# Properties

 Label 1568.2.i.k Level $1568$ Weight $2$ Character orbit 1568.i Analytic conductor $12.521$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1568.i (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.5205430369$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 2 - 2 \zeta_{6} ) q^{3} -\zeta_{6} q^{9} + ( 4 - 4 \zeta_{6} ) q^{11} -4 q^{13} + ( 2 - 2 \zeta_{6} ) q^{17} + 6 \zeta_{6} q^{19} -8 \zeta_{6} q^{23} + ( 5 - 5 \zeta_{6} ) q^{25} + 4 q^{27} + 2 q^{29} + ( 4 - 4 \zeta_{6} ) q^{31} -8 \zeta_{6} q^{33} -10 \zeta_{6} q^{37} + ( -8 + 8 \zeta_{6} ) q^{39} -10 q^{41} + 4 q^{43} -4 \zeta_{6} q^{47} -4 \zeta_{6} q^{51} + ( 2 - 2 \zeta_{6} ) q^{53} + 12 q^{57} + ( -10 + 10 \zeta_{6} ) q^{59} + 8 \zeta_{6} q^{61} + ( 8 - 8 \zeta_{6} ) q^{67} -16 q^{69} + ( 6 - 6 \zeta_{6} ) q^{73} -10 \zeta_{6} q^{75} + 16 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} + 2 q^{83} + ( 4 - 4 \zeta_{6} ) q^{87} -18 \zeta_{6} q^{89} -8 \zeta_{6} q^{93} -2 q^{97} -4 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{3} - q^{9} + O(q^{10})$$ $$2q + 2q^{3} - q^{9} + 4q^{11} - 8q^{13} + 2q^{17} + 6q^{19} - 8q^{23} + 5q^{25} + 8q^{27} + 4q^{29} + 4q^{31} - 8q^{33} - 10q^{37} - 8q^{39} - 20q^{41} + 8q^{43} - 4q^{47} - 4q^{51} + 2q^{53} + 24q^{57} - 10q^{59} + 8q^{61} + 8q^{67} - 32q^{69} + 6q^{73} - 10q^{75} + 16q^{79} + 11q^{81} + 4q^{83} + 4q^{87} - 18q^{89} - 8q^{93} - 4q^{97} - 8q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$$.

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 1.00000 + 1.73205i 0 0 0 0 0 −0.500000 + 0.866025i 0
1537.1 0 1.00000 1.73205i 0 0 0 0 0 −0.500000 0.866025i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.2.i.k 2
4.b odd 2 1 1568.2.i.b 2
7.b odd 2 1 1568.2.i.c 2
7.c even 3 1 224.2.a.a 1
7.c even 3 1 inner 1568.2.i.k 2
7.d odd 6 1 1568.2.a.h 1
7.d odd 6 1 1568.2.i.c 2
21.h odd 6 1 2016.2.a.e 1
28.d even 2 1 1568.2.i.j 2
28.f even 6 1 1568.2.a.b 1
28.f even 6 1 1568.2.i.j 2
28.g odd 6 1 224.2.a.b yes 1
28.g odd 6 1 1568.2.i.b 2
35.j even 6 1 5600.2.a.t 1
56.j odd 6 1 3136.2.a.f 1
56.k odd 6 1 448.2.a.b 1
56.m even 6 1 3136.2.a.y 1
56.p even 6 1 448.2.a.f 1
84.n even 6 1 2016.2.a.g 1
112.u odd 12 2 1792.2.b.b 2
112.w even 12 2 1792.2.b.f 2
140.p odd 6 1 5600.2.a.c 1
168.s odd 6 1 4032.2.a.p 1
168.v even 6 1 4032.2.a.z 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.2.a.a 1 7.c even 3 1
224.2.a.b yes 1 28.g odd 6 1
448.2.a.b 1 56.k odd 6 1
448.2.a.f 1 56.p even 6 1
1568.2.a.b 1 28.f even 6 1
1568.2.a.h 1 7.d odd 6 1
1568.2.i.b 2 4.b odd 2 1
1568.2.i.b 2 28.g odd 6 1
1568.2.i.c 2 7.b odd 2 1
1568.2.i.c 2 7.d odd 6 1
1568.2.i.j 2 28.d even 2 1
1568.2.i.j 2 28.f even 6 1
1568.2.i.k 2 1.a even 1 1 trivial
1568.2.i.k 2 7.c even 3 1 inner
1792.2.b.b 2 112.u odd 12 2
1792.2.b.f 2 112.w even 12 2
2016.2.a.e 1 21.h odd 6 1
2016.2.a.g 1 84.n even 6 1
3136.2.a.f 1 56.j odd 6 1
3136.2.a.y 1 56.m even 6 1
4032.2.a.p 1 168.s odd 6 1
4032.2.a.z 1 168.v even 6 1
5600.2.a.c 1 140.p odd 6 1
5600.2.a.t 1 35.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1568, [\chi])$$:

 $$T_{3}^{2} - 2 T_{3} + 4$$ $$T_{5}$$ $$T_{11}^{2} - 4 T_{11} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$4 - 2 T + T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$16 - 4 T + T^{2}$$
$13$ $$( 4 + T )^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$36 - 6 T + T^{2}$$
$23$ $$64 + 8 T + T^{2}$$
$29$ $$( -2 + T )^{2}$$
$31$ $$16 - 4 T + T^{2}$$
$37$ $$100 + 10 T + T^{2}$$
$41$ $$( 10 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$16 + 4 T + T^{2}$$
$53$ $$4 - 2 T + T^{2}$$
$59$ $$100 + 10 T + T^{2}$$
$61$ $$64 - 8 T + T^{2}$$
$67$ $$64 - 8 T + T^{2}$$
$71$ $$T^{2}$$
$73$ $$36 - 6 T + T^{2}$$
$79$ $$256 - 16 T + T^{2}$$
$83$ $$( -2 + T )^{2}$$
$89$ $$324 + 18 T + T^{2}$$
$97$ $$( 2 + T )^{2}$$