# Properties

 Label 7056.2.b.x Level $7056$ Weight $2$ Character orbit 7056.b Analytic conductor $56.342$ Analytic rank $0$ Dimension $8$ 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$56.3424436662$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.339738624.1 Defining polynomial: $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 2352) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -\beta_{4} + \beta_{6} ) q^{5} +O(q^{10})$$ $$q + ( -\beta_{4} + \beta_{6} ) q^{5} + ( \beta_{2} + \beta_{4} - \beta_{6} ) q^{11} + ( 2 \beta_{2} + \beta_{4} ) q^{13} + ( 2 \beta_{2} - \beta_{4} + \beta_{6} ) q^{17} + ( -2 \beta_{1} + \beta_{3} ) q^{19} + ( -\beta_{2} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{23} + ( -3 - \beta_{1} + 2 \beta_{5} ) q^{25} + ( -2 + \beta_{1} + \beta_{3} ) q^{29} + ( 4 - 2 \beta_{1} - \beta_{5} ) q^{31} + ( \beta_{1} + 2 \beta_{3} ) q^{37} + ( -\beta_{2} - \beta_{6} + \beta_{7} ) q^{41} + ( -2 \beta_{2} + 6 \beta_{4} + \beta_{7} ) q^{43} + ( -2 + 2 \beta_{1} - \beta_{5} ) q^{47} + ( -2 - 2 \beta_{1} - 2 \beta_{5} ) q^{53} + ( 8 + 2 \beta_{1} - \beta_{3} - 2 \beta_{5} ) q^{55} + ( 6 + 2 \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{59} + ( -\beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{61} + ( 2 + 3 \beta_{1} - 2 \beta_{3} - \beta_{5} ) q^{65} + ( 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{7} ) q^{67} + ( 3 \beta_{2} + 3 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{71} + ( 2 \beta_{2} - \beta_{4} - 2 \beta_{6} + 2 \beta_{7} ) q^{73} + ( -4 \beta_{2} - 4 \beta_{4} - \beta_{7} ) q^{79} + ( -2 \beta_{1} - 3 \beta_{3} - \beta_{5} ) q^{83} + ( -8 + \beta_{1} - 2 \beta_{3} + 2 \beta_{5} ) q^{85} + ( 2 \beta_{2} + 7 \beta_{4} + 3 \beta_{6} + 2 \beta_{7} ) q^{89} + ( 8 \beta_{2} + 2 \beta_{4} + 3 \beta_{7} ) q^{95} + ( -6 \beta_{2} + \beta_{4} + 2 \beta_{6} - 2 \beta_{7} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q + O(q^{10})$$ $$8q - 24q^{25} - 16q^{29} + 32q^{31} - 16q^{47} - 16q^{53} + 64q^{55} + 48q^{59} + 16q^{65} - 64q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{6} + 20$$$$)/14$$ $$\beta_{2}$$ $$=$$ $$($$$$-2 \nu^{7} + 7 \nu^{5} - 28 \nu^{3} + 2 \nu$$$$)/14$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{7} + 7 \nu^{5} - 21 \nu^{3} + 22 \nu$$$$)/7$$ $$\beta_{4}$$ $$=$$ $$($$$$-6 \nu^{7} + 21 \nu^{5} - 70 \nu^{3} + 6 \nu$$$$)/14$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 14 \nu^{5} - 49 \nu^{3} + 52 \nu$$$$)/7$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{6} + 4 \nu^{4} - 12 \nu^{2} + 4$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$4 \nu^{6} - 14 \nu^{4} + 56 \nu^{2} - 18$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{5} - \beta_{3} - 2 \beta_{2}$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{7} + \beta_{6} - \beta_{1} + 2$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{4} - 3 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$($$$$3 \beta_{7} + 4 \beta_{6} + 4 \beta_{1} - 6$$$$)/2$$ $$\nu^{5}$$ $$=$$ $$($$$$-3 \beta_{5} + 4 \beta_{4} + 7 \beta_{3} - 10 \beta_{2}$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$14 \beta_{1} - 20$$ $$\nu^{7}$$ $$=$$ $$-5 \beta_{5} - 7 \beta_{4} + 12 \beta_{3} + 17 \beta_{2}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7056\mathbb{Z}\right)^\times$$.

