Properties

 Label 7056.2.a.z Level 7056 Weight 2 Character orbit 7056.a Self dual yes Analytic conductor 56.342 Analytic rank 1 Dimension 1 CM discriminant -3 Inner twists 2

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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 252) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + O(q^{10})$$ $$q - 5q^{13} - q^{19} - 5q^{25} + 11q^{31} + 11q^{37} + 13q^{43} - 14q^{61} - 5q^{67} - 17q^{73} - 17q^{79} - 14q^{97} + O(q^{100})$$

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 0 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.a.z 1
3.b odd 2 1 CM 7056.2.a.z 1
4.b odd 2 1 1764.2.a.d 1
7.b odd 2 1 7056.2.a.be 1
7.d odd 6 2 1008.2.s.i 2
12.b even 2 1 1764.2.a.d 1
21.c even 2 1 7056.2.a.be 1
21.g even 6 2 1008.2.s.i 2
28.d even 2 1 1764.2.a.f 1
28.f even 6 2 252.2.k.b 2
28.g odd 6 2 1764.2.k.f 2
84.h odd 2 1 1764.2.a.f 1
84.j odd 6 2 252.2.k.b 2
84.n even 6 2 1764.2.k.f 2
252.n even 6 2 2268.2.l.e 2
252.r odd 6 2 2268.2.i.c 2
252.bj even 6 2 2268.2.i.c 2
252.bn odd 6 2 2268.2.l.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.k.b 2 28.f even 6 2
252.2.k.b 2 84.j odd 6 2
1008.2.s.i 2 7.d odd 6 2
1008.2.s.i 2 21.g even 6 2
1764.2.a.d 1 4.b odd 2 1
1764.2.a.d 1 12.b even 2 1
1764.2.a.f 1 28.d even 2 1
1764.2.a.f 1 84.h odd 2 1
1764.2.k.f 2 28.g odd 6 2
1764.2.k.f 2 84.n even 6 2
2268.2.i.c 2 252.r odd 6 2
2268.2.i.c 2 252.bj even 6 2
2268.2.l.e 2 252.n even 6 2
2268.2.l.e 2 252.bn odd 6 2
7056.2.a.z 1 1.a even 1 1 trivial
7056.2.a.z 1 3.b odd 2 1 CM
7056.2.a.be 1 7.b odd 2 1
7056.2.a.be 1 21.c even 2 1

Hecke kernels

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

 $$T_{5}$$ $$T_{11}$$ $$T_{13} + 5$$ $$T_{17}$$ $$T_{23}$$

Hecke characteristic polynomials

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