# Properties

 Label 6720.2.a.e 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

# Learn more about

## 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

 $$f(q)$$ $$=$$ $$q - q^{3} - q^{5} - q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} - q^{5} - q^{7} + q^{9} - 2q^{11} - 4q^{13} + q^{15} + 2q^{17} + 2q^{19} + q^{21} + 4q^{23} + q^{25} - q^{27} + 2q^{29} - 6q^{31} + 2q^{33} + q^{35} + 6q^{37} + 4q^{39} + 6q^{41} + 4q^{43} - q^{45} + q^{49} - 2q^{51} - 8q^{53} + 2q^{55} - 2q^{57} + 10q^{61} - q^{63} + 4q^{65} + 12q^{67} - 4q^{69} - 14q^{71} + 4q^{73} - q^{75} + 2q^{77} - 8q^{79} + q^{81} - 12q^{83} - 2q^{85} - 2q^{87} - 14q^{89} + 4q^{91} + 6q^{93} - 2q^{95} + 8q^{97} - 2q^{99} + O(q^{100})$$

## 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

## 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.e 1
4.b odd 2 1 6720.2.a.bs 1
8.b even 2 1 420.2.a.d 1
8.d odd 2 1 1680.2.a.i 1
24.f even 2 1 5040.2.a.r 1
24.h odd 2 1 1260.2.a.a 1
40.e odd 2 1 8400.2.a.bq 1
40.f even 2 1 2100.2.a.h 1
40.i odd 4 2 2100.2.k.h 2
56.h odd 2 1 2940.2.a.c 1
56.j odd 6 2 2940.2.q.l 2
56.p even 6 2 2940.2.q.b 2
120.i odd 2 1 6300.2.a.u 1
120.w even 4 2 6300.2.k.f 2
168.i even 2 1 8820.2.a.s 1

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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$

## Hecke kernels

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

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ 
$3$ $$1 + T$$
$5$ $$1 + T$$
$7$ $$1 + T$$
$11$ $$1 + 2 T + 11 T^{2}$$
$13$ $$1 + 4 T + 13 T^{2}$$
$17$ $$1 - 2 T + 17 T^{2}$$
$19$ $$1 - 2 T + 19 T^{2}$$
$23$ $$1 - 4 T + 23 T^{2}$$
$29$ $$1 - 2 T + 29 T^{2}$$
$31$ $$1 + 6 T + 31 T^{2}$$
$37$ $$1 - 6 T + 37 T^{2}$$
$41$ $$1 - 6 T + 41 T^{2}$$
$43$ $$1 - 4 T + 43 T^{2}$$
$47$ $$1 + 47 T^{2}$$
$53$ $$1 + 8 T + 53 T^{2}$$
$59$ $$1 + 59 T^{2}$$
$61$ $$1 - 10 T + 61 T^{2}$$
$67$ $$1 - 12 T + 67 T^{2}$$
$71$ $$1 + 14 T + 71 T^{2}$$
$73$ $$1 - 4 T + 73 T^{2}$$
$79$ $$1 + 8 T + 79 T^{2}$$
$83$ $$1 + 12 T + 83 T^{2}$$
$89$ $$1 + 14 T + 89 T^{2}$$
$97$ $$1 - 8 T + 97 T^{2}$$
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