Newspace parameters
| Level: | \( N \) | \(=\) | \( 39 = 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 39.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(20.0863976104\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 301x - 420 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{3} - 301x - 420 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3\nu - 201 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 2\beta_{2} + 3\beta _1 + 402 ) / 2 \)
|
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 |
|
−39.2088 | 81.0000 | 1025.33 | 358.916 | −3175.91 | −8025.48 | −20126.9 | 6561.00 | −14072.6 | |||||||||||||||||||||||||||
| 1.2 | −8.80911 | 81.0000 | −434.400 | 374.523 | −713.538 | 1691.69 | 8336.94 | 6561.00 | −3299.21 | ||||||||||||||||||||||||||||
| 1.3 | 30.0179 | 81.0000 | 389.073 | −1621.44 | 2431.45 | −8072.22 | −3690.02 | 6561.00 | −48672.1 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(13\) | \( -1 \) |
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 | 39.10.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 117.10.a.a | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 39.10.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 117.10.a.a | 3 | 3.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 18T_{2}^{2} - 1096T_{2} - 10368 \)
acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(39))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 18 T^{2} + \cdots - 10368 \)
|
| $3$ |
\( (T - 81)^{3} \)
|
| $5$ |
\( T^{3} + 888 T^{2} + \cdots + 217957440 \)
|
| $7$ |
\( T^{3} + \cdots - 109593639680 \)
|
| $11$ |
\( T^{3} + \cdots - 56844752394240 \)
|
| $13$ |
\( (T - 28561)^{3} \)
|
| $17$ |
\( T^{3} + \cdots + 16\!\cdots\!00 \)
|
| $19$ |
\( T^{3} + \cdots + 78\!\cdots\!36 \)
|
| $23$ |
\( T^{3} + \cdots + 35\!\cdots\!00 \)
|
| $29$ |
\( T^{3} + \cdots - 50\!\cdots\!20 \)
|
| $31$ |
\( T^{3} + \cdots - 34\!\cdots\!08 \)
|
| $37$ |
\( T^{3} + \cdots + 64\!\cdots\!56 \)
|
| $41$ |
\( T^{3} + \cdots - 11\!\cdots\!12 \)
|
| $43$ |
\( T^{3} + \cdots - 58\!\cdots\!08 \)
|
| $47$ |
\( T^{3} + \cdots + 31\!\cdots\!60 \)
|
| $53$ |
\( T^{3} + \cdots + 84\!\cdots\!64 \)
|
| $59$ |
\( T^{3} + \cdots - 11\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots + 25\!\cdots\!68 \)
|
| $67$ |
\( T^{3} + \cdots - 19\!\cdots\!28 \)
|
| $71$ |
\( T^{3} + \cdots - 42\!\cdots\!00 \)
|
| $73$ |
\( T^{3} + \cdots + 74\!\cdots\!76 \)
|
| $79$ |
\( T^{3} + \cdots - 27\!\cdots\!88 \)
|
| $83$ |
\( T^{3} + \cdots - 23\!\cdots\!36 \)
|
| $89$ |
\( T^{3} + \cdots - 46\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 24\!\cdots\!64 \)
|
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