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 $4$

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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:

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

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