# Properties

 Label 7056.2.a.ct Level $7056$ Weight $2$ Character orbit 7056.a Self dual yes Analytic conductor $56.342$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 392) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{5} +O(q^{10})$$ $$q + 2 \beta q^{5} + 6 q^{11} + 4 \beta q^{13} + \beta q^{17} -3 \beta q^{19} + 4 q^{23} + 3 q^{25} + 6 q^{29} + 2 \beta q^{31} + 2 q^{37} -\beta q^{41} -10 q^{43} + 2 \beta q^{47} + 2 q^{53} + 12 \beta q^{55} -\beta q^{59} + 6 \beta q^{61} + 16 q^{65} -4 q^{67} -12 q^{71} + 7 \beta q^{73} + 4 q^{79} -\beta q^{83} + 4 q^{85} -3 \beta q^{89} -12 q^{95} -9 \beta q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 12q^{11} + 8q^{23} + 6q^{25} + 12q^{29} + 4q^{37} - 20q^{43} + 4q^{53} + 32q^{65} - 8q^{67} - 24q^{71} + 8q^{79} + 8q^{85} - 24q^{95} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 0 0 −2.82843 0 0 0 0 0
1.2 0 0 0 2.82843 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 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7056.2.a.ct 2
3.b odd 2 1 784.2.a.k 2
4.b odd 2 1 3528.2.a.be 2
7.b odd 2 1 inner 7056.2.a.ct 2
12.b even 2 1 392.2.a.g 2
21.c even 2 1 784.2.a.k 2
21.g even 6 2 784.2.i.n 4
21.h odd 6 2 784.2.i.n 4
24.f even 2 1 3136.2.a.bk 2
24.h odd 2 1 3136.2.a.bp 2
28.d even 2 1 3528.2.a.be 2
28.f even 6 2 3528.2.s.bj 4
28.g odd 6 2 3528.2.s.bj 4
60.h even 2 1 9800.2.a.bv 2
84.h odd 2 1 392.2.a.g 2
84.j odd 6 2 392.2.i.h 4
84.n even 6 2 392.2.i.h 4
168.e odd 2 1 3136.2.a.bk 2
168.i even 2 1 3136.2.a.bp 2
420.o odd 2 1 9800.2.a.bv 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.2.a.g 2 12.b even 2 1
392.2.a.g 2 84.h odd 2 1
392.2.i.h 4 84.j odd 6 2
392.2.i.h 4 84.n even 6 2
784.2.a.k 2 3.b odd 2 1
784.2.a.k 2 21.c even 2 1
784.2.i.n 4 21.g even 6 2
784.2.i.n 4 21.h odd 6 2
3136.2.a.bk 2 24.f even 2 1
3136.2.a.bk 2 168.e odd 2 1
3136.2.a.bp 2 24.h odd 2 1
3136.2.a.bp 2 168.i even 2 1
3528.2.a.be 2 4.b odd 2 1
3528.2.a.be 2 28.d even 2 1
3528.2.s.bj 4 28.f even 6 2
3528.2.s.bj 4 28.g odd 6 2
7056.2.a.ct 2 1.a even 1 1 trivial
7056.2.a.ct 2 7.b odd 2 1 inner
9800.2.a.bv 2 60.h even 2 1
9800.2.a.bv 2 420.o odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 8$$ $$T_{11} - 6$$ $$T_{13}^{2} - 32$$ $$T_{17}^{2} - 2$$ $$T_{23} - 4$$