Newspace parameters
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 22 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(201.223687887\) |
| 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: |
\( x^{3} - x^{2} - 12529199x - 17012391021 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{21}\cdot 3^{4}\cdot 7 \) |
| Twist minimal: | no (minimal twist has level 24) |
| 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
\( x^{3} - x^{2} - 12529199x - 17012391021 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 576\nu^{2} - 1212288\nu - 4810808512 ) / 11 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 8064\nu^{2} - 13187328\nu - 67352580736 ) / 11 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 14\beta _1 + 114688 ) / 344064 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 451\beta_{2} - 4906\beta _1 + 615835213824 ) / 73728 \)
|
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 | −2.53316e7 | 0 | 1.44995e9 | 0 | 0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | 0 | 0 | −1.51338e7 | 0 | −8.40758e8 | 0 | 0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 0 | 0 | 3.51850e7 | 0 | 2.43346e8 | 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 | 72.22.a.c | 3 | |
| 3.b | odd | 2 | 1 | 24.22.a.d | ✓ | 3 | |
| 12.b | even | 2 | 1 | 48.22.a.j | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.22.a.d | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 48.22.a.j | 3 | 12.b | even | 2 | 1 | ||
| 72.22.a.c | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{3} + 5280498T_{5}^{2} - 1040411335225620T_{5} - 13488669456493277401000 \)
acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(72))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} \)
$3$
\( T^{3} \)
$5$
\( T^{3} + 5280498 T^{2} + \cdots - 13\!\cdots\!00 \)
$7$
\( T^{3} - 852542376 T^{2} + \cdots + 29\!\cdots\!00 \)
$11$
\( T^{3} + 62490757668 T^{2} + \cdots + 18\!\cdots\!60 \)
$13$
\( T^{3} - 203765207802 T^{2} + \cdots - 64\!\cdots\!24 \)
$17$
\( T^{3} + 695827819926 T^{2} + \cdots - 13\!\cdots\!12 \)
$19$
\( T^{3} - 4955504123196 T^{2} + \cdots - 18\!\cdots\!64 \)
$23$
\( T^{3} + 150867407938152 T^{2} + \cdots + 60\!\cdots\!52 \)
$29$
\( T^{3} + \cdots - 20\!\cdots\!76 \)
$31$
\( T^{3} - 652508601550896 T^{2} + \cdots - 88\!\cdots\!20 \)
$37$
\( T^{3} + \cdots - 10\!\cdots\!76 \)
$41$
\( T^{3} + \cdots + 91\!\cdots\!60 \)
$43$
\( T^{3} + \cdots - 11\!\cdots\!04 \)
$47$
\( T^{3} + \cdots + 45\!\cdots\!00 \)
$53$
\( T^{3} + \cdots + 78\!\cdots\!20 \)
$59$
\( T^{3} + \cdots - 25\!\cdots\!60 \)
$61$
\( T^{3} + \cdots + 12\!\cdots\!96 \)
$67$
\( T^{3} + \cdots + 46\!\cdots\!16 \)
$71$
\( T^{3} + \cdots + 43\!\cdots\!20 \)
$73$
\( T^{3} + \cdots + 27\!\cdots\!00 \)
$79$
\( T^{3} + \cdots - 33\!\cdots\!64 \)
$83$
\( T^{3} + \cdots + 54\!\cdots\!88 \)
$89$
\( T^{3} + \cdots - 14\!\cdots\!88 \)
$97$
\( T^{3} + \cdots + 12\!\cdots\!08 \)
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