Newspace parameters
Level: | \( N \) | \(=\) | \( 168 = 2^{3} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 168.q (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.34148675396\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-19})\) |
Defining polynomial: |
\( x^{4} - x^{3} - 4x^{2} - 5x + 25 \)
|
Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - x^{3} - 4x^{2} - 5x + 25 \)
:
\(\beta_{1}\) | \(=\) |
\( \nu \)
|
\(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 4\nu^{2} - 4\nu - 5 ) / 20 \)
|
\(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 4\nu + 5 ) / 4 \)
|
\(\nu\) | \(=\) |
\( \beta_1 \)
|
\(\nu^{2}\) | \(=\) |
\( \beta_{3} + 5\beta_{2} \)
|
\(\nu^{3}\) | \(=\) |
\( -4\beta_{3} + 4\beta _1 + 5 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).
\(n\) | \(73\) | \(85\) | \(113\) | \(127\) |
\(\chi(n)\) | \(-\beta_{2}\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
25.1 |
|
0 | 0.500000 | − | 0.866025i | 0 | −1.63746 | − | 2.83616i | 0 | −1.50000 | − | 2.17945i | 0 | −0.500000 | − | 0.866025i | 0 | ||||||||||||||||||||||
25.2 | 0 | 0.500000 | − | 0.866025i | 0 | 2.13746 | + | 3.70219i | 0 | −1.50000 | + | 2.17945i | 0 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||||||||||
121.1 | 0 | 0.500000 | + | 0.866025i | 0 | −1.63746 | + | 2.83616i | 0 | −1.50000 | + | 2.17945i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||
121.2 | 0 | 0.500000 | + | 0.866025i | 0 | 2.13746 | − | 3.70219i | 0 | −1.50000 | − | 2.17945i | 0 | −0.500000 | + | 0.866025i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 168.2.q.c | ✓ | 4 |
3.b | odd | 2 | 1 | 504.2.s.i | 4 | ||
4.b | odd | 2 | 1 | 336.2.q.g | 4 | ||
7.b | odd | 2 | 1 | 1176.2.q.l | 4 | ||
7.c | even | 3 | 1 | inner | 168.2.q.c | ✓ | 4 |
7.c | even | 3 | 1 | 1176.2.a.k | 2 | ||
7.d | odd | 6 | 1 | 1176.2.a.n | 2 | ||
7.d | odd | 6 | 1 | 1176.2.q.l | 4 | ||
8.b | even | 2 | 1 | 1344.2.q.w | 4 | ||
8.d | odd | 2 | 1 | 1344.2.q.x | 4 | ||
12.b | even | 2 | 1 | 1008.2.s.r | 4 | ||
21.c | even | 2 | 1 | 3528.2.s.bk | 4 | ||
21.g | even | 6 | 1 | 3528.2.a.bd | 2 | ||
21.g | even | 6 | 1 | 3528.2.s.bk | 4 | ||
21.h | odd | 6 | 1 | 504.2.s.i | 4 | ||
21.h | odd | 6 | 1 | 3528.2.a.bk | 2 | ||
28.d | even | 2 | 1 | 2352.2.q.bf | 4 | ||
28.f | even | 6 | 1 | 2352.2.a.ba | 2 | ||
28.f | even | 6 | 1 | 2352.2.q.bf | 4 | ||
28.g | odd | 6 | 1 | 336.2.q.g | 4 | ||
28.g | odd | 6 | 1 | 2352.2.a.bf | 2 | ||
56.j | odd | 6 | 1 | 9408.2.a.dj | 2 | ||
56.k | odd | 6 | 1 | 1344.2.q.x | 4 | ||
56.k | odd | 6 | 1 | 9408.2.a.dp | 2 | ||
56.m | even | 6 | 1 | 9408.2.a.dw | 2 | ||
56.p | even | 6 | 1 | 1344.2.q.w | 4 | ||
