Properties

 Label 884.1.cn.a Level $884$ Weight $1$ Character orbit 884.cn 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.cn (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.

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

Character values

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

 $$n$$ $$105$$ $$443$$ $$613$$ $$\chi(n)$$ $$\zeta_{48}^{9}$$ $$-1$$ $$\zeta_{48}^{4}$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
63.1
 −0.608761 − 0.793353i −0.793353 + 0.608761i 0.793353 + 0.608761i 0.991445 − 0.130526i 0.608761 − 0.793353i −0.130526 − 0.991445i 0.130526 − 0.991445i −0.608761 + 0.793353i 0.793353 − 0.608761i −0.991445 − 0.130526i −0.793353 − 0.608761i 0.608761 + 0.793353i 0.991445 + 0.130526i 0.130526 + 0.991445i −0.130526 + 0.991445i −0.991445 + 0.130526i
0.991445 0.130526i 0 0.965926 0.258819i −1.05217 1.57469i 0 0 0.923880 0.382683i 0.608761 + 0.793353i −1.24871 1.42388i
167.1 −0.130526 0.991445i 0 −0.965926 + 0.258819i 0.867580 + 0.172572i 0 0 0.382683 + 0.923880i 0.793353 0.608761i 0.0578541 0.882683i
227.1 0.130526 0.991445i 0 −0.965926 0.258819i −0.349942 1.75928i 0 0 −0.382683 + 0.923880i −0.793353 0.608761i −1.78990 + 0.117317i
267.1 −0.608761 0.793353i 0 −0.258819 + 0.965926i 0.835400 + 1.25026i 0 0 0.923880 0.382683i −0.991445 + 0.130526i 0.483342 1.42388i
279.1 −0.991445 0.130526i 0 0.965926 + 0.258819i 0.534534 + 0.357164i 0 0 −0.923880 0.382683i −0.608761 + 0.793353i −0.483342 0.423880i
371.1 −0.793353 + 0.608761i 0 0.258819 0.965926i −1.95737 0.389345i 0 0 0.382683 + 0.923880i 0.130526 + 0.991445i 1.78990 0.882683i
431.1 0.793353 + 0.608761i 0 0.258819 + 0.965926i 0.0255190 + 0.128293i 0 0 −0.382683 + 0.923880i −0.130526 + 0.991445i −0.0578541 + 0.117317i
435.1 0.991445 + 0.130526i 0 0.965926 + 0.258819i −1.05217 + 1.57469i 0 0 0.923880 + 0.382683i 0.608761 0.793353i −1.24871 + 1.42388i
479.1 0.130526 + 0.991445i 0 −0.965926 + 0.258819i −0.349942 + 1.75928i 0 0 −0.382683 0.923880i −0.793353 + 0.608761i −1.78990 0.117317i
483.1 0.608761 0.793353i 0 −0.258819 0.965926i 1.09645 + 0.732626i 0 0 −0.923880 0.382683i 0.991445 + 0.130526i 1.24871 0.423880i
487.1 −0.130526 + 0.991445i 0 −0.965926 0.258819i 0.867580 0.172572i 0 0 0.382683 0.923880i 0.793353 + 0.608761i 0.0578541 + 0.882683i
583.1 −0.991445 + 0.130526i 0 0.965926 0.258819i 0.534534 0.357164i 0 0 −0.923880 + 0.382683i −0.608761 0.793353i −0.483342 + 0.423880i
639.1 −0.608761 + 0.793353i 0 −0.258819 0.965926i 0.835400 1.25026i 0 0 0.923880 + 0.382683i −0.991445 0.130526i 0.483342 + 1.42388i
683.1 0.793353 0.608761i 0 0.258819 0.965926i 0.0255190 0.128293i 0 0 −0.382683 0.923880i −0.130526 0.991445i −0.0578541 0.117317i
691.1 −0.793353 0.608761i 0 0.258819 + 0.965926i −1.95737 + 0.389345i 0 0 0.382683 0.923880i 0.130526 0.991445i 1.78990 + 0.882683i
787.1 0.608761 + 0.793353i 0 −0.258819 + 0.965926i 1.09645 0.732626i 0 0 −0.923880 + 0.382683i 0.991445 0.130526i 1.24871 + 0.423880i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 787.1 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.bh even 48 1 inner
884.cn 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.cn.a 16
4.b odd 2 1 CM 884.1.cn.a 16
13.f odd 12 1 884.1.ct.a yes 16
17.e odd 16 1 884.1.ct.a yes 16
52.l even 12 1 884.1.ct.a yes 16
68.i even 16 1 884.1.ct.a yes 16
221.bh even 48 1 inner 884.1.cn.a 16
884.cn odd 48 1 inner 884.1.cn.a 16

