Newspace parameters
| Level: | \( N \) | \(=\) | \( 130 = 2 \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 130.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(1.03805522628\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} - x^{3} - 2x^{2} - 3x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 2\nu^{2} + \nu - 6 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - \nu^{2} + \nu - 3 ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} - \nu^{2} + 4\nu + 9 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{2} + \beta _1 + 1 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{3} + 3\beta_{2} + \beta _1 + 8 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).
| \(n\) | \(27\) | \(41\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 129.1 |
|
1.00000 | − | 2.52434i | 1.00000 | −2.18614 | + | 0.469882i | − | 2.52434i | 2.37228 | 1.00000 | −3.37228 | −2.18614 | + | 0.469882i | ||||||||||||||||||||||||
| 129.2 | 1.00000 | − | 0.792287i | 1.00000 | 0.686141 | − | 2.12819i | − | 0.792287i | −3.37228 | 1.00000 | 2.37228 | 0.686141 | − | 2.12819i | |||||||||||||||||||||||||
| 129.3 | 1.00000 | 0.792287i | 1.00000 | 0.686141 | + | 2.12819i | 0.792287i | −3.37228 | 1.00000 | 2.37228 | 0.686141 | + | 2.12819i | |||||||||||||||||||||||||||
| 129.4 | 1.00000 | 2.52434i | 1.00000 | −2.18614 | − | 0.469882i | 2.52434i | 2.37228 | 1.00000 | −3.37228 | −2.18614 | − | 0.469882i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 65.d | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 130.2.c.b | yes | 4 |
| 3.b | odd | 2 | 1 | 1170.2.f.a | 4 | ||
| 4.b | odd | 2 | 1 | 1040.2.f.c | 4 | ||
| 5.b | even | 2 | 1 | 130.2.c.a | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 650.2.d.e | 8 | ||
| 13.b | even | 2 | 1 | 130.2.c.a | ✓ | 4 | |
| 13.d | odd | 4 | 2 | 1690.2.b.d | 8 | ||
| 15.d | odd | 2 | 1 | 1170.2.f.b | 4 | ||
| 20.d | odd | 2 | 1 | 1040.2.f.d | 4 | ||
| 39.d | odd | 2 | 1 | 1170.2.f.b | 4 | ||
| 52.b | odd | 2 | 1 | 1040.2.f.d | 4 | ||
| 65.d | even | 2 | 1 | inner | 130.2.c.b | yes | 4 |
| 65.f | even | 4 | 2 | 8450.2.a.cj | 4 | ||
| 65.g | odd | 4 | 2 | 1690.2.b.d | 8 | ||
| 65.h | odd | 4 | 2 | 650.2.d.e | 8 | ||
| 65.k | even | 4 | 2 | 8450.2.a.cn | 4 | ||
| 195.e | odd | 2 | 1 | 1170.2.f.a | 4 | ||
| 260.g | odd | 2 | 1 | 1040.2.f.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 130.2.c.a | ✓ | 4 | 5.b | even | 2 | 1 | |
| 130.2.c.a | ✓ | 4 | 13.b | even | 2 | 1 | |
| 130.2.c.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 130.2.c.b | yes | 4 | 65.d | even | 2 | 1 | inner |
| 650.2.d.e | 8 | 5.c | odd | 4 | 2 | ||
| 650.2.d.e | 8 | 65.h | odd | 4 | 2 | ||
| 1040.2.f.c | 4 | 4.b | odd | 2 | 1 | ||
| 1040.2.f.c | 4 | 260.g | odd | 2 | 1 | ||
| 1040.2.f.d | 4 | 20.d | odd | 2 | 1 | ||
| 1040.2.f.d | 4 | 52.b | odd | 2 | 1 | ||
| 1170.2.f.a | 4 | 3.b | odd | 2 | 1 | ||
| 1170.2.f.a | 4 | 195.e | odd | 2 | 1 | ||
| 1170.2.f.b | 4 | 15.d | odd | 2 | 1 | ||
| 1170.2.f.b | 4 | 39.d | odd | 2 | 1 | ||
| 1690.2.b.d | 8 | 13.d | odd | 4 | 2 | ||
| 1690.2.b.d | 8 | 65.g | odd | 4 | 2 | ||
| 8450.2.a.cj | 4 | 65.f | even | 4 | 2 | ||
| 8450.2.a.cn | 4 | 65.k | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + T_{7} - 8 \)
acting on \(S_{2}^{\mathrm{new}}(130, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
|
| $3$ |
\( T^{4} + 7T^{2} + 4 \)
|
| $5$ |
\( T^{4} + 3 T^{3} + \cdots + 25 \)
|
| $7$ |
\( (T^{2} + T - 8)^{2} \)
|
| $11$ |
\( T^{4} + 28T^{2} + 64 \)
|
| $13$ |
\( (T^{2} + 2 T + 13)^{2} \)
|
| $17$ |
\( T^{4} + 43T^{2} + 256 \)
|
| $19$ |
\( (T^{2} + 12)^{2} \)
|
| $23$ |
\( (T^{2} + 44)^{2} \)
|
| $29$ |
\( (T^{2} + 6 T - 24)^{2} \)
|
| $31$ |
\( (T^{2} + 12)^{2} \)
|
| $37$ |
\( (T^{2} + T - 74)^{2} \)
|
| $41$ |
\( T^{4} + 112T^{2} + 1024 \)
|
| $43$ |
\( T^{4} + 87T^{2} + 36 \)
|
| $47$ |
\( (T^{2} + 15 T + 48)^{2} \)
|
| $53$ |
\( T^{4} + 28T^{2} + 64 \)
|
| $59$ |
\( (T^{2} + 44)^{2} \)
|
| $61$ |
\( (T^{2} + 2 T - 32)^{2} \)
|
| $67$ |
\( (T - 4)^{4} \)
|
| $71$ |
\( T^{4} + 175T^{2} + 2500 \)
|
| $73$ |
\( (T - 10)^{4} \)
|
| $79$ |
\( (T^{2} + 2 T - 32)^{2} \)
|
| $83$ |
\( (T^{2} - 6 T - 24)^{2} \)
|
| $89$ |
\( T^{4} + 76T^{2} + 256 \)
|
| $97$ |
\( (T^{2} - 2 T - 32)^{2} \)
|
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