This is the discriminant modular form $\Delta=\sum \tau(n)q^n$, where $\tau$ is the Ramanujan tau function [A000594 ]. It is the minimal weight newform of level $1$.
Newspace parameters
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.768343180560\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Expression as an eta quotient
\(f(z) = \eta(z)^{24}=q\prod_{n=1}^\infty(1 - q^{n})^{24}\)
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 |
|
−24.0000 | 252.000 | −1472.00 | 4830.00 | −6048.00 | −16744.0 | 84480.0 | −113643. | −115920. | |||||||||||||||||||||
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 | 1.12.a.a | ✓ | 1 |
| 3.b | odd | 2 | 1 | 9.12.a.b | 1 | ||
| 4.b | odd | 2 | 1 | 16.12.a.a | 1 | ||
| 5.b | even | 2 | 1 | 25.12.a.b | 1 | ||
| 5.c | odd | 4 | 2 | 25.12.b.b | 2 | ||
| 7.b | odd | 2 | 1 | 49.12.a.a | 1 | ||
| 7.c | even | 3 | 2 | 49.12.c.b | 2 | ||
| 7.d | odd | 6 | 2 | 49.12.c.c | 2 | ||
| 8.b | even | 2 | 1 | 64.12.a.b | 1 | ||
| 8.d | odd | 2 | 1 | 64.12.a.f | 1 | ||
| 9.c | even | 3 | 2 | 81.12.c.d | 2 | ||
| 9.d | odd | 6 | 2 | 81.12.c.b | 2 | ||
| 11.b | odd | 2 | 1 | 121.12.a.b | 1 | ||
| 12.b | even | 2 | 1 | 144.12.a.d | 1 | ||
| 13.b | even | 2 | 1 | 169.12.a.a | 1 | ||
| 15.d | odd | 2 | 1 | 225.12.a.b | 1 | ||
| 15.e | even | 4 | 2 | 225.12.b.d | 2 | ||
| 16.e | even | 4 | 2 | 256.12.b.e | 2 | ||
| 16.f | odd | 4 | 2 | 256.12.b.c | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1.12.a.a | ✓ | 1 | 1.a | even | 1 | 1 | trivial |
| 9.12.a.b | 1 | 3.b | odd | 2 | 1 | ||
| 16.12.a.a | 1 | 4.b | odd | 2 | 1 | ||
| 25.12.a.b | 1 | 5.b | even | 2 | 1 | ||
| 25.12.b.b | 2 | 5.c | odd | 4 | 2 | ||
| 49.12.a.a | 1 | 7.b | odd | 2 | 1 | ||
| 49.12.c.b | 2 | 7.c | even | 3 | 2 | ||
| 49.12.c.c | 2 | 7.d | odd | 6 | 2 | ||
| 64.12.a.b | 1 | 8.b | even | 2 | 1 | ||
| 64.12.a.f | 1 | 8.d | odd | 2 | 1 | ||
| 81.12.c.b | 2 | 9.d | odd | 6 | 2 | ||
| 81.12.c.d | 2 | 9.c | even | 3 | 2 | ||
| 121.12.a.b | 1 | 11.b | odd | 2 | 1 | ||
| 144.12.a.d | 1 | 12.b | even | 2 | 1 | ||
| 169.12.a.a | 1 | 13.b | even | 2 | 1 | ||
| 225.12.a.b | 1 | 15.d | odd | 2 | 1 | ||
| 225.12.b.d | 2 | 15.e | even | 4 | 2 | ||
| 256.12.b.c | 2 | 16.f | odd | 4 | 2 | ||
| 256.12.b.e | 2 | 16.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ | \( 24 + T \) |
| $3$ | \( -252 + T \) |
| $5$ | \( -4830 + T \) |
| $7$ | \( 16744 + T \) |
| $11$ | \( -534612 + T \) |
| $13$ | \( 577738 + T \) |
| $17$ | \( 6905934 + T \) |
| $19$ | \( -10661420 + T \) |
| $23$ | \( -18643272 + T \) |
| $29$ | \( -128406630 + T \) |
| $31$ | \( 52843168 + T \) |
| $37$ | \( 182213314 + T \) |
| $41$ | \( -308120442 + T \) |
| $43$ | \( 17125708 + T \) |
| $47$ | \( -2687348496 + T \) |
| $53$ | \( 1596055698 + T \) |
| $59$ | \( 5189203740 + T \) |
| $61$ | \( -6956478662 + T \) |
| $67$ | \( 15481826884 + T \) |
| $71$ | \( -9791485272 + T \) |
| $73$ | \( -1463791322 + T \) |
| $79$ | \( -38116845680 + T \) |
| $83$ | \( 29335099668 + T \) |
| $89$ | \( 24992917110 + T \) |
| $97$ | \( -75013568546 + T \) |
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Additional information
\(\displaystyle q\prod_{n\geq1}(1-q^n)^{24} \)
\(\eta(z)^{24}\)