# Properties

 Label 884.1.ct.a Level $884$ Weight $1$ Character orbit 884.ct Analytic conductor $0.441$ Analytic rank $0$ Dimension $16$ Projective image $D_{48}$ CM discriminant -4 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$884 = 2^{2} \cdot 13 \cdot 17$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 884.ct (of order $$48$$, degree $$16$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.441173471168$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{48}^{17} q^{2} -\zeta_{48}^{10} q^{4} + ( \zeta_{48}^{2} - \zeta_{48}^{19} ) q^{5} + \zeta_{48}^{3} q^{8} + \zeta_{48}^{13} q^{9} +O(q^{10})$$ $$q + \zeta_{48}^{17} q^{2} -\zeta_{48}^{10} q^{4} + ( \zeta_{48}^{2} - \zeta_{48}^{19} ) q^{5} + \zeta_{48}^{3} q^{8} + \zeta_{48}^{13} q^{9} + ( \zeta_{48}^{12} + \zeta_{48}^{19} ) q^{10} + \zeta_{48}^{23} q^{13} + \zeta_{48}^{20} q^{16} -\zeta_{48}^{18} q^{17} -\zeta_{48}^{6} q^{18} + ( -\zeta_{48}^{5} - \zeta_{48}^{12} ) q^{20} + ( \zeta_{48}^{4} - \zeta_{48}^{14} - \zeta_{48}^{21} ) q^{25} -\zeta_{48}^{16} q^{26} + ( -1 + \zeta_{48} ) q^{29} -\zeta_{48}^{13} q^{32} + \zeta_{48}^{11} q^{34} -\zeta_{48}^{23} q^{36} + ( \zeta_{48}^{10} - \zeta_{48}^{15} ) q^{37} + ( \zeta_{48}^{5} - \zeta_{48}^{22} ) q^{40} + ( -\zeta_{48}^{3} + \zeta_{48}^{16} ) q^{41} + ( \zeta_{48}^{8} + \zeta_{48}^{15} ) q^{45} -\zeta_{48}^{11} q^{49} + ( \zeta_{48}^{7} + \zeta_{48}^{14} + \zeta_{48}^{21} ) q^{50} + \zeta_{48}^{9} q^{52} + ( -\zeta_{48}^{8} + \zeta_{48}^{22} ) q^{53} + ( -\zeta_{48}^{17} + \zeta_{48}^{18} ) q^{58} + ( \zeta_{48}^{9} + \zeta_{48}^{14} ) q^{61} + \zeta_{48}^{6} q^{64} + ( -\zeta_{48} + \zeta_{48}^{18} ) q^{65} -\zeta_{48}^{4} q^{68} + \zeta_{48}^{16} q^{72} + ( -\zeta_{48}^{4} - \zeta_{48}^{17} ) q^{73} + ( -\zeta_{48}^{3} + \zeta_{48}^{8} ) q^{74} + ( \zeta_{48}^{15} + \zeta_{48}^{22} ) q^{80} -\zeta_{48}^{2} q^{81} + ( -\zeta_{48}^{9} - \zeta_{48}^{20} ) q^{82} + ( -\zeta_{48}^{13} - \zeta_{48}^{20} ) q^{85} + ( \zeta_{48}^{5} + \zeta_{48}^{11} ) q^{89} + ( -\zeta_{48} - \zeta_{48}^{8} ) q^{90} + ( \zeta_{48}^{7} - \zeta_{48}^{22} ) q^{97} + \zeta_{48}^{4} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q + 8q^{26} - 16q^{29} - 8q^{41} + 8q^{45} - 8q^{53} - 8q^{72} + 8q^{74} - 8q^{90} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/884\mathbb{Z}\right)^\times$$.

 $$n$$ $$105$$ $$443$$ $$613$$ $$\chi(n)$$ $$-\zeta_{48}^{21}$$ $$-1$$ $$-\zeta_{48}^{4}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 0.793353 + 0.608761i −0.991445 − 0.130526i 0.991445 − 0.130526i −0.793353 − 0.608761i 0.130526 + 0.991445i 0.608761 − 0.793353i −0.608761 − 0.793353i 0.793353 − 0.608761i −0.130526 − 0.991445i 0.991445 + 0.130526i −0.991445 + 0.130526i 0.130526 − 0.991445i −0.793353 + 0.608761i −0.608761 + 0.793353i −0.130526 + 0.991445i 0.608761 + 0.793353i
0.130526 0.991445i 0 −0.965926 0.258819i −0.732626 + 1.09645i 0 0 −0.382683 + 0.923880i −0.608761 + 0.793353i 0.991445 + 0.869474i
11.1 0.608761 0.793353i 0 −0.258819 0.965926i 0.172572 + 0.867580i 0 0 −0.923880 0.382683i 0.130526 0.991445i 0.793353 + 0.391239i
71.1 −0.608761 0.793353i 0 −0.258819 + 0.965926i 1.75928 + 0.349942i 0 0 0.923880 0.382683i −0.130526 0.991445i −0.793353 1.60876i
163.1 −0.130526 + 0.991445i 0 −0.965926 0.258819i 1.25026 + 0.835400i 0 0 0.382683 0.923880i 0.608761 0.793353i −0.991445 + 1.13053i
175.1 0.793353 0.608761i 0 0.258819 0.965926i −0.357164 0.534534i 0 0 −0.382683 0.923880i 0.991445 0.130526i −0.608761 0.206647i
215.1 −0.991445 0.130526i 0 0.965926 + 0.258819i −0.389345 1.95737i 0 0 −0.923880 0.382683i 0.793353 + 0.608761i 0.130526 + 1.99144i
275.1 0.991445 0.130526i 0 0.965926 0.258819i −0.128293 0.0255190i 0 0 0.923880 0.382683i −0.793353 + 0.608761i −0.130526 0.00855514i
