Properties

Label 1176.1.s.a
Level $1176$
Weight $1$
Character orbit 1176.s
Analytic conductor $0.587$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1176.s (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.586900454856\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1176.1
Artin image: $C_6\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} - q^{6} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6}^{2} q^{5} - q^{6} + q^{8} + \zeta_{6}^{2} q^{9} -\zeta_{6} q^{10} + \zeta_{6} q^{11} -\zeta_{6}^{2} q^{12} - q^{15} + \zeta_{6}^{2} q^{16} -\zeta_{6} q^{18} + q^{20} - q^{22} + \zeta_{6} q^{24} - q^{27} - q^{29} -\zeta_{6}^{2} q^{30} -\zeta_{6} q^{31} -\zeta_{6} q^{32} + \zeta_{6}^{2} q^{33} + q^{36} + \zeta_{6}^{2} q^{40} -\zeta_{6}^{2} q^{44} -\zeta_{6} q^{45} - q^{48} + \zeta_{6} q^{53} -\zeta_{6}^{2} q^{54} - q^{55} -\zeta_{6}^{2} q^{58} -\zeta_{6} q^{59} + \zeta_{6} q^{60} + q^{62} + q^{64} -\zeta_{6} q^{66} + \zeta_{6}^{2} q^{72} + 2 \zeta_{6} q^{73} -\zeta_{6}^{2} q^{79} -\zeta_{6} q^{80} -\zeta_{6} q^{81} + q^{83} -\zeta_{6} q^{87} + \zeta_{6} q^{88} + q^{90} -\zeta_{6}^{2} q^{93} -\zeta_{6}^{2} q^{96} + q^{97} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{3} - q^{4} - q^{5} - 2 q^{6} + 2 q^{8} - q^{9} + O(q^{10}) \) \( 2 q - q^{2} + q^{3} - q^{4} - q^{5} - 2 q^{6} + 2 q^{8} - q^{9} - q^{10} + q^{11} + q^{12} - 2 q^{15} - q^{16} - q^{18} + 2 q^{20} - 2 q^{22} + q^{24} - 2 q^{27} - 2 q^{29} + q^{30} - q^{31} - q^{32} - q^{33} + 2 q^{36} - q^{40} + q^{44} - q^{45} - 2 q^{48} + q^{53} + q^{54} - 2 q^{55} + q^{58} - q^{59} + q^{60} + 2 q^{62} + 2 q^{64} - q^{66} - q^{72} + 2 q^{73} + q^{79} - q^{80} - q^{81} + 2 q^{83} - q^{87} + q^{88} + 2 q^{90} + q^{93} + q^{96} + 2 q^{97} - 2 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times\).

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 0 1.00000 −0.500000 0.866025i −0.500000 + 0.866025i
1157.1 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 0 1.00000 −0.500000 + 0.866025i −0.500000 0.866025i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
7.c even 3 1 inner
168.s odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1176.1.s.a 2
3.b odd 2 1 1176.1.s.b 2
7.b odd 2 1 168.1.s.a 2
7.c even 3 1 1176.1.n.c 1
7.c even 3 1 inner 1176.1.s.a 2
7.d odd 6 1 168.1.s.a 2
7.d odd 6 1 1176.1.n.d 1
8.b even 2 1 1176.1.s.b 2
21.c even 2 1 168.1.s.b yes 2
21.g even 6 1 168.1.s.b yes 2
21.g even 6 1 1176.1.n.a 1
21.h odd 6 1 1176.1.n.b 1
21.h odd 6 1 1176.1.s.b 2
24.h odd 2 1 CM 1176.1.s.a 2
28.d even 2 1 672.1.ba.b 2
28.f even 6 1 672.1.ba.b 2
56.e even 2 1 672.1.ba.a 2
56.h odd 2 1 168.1.s.b yes 2
56.j odd 6 1 168.1.s.b yes 2
56.j odd 6 1 1176.1.n.a 1
56.m even 6 1 672.1.ba.a 2
56.p even 6 1 1176.1.n.b 1
56.p even 6 1 1176.1.s.b 2
84.h odd 2 1 672.1.ba.a 2
84.j odd 6 1 672.1.ba.a 2
168.e odd 2 1 672.1.ba.b 2
168.i even 2 1 168.1.s.a 2
168.s odd 6 1 1176.1.n.c 1
168.s odd 6 1 inner 1176.1.s.a 2
168.ba even 6 1 168.1.s.a 2
168.ba even 6 1 1176.1.n.d 1
168.be odd 6 1 672.1.ba.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.1.s.a 2 7.b odd 2 1
168.1.s.a 2 7.d odd 6 1
168.1.s.a 2 168.i even 2 1
168.1.s.a 2 168.ba even 6 1
168.1.s.b yes 2 21.c even 2 1
168.1.s.b yes 2 21.g even 6 1
168.1.s.b yes 2 56.h odd 2 1
168.1.s.b yes 2 56.j odd 6 1
672.1.ba.a 2 56.e even 2 1
672.1.ba.a 2 56.m even 6 1
672.1.ba.a 2 84.h odd 2 1
672.1.ba.a 2 84.j odd 6 1
672.1.ba.b 2 28.d even 2 1
672.1.ba.b 2 28.f even 6 1
672.1.ba.b 2 168.e odd 2 1
672.1.ba.b 2 168.be odd 6 1
1176.1.n.a 1 21.g even 6 1
1176.1.n.a 1 56.j odd 6 1
1176.1.n.b 1 21.h odd 6 1
1176.1.n.b 1 56.p even 6 1
1176.1.n.c 1 7.c even 3 1
1176.1.n.c 1 168.s odd 6 1
1176.1.n.d 1 7.d odd 6 1
1176.1.n.d 1 168.ba even 6 1
1176.1.s.a 2 1.a even 1 1 trivial
1176.1.s.a 2 7.c even 3 1 inner
1176.1.s.a 2 24.h odd 2 1 CM
1176.1.s.a 2 168.s odd 6 1 inner
1176.1.s.b 2 3.b odd 2 1
1176.1.s.b 2 8.b even 2 1
1176.1.s.b 2 21.h odd 6 1
1176.1.s.b 2 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + T_{5} + 1 \) acting on \(S_{1}^{\mathrm{new}}(1176, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 + T + T^{2} \)
$7$ \( T^{2} \)
$11$ \( 1 - T + T^{2} \)
$13$ \( T^{2} \)
$17$ \( T^{2} \)
$19$ \( T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 1 + T )^{2} \)
$31$ \( 1 + T + T^{2} \)
$37$ \( T^{2} \)
$41$ \( T^{2} \)
$43$ \( T^{2} \)
$47$ \( T^{2} \)
$53$ \( 1 - T + T^{2} \)
$59$ \( 1 + T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 - 2 T + T^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( ( -1 + T )^{2} \)
$89$ \( T^{2} \)
$97$ \( ( -1 + T )^{2} \)
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