Newspace parameters
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(55.3207090003\) |
| 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: |
\( x^{3} - 829x - 6375 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{4} \) |
| 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
\( x^{3} - 829x - 6375 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 96\nu^{2} + 768\nu - 53056 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -864\nu^{2} + 17280\nu + 477504 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 9\beta_1 ) / 3456 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{2} + 45\beta _1 + 477504 ) / 864 \)
|
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 | 0 | 0 | −7004.55 | 0 | −37139.0 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −1973.64 | 0 | 55801.0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 10562.2 | 0 | −36458.0 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
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 | 72.12.a.h | yes | 3 |
| 3.b | odd | 2 | 1 | 72.12.a.g | ✓ | 3 | |
| 4.b | odd | 2 | 1 | 144.12.a.t | 3 | ||
| 12.b | even | 2 | 1 | 144.12.a.s | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.12.a.g | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 72.12.a.h | yes | 3 | 1.a | even | 1 | 1 | trivial |
| 144.12.a.s | 3 | 12.b | even | 2 | 1 | ||
| 144.12.a.t | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} - 1584T_{5}^{2} - 81004800T_{5} - 146016358400 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(72))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 146016358400 \)
|
| $7$ |
\( T^{3} + \cdots - 75555237651264 \)
|
| $11$ |
\( T^{3} + \cdots + 18\!\cdots\!52 \)
|
| $13$ |
\( T^{3} + \cdots - 30\!\cdots\!72 \)
|
| $17$ |
\( T^{3} + \cdots + 18\!\cdots\!76 \)
|
| $19$ |
\( T^{3} + \cdots + 19\!\cdots\!04 \)
|
| $23$ |
\( T^{3} + \cdots + 50\!\cdots\!24 \)
|
| $29$ |
\( T^{3} + \cdots - 92\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 39\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 76\!\cdots\!40 \)
|
| $41$ |
\( T^{3} + \cdots - 92\!\cdots\!40 \)
|
| $43$ |
\( T^{3} + \cdots - 93\!\cdots\!00 \)
|
| $47$ |
\( T^{3} + \cdots + 70\!\cdots\!44 \)
|
| $53$ |
\( T^{3} + \cdots - 60\!\cdots\!48 \)
|
| $59$ |
\( T^{3} + \cdots - 77\!\cdots\!84 \)
|
| $61$ |
\( T^{3} + \cdots + 38\!\cdots\!36 \)
|
| $67$ |
\( T^{3} + \cdots + 61\!\cdots\!92 \)
|
| $71$ |
\( T^{3} + \cdots + 54\!\cdots\!08 \)
|
| $73$ |
\( T^{3} + \cdots - 23\!\cdots\!68 \)
|
| $79$ |
\( T^{3} + \cdots - 95\!\cdots\!68 \)
|
| $83$ |
\( T^{3} + \cdots + 86\!\cdots\!88 \)
|
| $89$ |
\( T^{3} + \cdots + 14\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots + 27\!\cdots\!52 \)
|
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