# Properties

 Label 7056.2.h.l Level $7056$ Weight $2$ Character orbit 7056.h 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.h (of order $$2$$, degree $$1$$, 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: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{8}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{5}+O(q^{10})$$ q - b1 * q^5 $$q - \beta_1 q^{5} - 3 \beta_{5} q^{11} + (\beta_{5} + 2 \beta_{3}) q^{13} - \beta_{4} q^{17} + (\beta_{7} - 3 \beta_{2}) q^{19} + (3 \beta_{5} + 2 \beta_{3}) q^{23} + ( - \beta_{6} + 1) q^{25} - \beta_{7} q^{29} + ( - \beta_{7} - 3 \beta_{2}) q^{31} - \beta_{6} q^{37} + (2 \beta_{4} + \beta_1) q^{41} + ( - 2 \beta_{4} + 4 \beta_1) q^{43} + \beta_{6} q^{47} + \beta_{2} q^{53} + (3 \beta_{7} - 3 \beta_{2}) q^{55} - 3 \beta_{6} q^{59} + (4 \beta_{5} + \beta_{3}) q^{61} + ( - 3 \beta_{7} + 7 \beta_{2}) q^{65} + ( - \beta_{4} + 5 \beta_1) q^{67} - 3 \beta_{5} q^{71} + ( - 4 \beta_{5} - \beta_{3}) q^{73} + (7 \beta_{4} - 5 \beta_1) q^{79} + (\beta_{6} - 6) q^{83} - 2 q^{85} + (2 \beta_{4} + 3 \beta_1) q^{89} + (6 \beta_{5} + 4 \beta_{3}) q^{95} + (2 \beta_{5} - 3 \beta_{3}) q^{97}+O(q^{100})$$ q - b1 * q^5 - 3*b5 * q^11 + (b5 + 2*b3) * q^13 - b4 * q^17 + (b7 - 3*b2) * q^19 + (3*b5 + 2*b3) * q^23 + (-b6 + 1) * q^25 - b7 * q^29 + (-b7 - 3*b2) * q^31 - b6 * q^37 + (2*b4 + b1) * q^41 + (-2*b4 + 4*b1) * q^43 + b6 * q^47 + b2 * q^53 + (3*b7 - 3*b2) * q^55 - 3*b6 * q^59 + (4*b5 + b3) * q^61 + (-3*b7 + 7*b2) * q^65 + (-b4 + 5*b1) * q^67 - 3*b5 * q^71 + (-4*b5 - b3) * q^73 + (7*b4 - 5*b1) * q^79 + (b6 - 6) * q^83 - 2 * q^85 + (2*b4 + 3*b1) * q^89 + (6*b5 + 4*b3) * q^95 + (2*b5 - 3*b3) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q+O(q^{10})$$ 8 * q $$8 q + 8 q^{25} - 48 q^{83} - 16 q^{85}+O(q^{100})$$ 8 * q + 8 * q^25 - 48 * q^83 - 16 * q^85

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{24}^{6} + 2\zeta_{24}^{4} - 1$$ v^6 + 2*v^4 - 1 $$\beta_{2}$$ $$=$$ $$-\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ -v^5 - v^3 + v $$\beta_{3}$$ $$=$$ $$-2\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -2*v^7 + v^5 + v^3 + v $$\beta_{4}$$ $$=$$ $$-\zeta_{24}^{6} + 2\zeta_{24}^{4} - 1$$ -v^6 + 2*v^4 - 1 $$\beta_{5}$$ $$=$$ $$-\zeta_{24}^{5} + \zeta_{24}^{3} + \zeta_{24}$$ -v^5 + v^3 + v $$\beta_{6}$$ $$=$$ $$-2\zeta_{24}^{6} + 4\zeta_{24}^{2}$$ -2*v^6 + 4*v^2 $$\beta_{7}$$ $$=$$ $$2\zeta_{24}^{7} + \zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24}$$ 2*v^7 + v^5 - v^3 + v
 $$\zeta_{24}$$ $$=$$ $$( \beta_{7} + \beta_{5} + \beta_{3} + \beta_{2} ) / 4$$ (b7 + b5 + b3 + b2) / 4 $$\zeta_{24}^{2}$$ $$=$$ $$( \beta_{6} - \beta_{4} + \beta_1 ) / 4$$ (b6 - b4 + b1) / 4 $$\zeta_{24}^{3}$$ $$=$$ $$( \beta_{5} - \beta_{2} ) / 2$$ (b5 - b2) / 2 $$\zeta_{24}^{4}$$ $$=$$ $$( \beta_{4} + \beta _1 + 2 ) / 4$$ (b4 + b1 + 2) / 4 $$\zeta_{24}^{5}$$ $$=$$ $$( \beta_{7} - \beta_{5} + \beta_{3} - \beta_{2} ) / 4$$ (b7 - b5 + b3 - b2) / 4 $$\zeta_{24}^{6}$$ $$=$$ $$( -\beta_{4} + \beta_1 ) / 2$$ (-b4 + b1) / 2 $$\zeta_{24}^{7}$$ $$=$$ $$( \beta_{7} + \beta_{5} - \beta_{3} - \beta_{2} ) / 4$$ (b7 + b5 - b3 - b2) / 4