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

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T^{8} + T^{16}$$
$3$ $$T^{16}$$
$5$ $$1 - 8 T + 84 T^{2} - 336 T^{3} + 658 T^{4} - 672 T^{5} + 336 T^{6} - 64 T^{7} + 2 T^{8} + 8 T^{9} - 20 T^{10} + 16 T^{11} - 2 T^{12} + T^{16}$$
$7$ $$T^{16}$$
$11$ $$T^{16}$$
$13$ $$1 - T^{8} + T^{16}$$
$17$ $$( 1 - T^{4} + T^{8} )^{2}$$
$19$ $$T^{16}$$
$23$ $$T^{16}$$
$29$ $$1 - 16 T + 92 T^{2} - 168 T^{3} + 196 T^{4} - 392 T^{5} + 756 T^{6} - 1024 T^{7} + 1109 T^{8} - 1016 T^{9} + 784 T^{10} - 504 T^{11} + 266 T^{12} - 112 T^{13} + 36 T^{14} - 8 T^{15} + T^{16}$$
$31$ $$T^{16}$$
$37$ $$1 + 8 T + 24 T^{2} - 144 T^{3} + 274 T^{4} - 288 T^{5} + 216 T^{6} - 32 T^{7} + 5 T^{8} + 16 T^{9} - 20 T^{10} - 16 T^{11} + 4 T^{12} + T^{16}$$
$41$ $$1 - 16 T + 68 T^{2} - 16 T^{3} - 106 T^{4} - 16 T^{5} + 116 T^{6} + 16 T^{7} + 69 T^{8} + 24 T^{9} + 56 T^{10} + 8 T^{11} + 28 T^{12} + 8 T^{14} + T^{16}$$
$43$ $$T^{16}$$
$47$ $$T^{16}$$
$53$ $$( 1 + 8 T + 22 T^{2} + 20 T^{3} + 18 T^{4} + 16 T^{5} + 10 T^{6} + 4 T^{7} + T^{8} )^{2}$$
$59$ $$T^{16}$$
$61$ $$1 + 8 T + 40 T^{2} - 96 T^{3} + 146 T^{4} - 24 T^{5} - 152 T^{6} + 96 T^{7} + 5 T^{8} + 8 T^{9} + 52 T^{10} + 4 T^{12} + 8 T^{13} + T^{16}$$
$67$ $$T^{16}$$
$71$ $$T^{16}$$
$73$ $$1 + 8 T + 8 T^{2} - 168 T^{3} + 406 T^{4} - 560 T^{5} + 756 T^{6} - 1000 T^{7} + 1106 T^{8} - 1016 T^{9} + 784 T^{10} - 504 T^{11} + 266 T^{12} - 112 T^{13} + 36 T^{14} - 8 T^{15} + T^{16}$$
$79$ $$T^{16}$$
$83$ $$T^{16}$$
$89$ $$( 4 - 8 T^{2} + 14 T^{4} - 4 T^{6} + T^{8} )^{2}$$
$97$ $$4 + 16 T + 40 T^{2} + 96 T^{3} + 140 T^{4} - 48 T^{5} + 40 T^{6} + 192 T^{7} + 2 T^{8} - 8 T^{9} + 88 T^{10} - 2 T^{12} + 16 T^{13} + T^{16}$$