Newspace parameters
| Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 117.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(6.90322347067\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.8112.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 5x^{2} + 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-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
\( x^{4} + 5x^{2} + 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 6\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} + 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( -6\nu^{3} - 18\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 6\beta_1 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 5 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{3} - 3\beta_1 ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(92\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 64.1 |
|
− | 4.42782i | 0 | −11.6056 | 20.3925i | 0 | 0 | 15.9647i | 0 | 90.2944 | |||||||||||||||||||||||||||||
| 64.2 | − | 3.52058i | 0 | −4.39445 | − | 9.17304i | 0 | 0 | − | 12.6936i | 0 | −32.2944 | ||||||||||||||||||||||||||||
| 64.3 | 3.52058i | 0 | −4.39445 | 9.17304i | 0 | 0 | 12.6936i | 0 | −32.2944 | |||||||||||||||||||||||||||||||
| 64.4 | 4.42782i | 0 | −11.6056 | − | 20.3925i | 0 | 0 | − | 15.9647i | 0 | 90.2944 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 39.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-39}) \) |
| 3.b | odd | 2 | 1 | inner |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 117.4.b.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 117.4.b.c | ✓ | 4 |
| 13.b | even | 2 | 1 | inner | 117.4.b.c | ✓ | 4 |
| 13.d | odd | 4 | 2 | 1521.4.a.y | 4 | ||
| 39.d | odd | 2 | 1 | CM | 117.4.b.c | ✓ | 4 |
| 39.f | even | 4 | 2 | 1521.4.a.y | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 117.4.b.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 117.4.b.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 117.4.b.c | ✓ | 4 | 13.b | even | 2 | 1 | inner |
| 117.4.b.c | ✓ | 4 | 39.d | odd | 2 | 1 | CM |
| 1521.4.a.y | 4 | 13.d | odd | 4 | 2 | ||
| 1521.4.a.y | 4 | 39.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 32T_{2}^{2} + 243 \)
acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 32T^{2} + 243 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 500 T^{2} + 34992 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 5324 T^{2} + 2056752 \)
|
| $13$ |
\( (T^{2} - 2197)^{2} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 9184890672 \)
|
| $43$ |
\( (T - 452)^{4} \)
|
| $47$ |
\( T^{4} + \cdots + 2077595568 \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( T^{4} + \cdots + 163107544752 \)
|
| $61$ |
\( (T^{2} - 892372)^{2} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} + \cdots + 21709693872 \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( (T^{2} - 174928)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 20406871728 \)
|
| $89$ |
\( T^{4} + \cdots + 67381252272 \)
|
| $97$ |
\( T^{4} \)
|
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