# Properties

 Label 7056.2.k.e 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: $$\Q(\zeta_{16})$$ Defining polynomial: $$x^{8} + 1$$ x^8 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 1764) 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 $$q - \beta_{5} q^{5} + \beta_1 q^{11} + (\beta_{4} + 2 \beta_{2}) q^{13} + 3 \beta_{7} q^{17} + 2 \beta_{4} q^{19} + (2 \beta_{3} + 3 \beta_1) q^{23} + (\beta_{6} - 3) q^{25} + ( - 2 \beta_{3} + 2 \beta_1) q^{29} + (2 \beta_{4} - 4 \beta_{2}) q^{31} + (3 \beta_{6} + 4) q^{37} + (\beta_{7} + 6 \beta_{5}) q^{41} + (2 \beta_{6} - 4) q^{43} + ( - 4 \beta_{7} + 6 \beta_{5}) q^{47} + (3 \beta_{3} - 4 \beta_1) q^{53} - 2 \beta_{2} q^{55} + ( - 8 \beta_{7} - 2 \beta_{5}) q^{59} + ( - 4 \beta_{4} - 5 \beta_{2}) q^{61} + ( - 3 \beta_{3} - 2 \beta_1) q^{65} + ( - 6 \beta_{6} - 4) q^{67} + (8 \beta_{3} + \beta_1) q^{71} + ( - 6 \beta_{4} + \beta_{2}) q^{73} - 8 \beta_{6} q^{79} + ( - 4 \beta_{7} - 4 \beta_{5}) q^{83} - 3 \beta_{6} q^{85} + ( - 2 \beta_{7} - 7 \beta_{5}) q^{89} - 2 \beta_{3} q^{95} + (6 \beta_{4} - \beta_{2}) q^{97}+O(q^{100})$$ q - b5 * q^5 + b1 * q^11 + (b4 + 2*b2) * q^13 + 3*b7 * q^17 + 2*b4 * q^19 + (2*b3 + 3*b1) * q^23 + (b6 - 3) * q^25 + (-2*b3 + 2*b1) * q^29 + (2*b4 - 4*b2) * q^31 + (3*b6 + 4) * q^37 + (b7 + 6*b5) * q^41 + (2*b6 - 4) * q^43 + (-4*b7 + 6*b5) * q^47 + (3*b3 - 4*b1) * q^53 - 2*b2 * q^55 + (-8*b7 - 2*b5) * q^59 + (-4*b4 - 5*b2) * q^61 + (-3*b3 - 2*b1) * q^65 + (-6*b6 - 4) * q^67 + (8*b3 + b1) * q^71 + (-6*b4 + b2) * q^73 - 8*b6 * q^79 + (-4*b7 - 4*b5) * q^83 - 3*b6 * q^85 + (-2*b7 - 7*b5) * q^89 - 2*b3 * q^95 + (6*b4 - b2) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q - 24 q^{25} + 32 q^{37} - 32 q^{43} - 32 q^{67}+O(q^{100})$$ 8 * q - 24 * q^25 + 32 * q^37 - 32 * q^43 - 32 * q^67

