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: | \(0\) |
| 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: |
\( x^{3} - x^{2} - 1025x - 1873 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| 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
\( x^{3} - x^{2} - 1025x - 1873 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 2\nu^{2} - 4\nu - 1366 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 5\beta_{2} + 2\beta _1 + 1368 ) / 2 \)
|
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 |
|
−4.00000 | 9.00000 | 16.0000 | −63.0946 | −36.0000 | −197.588 | −64.0000 | 81.0000 | 252.378 | |||||||||||||||||||||||||||
| 1.2 | −4.00000 | 9.00000 | 16.0000 | −5.67331 | −36.0000 | 254.708 | −64.0000 | 81.0000 | 22.6932 | ||||||||||||||||||||||||||||
| 1.3 | −4.00000 | 9.00000 | 16.0000 | 64.7679 | −36.0000 | −91.1204 | −64.0000 | 81.0000 | −259.072 | ||||||||||||||||||||||||||||
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.h | ✓ | 3 |
| 3.b | odd | 2 | 1 | 414.6.a.m | 3 | ||
| 4.b | odd | 2 | 1 | 1104.6.a.j | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 138.6.a.h | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 414.6.a.m | 3 | 3.b | odd | 2 | 1 | ||
| 1104.6.a.j | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 4T_{5}^{2} - 4096T_{5} - 23184 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(138))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 4)^{3} \)
|
| $3$ |
\( (T - 9)^{3} \)
|
| $5$ |
\( T^{3} + 4 T^{2} + \cdots - 23184 \)
|
| $7$ |
\( T^{3} + 34 T^{2} + \cdots - 4585832 \)
|
| $11$ |
\( T^{3} - 300 T^{2} + \cdots - 18434592 \)
|
| $13$ |
\( T^{3} - 998 T^{2} + \cdots - 16119176 \)
|
| $17$ |
\( T^{3} - 392 T^{2} + \cdots - 72900432 \)
|
| $19$ |
\( T^{3} + \cdots + 3608499640 \)
|
| $23$ |
\( (T + 529)^{3} \)
|
| $29$ |
\( T^{3} + \cdots + 26873648952 \)
|
| $31$ |
\( T^{3} + \cdots - 22999096000 \)
|
| $37$ |
\( T^{3} + \cdots - 795432448904 \)
|
| $41$ |
\( T^{3} + \cdots + 1169089349784 \)
|
| $43$ |
\( T^{3} + \cdots + 3317925025400 \)
|
| $47$ |
\( T^{3} + \cdots + 2069618590080 \)
|
| $53$ |
\( T^{3} + \cdots - 90672025008 \)
|
| $59$ |
\( T^{3} + \cdots - 206257275840 \)
|
| $61$ |
\( T^{3} + \cdots + 54318607384 \)
|
| $67$ |
\( T^{3} + \cdots - 13525331979384 \)
|
| $71$ |
\( T^{3} + \cdots - 79612179492096 \)
|
| $73$ |
\( T^{3} + \cdots + 31081392542392 \)
|
| $79$ |
\( T^{3} + \cdots + 357863106252424 \)
|
| $83$ |
\( T^{3} + \cdots - 18660485323872 \)
|
| $89$ |
\( T^{3} + \cdots + 50621579393712 \)
|
| $97$ |
\( T^{3} + \cdots + 571541083003976 \)
|
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