# Properties

 Label 6720.2.a.x Level $6720$ Weight $2$ Character orbit 6720.a Self dual yes Analytic conductor $53.659$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$6720 = 2^{6} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6720.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$53.6594701583$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 420) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} + q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{5} + q^{7} + q^{9} - 6 q^{11} + 4 q^{13} - q^{15} + 6 q^{17} - 2 q^{19} - q^{21} + q^{25} - q^{27} - 6 q^{29} - 10 q^{31} + 6 q^{33} + q^{35} - 2 q^{37} - 4 q^{39} - 6 q^{41} + 4 q^{43} + q^{45} + q^{49} - 6 q^{51} + 12 q^{53} - 6 q^{55} + 2 q^{57} - 14 q^{61} + q^{63} + 4 q^{65} + 4 q^{67} + 6 q^{71} - 4 q^{73} - q^{75} - 6 q^{77} - 16 q^{79} + q^{81} + 12 q^{83} + 6 q^{85} + 6 q^{87} + 6 q^{89} + 4 q^{91} + 10 q^{93} - 2 q^{95} - 16 q^{97} - 6 q^{99} + O(q^{100})$$

## 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 −1.00000 0 1.00000 0 1.00000 0 1.00000 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
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 6720.2.a.x 1
4.b odd 2 1 6720.2.a.ch 1
8.b even 2 1 420.2.a.c 1
8.d odd 2 1 1680.2.a.a 1
24.f even 2 1 5040.2.a.bc 1
24.h odd 2 1 1260.2.a.i 1
40.e odd 2 1 8400.2.a.cj 1
40.f even 2 1 2100.2.a.d 1
40.i odd 4 2 2100.2.k.j 2
56.h odd 2 1 2940.2.a.f 1
56.j odd 6 2 2940.2.q.i 2
56.p even 6 2 2940.2.q.e 2
120.i odd 2 1 6300.2.a.a 1
120.w even 4 2 6300.2.k.a 2
168.i even 2 1 8820.2.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.a.c 1 8.b even 2 1
1260.2.a.i 1 24.h odd 2 1
1680.2.a.a 1 8.d odd 2 1
2100.2.a.d 1 40.f even 2 1
2100.2.k.j 2 40.i odd 4 2
2940.2.a.f 1 56.h odd 2 1
2940.2.q.e 2 56.p even 6 2
2940.2.q.i 2 56.j odd 6 2
5040.2.a.bc 1 24.f even 2 1
6300.2.a.a 1 120.i odd 2 1
6300.2.k.a 2 120.w even 4 2
6720.2.a.x 1 1.a even 1 1 trivial
6720.2.a.ch 1 4.b odd 2 1
8400.2.a.cj 1 40.e odd 2 1
8820.2.a.b 1 168.i even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6720))$$:

 $$T_{11} + 6$$ $$T_{13} - 4$$ $$T_{17} - 6$$ $$T_{19} + 2$$ $$T_{23}$$ $$T_{29} + 6$$ $$T_{31} + 10$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$1 + T$$
$5$ $$-1 + T$$
$7$ $$-1 + T$$
$11$ $$6 + T$$
$13$ $$-4 + T$$
$17$ $$-6 + T$$
$19$ $$2 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$10 + T$$
$37$ $$2 + T$$
$41$ $$6 + T$$
$43$ $$-4 + T$$
$47$ $$T$$
$53$ $$-12 + T$$
$59$ $$T$$
$61$ $$14 + T$$
$67$ $$-4 + T$$
$71$ $$-6 + T$$
$73$ $$4 + T$$
$79$ $$16 + T$$
$83$ $$-12 + T$$
$89$ $$-6 + T$$
$97$ $$16 + T$$
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