# Properties

 Label 7056.2.k.f Level $7056$ Weight $2$ Character orbit 7056.k Analytic conductor $56.342$ Analytic rank $0$ Dimension $8$ 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.k (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$56.3424436662$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ x^8 - x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{6}\cdot 3^{4}$$ Twist minimal: no (minimal twist has level 126) 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.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{5}+O(q^{10})$$ q - b5 * q^5 $$q - \beta_{5} q^{5} + \beta_{7} q^{11} - \beta_{3} q^{13} + \beta_{6} q^{17} + ( - \beta_{3} + 2 \beta_1) q^{19} + \beta_{2} q^{23} + (2 \beta_{4} + 4) q^{25} + ( - \beta_{7} - \beta_{2}) q^{29} + (3 \beta_{3} - \beta_1) q^{31} + (\beta_{4} + 4) q^{37} + 2 \beta_{6} q^{41} + ( - \beta_{4} - 4) q^{43} - \beta_{6} q^{47} + ( - \beta_{7} - \beta_{2}) q^{53} + (3 \beta_{3} + 3 \beta_1) q^{55} + (\beta_{6} - 3 \beta_{5}) q^{59} + (\beta_{3} + 2 \beta_1) q^{61} + ( - 2 \beta_{7} + \beta_{2}) q^{65} + 10 q^{67} + ( - 2 \beta_{7} + \beta_{2}) q^{71} + ( - 2 \beta_{3} - 2 \beta_1) q^{73} + (\beta_{4} + 7) q^{79} + ( - \beta_{6} + \beta_{5}) q^{83} - \beta_{4} q^{85} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{89} - \beta_{2} q^{95} + (2 \beta_{3} - 5 \beta_1) q^{97}+O(q^{100})$$ q - b5 * q^5 + b7 * q^11 - b3 * q^13 + b6 * q^17 + (-b3 + 2*b1) * q^19 + b2 * q^23 + (2*b4 + 4) * q^25 + (-b7 - b2) * q^29 + (3*b3 - b1) * q^31 + (b4 + 4) * q^37 + 2*b6 * q^41 + (-b4 - 4) * q^43 - b6 * q^47 + (-b7 - b2) * q^53 + (3*b3 + 3*b1) * q^55 + (b6 - 3*b5) * q^59 + (b3 + 2*b1) * q^61 + (-2*b7 + b2) * q^65 + 10 * q^67 + (-2*b7 + b2) * q^71 + (-2*b3 - 2*b1) * q^73 + (b4 + 7) * q^79 + (-b6 + b5) * q^83 - b4 * q^85 + (-2*b6 - 2*b5) * q^89 - b2 * q^95 + (2*b3 - 5*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 32 q^{25} + 32 q^{37} - 32 q^{43} + 80 q^{67} + 56 q^{79}+O(q^{100})$$ 8 * q + 32 * q^25 + 32 * q^37 - 32 * q^43 + 80 * q^67 + 56 * q^79

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{24}^{4} - 1$$ 2*v^4 - 1 $$\beta_{2}$$ $$=$$ $$3\zeta_{24}^{5} + 3\zeta_{24}^{3} - 3\zeta_{24}$$ 3*v^5 + 3*v^3 - 3*v $$\beta_{3}$$ $$=$$ $$2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ 2*v^7 + v^5 - v^3 + v $$\beta_{4}$$ $$=$$ $$-3\zeta_{24}^{5} + 3\zeta_{24}^{3} + 3\zeta_{24}$$ -3*v^5 + 3*v^3 + 3*v $$\beta_{5}$$ $$=$$ $$-2\zeta_{24}^{7} - \zeta_{24}^{6} + \zeta_{24}^{5} + \zeta_{24}^{3} + 2\zeta_{24}^{2} + \zeta_{24}$$ -2*v^7 - v^6 + v^5 + v^3 + 2*v^2 + v $$\beta_{6}$$ $$=$$ $$2\zeta_{24}^{7} - 2\zeta_{24}^{6} - \zeta_{24}^{5} - \zeta_{24}^{3} + 4\zeta_{24}^{2} - \zeta_{24}$$ 2*v^7 - 2*v^6 - v^5 - v^3 + 4*v^2 - v $$\beta_{7}$$ $$=$$ $$-3\zeta_{24}^{6}$$ -3*v^6
