Properties

 Label 7056.2.a.bc Level $7056$ Weight $2$ Character orbit 7056.a Self dual yes Analytic conductor $56.342$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1176) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q+O(q^{10})$$ q $$q + 4 q^{13} - 4 q^{17} + 4 q^{19} + 4 q^{23} - 5 q^{25} - 2 q^{29} - 8 q^{31} - 6 q^{37} - 12 q^{41} - 4 q^{43} - 8 q^{47} - 6 q^{53} + 12 q^{59} + 4 q^{61} + 4 q^{67} - 12 q^{71} + 8 q^{73} + 16 q^{79} - 4 q^{83} - 4 q^{89} + 16 q^{97}+O(q^{100})$$ q + 4 * q^13 - 4 * q^17 + 4 * q^19 + 4 * q^23 - 5 * q^25 - 2 * q^29 - 8 * q^31 - 6 * q^37 - 12 * q^41 - 4 * q^43 - 8 * q^47 - 6 * q^53 + 12 * q^59 + 4 * q^61 + 4 * q^67 - 12 * q^71 + 8 * q^73 + 16 * q^79 - 4 * q^83 - 4 * q^89 + 16 * q^97

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

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.bc 1
3.b odd 2 1 2352.2.a.h 1
4.b odd 2 1 3528.2.a.n 1
7.b odd 2 1 7056.2.a.ba 1
12.b even 2 1 1176.2.a.h yes 1
21.c even 2 1 2352.2.a.r 1
21.g even 6 2 2352.2.q.h 2
21.h odd 6 2 2352.2.q.t 2
24.f even 2 1 9408.2.a.u 1
24.h odd 2 1 9408.2.a.ck 1
28.d even 2 1 3528.2.a.m 1
28.f even 6 2 3528.2.s.m 2
28.g odd 6 2 3528.2.s.n 2
84.h odd 2 1 1176.2.a.b 1
84.j odd 6 2 1176.2.q.h 2
84.n even 6 2 1176.2.q.c 2
168.e odd 2 1 9408.2.a.cl 1
168.i even 2 1 9408.2.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.b 1 84.h odd 2 1
1176.2.a.h yes 1 12.b even 2 1
1176.2.q.c 2 84.n even 6 2
1176.2.q.h 2 84.j odd 6 2
2352.2.a.h 1 3.b odd 2 1
2352.2.a.r 1 21.c even 2 1
2352.2.q.h 2 21.g even 6 2
2352.2.q.t 2 21.h odd 6 2
3528.2.a.m 1 28.d even 2 1
3528.2.a.n 1 4.b odd 2 1
3528.2.s.m 2 28.f even 6 2
3528.2.s.n 2 28.g odd 6 2
7056.2.a.ba 1 7.b odd 2 1
7056.2.a.bc 1 1.a even 1 1 trivial
9408.2.a.u 1 24.f even 2 1
9408.2.a.v 1 168.i even 2 1
9408.2.a.ck 1 24.h odd 2 1
9408.2.a.cl 1 168.e odd 2 1

Hecke kernels

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

 $$T_{5}$$ T5 $$T_{11}$$ T11 $$T_{13} - 4$$ T13 - 4 $$T_{17} + 4$$ T17 + 4 $$T_{23} - 4$$ T23 - 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T$$
$13$ $$T - 4$$
$17$ $$T + 4$$
$19$ $$T - 4$$
$23$ $$T - 4$$
$29$ $$T + 2$$
$31$ $$T + 8$$
$37$ $$T + 6$$
$41$ $$T + 12$$
$43$ $$T + 4$$
$47$ $$T + 8$$
$53$ $$T + 6$$
$59$ $$T - 12$$
$61$ $$T - 4$$
$67$ $$T - 4$$
$71$ $$T + 12$$
$73$ $$T - 8$$
$79$ $$T - 16$$
$83$ $$T + 4$$
$89$ $$T + 4$$
$97$ $$T - 16$$