Properties

Label 19.3.b.a
Level 19
Weight 3
Character orbit 19.b
Self dual Yes
Analytic conductor 0.518
Analytic rank 0
Dimension 1
CM disc. -19
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 19.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(0.517712502285\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut -\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 5q^{7} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut 3q^{11} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 15q^{17} \) \(\mathstrut -\mathstrut 19q^{19} \) \(\mathstrut -\mathstrut 36q^{20} \) \(\mathstrut -\mathstrut 30q^{23} \) \(\mathstrut +\mathstrut 56q^{25} \) \(\mathstrut -\mathstrut 20q^{28} \) \(\mathstrut +\mathstrut 45q^{35} \) \(\mathstrut +\mathstrut 36q^{36} \) \(\mathstrut -\mathstrut 85q^{43} \) \(\mathstrut +\mathstrut 12q^{44} \) \(\mathstrut -\mathstrut 81q^{45} \) \(\mathstrut +\mathstrut 75q^{47} \) \(\mathstrut -\mathstrut 24q^{49} \) \(\mathstrut -\mathstrut 27q^{55} \) \(\mathstrut +\mathstrut 103q^{61} \) \(\mathstrut -\mathstrut 45q^{63} \) \(\mathstrut +\mathstrut 64q^{64} \) \(\mathstrut +\mathstrut 60q^{68} \) \(\mathstrut -\mathstrut 25q^{73} \) \(\mathstrut -\mathstrut 76q^{76} \) \(\mathstrut -\mathstrut 15q^{77} \) \(\mathstrut -\mathstrut 144q^{80} \) \(\mathstrut +\mathstrut 81q^{81} \) \(\mathstrut +\mathstrut 90q^{83} \) \(\mathstrut -\mathstrut 135q^{85} \) \(\mathstrut -\mathstrut 120q^{92} \) \(\mathstrut +\mathstrut 171q^{95} \) \(\mathstrut +\mathstrut 27q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/19\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
0
0 0 4.00000 −9.00000 0 −5.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
19.b Odd 1 CM by \(\Q(\sqrt{-19}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) acting on \(S_{3}^{\mathrm{new}}(19, [\chi])\).