 $$\zeta_{24}$$ $$=$$ $$( -\beta_{6} + 2\beta_{5} + \beta_{4} + 3\beta_{3} - \beta_{2} ) / 12$$ (-b6 + 2*b5 + b4 + 3*b3 - b2) / 12 $$\zeta_{24}^{2}$$ $$=$$ $$( -\beta_{7} + \beta_{6} + \beta_{5} ) / 6$$ (-b7 + b6 + b5) / 6 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{4} + \beta_{2} ) / 6$$ (b4 + b2) / 6 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta _1 + 1 ) / 2$$ (b1 + 1) / 2 $$\zeta_{24}^{5}$$ $$=$$ $$( -\beta_{6} + 2\beta_{5} - \beta_{4} + 3\beta_{3} + \beta_{2} ) / 12$$ (-b6 + 2*b5 - b4 + 3*b3 + b2) / 12 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{7} ) / 3$$ (-b7) / 3 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{6} - 2\beta_{5} + \beta_{4} + 3\beta_{3} + \beta_{2} ) / 12$$ (b6 - 2*b5 + b4 + 3*b3 + b2) / 12

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 0.965926 + 0.258819i 0.965926 − 0.258819i 0.258819 + 0.965926i 0.258819 − 0.965926i −0.965926 − 0.258819i −0.965926 + 0.258819i −0.258819 − 0.965926i −0.258819 + 0.965926i
0 0 0 −4.18154 0 0 0 0 0
881.2 0 0 0 −4.18154 0 0 0 0 0
881.3 0 0 0 −0.717439 0 0 0 0 0
881.4 0 0 0 −0.717439 0 0 0 0 0
881.5 0 0 0 0.717439 0 0 0 0 0
881.6 0 0 0 0.717439 0 0 0 0 0
881.7 0 0 0 4.18154 0 0 0 0 0
881.8 0 0 0 4.18154 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.k.f 8
3.b odd 2 1 inner 7056.2.k.f 8
4.b odd 2 1 882.2.d.a 8
7.b odd 2 1 inner 7056.2.k.f 8
7.c even 3 1 1008.2.bt.c 8
7.d odd 6 1 1008.2.bt.c 8
12.b even 2 1 882.2.d.a 8
21.c even 2 1 inner 7056.2.k.f 8
21.g even 6 1 1008.2.bt.c 8
21.h odd 6 1 1008.2.bt.c 8
28.d even 2 1 882.2.d.a 8
28.f even 6 1 126.2.k.a 8
28.f even 6 1 882.2.k.a 8
28.g odd 6 1 126.2.k.a 8
28.g odd 6 1 882.2.k.a 8
84.h odd 2 1 882.2.d.a 8
84.j odd 6 1 126.2.k.a 8
84.j odd 6 1 882.2.k.a 8
84.n even 6 1 126.2.k.a 8
84.n even 6 1 882.2.k.a 8
140.p odd 6 1 3150.2.bf.a 8
140.s even 6 1 3150.2.bf.a 8
140.w even 12 1 3150.2.bp.b 8
140.w even 12 1 3150.2.bp.e 8
140.x odd 12 1 3150.2.bp.b 8
140.x odd 12 1 3150.2.bp.e 8
252.n even 6 1 1134.2.t.e 8
252.o even 6 1 1134.2.t.e 8
252.r odd 6 1 1134.2.l.f 8
252.u odd 6 1 1134.2.l.f 8
252.bb even 6 1 1134.2.l.f 8
