Newspace parameters
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(77.2062688454\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - 14629x - 625725 \)
|
| 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
\( x^{3} - 14629x - 625725 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 96\nu^{2} + 17760\nu - 936256 ) / 7 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 96\nu^{2} - 6432\nu - 936256 ) / 7 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{2} + \beta_1 ) / 3456 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 185\beta_{2} + 67\beta _1 + 33705216 ) / 3456 \)
|
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 | −63560.1 | 0 | 374836. | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −15022.7 | 0 | −480334. | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 28326.9 | 0 | 120030. | 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 | 72.14.a.g | ✓ | 3 |
| 3.b | odd | 2 | 1 | 72.14.a.h | yes | 3 | |
| 4.b | odd | 2 | 1 | 144.14.a.t | 3 | ||
| 12.b | even | 2 | 1 | 144.14.a.u | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.14.a.g | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 72.14.a.h | yes | 3 | 3.b | odd | 2 | 1 | |
| 144.14.a.t | 3 | 4.b | odd | 2 | 1 | ||
| 144.14.a.u | 3 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 50256T_{5}^{2} - 1271159040T_{5} - 27047793152000 \)
acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(72))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} \)
|
| $5$ |
\( T^{3} + \cdots - 27047793152000 \)
|
| $7$ |
\( T^{3} + \cdots + 21\!\cdots\!32 \)
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| $11$ |
\( T^{3} + \cdots - 90\!\cdots\!68 \)
|
| $13$ |
\( T^{3} + \cdots + 23\!\cdots\!44 \)
|
| $17$ |
\( T^{3} + \cdots - 74\!\cdots\!88 \)
|
| $19$ |
\( T^{3} + \cdots + 20\!\cdots\!16 \)
|
| $23$ |
\( T^{3} + \cdots + 49\!\cdots\!52 \)
|
| $29$ |
\( T^{3} + \cdots - 64\!\cdots\!20 \)
|
| $31$ |
\( T^{3} + \cdots - 13\!\cdots\!00 \)
|
| $37$ |
\( T^{3} + \cdots - 43\!\cdots\!20 \)
|
| $41$ |
\( T^{3} + \cdots + 26\!\cdots\!40 \)
|
| $43$ |
\( T^{3} + \cdots - 45\!\cdots\!00 \)
|
| $47$ |
\( T^{3} + \cdots - 32\!\cdots\!32 \)
|
| $53$ |
\( T^{3} + \cdots - 93\!\cdots\!44 \)
|
| $59$ |
\( T^{3} + \cdots - 69\!\cdots\!96 \)
|
| $61$ |
\( T^{3} + \cdots - 85\!\cdots\!64 \)
|
| $67$ |
\( T^{3} + \cdots + 52\!\cdots\!64 \)
|
| $71$ |
\( T^{3} + \cdots - 20\!\cdots\!32 \)
|
| $73$ |
\( T^{3} + \cdots - 63\!\cdots\!44 \)
|
| $79$ |
\( T^{3} + \cdots - 48\!\cdots\!72 \)
|
| $83$ |
\( T^{3} + \cdots - 21\!\cdots\!96 \)
|
| $89$ |
\( T^{3} + \cdots - 25\!\cdots\!00 \)
|
| $97$ |
\( T^{3} + \cdots - 95\!\cdots\!76 \)
|
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