# Properties

 Label 1568.1.bl.a Level $1568$ Weight $1$ Character orbit 1568.bl Analytic conductor $0.783$ Analytic rank $0$ Dimension $8$ Projective image $D_{8}$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1568 = 2^{5} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1568.bl (of order $$24$$, degree $$8$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.782533939809$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\Q(\zeta_{24})$$ Defining polynomial: $$x^{8} - x^{4} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 224) Projective image: $$D_{8}$$ Projective field: Galois closure of 8.0.5156108238848.3

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q -\zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} + \zeta_{24}^{3} q^{8} + \zeta_{24}^{11} q^{9} +O(q^{10})$$ $$q -\zeta_{24}^{5} q^{2} + \zeta_{24}^{10} q^{4} + \zeta_{24}^{3} q^{8} + \zeta_{24}^{11} q^{9} + ( \zeta_{24}^{7} - \zeta_{24}^{10} ) q^{11} -\zeta_{24}^{8} q^{16} + \zeta_{24}^{4} q^{18} + ( 1 - \zeta_{24}^{3} ) q^{22} + ( \zeta_{24}^{2} - \zeta_{24}^{8} ) q^{23} + \zeta_{24} q^{25} + ( \zeta_{24}^{6} - \zeta_{24}^{9} ) q^{29} -\zeta_{24} q^{32} -\zeta_{24}^{9} q^{36} + ( \zeta_{24}^{2} - \zeta_{24}^{11} ) q^{37} + ( -1 + \zeta_{24}^{9} ) q^{43} + ( -\zeta_{24}^{5} + \zeta_{24}^{8} ) q^{44} + ( -\zeta_{24} - \zeta_{24}^{7} ) q^{46} -\zeta_{24}^{6} q^{50} + ( \zeta_{24} + \zeta_{24}^{4} ) q^{53} + ( -\zeta_{24}^{2} - \zeta_{24}^{11} ) q^{58} + \zeta_{24}^{6} q^{64} + ( -\zeta_{24}^{4} - \zeta_{24}^{7} ) q^{67} -\zeta_{24}^{2} q^{72} + ( -\zeta_{24}^{4} - \zeta_{24}^{7} ) q^{74} + ( -\zeta_{24}^{5} + \zeta_{24}^{11} ) q^{79} -\zeta_{24}^{10} q^{81} + ( \zeta_{24}^{2} + \zeta_{24}^{5} ) q^{86} + ( \zeta_{24} + \zeta_{24}^{10} ) q^{88} + ( -1 + \zeta_{24}^{6} ) q^{92} + ( -\zeta_{24}^{6} + \zeta_{24}^{9} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + O(q^{10})$$ $$8 q + 4 q^{16} + 4 q^{18} + 8 q^{22} + 4 q^{23} - 8 q^{43} - 4 q^{44} + 4 q^{53} - 4 q^{67} - 4 q^{74} - 8 q^{92} + O(q^{100})$$