252.bj even 6 1 1134.2.l.f 8
252.bl odd 6 1 1134.2.t.e 8
252.bn odd 6 1 1134.2.t.e 8
420.ba even 6 1 3150.2.bf.a 8
420.be odd 6 1 3150.2.bf.a 8
420.bp odd 12 1 3150.2.bp.b 8
420.bp odd 12 1 3150.2.bp.e 8
420.br even 12 1 3150.2.bp.b 8
420.br even 12 1 3150.2.bp.e 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.2.k.a 8 28.f even 6 1
126.2.k.a 8 28.g odd 6 1
126.2.k.a 8 84.j odd 6 1
126.2.k.a 8 84.n even 6 1
882.2.d.a 8 4.b odd 2 1
882.2.d.a 8 12.b even 2 1
882.2.d.a 8 28.d even 2 1
882.2.d.a 8 84.h odd 2 1
882.2.k.a 8 28.f even 6 1
882.2.k.a 8 28.g odd 6 1
882.2.k.a 8 84.j odd 6 1
882.2.k.a 8 84.n even 6 1
1008.2.bt.c 8 7.c even 3 1
1008.2.bt.c 8 7.d odd 6 1
1008.2.bt.c 8 21.g even 6 1
1008.2.bt.c 8 21.h odd 6 1
1134.2.l.f 8 252.r odd 6 1
1134.2.l.f 8 252.u odd 6 1
1134.2.l.f 8 252.bb even 6 1
1134.2.l.f 8 252.bj even 6 1
1134.2.t.e 8 252.n even 6 1
1134.2.t.e 8 252.o even 6 1
1134.2.t.e 8 252.bl odd 6 1
1134.2.t.e 8 252.bn odd 6 1
3150.2.bf.a 8 140.p odd 6 1
3150.2.bf.a 8 140.s even 6 1
3150.2.bf.a 8 420.ba even 6 1
3150.2.bf.a 8 420.be odd 6 1
3150.2.bp.b 8 140.w even 12 1
3150.2.bp.b 8 140.x odd 12 1
3150.2.bp.b 8 420.bp odd 12 1
3150.2.bp.b 8 420.br even 12 1
3150.2.bp.e 8 140.w even 12 1
3150.2.bp.e 8 140.x odd 12 1
3150.2.bp.e 8 420.bp odd 12 1
3150.2.bp.e 8 420.br even 12 1
7056.2.k.f 8 1.a even 1 1 trivial
7056.2.k.f 8 3.b odd 2 1 inner
7056.2.k.f 8 7.b odd 2 1 inner
7056.2.k.f 8 21.c 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 $$S_{2}^{\mathrm{new}}(7056, [\chi])$$:

 $$T_{5}^{4} - 18T_{5}^{2} + 9$$ T5^4 - 18*T5^2 + 9 $$T_{11}^{2} + 9$$ T11^2 + 9 $$T_{13}^{2} + 6$$ T13^2 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 18 T^{2} + 9)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T^{2} + 9)^{4}$$
$13$ $$(T^{2} + 6)^{4}$$
$17$ $$(T^{4} - 36 T^{2} + 36)^{2}$$
$19$ $$(T^{4} + 36 T^{2} + 36)^{2}$$
$23$ $$(T^{2} + 18)^{4}$$
$29$ $$(T^{4} + 54 T^{2} + 81)^{2}$$
$31$ $$(T^{4} + 114 T^{2} + 2601)^{2}$$
$37$ $$(T^{2} - 8 T - 2)^{4}$$
$41$ $$(T^{4} - 144 T^{2} + 576)^{2}$$
$43$ $$(T^{2} + 8 T - 2)^{4}$$
$47$ $$(T^{4} - 36 T^{2} + 36)^{2}$$
$53$ $$(T^{4} + 54 T^{2} + 81)^{2}$$
$59$ $$(T^{4} - 198 T^{2} + 8649)^{2}$$
$61$ $$(T^{4} + 36 T^{2} + 36)^{2}$$
$67$ $$(T - 10)^{8}$$
$71$ $$(T^{4} + 108 T^{2} + 324)^{2}$$
$73$ $$(T^{4} + 72 T^{2} + 144)^{2}$$
$79$ $$(T^{2} - 14 T + 31)^{4}$$
$83$ $$(T^{4} - 54 T^{2} + 441)^{2}$$
$89$ $$(T^{2} - 108)^{4}$$
$97$ $$(T^{4} + 198 T^{2} + 2601)^{2}$$