# Properties

 Label 776.1.bp.a Level $776$ Weight $1$ Character orbit 776.bp Analytic conductor $0.387$ Analytic rank $0$ Dimension $16$ Projective image $D_{48}$ CM discriminant -8 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$776 = 2^{3} \cdot 97$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 776.bp (of order $$48$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.387274449803$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{48})$$ Defining polynomial: $$x^{16} - x^{8} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{48}$$ Projective field: Galois closure of $$\mathbb{Q}[x]/(x^{48} + \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q -\zeta_{48}^{19} q^{2} + ( \zeta_{48}^{8} - \zeta_{48}^{18} ) q^{3} -\zeta_{48}^{14} q^{4} + ( \zeta_{48}^{3} - \zeta_{48}^{13} ) q^{6} -\zeta_{48}^{9} q^{8} + ( \zeta_{48}^{2} - \zeta_{48}^{12} + \zeta_{48}^{16} ) q^{9} +O(q^{10})$$ $$q -\zeta_{48}^{19} q^{2} + ( \zeta_{48}^{8} - \zeta_{48}^{18} ) q^{3} -\zeta_{48}^{14} q^{4} + ( \zeta_{48}^{3} - \zeta_{48}^{13} ) q^{6} -\zeta_{48}^{9} q^{8} + ( \zeta_{48}^{2} - \zeta_{48}^{12} + \zeta_{48}^{16} ) q^{9} + ( \zeta_{48}^{15} + \zeta_{48}^{19} ) q^{11} + ( -\zeta_{48}^{8} - \zeta_{48}^{22} ) q^{12} -\zeta_{48}^{4} q^{16} + ( -\zeta_{48}^{2} + \zeta_{48}^{17} ) q^{17} + ( -\zeta_{48}^{7} + \zeta_{48}^{11} - \zeta_{48}^{21} ) q^{18} + ( \zeta_{48}^{12} - \zeta_{48}^{15} ) q^{19} + ( \zeta_{48}^{10} + \zeta_{48}^{14} ) q^{22} + ( -\zeta_{48}^{3} - \zeta_{48}^{17} ) q^{24} -\zeta_{48}^{11} q^{25} + ( -1 - \zeta_{48}^{6} + \zeta_{48}^{10} - \zeta_{48}^{20} ) q^{27} + \zeta_{48}^{23} q^{32} + ( -\zeta_{48}^{3} + \zeta_{48}^{9} + \zeta_{48}^{13} + \zeta_{48}^{23} ) q^{33} + ( \zeta_{48}^{12} + \zeta_{48}^{21} ) q^{34} + ( -\zeta_{48}^{2} + \zeta_{48}^{6} - \zeta_{48}^{16} ) q^{36} + ( \zeta_{48}^{7} - \zeta_{48}^{10} ) q^{38} + ( \zeta_{48} - \zeta_{48}^{22} ) q^{41} + ( -\zeta_{48}^{21} - \zeta_{48}^{23} ) q^{43} + ( \zeta_{48}^{5} + \zeta_{48}^{9} ) q^{44} + ( -\zeta_{48}^{12} + \zeta_{48}^{22} ) q^{48} -\zeta_{48}^{5} q^{49} -\zeta_{48}^{6} q^{50} + ( -\zeta_{48} - \zeta_{48}^{10} + \zeta_{48}^{11} + \zeta_{48}^{20} ) q^{51} + ( -\zeta_{48} + \zeta_{48}^{5} - \zeta_{48}^{15} + \zeta_{48}^{19} ) q^{54} + ( \zeta_{48}^{6} - \zeta_{48}^{9} + \zeta_{48}^{20} - \zeta_{48}^{23} ) q^{57} + ( \zeta_{48}^{4} - \zeta_{48}^{13} ) q^{59} + \zeta_{48}^{18} q^{64} + ( \zeta_{48}^{4} + \zeta_{48}^{8} + \zeta_{48}^{18} + \zeta_{48}^{22} ) q^{66} + ( \zeta_{48}^{13} + \zeta_{48}^{14} ) q^{67} + ( \zeta_{48}^{7} + \zeta_{48}^{16} ) q^{68} + ( \zeta_{48} - \zeta_{48}^{11} + \zeta_{48}^{21} ) q^{72} + ( -\zeta_{48}^{4} - \zeta_{48}^{16} ) q^{73} + ( -\zeta_{48}^{5} - \zeta_{48}^{19} ) q^{75} + ( \zeta_{48}^{2} - \zeta_{48}^{5} ) q^{76} + ( -1 + \zeta_{48}^{4} - \zeta_{48}^{8} - \zeta_{48}^{14} + \zeta_{48}^{18} ) q^{81} + ( -\zeta_{48}^{17} - \zeta_{48}^{20} ) q^{82} + ( \zeta_{48} + \zeta_{48}^{18} ) q^{83} + ( -\zeta_{48}^{16} - \zeta_{48}^{18} ) q^{86} + ( 1 + \zeta_{48}^{4} ) q^{88} + ( -\zeta_{48}^{20} + \zeta_{48}^{22} ) q^{89} + ( -\zeta_{48}^{7} + \zeta_{48}^{17} ) q^{96} -\zeta_{48} q^{97} - q^{98} + ( \zeta_{48}^{3} - \zeta_{48}^{11} + \zeta_{48}^{17} + \zeta_{48}^{21} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + 8q^{3} - 8q^{9} + O(q^{10})$$ $$16q + 8q^{3} - 8q^{9} - 8q^{12} - 16q^{27} + 8q^{36} + 8q^{66} - 8q^{68} + 8q^{73} - 24q^{81} + 8q^{86} + 16q^{88} - 16q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/776\mathbb{Z}\right)^\times$$.

