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 $4$

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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:

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

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