Newspace parameters
Level: | \( N \) | \(=\) | \( 168 = 2^{3} \cdot 3 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 168.j (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.34148675396\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(i, \sqrt{5})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} + 3x^{2} + 1 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{4} + 3x^{2} + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu^{2} + \nu + 1 \) |
\(\beta_{2}\) | \(=\) | \( \nu^{3} + 2\nu \) |
\(\beta_{3}\) | \(=\) | \( -\nu^{2} + \nu - 1 \) |
\(\nu\) | \(=\) | \( ( \beta_{3} + \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( -\beta_{3} + \beta _1 - 2 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( -\beta_{3} + \beta_{2} - \beta_1 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/168\mathbb{Z}\right)^\times\).
\(n\) | \(73\) | \(85\) | \(113\) | \(127\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
155.1 |
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1.00000 | − | 1.00000i | −1.61803 | − | 0.618034i | − | 2.00000i | −3.23607 | −2.23607 | + | 1.00000i | − | 1.00000i | −2.00000 | − | 2.00000i | 2.23607 | + | 2.00000i | −3.23607 | + | 3.23607i | ||||||||||||||||
155.2 | 1.00000 | − | 1.00000i | 0.618034 | + | 1.61803i | − | 2.00000i | 1.23607 | 2.23607 | + | 1.00000i | − | 1.00000i | −2.00000 | − | 2.00000i | −2.23607 | + | 2.00000i | 1.23607 | − | 1.23607i | |||||||||||||||||
155.3 | 1.00000 | + | 1.00000i | −1.61803 | + | 0.618034i | 2.00000i | −3.23607 | −2.23607 | − | 1.00000i | 1.00000i | −2.00000 | + | 2.00000i | 2.23607 | − | 2.00000i | −3.23607 | − | 3.23607i | |||||||||||||||||||
155.4 | 1.00000 | + | 1.00000i | 0.618034 | − | 1.61803i | 2.00000i | 1.23607 | 2.23607 | − | 1.00000i | 1.00000i | −2.00000 | + | 2.00000i | −2.23607 | − | 2.00000i | 1.23607 | + | 1.23607i | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
24.f | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 168.2.j.c | yes | 4 |
3.b | odd | 2 | 1 | 168.2.j.a | ✓ | 4 | |
4.b | odd | 2 | 1 | 672.2.j.b | 4 | ||
8.b | even | 2 | 1 | 672.2.j.c | 4 | ||
8.d | odd | 2 | 1 | 168.2.j.a | ✓ | 4 | |
12.b | even | 2 | 1 | 672.2.j.c | 4 | ||
24.f | even | 2 | 1 | inner | 168.2.j.c | yes | 4 |
24.h | odd | 2 | 1 | 672.2.j.b | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
168.2.j.a | ✓ | 4 | 3.b | odd | 2 | 1 | |
168.2.j.a | ✓ | 4 | 8.d | odd | 2 | 1 | |
168.2.j.c | yes | 4 | 1.a | even | 1 | 1 | trivial |
168.2.j.c | yes | 4 | 24.f | even | 2 | 1 | inner |
672.2.j.b | 4 | 4.b | odd | 2 | 1 | ||
672.2.j.b | 4 | 24.h | odd | 2 | 1 | ||
672.2.j.c | 4 | 8.b | even | 2 | 1 | ||
672.2.j.c | 4 | 12.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 2T_{5} - 4 \)
acting on \(S_{2}^{\mathrm{new}}(168, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T^{2} - 2 T + 2)^{2} \)
$3$
\( T^{4} + 2 T^{3} + 2 T^{2} + 6 T + 9 \)
$5$
\( (T^{2} + 2 T - 4)^{2} \)
$7$
\( (T^{2} + 1)^{2} \)
$11$
\( (T^{2} + 20)^{2} \)
$13$
\( T^{4} + 12T^{2} + 16 \)
$17$
\( (T^{2} + 4)^{2} \)
$19$
\( (T^{2} - 10 T + 20)^{2} \)
$23$
\( (T^{2} + 8 T - 4)^{2} \)
$29$
\( T^{4} \)
$31$
\( (T^{2} + 80)^{2} \)
$37$
\( T^{4} + 48T^{2} + 256 \)
$41$
\( (T^{2} + 20)^{2} \)
$43$
\( (T^{2} - 8 T - 4)^{2} \)
$47$
\( (T^{2} - 4 T - 16)^{2} \)
$53$
\( (T^{2} - 12 T + 16)^{2} \)
$59$
\( T^{4} + 12T^{2} + 16 \)
$61$
\( T^{4} + 140T^{2} + 400 \)
$67$
\( (T^{2} + 4 T - 76)^{2} \)
$71$
\( (T^{2} + 4 T - 76)^{2} \)
$73$
\( (T^{2} - 8 T - 4)^{2} \)
$79$
\( T^{4} + 192T^{2} + 4096 \)
$83$
\( T^{4} + 252 T^{2} + 13456 \)
$89$
\( (T^{2} + 36)^{2} \)
$97$
\( (T^{2} - 16 T + 44)^{2} \)
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