 $$n$$ $$389$$ $$393$$ $$583$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{48}^{11}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 0.991445 + 0.130526i 0.608761 − 0.793353i 0.793353 − 0.608761i 0.130526 − 0.991445i 0.130526 + 0.991445i 0.793353 + 0.608761i 0.991445 − 0.130526i −0.991445 + 0.130526i −0.793353 − 0.608761i −0.130526 − 0.991445i −0.130526 + 0.991445i −0.793353 + 0.608761i −0.608761 + 0.793353i −0.991445 − 0.130526i 0.608761 + 0.793353i −0.608761 − 0.793353i
0.793353 0.608761i 1.20711 + 0.158919i 0.258819 0.965926i 0 1.05441 0.608761i 0 −0.382683 0.923880i 0.465926 + 0.124844i 0
11.1 −0.130526 0.991445i 1.20711 1.57313i −0.965926 + 0.258819i 0 −1.71723 0.991445i 0 0.382683 + 0.923880i −0.758819 2.83195i 0
99.1 −0.991445 0.130526i −0.207107 + 0.158919i 0.965926 + 0.258819i 0 0.226078 0.130526i 0 −0.923880 0.382683i −0.241181 + 0.900100i 0
163.1 0.608761 + 0.793353i −0.207107 + 1.57313i −0.258819 + 0.965926i 0 −1.37413 + 0.793353i 0 −0.923880 + 0.382683i −1.46593 0.392794i 0
219.1 0.608761 0.793353i −0.207107 1.57313i −0.258819 0.965926i 0 −1.37413 0.793353i 0 −0.923880 0.382683i −1.46593 + 0.392794i 0
243.1 −0.991445 + 0.130526i −0.207107 0.158919i 0.965926 0.258819i 0 0.226078 + 0.130526i 0 −0.923880 + 0.382683i −0.241181 0.900100i 0
259.1 0.793353 + 0.608761i 1.20711 0.158919i 0.258819 + 0.965926i 0 1.05441 + 0.608761i 0 −0.382683 + 0.923880i 0.465926 0.124844i 0
323.1 −0.793353 0.608761i 1.20711 0.158919i 0.258819 + 0.965926i 0 −1.05441 0.608761i 0 0.382683 0.923880i 0.465926 0.124844i 0
339.1 0.991445 0.130526i −0.207107 0.158919i 0.965926 0.258819i 0 −0.226078 0.130526i 0 0.923880 0.382683i −0.241181 0.900100i 0
363.1 −0.608761 + 0.793353i −0.207107 1.57313i −0.258819 0.965926i 0 1.37413 + 0.793353i 0 0.923880 + 0.382683i −1.46593 + 0.392794i 0
419.1 −0.608761 0.793353i −0.207107 + 1.57313i −0.258819 + 0.965926i 0 1.37413 0.793353i 0 0.923880 0.382683i −1.46593 0.392794i 0
483.1 0.991445 + 0.130526i −0.207107 + 0.158919i 0.965926 + 0.258819i 0 −0.226078 + 0.130526i 0 0.923880 + 0.382683i −0.241181 + 0.900100i 0
571.1 0.130526 + 0.991445i 1.20711 1.57313i −0.965926 + 0.258819i 0 1.71723 + 0.991445i 0 −0.382683 0.923880i −0.758819 2.83195i 0
579.1 −0.793353 + 0.608761i 1.20711 + 0.158919i 0.258819 0.965926i 0 −1.05441 + 0.608761i 0 0.382683 + 0.923880i 0.465926 + 0.124844i 0
635.1 −0.130526 + 0.991445i 1.20711 + 1.57313i −0.965926 0.258819i 0 −1.71723 + 0.991445i 0 0.382683 0.923880i −0.758819 + 2.83195i 0
