Properties

Level 5
Weight 6
Character $\chi_{5}(4, \cdot)$
Label 5.6.4.a
Dimension of Galois orbit 2
Twist info Is minimal
CM No
Atkin-Lehner eigenvalues \( \omega_{ 5 } \) : -1

Related objects

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Show commands for: SageMath

magma: D := FullDirichletGroup(5);
magma: c := D![2];
magma: S:= CuspForms(c,6);
magma: N := Newforms(S);
sage: D = DirichletGroup(5)
sage: c = D.Element(D,vector([2]))
sage: N = Newforms(c,6,names="a")
sage: f = N[0]

q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field
\(q \) \(\mathstrut+\) \(\alpha q^{2} \) \(\mathstrut-\) \(3 \alpha q^{3} \) \(\mathstrut-\) \(12q^{4} \) \(\mathstrut+\) \(\bigl(5 \alpha \) \(\mathstrut- 45\bigr)q^{5} \) \(\mathstrut+\) \(132q^{6} \) \(\mathstrut-\) \(9 \alpha q^{7} \) \(\mathstrut+\) \(20 \alpha q^{8} \) \(\mathstrut-\) \(153q^{9} \) \(\mathstrut+O(q^{10}) \)
where
\(\alpha ^{2} \) \(\mathstrut +\mathstrut 44\)\(\mathstrut=0\).


(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 $\Q(\alpha)\cong\,$ \(\Q(\sqrt{-11}) \) where $ \alpha $ has minimal polynomial
\(x ^{2} \) \(\mathstrut +\mathstrut 44\)
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
\( v_{ 0 }(a(n)) \) -6.63324958071080i 19.8997487421324i -12.0000000000000 -45.0000000000000 - 33.1662479035540i
\( v_{ 1 }(a(n)) \) 6.63324958071080i -19.8997487421324i -12.0000000000000 -45.0000000000000 + 33.1662479035540i

Detailed data

Further Properties

Download this Newform

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