Newspace parameters
| Level: | \( N \) | \(=\) | \( 72 = 2^{3} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 9 \) |
| Character orbit: | \([\chi]\) | \(=\) | 72.p (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(29.3312599244\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{6}\cdot 7^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( 56\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( 28\nu^{3} \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 56 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} ) / 28 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(55\) | \(65\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 43.1 |
|
−8.00000 | + | 13.8564i | −77.0857 | − | 24.8755i | −128.000 | − | 221.703i | 0 | 961.371 | − | 869.127i | 0 | 4096.00 | 5323.41 | + | 3835.10i | 0 | ||||||||||||||||||||
| 43.2 | −8.00000 | + | 13.8564i | 60.0857 | + | 54.3204i | −128.000 | − | 221.703i | 0 | −1233.37 | + | 398.009i | 0 | 4096.00 | 659.586 | + | 6527.76i | 0 | |||||||||||||||||||||
| 67.1 | −8.00000 | − | 13.8564i | −77.0857 | + | 24.8755i | −128.000 | + | 221.703i | 0 | 961.371 | + | 869.127i | 0 | 4096.00 | 5323.41 | − | 3835.10i | 0 | |||||||||||||||||||||
| 67.2 | −8.00000 | − | 13.8564i | 60.0857 | − | 54.3204i | −128.000 | + | 221.703i | 0 | −1233.37 | − | 398.009i | 0 | 4096.00 | 659.586 | − | 6527.76i | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-2}) \) |
| 9.c | even | 3 | 1 | inner |
| 72.p | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 72.9.p.a | ✓ | 4 |
| 8.d | odd | 2 | 1 | CM | 72.9.p.a | ✓ | 4 |
| 9.c | even | 3 | 1 | inner | 72.9.p.a | ✓ | 4 |
| 72.p | odd | 6 | 1 | inner | 72.9.p.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 72.9.p.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 72.9.p.a | ✓ | 4 | 8.d | odd | 2 | 1 | CM |
| 72.9.p.a | ✓ | 4 | 9.c | even | 3 | 1 | inner |
| 72.9.p.a | ✓ | 4 | 72.p | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} \)
acting on \(S_{9}^{\mathrm{new}}(72, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 16 T + 256)^{2} \)
|
| $3$ |
\( T^{4} + 34 T^{3} + \cdots + 43046721 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + \cdots + 90\!\cdots\!69 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} + 162434 T + 5457532033)^{2} \)
|
| $19$ |
\( (T^{2} - 72286 T - 45725423327)^{2} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( T^{4} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} + \cdots + 51\!\cdots\!29 \)
|
| $43$ |
\( T^{4} + \cdots + 31\!\cdots\!01 \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} \)
|
| $59$ |
\( T^{4} + \cdots + 20\!\cdots\!41 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} + \cdots + 10\!\cdots\!29 \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( (T^{2} + \cdots - 12\!\cdots\!27)^{2} \)
|
| $79$ |
\( T^{4} \)
|
| $83$ |
\( (T^{2} + \cdots + 912641320682116)^{2} \)
|
| $89$ |
\( (T + 95519806)^{4} \)
|
| $97$ |
\( T^{4} + \cdots + 30\!\cdots\!21 \)
|
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