
$79$ $$T - 456$$
$83$ $$T - 434$$
$89$ $$T - 726$$
$97$ $$T - 1378$$
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