Newspace parameters
| Level: | \( N \) | \(=\) | \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1380.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(11.0193554789\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.3144.1 |
| Defining polynomial: |
\( x^{3} - x^{2} - 16x - 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 16x - 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 10 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 10 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 1.00000 | 0 | −1.00000 | 0 | −3.73549 | 0 | 1.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 1.00000 | 0 | −1.00000 | 0 | 1.52644 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 1.00000 | 0 | −1.00000 | 0 | 4.20905 | 0 | 1.00000 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(23\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1380.2.a.j | ✓ | 3 |
| 3.b | odd | 2 | 1 | 4140.2.a.s | 3 | ||
| 4.b | odd | 2 | 1 | 5520.2.a.bv | 3 | ||
| 5.b | even | 2 | 1 | 6900.2.a.x | 3 | ||
| 5.c | odd | 4 | 2 | 6900.2.f.r | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1380.2.a.j | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 4140.2.a.s | 3 | 3.b | odd | 2 | 1 | ||
| 5520.2.a.bv | 3 | 4.b | odd | 2 | 1 | ||
| 6900.2.a.x | 3 | 5.b | even | 2 | 1 | ||
| 6900.2.f.r | 6 | 5.c | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} - 2T_{7}^{2} - 15T_{7} + 24 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1380))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} \)
$3$
\( (T - 1)^{3} \)
$5$
\( (T + 1)^{3} \)
$7$
\( T^{3} - 2 T^{2} - 15 T + 24 \)
$11$
\( T^{3} - 4 T^{2} - 14 T + 48 \)
$13$
\( T^{3} - 2 T^{2} - 18 T - 12 \)
$17$
\( T^{3} - 21T + 18 \)
$19$
\( T^{3} - 10 T^{2} + 14 T + 52 \)
$23$
\( (T - 1)^{3} \)
$29$
\( T^{3} - 21T - 18 \)
$31$
\( T^{3} - 10 T^{2} + 17 T + 4 \)
$37$
\( T^{3} - 2 T^{2} - 63 T - 108 \)
$41$
\( T^{3} - 8 T^{2} - 101 T + 582 \)
$43$
\( T^{3} - 16 T^{2} + 20 T + 304 \)
$47$
\( T^{3} + 4 T^{2} - 50 T - 216 \)
$53$
\( T^{3} - 8 T^{2} - 101 T + 582 \)
$59$
\( T^{3} - 21T + 18 \)
$61$
\( T^{3} - 10 T^{2} - 22 T + 292 \)
$67$
\( T^{3} - 10 T^{2} + 17 T + 4 \)
$71$
\( T^{3} + 8 T^{2} - 101 T - 582 \)
$73$
\( T^{3} - 18 T^{2} - 66 T + 1492 \)
$79$
\( T^{3} - 14T^{2} + 96 \)
$83$
\( T^{3} - 10 T^{2} - 17 T + 228 \)
$89$
\( T^{3} - 2 T^{2} - 200 T + 912 \)
$97$
\( T^{3} + 20 T^{2} - 88 T - 2336 \)
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