# Properties

 Level 3 Weight 9 Character $\chi_{3}(2, \cdot)$ Label 3.9.2.a Dimension of Galois orbit 2 Twist info Is minimal CM No Atkin-Lehner eigenvalues $\omega_{ 3 }$ : -2

# Related objects

Show commands for: SageMath
magma: D := FullDirichletGroup(3);
magma: c := D![1];
magma: S:= CuspForms(c,9);
magma: N := Newforms(S);
sage: D = DirichletGroup(3)
sage: c = D.Element(D,vector([1]))
sage: N = Newforms(c,9,names="a")
sage: f = N[0]

## q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field
$q$ $\mathstrut+$ $\alpha q^{2}$ $\mathstrut+$ $\bigl(- 3 \alpha$ $\mathstrut+ 45\bigr)q^{3}$ $\mathstrut-$ $248q^{4}$ $\mathstrut-$ $10 \alpha q^{5}$ $\mathstrut+$ $\bigl(45 \alpha$ $\mathstrut+ 1512\bigr)q^{6}$ $\mathstrut-$ $1750q^{7}$ $\mathstrut+$ $8 \alpha q^{8}$ $\mathstrut+$ $\bigl(- 270 \alpha$ $\mathstrut- 2511\bigr)q^{9}$ $\mathstrut+O(q^{10})$
where
$\alpha ^{2}$ $\mathstrut +\mathstrut 504$$\mathstrut=0$.

### Coefficient field

sage: K = f.hecke_eigenvalue_field() # note that sage often uses an isomorphic number field
The coefficient field is $\Q(\alpha)\cong\,$ $\Q(\sqrt{-14})$ where $\alpha$ has minimal polynomial
$x ^{2}$ $\mathstrut +\mathstrut 504$
over $\Q$.
sage: K.absolute_polynomial()

### Embeddings

It is possible to embed the Fourier coefficients in the $q$-expansion above in the field of complex numbers. The different embeddings of the first few Fourier coefficients are shown in the table below. Note that these include embeddings that do not preserve the character.
$n$ 2 3 4 5
$v_{ 0 }(a(n))$ -22.4499443206436i 45.0000000000000 + 67.3498329619309i -248.000000000000 224.499443206436i
$v_{ 1 }(a(n))$ 22.4499443206436i 45.0000000000000 - 67.3498329619309i -248.000000000000 -224.499443206436i

## 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)