Properties

Level 1
Weight 12
Character $\chi_{1}(1, \cdot)$
Label 1.12.1.a
Dimension of Galois orbit 1
Twist info Is minimal
CM No

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The discriminant modular form $\Delta(q)$ is the unique normalized cusp form of weight 12 and level $\Gamma_0(1)$. Its Fourier coefficients define the Ramanujan tau function.

magma: S := CuspForms(1,12);
magma: N := Newforms(S);
sage: N = Newforms(1,12,names="a")
sage: f = N[0]

q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field
\(q \) \(\mathstrut-\) \(24q^{2} \) \(\mathstrut+\) \(252q^{3} \) \(\mathstrut-\) \(1472q^{4} \) \(\mathstrut+\) \(4830q^{5} \) \(\mathstrut-\) \(6048q^{6} \) \(\mathstrut-\) \(16744q^{7} \) \(\mathstrut+\) \(84480q^{8} \) \(\mathstrut-\) \(113643q^{9} \) \(\mathstrut+O(q^{10}) \)

(To download coefficients, see below.)

Coefficient field

sage: K = f.hecke_eigenvalue_field() # note that sage often uses an isomorphic number field
The coefficient field is \(\Q\)

Detailed data

The first few Satake parameters \(\alpha_p\) and angles \(\theta_p = \textrm{Arg}(\alpha_p) \) are

\( p \) 2 3 5 7
\(\alpha_{p}\) \( -0.265165042944955 + 0.964203038783845i \) \( 0.299366806246473 + 0.954138100757845i \) \( 0.345606666602367 + 0.938379471216203i \) \( -0.188273848279482 + 0.982116570501707i \)
\(\theta_{p}\) \( 1.83917141540925 \) \( 1.26676737097408 \) \( 1.21791111404877 \) \( 1.76020059288887 \)

Explicit Formulas

\(-\frac{ 1 }{ 1728 } \cdot E_6^{ 2 }+\frac{ 1 }{ 1728 } \cdot E_4^{ 3 } \)

\(\displaystyle \Delta(z)=q\prod_{n\geq1}(1-q^n)^{24} \)

Further Properties

Download this Newform

The database contains the coefficients of \(q^n\) for \(0 \le n\le 999999 \).
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Download coefficients of \(q^n\) for \(0\le n\le \) (maximum 999999)