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