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Properties

Label 168.1.s.a
Level $168$
Weight $1$
Character orbit 168.s
Analytic conductor $0.084$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -24
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 168.s (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.0838429221223\)
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: yes
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.1176.1
Artin image: $C_3\times S_3$
Artin field: Galois closure of 6.0.677376.1

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{3} -\zeta_{6} q^{4} -\zeta_{6}^{2} q^{5} + q^{6} -\zeta_{6} q^{7} + q^{8} + \zeta_{6}^{2} q^{9} +O(q^{10})\) \( q + \zeta_{6}^{2} q^{2} -\zeta_{6} q^{3} -\zeta_{6} q^{4} -\zeta_{6}^{2} q^{5} + q^{6} -\zeta_{6} q^{7} + q^{8} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{10} + \zeta_{6} q^{11} + \zeta_{6}^{2} q^{12} + q^{14} - q^{15} + \zeta_{6}^{2} q^{16} -\zeta_{6} q^{18} - q^{20} + \zeta_{6}^{2} q^{21} - q^{22} -\zeta_{6} q^{24} + q^{27} + \zeta_{6}^{2} q^{28} - q^{29} -\zeta_{6}^{2} q^{30} + \zeta_{6} q^{31} -\zeta_{6} q^{32} -\zeta_{6}^{2} q^{33} - q^{35} + q^{36} -\zeta_{6}^{2} q^{40} -\zeta_{6} q^{42} -\zeta_{6}^{2} q^{44} + \zeta_{6} q^{45} + q^{48} + \zeta_{6}^{2} q^{49} + \zeta_{6} q^{53} + \zeta_{6}^{2} q^{54} + q^{55} -\zeta_{6} q^{56} -\zeta_{6}^{2} q^{58} + \zeta_{6} q^{59} + \zeta_{6} q^{60} - q^{62} + q^{63} + q^{64} + \zeta_{6} q^{66} -\zeta_{6}^{2} q^{70} + \zeta_{6}^{2} q^{72} -2 \zeta_{6} q^{73} -\zeta_{6}^{2} q^{77} -\zeta_{6}^{2} q^{79} + \zeta_{6} q^{80} -\zeta_{6} q^{81} - q^{83} + q^{84} + \zeta_{6} q^{87} + \zeta_{6} q^{88} - q^{90} -\zeta_{6}^{2} q^{93} + \zeta_{6}^{2} q^{96} - q^{97} -\zeta_{6} q^{98} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{3} - q^{4} + q^{5} + 2q^{6} - q^{7} + 2q^{8} - q^{9} + O(q^{10}) \) \( 2q - q^{2} - q^{3} - q^{4} + q^{5} + 2q^{6} - q^{7} + 2q^{8} - q^{9} + q^{10} + q^{11} - q^{12} + 2q^{14} - 2q^{15} - q^{16} - q^{18} - 2q^{20} - q^{21} - 2q^{22} - q^{24} + 2q^{27} - q^{28} - 2q^{29} + q^{30} + q^{31} - q^{32} + q^{33} - 2q^{35} + 2q^{36} + q^{40} - q^{42} + q^{44} + q^{45} + 2q^{48} - q^{49} + q^{53} - q^{54} + 2q^{55} - q^{56} + q^{58} + q^{59} + q^{60} - 2q^{62} + 2q^{63} + 2q^{64} + q^{66} + q^{70} - q^{72} - 2q^{73} + q^{77} + q^{79} + q^{80} - q^{81} - 2q^{83} + 2q^{84} + q^{87} + q^{88} - 2q^{90} + q^{93} - q^{96} - 2q^{97} - q^{98} - 2q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(85\) \(113\) \(127\)
\(\chi(n)\) \(\zeta_{6}^{2}\) \(-1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
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 −0.500000 + 0.866025i 1.00000 −0.500000 − 0.866025i 0.500000 − 0.866025i
149.1 −0.500000 + 0.866025i −0.500000 − 0.866025i −0.500000 − 0.866025i 0.500000 − 0.866025i 1.00000 −0.500000 − 0.866025i 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 168.1.s.a ✓ 2
3.b odd 2 1 168.1.s.b yes 2
4.b odd 2 1 672.1.ba.b 2
7.b odd 2 1 1176.1.s.a 2
7.c even 3 1 inner 168.1.s.a ✓ 2
7.c even 3 1 1176.1.n.d 1
7.d odd 6 1 1176.1.n.c 1
7.d odd 6 1 1176.1.s.a 2
8.b even 2 1 168.1.s.b yes 2
8.d odd 2 1 672.1.ba.a 2
12.b even 2 1 672.1.ba.a 2
21.c even 2 1 1176.1.s.b 2
21.g even 6 1 1176.1.n.b 1
21.g even 6 1 1176.1.s.b 2
21.h odd 6 1 168.1.s.b yes 2
21.h odd 6 1 1176.1.n.a 1
24.f even 2 1 672.1.ba.b 2
24.h odd 2 1 CM 168.1.s.a ✓ 2
28.g odd 6 1 672.1.ba.b 2
56.h odd 2 1 1176.1.s.b 2
56.j odd 6 1 1176.1.n.b 1
56.j odd 6 1 1176.1.s.b 2
56.k odd 6 1 672.1.ba.a 2
56.p even 6 1 168.1.s.b yes 2
56.p even 6 1 1176.1.n.a 1
84.n even 6 1 672.1.ba.a 2
168.i even 2 1 1176.1.s.a 2
168.s odd 6 1 inner 168.1.s.a ✓ 2
168.s odd 6 1 1176.1.n.d 1
168.v even 6 1 672.1.ba.b 2
168.ba even 6 1 1176.1.n.c 1
168.ba even 6 1 1176.1.s.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.1.s.a ✓ 2 1.a even 1 1 trivial
168.1.s.a ✓ 2 7.c even 3 1 inner
168.1.s.a ✓ 2 24.h odd 2 1 CM
168.1.s.a ✓ 2 168.s odd 6 1 inner
168.1.s.b yes 2 3.b odd 2 1
168.1.s.b yes 2 8.b even 2 1
168.1.s.b yes 2 21.h odd 6 1
168.1.s.b yes 2 56.p even 6 1
672.1.ba.a 2 8.d odd 2 1
672.1.ba.a 2 12.b even 2 1
672.1.ba.a 2 56.k odd 6 1
672.1.ba.a 2 84.n even 6 1
672.1.ba.b 2 4.b odd 2 1
672.1.ba.b 2 24.f even 2 1
672.1.ba.b 2 28.g odd 6 1
672.1.ba.b 2 168.v even 6 1
1176.1.n.a 1 21.h odd 6 1
1176.1.n.a 1 56.p even 6 1
1176.1.n.b 1 21.g even 6 1
1176.1.n.b 1 56.j odd 6 1
1176.1.n.c 1 7.d odd 6 1
1176.1.n.c 1 168.ba even 6 1
1176.1.n.d 1 7.c even 3 1
1176.1.n.d 1 168.s odd 6 1
1176.1.s.a 2 7.b odd 2 1
1176.1.s.a 2 7.d odd 6 1
1176.1.s.a 2 168.i even 2 1
1176.1.s.a 2 168.ba even 6 1
1176.1.s.b 2 21.c even 2 1
1176.1.s.b 2 21.g even 6 1
1176.1.s.b 2 56.h odd 2 1
1176.1.s.b 2 56.j odd 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 \(S_{1}^{\mathrm{new}}(168, [\chi])\).

Hecke characteristic polynomials

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