Newspace parameters
| Level: | \( N \) | \(=\) | \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1380.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(81.4226358079\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.72332.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 43x + 51 \)
|
| 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
\( x^{3} - x^{2} - 43x + 51 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 29 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + 29 \)
|
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 | −3.00000 | 0 | 5.00000 | 0 | −11.1542 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | 5.00000 | 0 | 13.7499 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | 5.00000 | 0 | 19.4043 | 0 | 9.00000 | 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 | 1380.4.a.c | ✓ | 3 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1380.4.a.c | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} - 22T_{7}^{2} - 103T_{7} + 2976 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1380))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( (T - 5)^{3} \)
|
| $7$ |
\( T^{3} - 22 T^{2} + \cdots + 2976 \)
|
| $11$ |
\( T^{3} - 28 T^{2} + \cdots + 1396 \)
|
| $13$ |
\( T^{3} - 34 T^{2} + \cdots + 126992 \)
|
| $17$ |
\( T^{3} + 186 T^{2} + \cdots - 440246 \)
|
| $19$ |
\( T^{3} + 174 T^{2} + \cdots - 1069632 \)
|
| $23$ |
\( (T - 23)^{3} \)
|
| $29$ |
\( T^{3} + 172 T^{2} + \cdots - 194856 \)
|
| $31$ |
\( T^{3} + 170 T^{2} + \cdots - 1997904 \)
|
| $37$ |
\( T^{3} + 264 T^{2} + \cdots - 499318 \)
|
| $41$ |
\( T^{3} + 158 T^{2} + \cdots - 7885558 \)
|
| $43$ |
\( T^{3} - 30 T^{2} + \cdots + 9374184 \)
|
| $47$ |
\( T^{3} + 722 T^{2} + \cdots - 28744776 \)
|
| $53$ |
\( T^{3} - 250 T^{2} + \cdots + 67830710 \)
|
| $59$ |
\( T^{3} + 212 T^{2} + \cdots + 13606800 \)
|
| $61$ |
\( T^{3} - 804 T^{2} + \cdots + 4040772 \)
|
| $67$ |
\( T^{3} - 1076 T^{2} + \cdots - 41097822 \)
|
| $71$ |
\( T^{3} + 102 T^{2} + \cdots - 8713206 \)
|
| $73$ |
\( T^{3} + 856 T^{2} + \cdots - 108803508 \)
|
| $79$ |
\( T^{3} - 1792 T^{2} + \cdots + 303137024 \)
|
| $83$ |
\( T^{3} - 648 T^{2} + \cdots + 376881136 \)
|
| $89$ |
\( T^{3} + \cdots + 1791787776 \)
|
| $97$ |
\( T^{3} + 3316 T^{2} + \cdots + 914041792 \)
|
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