379.1 0.130526 + 0.991445i 0 −0.965926 + 0.258819i −0.732626 1.09645i 0 0 −0.382683 0.923880i −0.608761 0.793353i 0.991445 0.869474i
539.1 −0.793353 + 0.608761i 0 0.258819 0.965926i −1.57469 + 1.05217i 0 0 0.382683 + 0.923880i −0.991445 + 0.130526i 0.608761 1.79335i
635.1 −0.608761 + 0.793353i 0 −0.258819 0.965926i 1.75928 0.349942i 0 0 0.923880 + 0.382683i −0.130526 + 0.991445i −0.793353 + 1.60876i
643.1 0.608761 + 0.793353i 0 −0.258819 + 0.965926i 0.172572 0.867580i 0 0 −0.923880 + 0.382683i 0.130526 + 0.991445i 0.793353 0.391239i
687.1 0.793353 + 0.608761i 0 0.258819 + 0.965926i −0.357164 + 0.534534i 0 0 −0.382683 + 0.923880i 0.991445 + 0.130526i −0.608761 + 0.206647i
743.1 −0.130526 0.991445i 0 −0.965926 + 0.258819i 1.25026 0.835400i 0 0 0.382683 + 0.923880i 0.608761 + 0.793353i −0.991445 1.13053i
839.1 0.991445 + 0.130526i 0 0.965926 + 0.258819i −0.128293 + 0.0255190i 0 0 0.923880 + 0.382683i −0.793353 0.608761i −0.130526 + 0.00855514i
843.1 −0.793353 0.608761i 0 0.258819 + 0.965926i −1.57469 1.05217i 0 0 0.382683 0.923880i −0.991445 0.130526i 0.608761 + 1.79335i
847.1 −0.991445 + 0.130526i 0 0.965926 0.258819i −0.389345 + 1.95737i 0 0 −0.923880 + 0.382683i 0.793353 0.608761i 0.130526 1.99144i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 847.1 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 CM by $$\Q(\sqrt{-1})$$
221.bi even 48 1 inner
884.ct odd 48 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 884.1.ct.a yes 16
4.b odd 2 1 CM 884.1.ct.a yes 16
13.f odd 12 1 884.1.cn.a 16
17.e odd 16 1 884.1.cn.a 16
52.l even 12 1 884.1.cn.a 16
68.i even 16 1 884.1.cn.a 16
221.bi even 48 1 inner 884.1.ct.a yes 16
884.ct odd 48 1 inner 884.1.ct.a yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
884.1.cn.a 16 13.f odd 12 1
884.1.cn.a 16 17.e odd 16 1
884.1.cn.a 16 52.l even 12 1
884.1.cn.a 16 68.i even 16 1
884.1.ct.a yes 16 1.a even 1 1 trivial
884.1.ct.a yes 16 4.b odd 2 1 CM
884.1.ct.a yes 16 221.bi even 48 1 inner
884.1.ct.a yes 16 884.ct odd 48 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(884, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{8} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$1 + 16 T + 76 T^{2} + 96 T^{3} + 146 T^{4} + 24 T^{5} + 112 T^{6} - 96 T^{7} + 2 T^{8} - 32 T^{9} + 52 T^{10} - 2 T^{12} - 8 T^{13} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$1 - T^{8} + T^{16}$$
$17$ $$( 1 + T^{4} )^{4}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$1 + 8 T + 92 T^{2} + 504 T^{3} + 1750 T^{4} + 4312 T^{5} + 7980 T^{6} + 11432 T^{7} + 12869 T^{8} + 11440 T^{9} + 8008 T^{10} + 4368 T^{11} + 1820 T^{12} + 560 T^{13} + 120 T^{14} + 16 T^{15} + T^{16}$$
$31$ $$T^{16}$$
$37$ $$1 - 8 T + 76 T^{2} - 192 T^{3} + 140 T^{4} + 40 T^{6} - 96 T^{7} + 5 T^{8} + 40 T^{9} + 52 T^{10} - 2 T^{12} - 8 T^{13} + T^{16}$$
$41$ $$1 + 16 T + 92 T^{2} + 168 T^{3} + 196 T^{4} + 392 T^{5} + 756 T^{6} + 1024 T^{7} + 1109 T^{8} + 1016 T^{9} + 784 T^{10} + 504 T^{11} + 266 T^{12} + 112 T^{13} + 36 T^{14} + 8 T^{15} + T^{16}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$( 1 + 8 T + 22 T^{2} + 20 T^{3} + 18 T^{4} + 16 T^{5} + 10 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$59$ $$T^{16}$$
$61$ $$1 + 8 T + 76 T^{2} + 192 T^{3} + 140 T^{4} + 40 T^{6} + 96 T^{7} + 5 T^{8} - 40 T^{9} + 52 T^{10} - 2 T^{12} + 8 T^{13} + T^{16}$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$1 + 8 T + 56 T^{2} - 16 T^{3} + 86 T^{4} - 112 T^{5} + 44 T^{6} + 112 T^{7} + 18 T^{8} + 24 T^{9} - 16 T^{10} - 16 T^{11} + 10 T^{12} - 4 T^{14} + T^{16}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$( 4 + 8 T^{2} + 14 T^{4} + 4 T^{6} + T^{8} )^{2}$$
$97$ $$4 - 16 T + 24 T^{2} - 96 T^{3} + 268 T^{4} - 288 T^{5} + 216 T^{6} - 32 T^{7} + 2 T^{8} - 8 T^{9} + 40 T^{10} - 16 T^{11} - 2 T^{12} + T^{16}$$