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
comment: Compute space of new eigenforms
[N,k,chi] = [1,12,Mod(1,1)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1, base_ring=CyclotomicField(1))
chi = DirichletCharacter(H, H._module([]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1.a (trivial) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
| 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
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
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.
comment: embeddings in the coefficient field
gp: mfembed(f)
| 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
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T + 24 \)
|
| $3$ |
\( T - 252 \)
|
| $5$ |
\( T - 4830 \)
|
| $7$ |
\( T + 16744 \)
|
| $11$ |
\( T - 534612 \)
|
| $13$ |
\( T + 577738 \)
|
| $17$ |
\( T + 6905934 \)
|
| $19$ |
\( T - 10661420 \)
|
| $23$ |
\( T - 18643272 \)
|
| $29$ |
\( T - 128406630 \)
|
| $31$ |
\( T + 52843168 \)
|
| $37$ |
\( T + 182213314 \)
|
| $41$ |
\( T - 308120442 \)
|
| $43$ |
\( T + 17125708 \)
|
| $47$ |
\( T - 2687348496 \)
|
| $53$ |
\( T + 1596055698 \)
|
| $59$ |
\( T + 5189203740 \)
|
| $61$ |
\( T - 6956478662 \)
|
| $67$ |
\( T + 15481826884 \)
|
| $71$ |
\( T - 9791485272 \)
|
| $73$ |
\( T - 1463791322 \)
|
| $79$ |
\( T - 38116845680 \)
|
| $83$ |
\( T + 29335099668 \)
|
| $89$ |
\( T + 24992917110 \)
|
| $97$ |
\( T - 75013568546 \)
|
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Additional information
\(\displaystyle q\prod_{n\geq1}(1-q^n)^{24} \)
\(\eta(z)^{24}\)