# Properties

 Label 7056.2.a.v Level 7056 Weight 2 Character orbit 7056.a Self dual yes Analytic conductor 56.342 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$7056 = 2^{4} \cdot 3^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 7056.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$56.3424436662$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 504) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{5} + O(q^{10})$$ $$q - q^{5} + 5q^{11} - 2q^{13} + 6q^{17} + 2q^{19} - 6q^{23} - 4q^{25} + 3q^{29} + 5q^{31} - 2q^{37} + 8q^{41} + 4q^{43} - 4q^{47} - 9q^{53} - 5q^{55} + 3q^{59} + 12q^{61} + 2q^{65} - 2q^{67} - 8q^{71} + 14q^{73} - q^{79} - 17q^{83} - 6q^{85} + 18q^{89} - 2q^{95} - 3q^{97} + O(q^{100})$$

## 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 0 0 0 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 7056.2.a.v 1
3.b odd 2 1 7056.2.a.bh 1
4.b odd 2 1 3528.2.a.h 1
7.b odd 2 1 7056.2.a.bm 1
7.d odd 6 2 1008.2.s.h 2
12.b even 2 1 3528.2.a.s 1
21.c even 2 1 7056.2.a.r 1
21.g even 6 2 1008.2.s.l 2
28.d even 2 1 3528.2.a.o 1
28.f even 6 2 504.2.s.b 2
28.g odd 6 2 3528.2.s.r 2
84.h odd 2 1 3528.2.a.l 1
84.j odd 6 2 504.2.s.f yes 2
84.n even 6 2 3528.2.s.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.s.b 2 28.f even 6 2
504.2.s.f yes 2 84.j odd 6 2
1008.2.s.h 2 7.d odd 6 2
1008.2.s.l 2 21.g even 6 2
3528.2.a.h 1 4.b odd 2 1
3528.2.a.l 1 84.h odd 2 1
3528.2.a.o 1 28.d even 2 1
3528.2.a.s 1 12.b even 2 1
3528.2.s.l 2 84.n even 6 2
3528.2.s.r 2 28.g odd 6 2
7056.2.a.r 1 21.c even 2 1
7056.2.a.v 1 1.a even 1 1 trivial
7056.2.a.bh 1 3.b odd 2 1
7056.2.a.bm 1 7.b odd 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(7056))$$:

 $$T_{5} + 1$$ $$T_{11} - 5$$ $$T_{13} + 2$$ $$T_{17} - 6$$ $$T_{23} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 + T + 5 T^{2}$$
$7$ 1
$11$ $$1 - 5 T + 11 T^{2}$$
$13$ $$1 + 2 T + 13 T^{2}$$
$17$ $$1 - 6 T + 17 T^{2}$$
$19$ $$1 - 2 T + 19 T^{2}$$
$23$ $$1 + 6 T + 23 T^{2}$$
$29$ $$1 - 3 T + 29 T^{2}$$
$31$ $$1 - 5 T + 31 T^{2}$$
$37$ $$1 + 2 T + 37 T^{2}$$
$41$ $$1 - 8 T + 41 T^{2}$$
$43$ $$1 - 4 T + 43 T^{2}$$
$47$ $$1 + 4 T + 47 T^{2}$$
$53$ $$1 + 9 T + 53 T^{2}$$
$59$ $$1 - 3 T + 59 T^{2}$$
$61$ $$1 - 12 T + 61 T^{2}$$
$67$ $$1 + 2 T + 67 T^{2}$$
$71$ $$1 + 8 T + 71 T^{2}$$
$73$ $$1 - 14 T + 73 T^{2}$$
$79$ $$1 + T + 79 T^{2}$$
$83$ $$1 + 17 T + 83 T^{2}$$
$89$ $$1 - 18 T + 89 T^{2}$$
$97$ $$1 + 3 T + 97 T^{2}$$