Properties

Label 876.1.br.a
Level $876$
Weight $1$
Character orbit 876.br
Analytic conductor $0.437$
Analytic rank $0$
Dimension $12$
Projective image $D_{36}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 876 = 2^{2} \cdot 3 \cdot 73 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 876.br (of order \(36\), degree \(12\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.437180951067\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\Q(\zeta_{36})\)
Defining polynomial: \(x^{12} - x^{6} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{36}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{36} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{36}^{15} q^{3} + ( \zeta_{36}^{10} + \zeta_{36}^{11} ) q^{7} -\zeta_{36}^{12} q^{9} +O(q^{10})\) \( q -\zeta_{36}^{15} q^{3} + ( \zeta_{36}^{10} + \zeta_{36}^{11} ) q^{7} -\zeta_{36}^{12} q^{9} + ( \zeta_{36}^{2} - \zeta_{36}^{11} ) q^{13} + ( -\zeta_{36}^{13} + \zeta_{36}^{15} ) q^{19} + ( \zeta_{36}^{7} + \zeta_{36}^{8} ) q^{21} + \zeta_{36}^{17} q^{25} -\zeta_{36}^{9} q^{27} + ( \zeta_{36} + \zeta_{36}^{6} ) q^{31} + ( -\zeta_{36} - \zeta_{36}^{7} ) q^{37} + ( -\zeta_{36}^{8} - \zeta_{36}^{17} ) q^{39} + ( -\zeta_{36}^{5} + \zeta_{36}^{16} ) q^{43} + ( -\zeta_{36}^{2} - \zeta_{36}^{3} - \zeta_{36}^{4} ) q^{49} + ( -\zeta_{36}^{10} + \zeta_{36}^{12} ) q^{57} + ( -1 - \zeta_{36}^{14} ) q^{61} + ( \zeta_{36}^{4} + \zeta_{36}^{5} ) q^{63} + ( -\zeta_{36}^{6} - \zeta_{36}^{16} ) q^{67} + \zeta_{36}^{9} q^{73} + \zeta_{36}^{14} q^{75} + ( \zeta_{36}^{9} + \zeta_{36}^{13} ) q^{79} -\zeta_{36}^{6} q^{81} + ( \zeta_{36}^{3} + \zeta_{36}^{4} + \zeta_{36}^{12} + \zeta_{36}^{13} ) q^{91} + ( \zeta_{36}^{3} - \zeta_{36}^{16} ) q^{93} + ( -\zeta_{36}^{10} + \zeta_{36}^{14} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12q + 6q^{9} + O(q^{10}) \) \( 12q + 6q^{9} + 6q^{31} - 6q^{57} - 12q^{61} - 6q^{67} - 6q^{81} - 6q^{91} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/876\mathbb{Z}\right)^\times\).