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{16}^{4}$$ 2*v^4 $$\beta_{2}$$ $$=$$ $$\zeta_{16}^{5} + \zeta_{16}^{3}$$ v^5 + v^3 $$\beta_{3}$$ $$=$$ $$\zeta_{16}^{6} + \zeta_{16}^{2}$$ v^6 + v^2 $$\beta_{4}$$ $$=$$ $$\zeta_{16}^{7} + \zeta_{16}$$ v^7 + v $$\beta_{5}$$ $$=$$ $$-\zeta_{16}^{7} + \zeta_{16}$$ -v^7 + v $$\beta_{6}$$ $$=$$ $$-\zeta_{16}^{6} + \zeta_{16}^{2}$$ -v^6 + v^2 $$\beta_{7}$$ $$=$$ $$-\zeta_{16}^{5} + \zeta_{16}^{3}$$ -v^5 + v^3
 $$\zeta_{16}$$ $$=$$ $$( \beta_{5} + \beta_{4} ) / 2$$ (b5 + b4) / 2 $$\zeta_{16}^{2}$$ $$=$$ $$( \beta_{6} + \beta_{3} ) / 2$$ (b6 + b3) / 2 $$\zeta_{16}^{3}$$ $$=$$ $$( \beta_{7} + \beta_{2} ) / 2$$ (b7 + b2) / 2 $$\zeta_{16}^{4}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{16}^{5}$$ $$=$$ $$( -\beta_{7} + \beta_{2} ) / 2$$ (-b7 + b2) / 2 $$\zeta_{16}^{6}$$ $$=$$ $$( -\beta_{6} + \beta_{3} ) / 2$$ (-b6 + b3) / 2 $$\zeta_{16}^{7}$$ $$=$$ $$( -\beta_{5} + \beta_{4} ) / 2$$ (-b5 + b4) / 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 0.923880 − 0.382683i 0.923880 + 0.382683i 0.382683 + 0.923880i 0.382683 − 0.923880i −0.382683 − 0.923880i −0.382683 + 0.923880i −0.923880 + 0.382683i −0.923880 − 0.382683i
0 0 0 −1.84776 0 0 0 0 0
881.2 0 0 0 −1.84776 0 0 0 0 0
881.3 0 0 0 −0.765367 0 0 0 0 0
881.4 0 0 0 −0.765367 0 0 0 0 0
881.5 0 0 0 0.765367 0 0 0 0 0
881.6 0 0 0 0.765367 0 0 0 0 0
881.7 0 0 0 1.84776 0 0 0 0 0
881.8 0 0 0 1.84776 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.e 8
3.b odd 2 1 inner 7056.2.k.e 8
4.b odd 2 1 1764.2.f.b 8
7.b odd 2 1 inner 7056.2.k.e 8
12.b even 2 1 1764.2.f.b 8
21.c even 2 1 inner 7056.2.k.e 8
28.d even 2 1 1764.2.f.b 8
28.f even 6 2 1764.2.t.c 16
28.g odd 6 2 1764.2.t.c 16
84.h odd 2 1 1764.2.f.b 8
84.j odd 6 2 1764.2.t.c 16
84.n even 6 2 1764.2.t.c 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1764.2.f.b 8 4.b odd 2 1
1764.2.f.b 8 12.b even 2 1
1764.2.f.b 8 28.d even 2 1
1764.2.f.b 8 84.h odd 2 1
1764.2.t.c 16 28.f even 6 2
1764.2.t.c 16 28.g odd 6 2
1764.2.t.c 16 84.j odd 6 2
1764.2.t.c 16 84.n even 6 2
7056.2.k.e 8 1.a even 1 1 trivial
7056.2.k.e 8 3.b odd 2 1 inner
7056.2.k.e 8 7.b odd 2 1 inner
7056.2.k.e 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 $$S_{2}^{\mathrm{new}}(7056, [\chi])$$:

 $$T_{5}^{4} - 4T_{5}^{2} + 2$$ T5^4 - 4*T5^2 + 2 $$T_{11}^{2} + 4$$ T11^2 + 4 $$T_{13}^{4} + 20T_{13}^{2} + 2$$ T13^4 + 20*T13^2 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} - 4 T^{2} + 2)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T^{2} + 4)^{4}$$
$13$ $$(T^{4} + 20 T^{2} + 2)^{2}$$
$17$ $$(T^{4} - 36 T^{2} + 162)^{2}$$
$19$ $$(T^{4} + 16 T^{2} + 32)^{2}$$
$23$ $$(T^{4} + 88 T^{2} + 784)^{2}$$
$29$ $$(T^{4} + 48 T^{2} + 64)^{2}$$
$31$ $$(T^{4} + 80 T^{2} + 1568)^{2}$$
$37$ $$(T^{2} - 8 T - 2)^{4}$$
$41$ $$(T^{4} - 148 T^{2} + 1058)^{2}$$
$43$ $$(T^{2} + 8 T + 8)^{4}$$
$47$ $$(T^{4} - 208 T^{2} + 9248)^{2}$$
$53$ $$(T^{4} + 164 T^{2} + 2116)^{2}$$
$59$ $$(T^{4} - 272 T^{2} + 16928)^{2}$$
$61$ $$(T^{4} + 164 T^{2} + 1922)^{2}$$
$67$ $$(T^{2} + 8 T - 56)^{4}$$
$71$ $$(T^{4} + 264 T^{2} + 15376)^{2}$$
$73$ $$(T^{4} + 148 T^{2} + 1058)^{2}$$
$79$ $$(T^{2} - 128)^{4}$$
$83$ $$(T^{4} - 128 T^{2} + 2048)^{2}$$
$89$ $$(T^{4} - 212 T^{2} + 578)^{2}$$
$97$ $$(T^{4} + 148 T^{2} + 1058)^{2}$$