Properties

 Label 7056.2.a.bz 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 42) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 3.00000 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.bz 1
3.b odd 2 1 2352.2.a.n 1
4.b odd 2 1 882.2.a.k 1
7.b odd 2 1 7056.2.a.g 1
7.d odd 6 2 1008.2.s.n 2
12.b even 2 1 294.2.a.a 1
21.c even 2 1 2352.2.a.m 1
21.g even 6 2 336.2.q.d 2
21.h odd 6 2 2352.2.q.m 2
24.f even 2 1 9408.2.a.db 1
24.h odd 2 1 9408.2.a.bm 1
28.d even 2 1 882.2.a.g 1
28.f even 6 2 126.2.g.b 2
28.g odd 6 2 882.2.g.b 2
60.h even 2 1 7350.2.a.cw 1
84.h odd 2 1 294.2.a.d 1
84.j odd 6 2 42.2.e.b 2
84.n even 6 2 294.2.e.f 2
168.e odd 2 1 9408.2.a.d 1
168.i even 2 1 9408.2.a.bu 1
168.ba even 6 2 1344.2.q.j 2
168.be odd 6 2 1344.2.q.v 2
252.n even 6 2 1134.2.h.a 2
252.r odd 6 2 1134.2.e.a 2
252.bj even 6 2 1134.2.e.p 2
252.bn odd 6 2 1134.2.h.p 2
420.o odd 2 1 7350.2.a.ce 1
420.be odd 6 2 1050.2.i.e 2
420.br even 12 4 1050.2.o.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.2.e.b 2 84.j odd 6 2
126.2.g.b 2 28.f even 6 2
294.2.a.a 1 12.b even 2 1
294.2.a.d 1 84.h odd 2 1
294.2.e.f 2 84.n even 6 2
336.2.q.d 2 21.g even 6 2
882.2.a.g 1 28.d even 2 1
882.2.a.k 1 4.b odd 2 1
882.2.g.b 2 28.g odd 6 2
1008.2.s.n 2 7.d odd 6 2
1050.2.i.e 2 420.be odd 6 2
1050.2.o.b 4 420.br even 12 4
1134.2.e.a 2 252.r odd 6 2
1134.2.e.p 2 252.bj even 6 2
1134.2.h.a 2 252.n even 6 2
1134.2.h.p 2 252.bn odd 6 2
1344.2.q.j 2 168.ba even 6 2
1344.2.q.v 2 168.be odd 6 2
2352.2.a.m 1 21.c even 2 1
2352.2.a.n 1 3.b odd 2 1
2352.2.q.m 2 21.h odd 6 2
7056.2.a.g 1 7.b odd 2 1
7056.2.a.bz 1 1.a even 1 1 trivial
7350.2.a.ce 1 420.o odd 2 1
7350.2.a.cw 1 60.h even 2 1
9408.2.a.d 1 168.e odd 2 1
9408.2.a.bm 1 24.h odd 2 1
9408.2.a.bu 1 168.i even 2 1
9408.2.a.db 1 24.f even 2 1

Hecke kernels

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

 $$T_{5} - 3$$ $$T_{11} - 3$$ $$T_{13} - 4$$ $$T_{17}$$ $$T_{23}$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 3 T + 5 T^{2}$$
$7$ 1
$11$ $$1 - 3 T + 11 T^{2}$$
$13$ $$1 - 4 T + 13 T^{2}$$
$17$ $$1 + 17 T^{2}$$
$19$ $$1 + 4 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 + 9 T + 29 T^{2}$$
$31$ $$1 + T + 31 T^{2}$$
$37$ $$1 - 8 T + 37 T^{2}$$
$41$ $$1 + 41 T^{2}$$
$43$ $$1 - 10 T + 43 T^{2}$$
$47$ $$1 - 6 T + 47 T^{2}$$
$53$ $$1 - 3 T + 53 T^{2}$$
$59$ $$1 + 3 T + 59 T^{2}$$
$61$ $$1 - 10 T + 61 T^{2}$$
$67$ $$1 - 10 T + 67 T^{2}$$
$71$ $$1 + 6 T + 71 T^{2}$$
$73$ $$1 + 2 T + 73 T^{2}$$
$79$ $$1 - T + 79 T^{2}$$
$83$ $$1 - 9 T + 83 T^{2}$$
$89$ $$1 - 6 T + 89 T^{2}$$
$97$ $$1 - T + 97 T^{2}$$