# Properties

 Label 7056.2.b.s 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: $$\Q(\sqrt{-3}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 7 x^{2} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 112) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{5} +O(q^{10})$$ $$q + \beta_{2} q^{5} + \beta_{3} q^{11} + 2 \beta_{2} q^{13} + 3 \beta_{2} q^{17} -\beta_{1} q^{19} -\beta_{3} q^{23} + 2 q^{25} + \beta_{1} q^{31} + 7 q^{37} + 2 \beta_{2} q^{41} + 2 \beta_{3} q^{43} -3 \beta_{1} q^{47} -3 q^{53} + 3 \beta_{1} q^{55} -3 \beta_{1} q^{59} + \beta_{2} q^{61} -6 q^{65} + \beta_{3} q^{67} -2 \beta_{3} q^{71} + 3 \beta_{2} q^{73} + \beta_{3} q^{79} -9 q^{85} -\beta_{2} q^{89} + \beta_{3} q^{95} -2 \beta_{2} q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + O(q^{10})$$ $$4q + 8q^{25} + 28q^{37} - 12q^{53} - 24q^{65} - 36q^{85} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7 x^{2} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3}$$$$/7$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{2} + 7$$$$)/7$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 14 \nu$$$$)/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$7 \beta_{2} - 7$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{1}$$

## 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
 1.32288 − 2.29129i −1.32288 + 2.29129i −1.32288 − 2.29129i 1.32288 + 2.29129i
0 0 0 1.73205i 0 0 0 0 0
1567.2 0 0 0 1.73205i 0 0 0 0 0
1567.3 0 0 0 1.73205i 0 0 0 0 0
1567.4 0 0 0 1.73205i 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
4.b odd 2 1 inner
7.b odd 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.b.s 4
3.b odd 2 1 784.2.f.d 4
4.b odd 2 1 inner 7056.2.b.s 4
7.b odd 2 1 inner 7056.2.b.s 4
7.c even 3 1 1008.2.cs.q 4
7.d odd 6 1 1008.2.cs.q 4
12.b even 2 1 784.2.f.d 4
21.c even 2 1 784.2.f.d 4
21.g even 6 1 112.2.p.c 4
21.g even 6 1 784.2.p.g 4
21.h odd 6 1 112.2.p.c 4
21.h odd 6 1 784.2.p.g 4
24.f even 2 1 3136.2.f.f 4
24.h odd 2 1 3136.2.f.f 4
28.d even 2 1 inner 7056.2.b.s 4
28.f even 6 1 1008.2.cs.q 4
28.g odd 6 1 1008.2.cs.q 4
84.h odd 2 1 784.2.f.d 4
84.j odd 6 1 112.2.p.c 4
84.j odd 6 1 784.2.p.g 4
84.n even 6 1 112.2.p.c 4
84.n even 6 1 784.2.p.g 4
168.e odd 2 1 3136.2.f.f 4
168.i even 2 1 3136.2.f.f 4
168.s odd 6 1 448.2.p.c 4
168.v even 6 1 448.2.p.c 4
168.ba even 6 1 448.2.p.c 4
168.be odd 6 1 448.2.p.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
112.2.p.c 4 21.g even 6 1
112.2.p.c 4 21.h odd 6 1
112.2.p.c 4 84.j odd 6 1
112.2.p.c 4 84.n even 6 1
448.2.p.c 4 168.s odd 6 1
448.2.p.c 4 168.v even 6 1
448.2.p.c 4 168.ba even 6 1
448.2.p.c 4 168.be odd 6 1
784.2.f.d 4 3.b odd 2 1
784.2.f.d 4 12.b even 2 1
784.2.f.d 4 21.c even 2 1
784.2.f.d 4 84.h odd 2 1
784.2.p.g 4 21.g even 6 1
784.2.p.g 4 21.h odd 6 1
784.2.p.g 4 84.j odd 6 1
784.2.p.g 4 84.n even 6 1
1008.2.cs.q 4 7.c even 3 1
1008.2.cs.q 4 7.d odd 6 1
1008.2.cs.q 4 28.f even 6 1
1008.2.cs.q 4 28.g odd 6 1
3136.2.f.f 4 24.f even 2 1
3136.2.f.f 4 24.h odd 2 1
3136.2.f.f 4 168.e odd 2 1
3136.2.f.f 4 168.i even 2 1
7056.2.b.s 4 1.a even 1 1 trivial
7056.2.b.s 4 4.b odd 2 1 inner
7056.2.b.s 4 7.b odd 2 1 inner
7056.2.b.s 4 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 $$S_{2}^{\mathrm{new}}(7056, [\chi])$$:

 $$T_{5}^{2} + 3$$ $$T_{11}^{2} + 21$$ $$T_{13}^{2} + 12$$ $$T_{17}^{2} + 27$$ $$T_{19}^{2} - 7$$ $$T_{31}^{2} - 7$$ $$T_{53} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ 1
$5$ $$( 1 - 7 T^{2} + 25 T^{4} )^{2}$$
$7$ 1
$11$ $$( 1 - T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 14 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 7 T^{2} + 289 T^{4} )^{2}$$
$19$ $$( 1 + 31 T^{2} + 361 T^{4} )^{2}$$
$23$ $$( 1 - 25 T^{2} + 529 T^{4} )^{2}$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$( 1 + 55 T^{2} + 961 T^{4} )^{2}$$
$37$ $$( 1 - 7 T + 37 T^{2} )^{4}$$
$41$ $$( 1 - 70 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 2 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 31 T^{2} + 2209 T^{4} )^{2}$$
$53$ $$( 1 + 3 T + 53 T^{2} )^{4}$$
$59$ $$( 1 + 55 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 119 T^{2} + 3721 T^{4} )^{2}$$
$67$ $$( 1 - 113 T^{2} + 4489 T^{4} )^{2}$$
$71$ $$( 1 - 58 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 119 T^{2} + 5329 T^{4} )^{2}$$
$79$ $$( 1 - 137 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$( 1 + 83 T^{2} )^{4}$$
$89$ $$( 1 - 175 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 182 T^{2} + 9409 T^{4} )^{2}$$