Properties

 Label 7056.2.a.co Level 7056 Weight 2 Character orbit 7056.a Self dual yes Analytic conductor 56.342 Analytic rank 1 Dimension 2 CM discriminant -7 Inner twists 4

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: $$2$$ Coefficient field: $$\Q(\sqrt{7})$$ Defining polynomial: $$x^{2} - 7$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 441) Fricke sign: $$1$$ Sato-Tate group: $N(\mathrm{U}(1))$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{7}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q +O(q^{10})$$ $$q + \beta q^{11} -\beta q^{23} -5 q^{25} -2 \beta q^{29} + 6 q^{37} -12 q^{43} -2 \beta q^{53} -4 q^{67} + \beta q^{71} -8 q^{79} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q - 10q^{25} + 12q^{37} - 24q^{43} - 8q^{67} - 16q^{79} + O(q^{100})$$

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
 −2.64575 2.64575
0 0 0 0 0 0 0 0 0
1.2 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
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
3.b odd 2 1 inner
21.c even 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.a.co 2
3.b odd 2 1 inner 7056.2.a.co 2
4.b odd 2 1 441.2.a.h 2
7.b odd 2 1 CM 7056.2.a.co 2
12.b even 2 1 441.2.a.h 2
21.c even 2 1 inner 7056.2.a.co 2
28.d even 2 1 441.2.a.h 2
28.f even 6 2 441.2.e.h 4
28.g odd 6 2 441.2.e.h 4
84.h odd 2 1 441.2.a.h 2
84.j odd 6 2 441.2.e.h 4
84.n even 6 2 441.2.e.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
441.2.a.h 2 4.b odd 2 1
441.2.a.h 2 12.b even 2 1
441.2.a.h 2 28.d even 2 1
441.2.a.h 2 84.h odd 2 1
441.2.e.h 4 28.f even 6 2
441.2.e.h 4 28.g odd 6 2
441.2.e.h 4 84.j odd 6 2
441.2.e.h 4 84.n even 6 2
7056.2.a.co 2 1.a even 1 1 trivial
7056.2.a.co 2 3.b odd 2 1 inner
7056.2.a.co 2 7.b odd 2 1 CM
7056.2.a.co 2 21.c even 2 1 inner

Hecke kernels

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

 $$T_{5}$$ $$T_{11}^{2} - 28$$ $$T_{13}$$ $$T_{17}$$ $$T_{23}^{2} - 28$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 + 5 T^{2} )^{2}$$
$7$ 1
$11$ $$1 - 6 T^{2} + 121 T^{4}$$
$13$ $$( 1 + 13 T^{2} )^{2}$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$( 1 + 19 T^{2} )^{2}$$
$23$ $$1 + 18 T^{2} + 529 T^{4}$$
$29$ $$1 - 54 T^{2} + 841 T^{4}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$( 1 - 6 T + 37 T^{2} )^{2}$$
$41$ $$( 1 + 41 T^{2} )^{2}$$
$43$ $$( 1 + 12 T + 43 T^{2} )^{2}$$
$47$ $$( 1 + 47 T^{2} )^{2}$$
$53$ $$1 - 6 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 59 T^{2} )^{2}$$
$61$ $$( 1 + 61 T^{2} )^{2}$$
$67$ $$( 1 + 4 T + 67 T^{2} )^{2}$$
$71$ $$1 + 114 T^{2} + 5041 T^{4}$$
$73$ $$( 1 + 73 T^{2} )^{2}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{2}$$
$89$ $$( 1 + 89 T^{2} )^{2}$$
$97$ $$( 1 + 97 T^{2} )^{2}$$