Newspace parameters
| Level: | \( N \) | \(=\) | \( 138 = 2 \cdot 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 138.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(22.1329671342\) |
| 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: |
\( x^{3} - x^{2} - 1383x - 16813 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{3} - x^{2} - 1383x - 16813 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 19\nu - 916 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} + 19\beta _1 + 916 \)
|
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 |
|
−4.00000 | −9.00000 | 16.0000 | −73.1526 | 36.0000 | 90.1716 | −64.0000 | 81.0000 | 292.610 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | −9.00000 | 16.0000 | −24.6530 | 36.0000 | 62.9650 | −64.0000 | 81.0000 | 98.6122 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | −9.00000 | 16.0000 | 79.8056 | 36.0000 | −203.137 | −64.0000 | 81.0000 | −319.222 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(1\) |
| \(23\) | \(1\) |
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 | 138.6.a.g | ✓ | 3 |
| 3.b | odd | 2 | 1 | 414.6.a.n | 3 | ||
| 4.b | odd | 2 | 1 | 1104.6.a.k | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 138.6.a.g | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 414.6.a.n | 3 | 3.b | odd | 2 | 1 | ||
| 1104.6.a.k | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 18T_{5}^{2} - 6002T_{5} - 143924 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(138))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
|
| $3$ |
\( (T + 9)^{3} \)
|
| $5$ |
\( T^{3} + 18 T^{2} + \cdots - 143924 \)
|
| $7$ |
\( T^{3} + 50 T^{2} + \cdots + 1153340 \)
|
| $11$ |
\( T^{3} - 6 T^{2} + \cdots + 41102592 \)
|
| $13$ |
\( T^{3} - 790 T^{2} + \cdots - 17724640 \)
|
| $17$ |
\( T^{3} + \cdots + 1251288504 \)
|
| $19$ |
\( T^{3} - 2692 T^{2} + \cdots + 566361936 \)
|
| $23$ |
\( (T + 529)^{3} \)
|
| $29$ |
\( T^{3} + \cdots - 33999878760 \)
|
| $31$ |
\( T^{3} + \cdots - 140877524608 \)
|
| $37$ |
\( T^{3} + \cdots - 383183552304 \)
|
| $41$ |
\( T^{3} + \cdots + 1113643632728 \)
|
| $43$ |
\( T^{3} + \cdots - 163187296392 \)
|
| $47$ |
\( T^{3} + \cdots - 1511593331104 \)
|
| $53$ |
\( T^{3} + \cdots - 2969882763452 \)
|
| $59$ |
\( T^{3} + \cdots - 25961931296880 \)
|
| $61$ |
\( T^{3} + \cdots - 15216738764992 \)
|
| $67$ |
\( T^{3} + \cdots - 19674353996712 \)
|
| $71$ |
\( T^{3} + \cdots - 39250356203520 \)
|
| $73$ |
\( T^{3} + \cdots - 24824926300760 \)
|
| $79$ |
\( T^{3} + \cdots + 44957071020660 \)
|
| $83$ |
\( T^{3} + \cdots - 200825772591136 \)
|
| $89$ |
\( T^{3} + \cdots + 460268929768536 \)
|
| $97$ |
\( T^{3} + \cdots - 14\!\cdots\!36 \)
|
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