# Properties

 Label 7056.2.a.j 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

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

## 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 −2.00000 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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 7056.2.a.j 1
3.b odd 2 1 7056.2.a.bv 1
4.b odd 2 1 3528.2.a.g 1
7.b odd 2 1 1008.2.a.i 1
12.b even 2 1 3528.2.a.t 1
21.c even 2 1 1008.2.a.c 1
28.d even 2 1 504.2.a.g yes 1
28.f even 6 2 3528.2.s.c 2
28.g odd 6 2 3528.2.s.s 2
56.e even 2 1 4032.2.a.j 1
56.h odd 2 1 4032.2.a.i 1
84.h odd 2 1 504.2.a.d 1
84.j odd 6 2 3528.2.s.z 2
84.n even 6 2 3528.2.s.k 2
168.e odd 2 1 4032.2.a.bl 1
168.i even 2 1 4032.2.a.ba 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.a.d 1 84.h odd 2 1
504.2.a.g yes 1 28.d even 2 1
1008.2.a.c 1 21.c even 2 1
1008.2.a.i 1 7.b odd 2 1
3528.2.a.g 1 4.b odd 2 1
3528.2.a.t 1 12.b even 2 1
3528.2.s.c 2 28.f even 6 2
3528.2.s.k 2 84.n even 6 2
3528.2.s.s 2 28.g odd 6 2
3528.2.s.z 2 84.j odd 6 2
4032.2.a.i 1 56.h odd 2 1
4032.2.a.j 1 56.e even 2 1
4032.2.a.ba 1 168.i even 2 1
4032.2.a.bl 1 168.e odd 2 1
7056.2.a.j 1 1.a even 1 1 trivial
7056.2.a.bv 1 3.b odd 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

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