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: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 45x^{2} - 97x - 18 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 45x^{2} - 97x - 18 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 5\nu - 22 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 3\nu^{2} - 37\nu - 29 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{2} + 5\beta _1 + 71 ) / 3 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 3\beta_{3} + 9\beta_{2} + 52\beta _1 + 337 ) / 3 \)
|
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 | −19.9612 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −3.00000 | 0 | −5.00000 | 0 | 4.81437 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −3.00000 | 0 | −5.00000 | 0 | 19.5952 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −3.00000 | 0 | −5.00000 | 0 | 30.5516 | 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.d | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1380.4.a.d | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} - 35T_{7}^{3} - 257T_{7}^{2} + 13887T_{7} - 57532 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1380))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 3)^{4} \)
|
| $5$ |
\( (T + 5)^{4} \)
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| $7$ |
\( T^{4} - 35 T^{3} + \cdots - 57532 \)
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| $11$ |
\( T^{4} + 56 T^{3} + \cdots + 184320 \)
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| $13$ |
\( T^{4} - 42 T^{3} + \cdots + 58320 \)
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| $17$ |
\( T^{4} + 139 T^{3} + \cdots - 5640238 \)
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| $19$ |
\( T^{4} - 100 T^{3} + \cdots - 2239200 \)
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| $23$ |
\( (T + 23)^{4} \)
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| $29$ |
\( T^{4} - 125 T^{3} + \cdots - 33207514 \)
|
| $31$ |
\( T^{4} + 3 T^{3} + \cdots - 73021240 \)
|
| $37$ |
\( T^{4} - 243 T^{3} + \cdots + 880043102 \)
|
| $41$ |
\( T^{4} + \cdots - 6104196830 \)
|
| $43$ |
\( T^{4} - 652 T^{3} + \cdots + 713784384 \)
|
| $47$ |
\( T^{4} + \cdots + 1539305568 \)
|
| $53$ |
\( T^{4} + \cdots - 1293810282 \)
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| $59$ |
\( T^{4} + \cdots - 7050280376 \)
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| $61$ |
\( T^{4} + \cdots + 4430279704 \)
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| $67$ |
\( T^{4} + \cdots + 36841049100 \)
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| $71$ |
\( T^{4} + \cdots + 7398692800 \)
|
| $73$ |
\( T^{4} + \cdots - 7586418496 \)
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| $79$ |
\( T^{4} + \cdots - 43066368000 \)
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| $83$ |
\( T^{4} + \cdots + 498259363484 \)
|
| $89$ |
\( T^{4} + \cdots - 6317654400 \)
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| $97$ |
\( T^{4} + \cdots - 166452103520 \)
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