Properties

Label 1176.1.s.c
Level $1176$
Weight $1$
Character orbit 1176.s
Analytic conductor $0.587$
Analytic rank $0$
Dimension $8$
Projective image $D_{4}$
CM discriminant -56
Inner twists $16$

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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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{4}\)
Projective field: Galois closure of 4.2.4032.1

$q$-expansion

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

\(f(q)\) \(=\) \( q - \zeta_{24}^{10} q^{2} - \zeta_{24}^{5} q^{3} - \zeta_{24}^{8} q^{4} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{5} - \zeta_{24}^{3} q^{6} - \zeta_{24}^{6} q^{8} + \zeta_{24}^{10} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{24}^{10} q^{2} - \zeta_{24}^{5} q^{3} - \zeta_{24}^{8} q^{4} + ( - \zeta_{24}^{7} - \zeta_{24}) q^{5} - \zeta_{24}^{3} q^{6} - \zeta_{24}^{6} q^{8} + \zeta_{24}^{10} q^{9} + (\zeta_{24}^{11} - \zeta_{24}^{5}) q^{10} - \zeta_{24} q^{12} + (\zeta_{24}^{9} + \zeta_{24}^{3}) q^{13} + (\zeta_{24}^{6} - 1) q^{15} - \zeta_{24}^{4} q^{16} + \zeta_{24}^{8} q^{18} + ( - \zeta_{24}^{7} + \zeta_{24}) q^{19} + (\zeta_{24}^{9} - \zeta_{24}^{3}) q^{20} + \zeta_{24}^{11} q^{24} + \zeta_{24}^{8} q^{25} + (\zeta_{24}^{7} + \zeta_{24}) q^{26} + \zeta_{24}^{3} q^{27} + (\zeta_{24}^{10} + \zeta_{24}^{4}) q^{30} - \zeta_{24}^{2} q^{32} + \zeta_{24}^{6} q^{36} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{38} + ( - \zeta_{24}^{8} + \zeta_{24}^{2}) q^{39} + (\zeta_{24}^{7} - \zeta_{24}) q^{40} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{45} + \zeta_{24}^{9} q^{48} + \zeta_{24}^{6} q^{50} + ( - \zeta_{24}^{11} + \zeta_{24}^{5}) q^{52} + \zeta_{24} q^{54} + ( - \zeta_{24}^{6} - 1) q^{57} + ( - \zeta_{24}^{11} - \zeta_{24}^{5}) q^{59} + (\zeta_{24}^{8} + \zeta_{24}^{2}) q^{60} + (\zeta_{24}^{7} - \zeta_{24}) q^{61} - q^{64} + ( - 2 \zeta_{24}^{10} + \zeta_{24}^{4}) q^{65} - \zeta_{24}^{6} q^{71} + \zeta_{24}^{4} q^{72} + \zeta_{24} q^{75} + ( - \zeta_{24}^{9} - \zeta_{24}^{3}) q^{76} + ( - \zeta_{24}^{6} + 1) q^{78} + (\zeta_{24}^{11} + \zeta_{24}^{5}) q^{80} - \zeta_{24}^{8} q^{81} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{83} + ( - \zeta_{24}^{9} + \zeta_{24}^{3}) q^{90} - \zeta_{24}^{2} q^{95} + \zeta_{24}^{7} q^{96} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{4} - 8 q^{15} - 4 q^{16} - 4 q^{18} - 4 q^{25} + 4 q^{30} + 4 q^{39} - 8 q^{57} - 4 q^{60} - 8 q^{64} + 4 q^{72} + 8 q^{78} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(295\) \(589\) \(785\) \(1081\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\zeta_{24}^{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
0.258819 0.965926i
−0.258819 + 0.965926i
0.965926 + 0.258819i
−0.965926 0.258819i
0.258819 + 0.965926i
−0.258819 0.965926i
0.965926 0.258819i
−0.965926 + 0.258819i
−0.866025 + 0.500000i −0.965926 + 0.258819i 0.500000 0.866025i 0.707107 + 1.22474i 0.707107 0.707107i 0 1.00000i 0.866025 0.500000i −1.22474 0.707107i
557.2 −0.866025 + 0.500000i 0.965926 0.258819i 0.500000 0.866025i −0.707107 1.22474i −0.707107 + 0.707107i 0 1.00000i 0.866025 0.500000i 1.22474 + 0.707107i
