Newspace parameters
| Level: | \( N \) | \(=\) | \( 117 = 3^{2} \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 117.q (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(0.934249703649\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 5x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).
| \(n\) | \(28\) | \(92\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 10.1 |
|
−1.93649 | + | 1.11803i | 0 | 1.50000 | − | 2.59808i | − | 2.23607i | 0 | −3.00000 | − | 1.73205i | 2.23607i | 0 | 2.50000 | + | 4.33013i | |||||||||||||||||||||
| 10.2 | 1.93649 | − | 1.11803i | 0 | 1.50000 | − | 2.59808i | 2.23607i | 0 | −3.00000 | − | 1.73205i | − | 2.23607i | 0 | 2.50000 | + | 4.33013i | ||||||||||||||||||||||
| 82.1 | −1.93649 | − | 1.11803i | 0 | 1.50000 | + | 2.59808i | 2.23607i | 0 | −3.00000 | + | 1.73205i | − | 2.23607i | 0 | 2.50000 | − | 4.33013i | ||||||||||||||||||||||
| 82.2 | 1.93649 | + | 1.11803i | 0 | 1.50000 | + | 2.59808i | − | 2.23607i | 0 | −3.00000 | + | 1.73205i | 2.23607i | 0 | 2.50000 | − | 4.33013i | ||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 13.e | even | 6 | 1 | inner |
| 39.h | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 117.2.q.d | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 117.2.q.d | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1872.2.by.l | 4 | ||
| 12.b | even | 2 | 1 | 1872.2.by.l | 4 | ||
| 13.c | even | 3 | 1 | 1521.2.b.g | 4 | ||
| 13.e | even | 6 | 1 | inner | 117.2.q.d | ✓ | 4 |
| 13.e | even | 6 | 1 | 1521.2.b.g | 4 | ||
| 13.f | odd | 12 | 2 | 1521.2.a.u | 4 | ||
| 39.h | odd | 6 | 1 | inner | 117.2.q.d | ✓ | 4 |
| 39.h | odd | 6 | 1 | 1521.2.b.g | 4 | ||
| 39.i | odd | 6 | 1 | 1521.2.b.g | 4 | ||
| 39.k | even | 12 | 2 | 1521.2.a.u | 4 | ||
| 52.i | odd | 6 | 1 | 1872.2.by.l | 4 | ||
| 156.r | even | 6 | 1 | 1872.2.by.l | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 117.2.q.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 117.2.q.d | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 117.2.q.d | ✓ | 4 | 13.e | even | 6 | 1 | inner |
| 117.2.q.d | ✓ | 4 | 39.h | odd | 6 | 1 | inner |
| 1521.2.a.u | 4 | 13.f | odd | 12 | 2 | ||
| 1521.2.a.u | 4 | 39.k | even | 12 | 2 | ||
| 1521.2.b.g | 4 | 13.c | even | 3 | 1 | ||
| 1521.2.b.g | 4 | 13.e | even | 6 | 1 | ||
| 1521.2.b.g | 4 | 39.h | odd | 6 | 1 | ||
| 1521.2.b.g | 4 | 39.i | odd | 6 | 1 | ||
| 1872.2.by.l | 4 | 4.b | odd | 2 | 1 | ||
| 1872.2.by.l | 4 | 12.b | even | 2 | 1 | ||
| 1872.2.by.l | 4 | 52.i | odd | 6 | 1 | ||
| 1872.2.by.l | 4 | 156.r | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(117, [\chi])\):
|
\( T_{2}^{4} - 5T_{2}^{2} + 25 \)
|
|
\( T_{5}^{2} + 5 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 5T^{2} + 25 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 5)^{2} \)
|
| $7$ |
\( (T^{2} + 6 T + 12)^{2} \)
|
| $11$ |
\( T^{4} - 20T^{2} + 400 \)
|
| $13$ |
\( (T^{2} - 5 T + 13)^{2} \)
|
| $17$ |
\( T^{4} + 15T^{2} + 225 \)
|
| $19$ |
\( (T^{2} + 6 T + 12)^{2} \)
|
| $23$ |
\( T^{4} + 60T^{2} + 3600 \)
|
| $29$ |
\( T^{4} + 15T^{2} + 225 \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( (T^{2} - 3 T + 3)^{2} \)
|
| $41$ |
\( T^{4} - 5T^{2} + 25 \)
|
| $43$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $47$ |
\( (T^{2} + 20)^{2} \)
|
| $53$ |
\( (T^{2} - 135)^{2} \)
|
| $59$ |
\( T^{4} - 80T^{2} + 6400 \)
|
| $61$ |
\( (T^{2} - 7 T + 49)^{2} \)
|
| $67$ |
\( (T^{2} + 6 T + 12)^{2} \)
|
| $71$ |
\( T^{4} - 20T^{2} + 400 \)
|
| $73$ |
\( (T^{2} + 243)^{2} \)
|
| $79$ |
\( (T - 8)^{4} \)
|
| $83$ |
\( (T^{2} + 20)^{2} \)
|
| $89$ |
\( T^{4} - 20T^{2} + 400 \)
|
| $97$ |
\( (T^{2} - 12 T + 48)^{2} \)
|
| show more | |
| show less |