 $$n$$ $$785$$ $$1765$$ $$4609$$ $$6175$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1567.1
 −0.662827 − 0.382683i −1.60021 + 0.923880i 1.60021 − 0.923880i 0.662827 + 0.382683i 0.662827 − 0.382683i 1.60021 + 0.923880i −1.60021 − 0.923880i −0.662827 + 0.382683i
0 0 0 4.29725i 0 0 0 0 0
1567.2 0 0 0 3.21486i 0 0 0 0 0
1567.3 0 0 0 1.68412i 0 0 0 0 0
1567.4 0 0 0 0.601731i 0 0 0 0 0
1567.5 0 0 0 0.601731i 0 0 0 0 0
1567.6 0 0 0 1.68412i 0 0 0 0 0
1567.7 0 0 0 3.21486i 0 0 0 0 0
1567.8 0 0 0 4.29725i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1567.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d 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.b.x 8
3.b odd 2 1 2352.2.b.k 8
4.b odd 2 1 7056.2.b.w 8
7.b odd 2 1 7056.2.b.w 8
12.b even 2 1 2352.2.b.l yes 8
21.c even 2 1 2352.2.b.l yes 8
21.g even 6 1 2352.2.bl.o 8
21.g even 6 1 2352.2.bl.q 8
21.h odd 6 1 2352.2.bl.r 8
21.h odd 6 1 2352.2.bl.t 8
28.d even 2 1 inner 7056.2.b.x 8
84.h odd 2 1 2352.2.b.k 8
84.j odd 6 1 2352.2.bl.r 8
84.j odd 6 1 2352.2.bl.t 8
84.n even 6 1 2352.2.bl.o 8
84.n even 6 1 2352.2.bl.q 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.b.k 8 3.b odd 2 1
2352.2.b.k 8 84.h odd 2 1
2352.2.b.l yes 8 12.b even 2 1
2352.2.b.l yes 8 21.c even 2 1
2352.2.bl.o 8 21.g even 6 1
2352.2.bl.o 8 84.n even 6 1
2352.2.bl.q 8 21.g even 6 1
2352.2.bl.q 8 84.n even 6 1
2352.2.bl.r 8 21.h odd 6 1
2352.2.bl.r 8 84.j odd 6 1
2352.2.bl.t 8 21.h odd 6 1
2352.2.bl.t 8 84.j odd 6 1
7056.2.b.w 8 4.b odd 2 1
7056.2.b.w 8 7.b odd 2 1
7056.2.b.x 8 1.a even 1 1 trivial
7056.2.b.x 8 28.d 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 $$S_{2}^{\mathrm{new}}(7056, [\chi])$$:

 $$T_{5}^{8} + 32 T_{5}^{6} + 284 T_{5}^{4} + 640 T_{5}^{2} + 196$$ $$T_{11}^{8} + 40 T_{11}^{6} + 392 T_{11}^{4} + 608 T_{11}^{2} + 16$$ $$T_{13}^{4} + 20 T_{13}^{2} + 98$$ $$T_{17}^{8} + 64 T_{17}^{6} + 860 T_{17}^{4} + 4160 T_{17}^{2} + 6724$$ $$T_{19}^{4} - 40 T_{19}^{2} - 96 T_{19} - 56$$ $$T_{31}^{4} - 16 T_{31}^{3} + 56 T_{31}^{2} + 160 T_{31} - 824$$ $$T_{53}^{4} + 8 T_{53}^{3} - 88 T_{53}^{2} - 32 T_{53} + 784$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$196 + 640 T^{2} + 284 T^{4} + 32 T^{6} + T^{8}$$
$7$ $$T^{8}$$
$11$ $$16 + 608 T^{2} + 392 T^{4} + 40 T^{6} + T^{8}$$
$13$ $$( 98 + 20 T^{2} + T^{4} )^{2}$$
$17$ $$6724 + 4160 T^{2} + 860 T^{4} + 64 T^{6} + T^{8}$$
$19$ $$( -56 - 96 T - 40 T^{2} + T^{4} )^{2}$$
$23$ $$38416 + 20384 T^{2} + 2312 T^{4} + 88 T^{6} + T^{8}$$
$29$ $$( 28 - 32 T - 4 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$31$ $$( -824 + 160 T + 56 T^{2} - 16 T^{3} + T^{4} )^{2}$$
$37$ $$( 964 + 192 T - 100 T^{2} + T^{4} )^{2}$$
$41$ $$6724 + 10912 T^{2} + 1820 T^{4} + 80 T^{6} + T^{8}$$
$43$ $$20214016 + 1679104 T^{2} + 42848 T^{4} + 368 T^{6} + T^{8}$$
$47$ $$( -392 - 224 T - 16 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$53$ $$( 784 - 32 T - 88 T^{2} + 8 T^{3} + T^{4} )^{2}$$
$59$ $$( -8456 + 1440 T + 80 T^{2} - 24 T^{3} + T^{4} )^{2}$$
$61$ $$9604 + 292432 T^{2} + 23636 T^{4} + 296 T^{6} + T^{8}$$
$67$ $$802816 + 327680 T^{2} + 18176 T^{4} + 256 T^{6} + T^{8}$$
$71$ $$10265616 + 1314144 T^{2} + 39816 T^{4} + 360 T^{6} + T^{8}$$
$73$ $$4866436 + 1008272 T^{2} + 33236 T^{4} + 328 T^{6} + T^{8}$$
$79$ $$430336 + 592640 T^{2} + 24416 T^{4} + 304 T^{6} + T^{8}$$
$83$ $$( 12256 - 192 T - 256 T^{2} + T^{4} )^{2}$$
$89$ $$1766857156 + 34989632 T^{2} + 257180 T^{4} + 832 T^{6} + T^{8}$$
$97$ $$325658116 + 10546576 T^{2} + 121364 T^{4} + 584 T^{6} + T^{8}$$