# Properties

 Label 7056.2.b.o Level $7056$ Weight $2$ Character orbit 7056.b Analytic conductor $56.342$ Analytic rank $0$ Dimension $4$ 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.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$56.3424436662$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{5})$$ Defining polynomial: $$x^{4} - x^{3} + 2 x^{2} + x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: no (minimal twist has level 1008) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{5} +O(q^{10})$$ $$q -\beta_{2} q^{5} + \beta_{2} q^{11} + 2 \beta_{1} q^{13} -4 q^{19} + 2 \beta_{2} q^{23} -10 q^{25} -\beta_{3} q^{29} + q^{31} + 4 q^{37} -2 \beta_{2} q^{41} -4 \beta_{1} q^{43} + 2 \beta_{3} q^{47} + \beta_{3} q^{53} + 15 q^{55} + \beta_{3} q^{59} -6 \beta_{1} q^{61} + 2 \beta_{3} q^{65} -4 \beta_{1} q^{67} -2 \beta_{2} q^{71} -4 \beta_{1} q^{73} -7 \beta_{1} q^{79} + \beta_{3} q^{83} -2 \beta_{2} q^{89} + 4 \beta_{2} q^{95} -3 \beta_{1} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q - 16q^{19} - 40q^{25} + 4q^{31} + 16q^{37} + 60q^{55} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 2 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$-\nu^{3} + 2 \nu^{2} - 2 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} - 2 \nu^{2} + 6 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$-3 \nu^{3} - 6$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 3 \beta_{2} + 3 \beta_{1} + 3$$$$)/12$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + 3 \beta_{2} + 9 \beta_{1} - 9$$$$)/12$$ $$\nu^{3}$$ $$=$$ $$($$$$-\beta_{3} - 6$$$$)/3$$

## 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
 −0.309017 + 0.535233i 0.809017 + 1.40126i 0.809017 − 1.40126i −0.309017 − 0.535233i
0 0 0 3.87298i 0 0 0 0 0
1567.2 0 0 0 3.87298i 0 0 0 0 0
1567.3 0 0 0 3.87298i 0 0 0 0 0
1567.4 0 0 0 3.87298i 0 0 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 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
28.d even 2 1 inner
84.h odd 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.o 4
3.b odd 2 1 inner 7056.2.b.o 4
4.b odd 2 1 7056.2.b.v 4
7.b odd 2 1 7056.2.b.v 4
7.c even 3 1 1008.2.cs.p yes 4
7.d odd 6 1 1008.2.cs.o 4
12.b even 2 1 7056.2.b.v 4
21.c even 2 1 7056.2.b.v 4
21.g even 6 1 1008.2.cs.o 4
21.h odd 6 1 1008.2.cs.p yes 4
28.d even 2 1 inner 7056.2.b.o 4
28.f even 6 1 1008.2.cs.p yes 4
28.g odd 6 1 1008.2.cs.o 4
84.h odd 2 1 inner 7056.2.b.o 4
84.j odd 6 1 1008.2.cs.p yes 4
84.n even 6 1 1008.2.cs.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1008.2.cs.o 4 7.d odd 6 1
1008.2.cs.o 4 21.g even 6 1
1008.2.cs.o 4 28.g odd 6 1
1008.2.cs.o 4 84.n even 6 1
1008.2.cs.p yes 4 7.c even 3 1
1008.2.cs.p yes 4 21.h odd 6 1
1008.2.cs.p yes 4 28.f even 6 1
1008.2.cs.p yes 4 84.j odd 6 1
7056.2.b.o 4 1.a even 1 1 trivial
7056.2.b.o 4 3.b odd 2 1 inner
7056.2.b.o 4 28.d even 2 1 inner
7056.2.b.o 4 84.h odd 2 1 inner
7056.2.b.v 4 4.b odd 2 1
7056.2.b.v 4 7.b odd 2 1
7056.2.b.v 4 12.b even 2 1
7056.2.b.v 4 21.c even 2 1

## Hecke kernels

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

 $$T_{5}^{2} + 15$$ $$T_{11}^{2} + 15$$ $$T_{13}^{2} + 12$$ $$T_{17}$$ $$T_{19} + 4$$ $$T_{31} - 1$$ $$T_{53}^{2} - 45$$