# Properties

 Level 7 Weight 3 Character $\chi_{7}(6, \cdot)$ Label 7.3.6.a Dimension of Galois orbit 1 Twist info Is minimal CM Yes Atkin-Lehner eigenvalues $\omega_{ 7 }$ : -3

# Related objects

Show commands for: SageMath
magma: D := FullDirichletGroup(7);
magma: c := D![3];
magma: S:= CuspForms(c,3);
magma: N := Newforms(S);
sage: D = DirichletGroup(7)
sage: c = D.Element(D,vector([3]))
sage: N = Newforms(c,3,names="a")
sage: f = N[0]

## q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field
$q$ $\mathstrut-$ $3q^{2}$ $\mathstrut+$ $5q^{4}$ $\mathstrut-$ $7q^{7}$ $\mathstrut-$ $3q^{8}$ $\mathstrut+$ $9q^{9}$ $\mathstrut+O(q^{10})$

### 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
$\alpha_{p}$ $-0.750000000000000 + 0.661437827766148i$ $-1.00000000000000$ $-1.00000000000000$
$\theta_{p}$ $2.41885840577638$ $3.14159265358979$ $3.14159265358979$

This cusp form has an eta product $\eta(z)^3\eta(7z)^3=q\prod_{n=1}^\infty (1-q^n)^3(1-q^{7n})^3$, where $q=\exp(2\pi i z)$.

## Further Properties

The database contains the coefficients of $q^n$ for $0 \le n\le 499$.
 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 499)