## Character values

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

 $$n$$ $$197$$ $$1471$$ $$1473$$ $$\chi(n)$$ $$\zeta_{24}^{9}$$ $$1$$ $$-\zeta_{24}^{8}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
117.1
 −0.258819 + 0.965926i −0.965926 + 0.258819i 0.965926 + 0.258819i 0.258819 + 0.965926i 0.258819 − 0.965926i 0.965926 − 0.258819i −0.965926 − 0.258819i −0.258819 − 0.965926i
0.965926 0.258819i 0 0.866025 0.500000i 0 0 0 0.707107 0.707107i 0.258819 + 0.965926i 0
325.1 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 0 −0.707107 + 0.707107i 0.965926 + 0.258819i 0
509.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 0 0.707107 + 0.707107i −0.965926 + 0.258819i 0
717.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 0 −0.707107 0.707107i −0.258819 + 0.965926i 0
901.1 −0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 0 −0.707107 + 0.707107i −0.258819 0.965926i 0
1109.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 0 0.707107 0.707107i −0.965926 0.258819i 0
1293.1 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 0 −0.707107 0.707107i 0.965926 0.258819i 0
1501.1 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 0 0.707107 + 0.707107i 0.258819 0.965926i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1501.1 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.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
7.c even 3 1 inner
7.d odd 6 1 inner
32.g even 8 1 inner
224.v odd 8 1 inner
224.bc odd 24 1 inner
224.bd even 24 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.1.bl.a 8
7.b odd 2 1 CM 1568.1.bl.a 8
7.c even 3 1 224.1.v.a 4
7.c even 3 1 inner 1568.1.bl.a 8
7.d odd 6 1 224.1.v.a 4
7.d odd 6 1 inner 1568.1.bl.a 8
21.g even 6 1 2016.1.dp.b 4
21.h odd 6 1 2016.1.dp.b 4
28.f even 6 1 896.1.v.a 4
28.g odd 6 1 896.1.v.a 4
32.g even 8 1 inner 1568.1.bl.a 8
56.j odd 6 1 1792.1.v.a 4
56.k odd 6 1 1792.1.v.b 4
56.m even 6 1 1792.1.v.b 4
56.p even 6 1 1792.1.v.a 4
112.u odd 12 1 3584.1.v.a 4
112.u odd 12 1 3584.1.v.c 4
112.v even 12 1 3584.1.v.a 4
112.v even 12 1 3584.1.v.c 4
112.w even 12 1 3584.1.v.b 4
112.w even 12 1 3584.1.v.d 4
112.x odd 12 1 3584.1.v.b 4
112.x odd 12 1 3584.1.v.d 4
224.v odd 8 1 inner 1568.1.bl.a 8
224.bc odd 24 1 224.1.v.a 4
224.bc odd 24 1 inner 1568.1.bl.a 8
224.bc odd 24 1 1792.1.v.a 4
224.bc odd 24 1 3584.1.v.b 4
224.bc odd 24 1 3584.1.v.d 4
224.bd even 24 1 224.1.v.a 4
224.bd even 24 1 inner 1568.1.bl.a 8
224.bd even 24 1 1792.1.v.a 4
224.bd even 24 1 3584.1.v.b 4
224.bd even 24 1 3584.1.v.d 4
224.be even 24 1 896.1.v.a 4
224.be even 24 1 1792.1.v.b 4
224.be even 24 1 3584.1.v.a 4
224.be even 24 1 3584.1.v.c 4
224.bf odd 24 1 896.1.v.a 4
224.bf odd 24 1 1792.1.v.b 4
224.bf odd 24 1 3584.1.v.a 4
224.bf odd 24 1 3584.1.v.c 4
672.ce odd 24 1 2016.1.dp.b 4
672.cl even 24 1 2016.1.dp.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
224.1.v.a 4 7.c even 3 1
224.1.v.a 4 7.d odd 6 1
224.1.v.a 4 224.bc odd 24 1
224.1.v.a 4 224.bd even 24 1
896.1.v.a 4 28.f even 6 1
896.1.v.a 4 28.g odd 6 1
896.1.v.a 4 224.be even 24 1
896.1.v.a 4 224.bf odd 24 1
1568.1.bl.a 8 1.a even 1 1 trivial
1568.1.bl.a 8 7.b odd 2 1 CM
1568.1.bl.a 8 7.c even 3 1 inner
1568.1.bl.a 8 7.d odd 6 1 inner
1568.1.bl.a 8 32.g even 8 1 inner
1568.1.bl.a 8 224.v odd 8 1 inner
1568.1.bl.a 8 224.bc odd 24 1 inner
1568.1.bl.a 8 224.bd even 24 1 inner
1792.1.v.a 4 56.j odd 6 1
1792.1.v.a 4 56.p even 6 1
1792.1.v.a 4 224.bc odd 24 1
1792.1.v.a 4 224.bd even 24 1
1792.1.v.b 4 56.k odd 6 1
1792.1.v.b 4 56.m even 6 1
1792.1.v.b 4 224.be even 24 1
1792.1.v.b 4 224.bf odd 24 1
2016.1.dp.b 4 21.g even 6 1
2016.1.dp.b 4 21.h odd 6 1
2016.1.dp.b 4 672.ce odd 24 1
2016.1.dp.b 4 672.cl even 24 1
3584.1.v.a 4 112.u odd 12 1
3584.1.v.a 4 112.v even 12 1
3584.1.v.a 4 224.be even 24 1
3584.1.v.a 4 224.bf odd 24 1
3584.1.v.b 4 112.w even 12 1
3584.1.v.b 4 112.x odd 12 1
3584.1.v.b 4 224.bc odd 24 1
3584.1.v.b 4 224.bd even 24 1
3584.1.v.c 4 112.u odd 12 1
3584.1.v.c 4 112.v even 12 1
3584.1.v.c 4 224.be even 24 1
3584.1.v.c 4 224.bf odd 24 1
3584.1.v.d 4 112.w even 12 1
3584.1.v.d 4 112.x odd 12 1
3584.1.v.d 4 224.bc odd 24 1
3584.1.v.d 4 224.bd even 24 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1568, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{4} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$T^{8}$$
$11$ $$4 - 8 T + 12 T^{2} - 8 T^{3} + 2 T^{4} + 8 T^{5} - 2 T^{6} + T^{8}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{2}$$
$29$ $$( 2 - 4 T + 2 T^{2} + T^{4} )^{2}$$
$31$ $$T^{8}$$
$37$ $$4 + 8 T + 12 T^{2} + 8 T^{3} + 2 T^{4} - 8 T^{5} - 2 T^{6} + T^{8}$$
$41$ $$T^{8}$$
$43$ $$( 2 + 4 T + 6 T^{2} + 4 T^{3} + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$4 - 8 T + 4 T^{2} - 8 T^{3} + 18 T^{4} - 16 T^{5} + 10 T^{6} - 4 T^{7} + T^{8}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$4 + 8 T + 4 T^{2} + 8 T^{3} + 18 T^{4} + 16 T^{5} + 10 T^{6} + 4 T^{7} + T^{8}$$
$71$ $$T^{8}$$
$73$ $$T^{8}$$
$79$ $$( 4 - 2 T^{2} + T^{4} )^{2}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$