56.p | even | 6 | 1 | 9408.2.a.ec | 2 | ||
84.j | odd | 6 | 1 | 7056.2.a.ch | 2 | ||
84.n | even | 6 | 1 | 1008.2.s.r | 4 | ||
84.n | even | 6 | 1 | 7056.2.a.cu | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
168.2.q.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
168.2.q.c | ✓ | 4 | 7.c | even | 3 | 1 | inner |
336.2.q.g | 4 | 4.b | odd | 2 | 1 | ||
336.2.q.g | 4 | 28.g | odd | 6 | 1 | ||
504.2.s.i | 4 | 3.b | odd | 2 | 1 | ||
504.2.s.i | 4 | 21.h | odd | 6 | 1 | ||
1008.2.s.r | 4 | 12.b | even | 2 | 1 | ||
1008.2.s.r | 4 | 84.n | even | 6 | 1 | ||
1176.2.a.k | 2 | 7.c | even | 3 | 1 | ||
1176.2.a.n | 2 | 7.d | odd | 6 | 1 | ||
1176.2.q.l | 4 | 7.b | odd | 2 | 1 | ||
1176.2.q.l | 4 | 7.d | odd | 6 | 1 | ||
1344.2.q.w | 4 | 8.b | even | 2 | 1 | ||
1344.2.q.w | 4 | 56.p | even | 6 | 1 | ||
1344.2.q.x | 4 | 8.d | odd | 2 | 1 | ||
1344.2.q.x | 4 | 56.k | odd | 6 | 1 | ||
2352.2.a.ba | 2 | 28.f | even | 6 | 1 | ||
2352.2.a.bf | 2 | 28.g | odd | 6 | 1 | ||
2352.2.q.bf | 4 | 28.d | even | 2 | 1 | ||
2352.2.q.bf | 4 | 28.f | even | 6 | 1 | ||
3528.2.a.bd | 2 | 21.g | even | 6 | 1 | ||
3528.2.a.bk | 2 | 21.h | odd | 6 | 1 | ||
3528.2.s.bk | 4 | 21.c | even | 2 | 1 | ||
3528.2.s.bk | 4 | 21.g | even | 6 | 1 | ||
7056.2.a.ch | 2 | 84.j | odd | 6 | 1 | ||
7056.2.a.cu | 2 | 84.n | even | 6 | 1 | ||
9408.2.a.dj | 2 | 56.j | odd | 6 | 1 | ||
9408.2.a.dp | 2 | 56.k | odd | 6 | 1 | ||
9408.2.a.dw | 2 | 56.m | even | 6 | 1 | ||
9408.2.a.ec | 2 | 56.p | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - T_{5}^{3} + 15T_{5}^{2} + 14T_{5} + 196 \)
acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( (T^{2} - T + 1)^{2} \)
$5$
\( T^{4} - T^{3} + 15 T^{2} + 14 T + 196 \)
$7$
\( (T^{2} + 3 T + 7)^{2} \)
$11$
\( T^{4} - T^{3} + 15 T^{2} + 14 T + 196 \)
$13$
\( (T^{2} - 5 T - 8)^{2} \)
$17$
\( (T^{2} - 4 T + 16)^{2} \)
$19$
\( T^{4} - 5 T^{3} + 33 T^{2} + 40 T + 64 \)
$23$
\( (T^{2} + 4 T + 16)^{2} \)
$29$
\( (T^{2} - 3 T - 12)^{2} \)
$31$
\( (T^{2} - T + 1)^{2} \)
$37$
\( T^{4} + 3 T^{3} + 21 T^{2} - 36 T + 144 \)
$41$
\( (T^{2} - 6 T - 48)^{2} \)
$43$
\( (T^{2} + 7 T - 2)^{2} \)
$47$
\( (T^{2} - 6 T + 36)^{2} \)
$53$
\( T^{4} + 11 T^{3} + 105 T^{2} + \cdots + 256 \)
$59$
\( T^{4} + 5 T^{3} + 33 T^{2} - 40 T + 64 \)
$61$
\( (T^{2} + 10 T + 100)^{2} \)
$67$
\( T^{4} + 7 T^{3} + 51 T^{2} - 14 T + 4 \)
$71$
\( (T - 2)^{4} \)
$73$
\( T^{4} - T^{3} + 15 T^{2} + 14 T + 196 \)
$79$
\( T^{4} + 8 T^{3} + 105 T^{2} + \cdots + 1681 \)
$83$
\( (T^{2} - 7 T - 2)^{2} \)
$89$
\( T^{4} - 6 T^{3} + 84 T^{2} + \cdots + 2304 \)
$97$
\( (T^{2} - 25 T + 142)^{2} \)
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