# Properties

 Label 1067.1.bg.a Level $1067$ Weight $1$ Character orbit 1067.bg Analytic conductor $0.533$ Analytic rank $0$ Dimension $16$ Projective image $D_{48}$ CM discriminant -11 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1067 = 11 \cdot 97$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1067.bg (of order $$48$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.532502368479$$ Analytic rank: $$0$$ Dimension: $$16$$ Coefficient field: $$\Q(\zeta_{48})$$ Defining polynomial: $$x^{16} - x^{8} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{48}^{5} - \zeta_{48}^{9} ) q^{3} + \zeta_{48}^{2} q^{4} + ( -\zeta_{48}^{3} + \zeta_{48}^{14} ) q^{5} + ( \zeta_{48}^{10} - \zeta_{48}^{14} + \zeta_{48}^{18} ) q^{9} +O(q^{10})$$ $$q + ( \zeta_{48}^{5} - \zeta_{48}^{9} ) q^{3} + \zeta_{48}^{2} q^{4} + ( -\zeta_{48}^{3} + \zeta_{48}^{14} ) q^{5} + ( \zeta_{48}^{10} - \zeta_{48}^{14} + \zeta_{48}^{18} ) q^{9} + \zeta_{48}^{11} q^{11} + ( \zeta_{48}^{7} - \zeta_{48}^{11} ) q^{12} + ( -\zeta_{48}^{8} + \zeta_{48}^{12} + \zeta_{48}^{19} - \zeta_{48}^{23} ) q^{15} + \zeta_{48}^{4} q^{16} + ( -\zeta_{48}^{5} + \zeta_{48}^{16} ) q^{20} + ( \zeta_{48}^{3} - \zeta_{48}^{10} ) q^{23} + ( -\zeta_{48}^{4} + \zeta_{48}^{6} - \zeta_{48}^{17} ) q^{25} + ( \zeta_{48}^{3} + \zeta_{48}^{15} - \zeta_{48}^{19} + \zeta_{48}^{23} ) q^{27} + ( -\zeta_{48}^{16} + \zeta_{48}^{22} ) q^{31} + ( \zeta_{48}^{16} - \zeta_{48}^{20} ) q^{33} + ( \zeta_{48}^{12} - \zeta_{48}^{16} + \zeta_{48}^{20} ) q^{36} + ( -\zeta_{48}^{17} - \zeta_{48}^{18} ) q^{37} + \zeta_{48}^{13} q^{44} + ( -1 + \zeta_{48}^{4} - \zeta_{48}^{8} - \zeta_{48}^{13} + \zeta_{48}^{17} - \zeta_{48}^{21} ) q^{45} + \zeta_{48}^{18} q^{47} + ( \zeta_{48}^{9} - \zeta_{48}^{13} ) q^{48} + \zeta_{48}^{23} q^{49} + ( -\zeta_{48}^{6} - \zeta_{48}^{20} ) q^{53} + ( -\zeta_{48} - \zeta_{48}^{14} ) q^{55} + ( -1 - \zeta_{48}^{11} ) q^{59} + ( \zeta_{48} - \zeta_{48}^{10} + \zeta_{48}^{14} + \zeta_{48}^{21} ) q^{60} + \zeta_{48}^{6} q^{64} + ( \zeta_{48}^{2} - \zeta_{48}^{7} ) q^{67} + ( \zeta_{48}^{8} - \zeta_{48}^{12} - \zeta_{48}^{15} + \zeta_{48}^{19} ) q^{69} + ( \zeta_{48}^{21} - \zeta_{48}^{22} ) q^{71} + ( -\zeta_{48}^{2} - \zeta_{48}^{9} + \zeta_{48}^{11} + \zeta_{48}^{13} - \zeta_{48}^{15} - \zeta_{48}^{22} ) q^{75} + ( -\zeta_{48}^{7} + \zeta_{48}^{18} ) q^{80} + ( 1 - \zeta_{48}^{4} + \zeta_{48}^{8} - \zeta_{48}^{12} + \zeta_{48}^{20} ) q^{81} + ( \zeta_{48}^{5} - \zeta_{48}^{12} ) q^{92} + ( -\zeta_{48} - \zeta_{48}^{3} + \zeta_{48}^{7} - \zeta_{48}^{21} ) q^{93} + \zeta_{48} q^{97} + ( \zeta_{48} - \zeta_{48}^{5} + \zeta_{48}^{21} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 8q^{15} - 8q^{20} + 8q^{31} - 8q^{33} + 8q^{36} - 24q^{45} - 16q^{59} + 8q^{69} + 24q^{81} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1067\mathbb{Z}\right)^\times$$.

