# Properties

 Label 7056.2.a.cf Level $7056$ Weight $2$ Character orbit 7056.a Self dual yes Analytic conductor $56.342$ Analytic rank $1$ 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: $$1$$ 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 147) 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.

 $$f(q)$$ $$=$$ $$q + ( -2 + \beta ) q^{5} +O(q^{10})$$ $$q + ( -2 + \beta ) q^{5} -2 q^{11} + ( 4 + \beta ) q^{13} + ( -2 - 3 \beta ) q^{17} -2 \beta q^{19} + ( -2 + 4 \beta ) q^{23} + ( 1 - 4 \beta ) q^{25} + ( 4 - 2 \beta ) q^{29} + ( 4 + 2 \beta ) q^{31} -4 q^{37} + ( -2 + 3 \beta ) q^{41} + 4 \beta q^{43} + 2 \beta q^{47} + 2 q^{53} + ( 4 - 2 \beta ) q^{55} + ( -4 - 2 \beta ) q^{59} + ( 8 - 3 \beta ) q^{61} + ( -6 + 2 \beta ) q^{65} -4 \beta q^{67} + ( -2 - 8 \beta ) q^{71} + ( 4 - 7 \beta ) q^{73} + ( -8 + 4 \beta ) q^{79} + ( -4 - 8 \beta ) q^{83} + ( -2 + 4 \beta ) q^{85} + ( 10 - 3 \beta ) q^{89} + ( -4 + 4 \beta ) q^{95} + ( 4 + \beta ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{5} + O(q^{10})$$ $$2q - 4q^{5} - 4q^{11} + 8q^{13} - 4q^{17} - 4q^{23} + 2q^{25} + 8q^{29} + 8q^{31} - 8q^{37} - 4q^{41} + 4q^{53} + 8q^{55} - 8q^{59} + 16q^{61} - 12q^{65} - 4q^{71} + 8q^{73} - 16q^{79} - 8q^{83} - 4q^{85} + 20q^{89} - 8q^{95} + 8q^{97} + O(q^{100})$$

## 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 −3.41421 0 0 0 0 0
1.2 0 0 0 −0.585786 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.cf 2
3.b odd 2 1 2352.2.a.bc 2
4.b odd 2 1 441.2.a.i 2
7.b odd 2 1 7056.2.a.cv 2
12.b even 2 1 147.2.a.e yes 2
21.c even 2 1 2352.2.a.be 2
21.g even 6 2 2352.2.q.bb 4
21.h odd 6 2 2352.2.q.bd 4
24.f even 2 1 9408.2.a.di 2
24.h odd 2 1 9408.2.a.dt 2
28.d even 2 1 441.2.a.j 2
28.f even 6 2 441.2.e.f 4
28.g odd 6 2 441.2.e.g 4
60.h even 2 1 3675.2.a.bd 2
84.h odd 2 1 147.2.a.d 2
84.j odd 6 2 147.2.e.e 4
84.n even 6 2 147.2.e.d 4
168.e odd 2 1 9408.2.a.ef 2
168.i even 2 1 9408.2.a.dq 2
420.o odd 2 1 3675.2.a.bf 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.2.a.d 2 84.h odd 2 1
147.2.a.e yes 2 12.b even 2 1
147.2.e.d 4 84.n even 6 2
147.2.e.e 4 84.j odd 6 2
441.2.a.i 2 4.b odd 2 1
441.2.a.j 2 28.d even 2 1
441.2.e.f 4 28.f even 6 2
441.2.e.g 4 28.g odd 6 2
2352.2.a.bc 2 3.b odd 2 1
2352.2.a.be 2 21.c even 2 1
2352.2.q.bb 4 21.g even 6 2
2352.2.q.bd 4 21.h odd 6 2
3675.2.a.bd 2 60.h even 2 1
3675.2.a.bf 2 420.o odd 2 1
7056.2.a.cf 2 1.a even 1 1 trivial
7056.2.a.cv 2 7.b odd 2 1
9408.2.a.di 2 24.f even 2 1
9408.2.a.dq 2 168.i even 2 1
9408.2.a.dt 2 24.h odd 2 1
9408.2.a.ef 2 168.e odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 4 T_{5} + 2$$ $$T_{11} + 2$$ $$T_{13}^{2} - 8 T_{13} + 14$$ $$T_{17}^{2} + 4 T_{17} - 14$$ $$T_{23}^{2} + 4 T_{23} - 28$$

## 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
$11$ $$( 1 + 2 T + 11 T^{2} )^{2}$$
$13$ $$1 - 8 T + 40 T^{2} - 104 T^{3} + 169 T^{4}$$
$17$ $$1 + 4 T + 20 T^{2} + 68 T^{3} + 289 T^{4}$$
$19$ $$1 + 30 T^{2} + 361 T^{4}$$
$23$ $$1 + 4 T + 18 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$1 - 8 T + 66 T^{2} - 232 T^{3} + 841 T^{4}$$
$31$ $$1 - 8 T + 70 T^{2} - 248 T^{3} + 961 T^{4}$$
$37$ $$( 1 + 4 T + 37 T^{2} )^{2}$$
$41$ $$1 + 4 T + 68 T^{2} + 164 T^{3} + 1681 T^{4}$$
$43$ $$1 + 54 T^{2} + 1849 T^{4}$$
$47$ $$1 + 86 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 2 T + 53 T^{2} )^{2}$$
$59$ $$1 + 8 T + 126 T^{2} + 472 T^{3} + 3481 T^{4}$$
$61$ $$1 - 16 T + 168 T^{2} - 976 T^{3} + 3721 T^{4}$$
$67$ $$1 + 102 T^{2} + 4489 T^{4}$$
$71$ $$1 + 4 T + 18 T^{2} + 284 T^{3} + 5041 T^{4}$$
$73$ $$1 - 8 T + 64 T^{2} - 584 T^{3} + 5329 T^{4}$$
$79$ $$1 + 16 T + 190 T^{2} + 1264 T^{3} + 6241 T^{4}$$
$83$ $$1 + 8 T + 54 T^{2} + 664 T^{3} + 6889 T^{4}$$
$89$ $$1 - 20 T + 260 T^{2} - 1780 T^{3} + 7921 T^{4}$$
$97$ $$1 - 8 T + 208 T^{2} - 776 T^{3} + 9409 T^{4}$$