# Properties

 Label 7056.2.a.cw Level 7056 Weight 2 Character orbit 7056.a Self dual yes Analytic conductor 56.342 Analytic rank 0 Dimension 2 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1176) 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.

 $$f(q)$$ $$=$$ $$q + ( 2 + \beta ) q^{5} +O(q^{10})$$ $$q + ( 2 + \beta ) q^{5} + ( -2 + 2 \beta ) q^{11} + 3 \beta q^{13} + ( 6 + \beta ) q^{17} + ( 4 + 2 \beta ) q^{19} + ( -2 - 2 \beta ) q^{23} + ( 1 + 4 \beta ) q^{25} -2 \beta q^{29} -2 \beta q^{31} + ( 4 - 4 \beta ) q^{37} + ( 6 + 3 \beta ) q^{41} + 8 \beta q^{43} + ( -4 + 6 \beta ) q^{47} + 2 q^{53} + 2 \beta q^{55} -6 \beta q^{59} + ( -4 - 5 \beta ) q^{61} + ( 6 + 6 \beta ) q^{65} -8 \beta q^{67} + ( -2 - 6 \beta ) q^{71} + ( -12 + 3 \beta ) q^{73} + ( -8 - 4 \beta ) q^{79} + 4 q^{83} + ( 14 + 8 \beta ) q^{85} + ( 10 - 3 \beta ) q^{89} + ( 12 + 8 \beta ) q^{95} + ( -4 + 3 \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 4q^{5} + O(q^{10})$$ $$2q + 4q^{5} - 4q^{11} + 12q^{17} + 8q^{19} - 4q^{23} + 2q^{25} + 8q^{37} + 12q^{41} - 8q^{47} + 4q^{53} - 8q^{61} + 12q^{65} - 4q^{71} - 24q^{73} - 16q^{79} + 8q^{83} + 28q^{85} + 20q^{89} + 24q^{95} - 8q^{97} + O(q^{100})$$

## 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
 −1.41421 1.41421
0 0 0 0.585786 0 0 0 0 0
1.2 0 0 0 3.41421 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

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.cw 2
3.b odd 2 1 2352.2.a.z 2
4.b odd 2 1 3528.2.a.bm 2
7.b odd 2 1 7056.2.a.ce 2
12.b even 2 1 1176.2.a.m yes 2
21.c even 2 1 2352.2.a.bg 2
21.g even 6 2 2352.2.q.ba 4
21.h odd 6 2 2352.2.q.bg 4
24.f even 2 1 9408.2.a.dr 2
24.h odd 2 1 9408.2.a.ed 2
28.d even 2 1 3528.2.a.bc 2
28.f even 6 2 3528.2.s.bl 4
28.g odd 6 2 3528.2.s.bc 4
84.h odd 2 1 1176.2.a.l 2
84.j odd 6 2 1176.2.q.n 4
84.n even 6 2 1176.2.q.m 4
168.e odd 2 1 9408.2.a.dv 2
168.i even 2 1 9408.2.a.dh 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.2.a.l 2 84.h odd 2 1
1176.2.a.m yes 2 12.b even 2 1
1176.2.q.m 4 84.n even 6 2
1176.2.q.n 4 84.j odd 6 2
2352.2.a.z 2 3.b odd 2 1
2352.2.a.bg 2 21.c even 2 1
2352.2.q.ba 4 21.g even 6 2
2352.2.q.bg 4 21.h odd 6 2
3528.2.a.bc 2 28.d even 2 1
3528.2.a.bm 2 4.b odd 2 1
3528.2.s.bc 4 28.g odd 6 2
3528.2.s.bl 4 28.f even 6 2
7056.2.a.ce 2 7.b odd 2 1
7056.2.a.cw 2 1.a even 1 1 trivial
9408.2.a.dh 2 168.i even 2 1
9408.2.a.dr 2 24.f even 2 1
9408.2.a.dv 2 168.e odd 2 1
9408.2.a.ed 2 24.h odd 2 1

## 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(7056))$$:

 $$T_{5}^{2} - 4 T_{5} + 2$$ $$T_{11}^{2} + 4 T_{11} - 4$$ $$T_{13}^{2} - 18$$ $$T_{17}^{2} - 12 T_{17} + 34$$ $$T_{23}^{2} + 4 T_{23} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$1 - 4 T + 12 T^{2} - 20 T^{3} + 25 T^{4}$$
$7$ 1
$11$ $$1 + 4 T + 18 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$1 + 8 T^{2} + 169 T^{4}$$
$17$ $$1 - 12 T + 68 T^{2} - 204 T^{3} + 289 T^{4}$$
$19$ $$1 - 8 T + 46 T^{2} - 152 T^{3} + 361 T^{4}$$
$23$ $$1 + 4 T + 42 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 + 50 T^{2} + 841 T^{4}$$
$31$ $$1 + 54 T^{2} + 961 T^{4}$$
$37$ $$1 - 8 T + 58 T^{2} - 296 T^{3} + 1369 T^{4}$$
$41$ $$1 - 12 T + 100 T^{2} - 492 T^{3} + 1681 T^{4}$$
$43$ $$1 - 42 T^{2} + 1849 T^{4}$$
$47$ $$1 + 8 T + 38 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$( 1 - 2 T + 53 T^{2} )^{2}$$
$59$ $$1 + 46 T^{2} + 3481 T^{4}$$
$61$ $$1 + 8 T + 88 T^{2} + 488 T^{3} + 3721 T^{4}$$
$67$ $$1 + 6 T^{2} + 4489 T^{4}$$
$71$ $$1 + 4 T + 74 T^{2} + 284 T^{3} + 5041 T^{4}$$
$73$ $$1 + 24 T + 272 T^{2} + 1752 T^{3} + 5329 T^{4}$$
$79$ $$1 + 16 T + 190 T^{2} + 1264 T^{3} + 6241 T^{4}$$
$83$ $$( 1 - 4 T + 83 T^{2} )^{2}$$
$89$ $$1 - 20 T + 260 T^{2} - 1780 T^{3} + 7921 T^{4}$$
$97$ $$1 + 8 T + 192 T^{2} + 776 T^{3} + 9409 T^{4}$$