557.3 0.866025 0.500000i −0.258819 0.965926i 0.500000 0.866025i −0.707107 1.22474i −0.707107 0.707107i 0 1.00000i −0.866025 + 0.500000i −1.22474 0.707107i
557.4 0.866025 0.500000i 0.258819 + 0.965926i 0.500000 0.866025i 0.707107 + 1.22474i 0.707107 + 0.707107i 0 1.00000i −0.866025 + 0.500000i 1.22474 + 0.707107i
1157.1 −0.866025 0.500000i −0.965926 0.258819i 0.500000 + 0.866025i 0.707107 1.22474i 0.707107 + 0.707107i 0 1.00000i 0.866025 + 0.500000i −1.22474 + 0.707107i
1157.2 −0.866025 0.500000i 0.965926 + 0.258819i 0.500000 + 0.866025i −0.707107 + 1.22474i −0.707107 0.707107i 0 1.00000i 0.866025 + 0.500000i 1.22474 0.707107i
1157.3 0.866025 + 0.500000i −0.258819 + 0.965926i 0.500000 + 0.866025i −0.707107 + 1.22474i −0.707107 + 0.707107i 0 1.00000i −0.866025 0.500000i −1.22474 + 0.707107i
1157.4 0.866025 + 0.500000i 0.258819 0.965926i 0.500000 + 0.866025i 0.707107 1.22474i 0.707107 0.707107i 0 1.00000i −0.866025 0.500000i 1.22474 0.707107i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1157.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
56.h odd 2 1 CM by \(\Q(\sqrt{-14}) \)
3.b odd 2 1 inner
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.c even 2 1 inner
21.g even 6 1 inner
21.h odd 6 1 inner
24.h odd 2 1 inner
56.j odd 6 1 inner
56.p even 6 1 inner
168.i even 2 1 inner
168.s odd 6 1 inner
168.ba even 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.c 8
3.b odd 2 1 inner 1176.1.s.c 8
7.b odd 2 1 inner 1176.1.s.c 8
7.c even 3 1 1176.1.n.e 4
7.c even 3 1 inner 1176.1.s.c 8
7.d odd 6 1 1176.1.n.e 4
7.d odd 6 1 inner 1176.1.s.c 8
8.b even 2 1 inner 1176.1.s.c 8
21.c even 2 1 inner 1176.1.s.c 8
21.g even 6 1 1176.1.n.e 4
21.g even 6 1 inner 1176.1.s.c 8
21.h odd 6 1 1176.1.n.e 4
21.h odd 6 1 inner 1176.1.s.c 8
24.h odd 2 1 inner 1176.1.s.c 8
56.h odd 2 1 CM 1176.1.s.c 8
56.j odd 6 1 1176.1.n.e 4
56.j odd 6 1 inner 1176.1.s.c 8
56.p even 6 1 1176.1.n.e 4
56.p even 6 1 inner 1176.1.s.c 8
168.i even 2 1 inner 1176.1.s.c 8
168.s odd 6 1 1176.1.n.e 4
168.s odd 6 1 inner 1176.1.s.c 8
168.ba even 6 1 1176.1.n.e 4
168.ba even 6 1 inner 1176.1.s.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1176.1.n.e 4 7.c even 3 1
1176.1.n.e 4 7.d odd 6 1
1176.1.n.e 4 21.g even 6 1
1176.1.n.e 4 21.h odd 6 1
1176.1.n.e 4 56.j odd 6 1
1176.1.n.e 4 56.p even 6 1
1176.1.n.e 4 168.s odd 6 1
1176.1.n.e 4 168.ba even 6 1
1176.1.s.c 8 1.a even 1 1 trivial
1176.1.s.c 8 3.b odd 2 1 inner
1176.1.s.c 8 7.b odd 2 1 inner
1176.1.s.c 8 7.c even 3 1 inner
1176.1.s.c 8 7.d odd 6 1 inner
1176.1.s.c 8 8.b even 2 1 inner
1176.1.s.c 8 21.c even 2 1 inner
1176.1.s.c 8 21.g even 6 1 inner
1176.1.s.c 8 21.h odd 6 1 inner
1176.1.s.c 8 24.h odd 2 1 inner
1176.1.s.c 8 56.h odd 2 1 CM
1176.1.s.c 8 56.j odd 6 1 inner
1176.1.s.c 8 56.p even 6 1 inner
1176.1.s.c 8 168.i even 2 1 inner
1176.1.s.c 8 168.s odd 6 1 inner
1176.1.s.c 8 168.ba even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 2T_{5}^{2} + 4 \) acting on \(S_{1}^{\mathrm{new}}(1176, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{4} + 1 \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$89$ \( T^{8} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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