\(n\) \(293\) \(439\) \(589\)
\(\chi(n)\) \(-1\) \(1\) \(\zeta_{36}^{17}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
257.1
−0.984808 0.173648i
−0.642788 + 0.766044i
−0.984808 + 0.173648i
−0.342020 0.939693i
0.342020 + 0.939693i
0.984808 0.173648i
0.642788 0.766044i
0.984808 + 0.173648i
−0.642788 0.766044i
0.342020 0.939693i
−0.342020 + 0.939693i
0.642788 + 0.766044i
0 −0.866025 + 0.500000i 0 0 0 0.168372 + 0.0451151i 0 0.500000 0.866025i 0
269.1 0 0.866025 0.500000i 0 0 0 0.218763 0.816436i 0 0.500000 0.866025i 0
317.1 0 −0.866025 0.500000i 0 0 0 0.168372 0.0451151i 0 0.500000 + 0.866025i 0
353.1 0 0.866025 0.500000i 0 0 0 0.296905 1.10806i 0 0.500000 0.866025i 0
377.1 0 −0.866025 + 0.500000i 0 0 0 1.58248 + 0.424024i 0 0.500000 0.866025i 0
413.1 0 0.866025 + 0.500000i 0 0 0 −0.515668 1.92450i 0 0.500000 + 0.866025i 0
461.1 0 −0.866025 + 0.500000i 0 0 0 −1.75085 0.469139i 0 0.500000 0.866025i 0
473.1 0 0.866025 0.500000i 0 0 0 −0.515668 + 1.92450i 0 0.500000 0.866025i 0
749.1 0 0.866025 + 0.500000i 0 0 0 0.218763 + 0.816436i 0 0.500000 + 0.866025i 0
797.1 0 −0.866025 0.500000i 0 0 0 1.58248 0.424024i 0 0.500000 + 0.866025i 0
809.1 0 0.866025 + 0.500000i 0 0 0 0.296905 + 1.10806i 0 0.500000 + 0.866025i 0
857.1 0 −0.866025 0.500000i 0 0 0 −1.75085 + 0.469139i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 857.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
73.k even 36 1 inner
219.v odd 36 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 876.1.br.a 12
3.b odd 2 1 CM 876.1.br.a 12
4.b odd 2 1 3504.1.fm.a 12
12.b even 2 1 3504.1.fm.a 12
73.k even 36 1 inner 876.1.br.a 12
219.v odd 36 1 inner 876.1.br.a 12
292.u odd 36 1 3504.1.fm.a 12
876.bq even 36 1 3504.1.fm.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
876.1.br.a 12 1.a even 1 1 trivial
876.1.br.a 12 3.b odd 2 1 CM
876.1.br.a 12 73.k even 36 1 inner
876.1.br.a 12 219.v odd 36 1 inner
3504.1.fm.a 12 4.b odd 2 1
3504.1.fm.a 12 12.b even 2 1
3504.1.fm.a 12 292.u odd 36 1
3504.1.fm.a 12 876.bq even 36 1

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(876, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{3} \)
$5$ \( T^{12} \)
$7$ \( 1 - 12 T + 45 T^{2} - 52 T^{3} + 69 T^{4} - 18 T^{5} + 2 T^{6} + 12 T^{7} - 9 T^{8} - 2 T^{9} + T^{12} \)
$11$ \( T^{12} \)
$13$ \( 64 + 32 T^{3} + 8 T^{6} + 4 T^{9} + T^{12} \)
$17$ \( T^{12} \)
$19$ \( 1 + 3 T^{2} + 69 T^{4} + 8 T^{6} - 6 T^{8} - 3 T^{10} + T^{12} \)
$23$ \( T^{12} \)
$29$ \( T^{12} \)
$31$ \( 1 - 12 T + 51 T^{2} - 70 T^{3} + 75 T^{4} - 114 T^{5} + 140 T^{6} - 126 T^{7} + 90 T^{8} - 50 T^{9} + 21 T^{10} - 6 T^{11} + T^{12} \)
$37$ \( 729 + 27 T^{6} + T^{12} \)
$41$ \( T^{12} \)
$43$ \( 1 + 12 T + 45 T^{2} + 52 T^{3} + 69 T^{4} + 18 T^{5} + 2 T^{6} - 12 T^{7} - 9 T^{8} + 2 T^{9} + T^{12} \)
$47$ \( T^{12} \)
$53$ \( T^{12} \)
$59$ \( T^{12} \)
$61$ \( ( 3 + 9 T + 18 T^{2} + 21 T^{3} + 15 T^{4} + 6 T^{5} + T^{6} )^{2} \)
$67$ \( ( 3 + 9 T + 9 T^{2} + 6 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} )^{2} \)
$71$ \( T^{12} \)
$73$ \( ( 1 + T^{2} )^{6} \)
$79$ \( 1 - 15 T^{2} + 60 T^{4} + 35 T^{6} + 21 T^{8} + 6 T^{10} + T^{12} \)
$83$ \( T^{12} \)
$89$ \( T^{12} \)
$97$ \( ( 3 + 9 T + 9 T^{2} - 3 T^{4} + T^{6} )^{2} \)
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