Properties

 Label 7056.2.a.bo 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 56) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{5}+O(q^{10})$$ q + 2 * q^5 $$q + 2 q^{5} - 4 q^{11} - 2 q^{13} - 6 q^{17} + 8 q^{19} - q^{25} - 6 q^{29} + 8 q^{31} - 2 q^{37} + 2 q^{41} + 4 q^{43} + 8 q^{47} - 6 q^{53} - 8 q^{55} + 6 q^{61} - 4 q^{65} + 4 q^{67} - 8 q^{71} - 10 q^{73} - 16 q^{79} - 8 q^{83} - 12 q^{85} - 6 q^{89} + 16 q^{95} + 6 q^{97}+O(q^{100})$$ q + 2 * q^5 - 4 * q^11 - 2 * q^13 - 6 * q^17 + 8 * q^19 - q^25 - 6 * q^29 + 8 * q^31 - 2 * q^37 + 2 * q^41 + 4 * q^43 + 8 * q^47 - 6 * q^53 - 8 * q^55 + 6 * q^61 - 4 * q^65 + 4 * q^67 - 8 * q^71 - 10 * q^73 - 16 * q^79 - 8 * q^83 - 12 * q^85 - 6 * q^89 + 16 * q^95 + 6 * 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 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.bo 1
3.b odd 2 1 784.2.a.e 1
4.b odd 2 1 3528.2.a.x 1
7.b odd 2 1 1008.2.a.d 1
12.b even 2 1 392.2.a.d 1
21.c even 2 1 112.2.a.b 1
21.g even 6 2 784.2.i.e 2
21.h odd 6 2 784.2.i.g 2
24.f even 2 1 3136.2.a.q 1
24.h odd 2 1 3136.2.a.p 1
28.d even 2 1 504.2.a.c 1
28.f even 6 2 3528.2.s.t 2
28.g odd 6 2 3528.2.s.e 2
56.e even 2 1 4032.2.a.bb 1
56.h odd 2 1 4032.2.a.bk 1
60.h even 2 1 9800.2.a.u 1
84.h odd 2 1 56.2.a.a 1
84.j odd 6 2 392.2.i.c 2
84.n even 6 2 392.2.i.d 2
105.g even 2 1 2800.2.a.p 1
105.k odd 4 2 2800.2.g.p 2
168.e odd 2 1 448.2.a.d 1
168.i even 2 1 448.2.a.e 1
336.v odd 4 2 1792.2.b.i 2
336.y even 4 2 1792.2.b.d 2
420.o odd 2 1 1400.2.a.g 1
420.w even 4 2 1400.2.g.g 2
924.n even 2 1 6776.2.a.g 1
1092.d odd 2 1 9464.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
56.2.a.a 1 84.h odd 2 1
112.2.a.b 1 21.c even 2 1
392.2.a.d 1 12.b even 2 1
392.2.i.c 2 84.j odd 6 2
392.2.i.d 2 84.n even 6 2
448.2.a.d 1 168.e odd 2 1
448.2.a.e 1 168.i even 2 1
504.2.a.c 1 28.d even 2 1
784.2.a.e 1 3.b odd 2 1
784.2.i.e 2 21.g even 6 2
784.2.i.g 2 21.h odd 6 2
1008.2.a.d 1 7.b odd 2 1
1400.2.a.g 1 420.o odd 2 1
1400.2.g.g 2 420.w even 4 2
1792.2.b.d 2 336.y even 4 2
1792.2.b.i 2 336.v odd 4 2
2800.2.a.p 1 105.g even 2 1
2800.2.g.p 2 105.k odd 4 2
3136.2.a.p 1 24.h odd 2 1
3136.2.a.q 1 24.f even 2 1
3528.2.a.x 1 4.b odd 2 1
3528.2.s.e 2 28.g odd 6 2
3528.2.s.t 2 28.f even 6 2
4032.2.a.bb 1 56.e even 2 1
4032.2.a.bk 1 56.h odd 2 1
6776.2.a.g 1 924.n even 2 1
7056.2.a.bo 1 1.a even 1 1 trivial
9464.2.a.c 1 1092.d odd 2 1
9800.2.a.u 1 60.h even 2 1

Hecke kernels

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

 $$T_{5} - 2$$ T5 - 2 $$T_{11} + 4$$ T11 + 4 $$T_{13} + 2$$ T13 + 2 $$T_{17} + 6$$ T17 + 6 $$T_{23}$$ T23

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T$$
$5$ $$T - 2$$
$7$ $$T$$
$11$ $$T + 4$$
$13$ $$T + 2$$
$17$ $$T + 6$$
$19$ $$T - 8$$
$23$ $$T$$
$29$ $$T + 6$$
$31$ $$T - 8$$
$37$ $$T + 2$$
$41$ $$T - 2$$
$43$ $$T - 4$$
$47$ $$T - 8$$
$53$ $$T + 6$$
$59$ $$T$$
$61$ $$T - 6$$
$67$ $$T - 4$$
$71$ $$T + 8$$
$73$ $$T + 10$$
$79$ $$T + 16$$
$83$ $$T + 8$$
$89$ $$T + 6$$
$97$ $$T - 6$$