# Properties

 Label 7056.2.a.cy Level 7056 Weight 2 Character orbit 7056.a Self dual yes Analytic conductor 56.342 Analytic rank 0 Dimension 4 CM no 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{7})$$ Defining polynomial: $$x^{4} - 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 1764) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} +O(q^{10})$$ $$q + \beta_{2} q^{5} + \beta_{3} q^{11} -3 \beta_{1} q^{13} + \beta_{2} q^{17} -2 \beta_{1} q^{19} -\beta_{3} q^{23} + 9 q^{25} + \beta_{3} q^{29} -6 \beta_{1} q^{31} + 4 q^{37} + \beta_{2} q^{41} -8 q^{43} + 2 \beta_{2} q^{47} + 2 \beta_{3} q^{53} + 14 \beta_{1} q^{55} + 2 \beta_{2} q^{59} + 7 \beta_{1} q^{61} -3 \beta_{3} q^{65} -12 q^{67} + 3 \beta_{3} q^{71} + \beta_{1} q^{73} + 4 q^{79} + 4 \beta_{2} q^{83} + 14 q^{85} -\beta_{2} q^{89} -2 \beta_{3} q^{95} -7 \beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 36q^{25} + 16q^{37} - 32q^{43} - 48q^{67} + 16q^{79} + 56q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 5 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 8$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{2} + 11 \beta_{1}$$$$)/2$$

## 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
 −1.16372 −2.57794 1.16372 2.57794
0 0 0 −3.74166 0 0 0 0 0
1.2 0 0 0 −3.74166 0 0 0 0 0
1.3 0 0 0 3.74166 0 0 0 0 0
1.4 0 0 0 3.74166 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.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.cy 4
3.b odd 2 1 inner 7056.2.a.cy 4
4.b odd 2 1 1764.2.a.m 4
7.b odd 2 1 inner 7056.2.a.cy 4
12.b even 2 1 1764.2.a.m 4
21.c even 2 1 inner 7056.2.a.cy 4
28.d even 2 1 1764.2.a.m 4
28.f even 6 2 1764.2.k.m 8
28.g odd 6 2 1764.2.k.m 8
84.h odd 2 1 1764.2.a.m 4
84.j odd 6 2 1764.2.k.m 8
84.n even 6 2 1764.2.k.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.a.m 4 4.b odd 2 1
1764.2.a.m 4 12.b even 2 1
1764.2.a.m 4 28.d even 2 1
1764.2.a.m 4 84.h odd 2 1
1764.2.k.m 8 28.f even 6 2
1764.2.k.m 8 28.g odd 6 2
1764.2.k.m 8 84.j odd 6 2
1764.2.k.m 8 84.n even 6 2
7056.2.a.cy 4 1.a even 1 1 trivial
7056.2.a.cy 4 3.b odd 2 1 inner
7056.2.a.cy 4 7.b odd 2 1 inner
7056.2.a.cy 4 21.c even 2 1 inner

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$1$$

## Hecke kernels

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

 $$T_{5}^{2} - 14$$ $$T_{11}^{2} - 28$$ $$T_{13}^{2} - 18$$ $$T_{17}^{2} - 14$$ $$T_{23}^{2} - 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 4 T^{2} + 25 T^{4} )^{2}$$
$7$ 1
$11$ $$( 1 - 6 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 8 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 + 20 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 30 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 + 18 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 30 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 - 10 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 4 T + 37 T^{2} )^{4}$$
$41$ $$( 1 + 68 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 + 8 T + 43 T^{2} )^{4}$$
$47$ $$( 1 + 38 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 6 T^{2} + 2809 T^{4} )^{2}$$
$59$ $$( 1 + 62 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 24 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 + 12 T + 67 T^{2} )^{4}$$
$71$ $$( 1 - 110 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 + 144 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 4 T + 79 T^{2} )^{4}$$
$83$ $$( 1 - 58 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 + 164 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 + 96 T^{2} + 9409 T^{4} )^{2}$$