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4607.1
 0.965926 + 0.258819i −0.965926 − 0.258819i −0.258819 + 0.965926i 0.258819 − 0.965926i −0.258819 − 0.965926i 0.258819 + 0.965926i 0.965926 − 0.258819i −0.965926 + 0.258819i
0 0 0 2.73205i 0 0 0 0 0
4607.2 0 0 0 2.73205i 0 0 0 0 0
4607.3 0 0 0 0.732051i 0 0 0 0 0
4607.4 0 0 0 0.732051i 0 0 0 0 0
4607.5 0 0 0 0.732051i 0 0 0 0 0
4607.6 0 0 0 0.732051i 0 0 0 0 0
4607.7 0 0 0 2.73205i 0 0 0 0 0
4607.8 0 0 0 2.73205i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4607.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
12.b even 2 1 inner
21.c even 2 1 inner
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.h.l 8
3.b odd 2 1 7056.2.h.m yes 8
4.b odd 2 1 7056.2.h.m yes 8
7.b odd 2 1 7056.2.h.m yes 8
12.b even 2 1 inner 7056.2.h.l 8
21.c even 2 1 inner 7056.2.h.l 8
28.d even 2 1 inner 7056.2.h.l 8
84.h odd 2 1 7056.2.h.m yes 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7056.2.h.l 8 1.a even 1 1 trivial
7056.2.h.l 8 12.b even 2 1 inner
7056.2.h.l 8 21.c even 2 1 inner
7056.2.h.l 8 28.d even 2 1 inner
7056.2.h.m yes 8 3.b odd 2 1
7056.2.h.m yes 8 4.b odd 2 1
7056.2.h.m yes 8 7.b odd 2 1
7056.2.h.m yes 8 84.h odd 2 1

## Hecke kernels

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

 $$T_{5}^{4} + 8T_{5}^{2} + 4$$ T5^4 + 8*T5^2 + 4 $$T_{11}^{2} - 18$$ T11^2 - 18 $$T_{13}^{4} - 52T_{13}^{2} + 484$$ T13^4 - 52*T13^2 + 484 $$T_{61}^{4} - 76T_{61}^{2} + 676$$ T61^4 - 76*T61^2 + 676 $$T_{83}^{2} + 12T_{83} + 24$$ T83^2 + 12*T83 + 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$T^{8}$$
$5$ $$(T^{4} + 8 T^{2} + 4)^{2}$$
$7$ $$T^{8}$$
$11$ $$(T^{2} - 18)^{4}$$
$13$ $$(T^{4} - 52 T^{2} + 484)^{2}$$
$17$ $$(T^{4} + 8 T^{2} + 4)^{2}$$
$19$ $$(T^{4} + 48 T^{2} + 144)^{2}$$
$23$ $$(T^{4} - 84 T^{2} + 36)^{2}$$
$29$ $$(T^{2} + 6)^{4}$$
$31$ $$(T^{4} + 48 T^{2} + 144)^{2}$$
$37$ $$(T^{2} - 12)^{4}$$
$41$ $$(T^{4} + 56 T^{2} + 676)^{2}$$
$43$ $$(T^{4} + 96 T^{2} + 576)^{2}$$
$47$ $$(T^{2} - 12)^{4}$$
$53$ $$(T^{2} + 2)^{4}$$
$59$ $$(T^{2} - 108)^{4}$$
$61$ $$(T^{4} - 76 T^{2} + 676)^{2}$$
$67$ $$(T^{4} + 168 T^{2} + 144)^{2}$$
$71$ $$(T^{2} - 18)^{4}$$
$73$ $$(T^{4} - 76 T^{2} + 676)^{2}$$
$79$ $$(T^{4} + 312 T^{2} + 17424)^{2}$$
$83$ $$(T^{2} + 12 T + 24)^{4}$$
$89$ $$(T^{4} + 152 T^{2} + 5476)^{2}$$
$97$ $$(T^{4} - 124 T^{2} + 2116)^{2}$$