 $$n$$ $$199$$ $$486$$ $$\chi(n)$$ $$-\zeta_{48}^{17}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
32.1
 −0.793353 − 0.608761i 0.793353 + 0.608761i 0.793353 − 0.608761i 0.608761 − 0.793353i −0.991445 + 0.130526i 0.991445 + 0.130526i −0.608761 − 0.793353i −0.130526 + 0.991445i 0.130526 + 0.991445i −0.130526 − 0.991445i 0.130526 − 0.991445i 0.608761 + 0.793353i −0.991445 − 0.130526i 0.991445 − 0.130526i −0.608761 + 0.793353i −0.793353 + 0.608761i
0 1.91532 0.252157i 0.258819 + 0.965926i −1.34861 + 1.18270i 0 0 0 2.63896 0.707107i 0
65.1 0 −1.91532 + 0.252157i 0.258819 + 0.965926i −0.583242 0.665060i 0 0 0 2.63896 0.707107i 0
197.1 0 −1.91532 0.252157i 0.258819 0.965926i −0.583242 + 0.665060i 0 0 0 2.63896 + 0.707107i 0
219.1 0 0.252157 + 1.91532i −0.258819 0.965926i 1.88981 + 0.123864i 0 0 0 −2.63896 + 0.707107i 0
340.1 0 −0.410670 0.315118i 0.965926 0.258819i 0.665060 1.34861i 0 0 0 −0.189469 0.707107i 0
483.1 0 0.410670 0.315118i 0.965926 + 0.258819i −1.18270 + 0.583242i 0 0 0 −0.189469 + 0.707107i 0
516.1 0 −0.252157 + 1.91532i −0.258819 + 0.965926i 0.0420463 + 0.641502i 0 0 0 −2.63896 0.707107i 0
538.1 0 0.315118 + 0.410670i −0.965926 0.258819i −0.123864 0.0420463i 0 0 0 0.189469 0.707107i 0
571.1 0 −0.315118 + 0.410670i −0.965926 + 0.258819i 0.641502 + 1.88981i 0 0 0 0.189469 + 0.707107i 0
593.1 0 0.315118 0.410670i −0.965926 + 0.258819i −0.123864 + 0.0420463i 0 0 0 0.189469 + 0.707107i 0
626.1 0 −0.315118 0.410670i −0.965926 0.258819i 0.641502 1.88981i 0 0 0 0.189469 0.707107i 0
648.1 0 0.252157 1.91532i −0.258819 + 0.965926i 1.88981 0.123864i 0 0 0 −2.63896 0.707107i 0
681.1 0 −0.410670 + 0.315118i 0.965926 + 0.258819i 0.665060 + 1.34861i 0 0 0 −0.189469 + 0.707107i 0
824.1 0 0.410670 + 0.315118i 0.965926 0.258819i −1.18270 0.583242i 0 0 0 −0.189469 0.707107i 0
945.1 0 −0.252157 1.91532i −0.258819 0.965926i 0.0420463 0.641502i 0 0 0 −2.63896 + 0.707107i 0
967.1 0 1.91532 + 0.252157i 0.258819 0.965926i −1.34861 1.18270i 0 0 0 2.63896 + 0.707107i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 967.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by $$\Q(\sqrt{-11})$$
97.k even 48 1 inner
1067.bg odd 48 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1067.1.bg.a 16
11.b odd 2 1 CM 1067.1.bg.a 16
97.k even 48 1 inner 1067.1.bg.a 16
1067.bg odd 48 1 inner 1067.1.bg.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1067.1.bg.a 16 1.a even 1 1 trivial
1067.1.bg.a 16 11.b odd 2 1 CM
1067.1.bg.a 16 97.k even 48 1 inner
1067.1.bg.a 16 1067.bg odd 48 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1067, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$1 + 24 T^{4} + 191 T^{8} - 24 T^{12} + T^{16}$$
$5$ $$1 + 16 T + 84 T^{2} + 144 T^{3} + 268 T^{4} + 288 T^{5} + 216 T^{6} + 32 T^{7} + 5 T^{8} + 8 T^{9} - 20 T^{10} - 32 T^{11} - 2 T^{12} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$1 - T^{8} + T^{16}$$
$13$ $$T^{16}$$
$17$ $$T^{16}$$
$19$ $$T^{16}$$
$23$ $$1 - 8 T + 76 T^{2} - 192 T^{3} + 140 T^{4} + 40 T^{6} - 96 T^{7} + 5 T^{8} + 40 T^{9} + 52 T^{10} - 2 T^{12} - 8 T^{13} + T^{16}$$
$29$ $$T^{16}$$
$31$ $$( 4 - 8 T + 4 T^{2} - 8 T^{3} + 18 T^{4} - 16 T^{5} + 10 T^{6} - 4 T^{7} + T^{8} )^{2}$$
$37$ $$1 - 8 T + 24 T^{2} + 144 T^{3} + 274 T^{4} + 288 T^{5} + 216 T^{6} + 32 T^{7} + 5 T^{8} - 16 T^{9} - 20 T^{10} + 16 T^{11} + 4 T^{12} + T^{16}$$
$41$ $$T^{16}$$
$43$ $$T^{16}$$
$47$ $$( 1 + T^{4} )^{4}$$
$53$ $$( 1 + 4 T + 12 T^{2} - 8 T^{3} + 5 T^{4} - 4 T^{5} - 2 T^{6} + T^{8} )^{2}$$
$59$ $$1 + 8 T + 92 T^{2} + 504 T^{3} + 1750 T^{4} + 4312 T^{5} + 7980 T^{6} + 11432 T^{7} + 12869 T^{8} + 11440 T^{9} + 8008 T^{10} + 4368 T^{11} + 1820 T^{12} + 560 T^{13} + 120 T^{14} + 16 T^{15} + T^{16}$$
$61$ $$T^{16}$$
$67$ $$1 - 16 T + 76 T^{2} - 96 T^{3} + 146 T^{4} - 24 T^{5} + 112 T^{6} + 96 T^{7} + 2 T^{8} + 32 T^{9} + 52 T^{10} - 2 T^{12} + 8 T^{13} + T^{16}$$
$71$ $$1 - 8 T + 76 T^{2} - 192 T^{3} + 140 T^{4} + 40 T^{6} - 96 T^{7} + 5 T^{8} + 40 T^{9} + 52 T^{10} - 2 T^{12} - 8 T^{13} + T^{16}$$
$73$ $$T^{16}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$T^{16}$$
$97$ $$1 - T^{8} + T^{16}$$