723.1 0.130526 0.991445i 1.20711 + 1.57313i −0.965926 0.258819i 0 1.71723 0.991445i 0 −0.382683 + 0.923880i −0.758819 + 2.83195i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 723.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
97.k even 48 1 inner
776.bp odd 48 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 776.1.bp.a 16
4.b odd 2 1 3104.1.et.a 16
8.b even 2 1 3104.1.et.a 16
8.d odd 2 1 CM 776.1.bp.a 16
97.k even 48 1 inner 776.1.bp.a 16
388.v odd 48 1 3104.1.et.a 16
776.bp odd 48 1 inner 776.1.bp.a 16
776.br even 48 1 3104.1.et.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
776.1.bp.a 16 1.a even 1 1 trivial
776.1.bp.a 16 8.d odd 2 1 CM
776.1.bp.a 16 97.k even 48 1 inner
776.1.bp.a 16 776.bp odd 48 1 inner
3104.1.et.a 16 4.b odd 2 1
3104.1.et.a 16 8.b even 2 1
3104.1.et.a 16 388.v odd 48 1
3104.1.et.a 16 776.br even 48 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(776, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{8} + T^{16}$$
$3$ $$( 1 + 4 T + 4 T^{2} - 20 T^{3} + 21 T^{4} - 16 T^{5} + 10 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$5$ $$T^{16}$$
$7$ $$T^{16}$$
$11$ $$1 - 24 T^{4} + 191 T^{8} + 24 T^{12} + T^{16}$$
$13$ $$T^{16}$$
$17$ $$4 + 16 T + 24 T^{2} + 96 T^{3} + 268 T^{4} + 288 T^{5} + 216 T^{6} + 32 T^{7} + 2 T^{8} + 8 T^{9} + 40 T^{10} + 16 T^{11} - 2 T^{12} + T^{16}$$
$19$ $$( 2 - 8 T + 4 T^{2} + 8 T^{3} + 6 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$23$ $$T^{16}$$
$29$ $$T^{16}$$
$31$ $$T^{16}$$
$37$ $$T^{16}$$
$41$ $$4 - 16 T + 40 T^{2} - 96 T^{3} + 140 T^{4} + 48 T^{5} + 40 T^{6} - 192 T^{7} + 2 T^{8} + 8 T^{9} + 88 T^{10} - 2 T^{12} - 16 T^{13} + T^{16}$$
$43$ $$1 - 24 T^{2} + 180 T^{4} + 288 T^{6} + 143 T^{8} - 12 T^{12} + T^{16}$$
$47$ $$T^{16}$$
$53$ $$T^{16}$$
$59$ $$4 + 16 T + 56 T^{2} + 64 T^{3} + 68 T^{4} - 128 T^{5} + 56 T^{6} - 16 T^{7} + 18 T^{8} + 72 T^{9} - 16 T^{10} - 8 T^{11} + 10 T^{12} - 4 T^{14} + T^{16}$$
$61$ $$T^{16}$$
$67$ $$1 + 16 T + 76 T^{2} + 96 T^{3} + 146 T^{4} + 24 T^{5} + 112 T^{6} - 96 T^{7} + 2 T^{8} - 32 T^{9} + 52 T^{10} - 2 T^{12} - 8 T^{13} + T^{16}$$
$71$ $$T^{16}$$
$73$ $$( 4 - 4 T + 2 T^{2} - 2 T^{3} + T^{4} )^{4}$$
$79$ $$T^{16}$$
$83$ $$1 + 8 T + 24 T^{2} - 144 T^{3} + 274 T^{4} - 288 T^{5} + 216 T^{6} - 32 T^{7} + 5 T^{8} + 16 T^{9} - 20 T^{10} - 16 T^{11} + 4 T^{12} + T^{16}$$
$89$ $$( 1 + 4 T + 18 T^{2} + 16 T^{3} + 2 T^{4} - 4 T^{5} - 2 T^{6} + T^{8} )^{2}$$
$97$ $$1 - T^{8} + T^{16}$$