# 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

# Related objects

Show commands for: SageMath

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
$q$ $\mathstrut-$ $24q^{2}$ $\mathstrut+$ $252q^{3}$ $\mathstrut-$ $1472q^{4}$ $\mathstrut+$ $4830q^{5}$ $\mathstrut-$ $6048q^{6}$ $\mathstrut-$ $16744q^{7}$ $\mathstrut+$ $84480q^{8}$ $\mathstrut-$ $113643q^{9}$ $\mathstrut+O(q^{10})$

### 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

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

$\displaystyle q\prod_{n\geq1}(1-q^n)^{24}$

## Further Properties

The database contains the coefficients of $q^n$ for $0 \le n\le 999999$.
 Choose format to download: .sage file (contains more information) .sobj file for sage (only coefficients) text file of the algebraic coefficients in a table text file of the complex coefficients in double precision text file of the q-expansion Download coefficients of $q^n$ for $0\le n\le$ (maximum 999999)