Properties

Label 19.9.b.a
Level 19
Weight 9
Character orbit 19.b
Self dual Yes
Analytic conductor 7.740
Analytic rank 0
Dimension 1
CM disc. -19
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 9 \)
Character orbit: \([\chi]\) = 19.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(7.74019359116\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut -\mathstrut 289q^{5} \) \(\mathstrut +\mathstrut 527q^{7} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 256q^{4} \) \(\mathstrut -\mathstrut 289q^{5} \) \(\mathstrut +\mathstrut 527q^{7} \) \(\mathstrut +\mathstrut 6561q^{9} \) \(\mathstrut +\mathstrut 25007q^{11} \) \(\mathstrut +\mathstrut 65536q^{16} \) \(\mathstrut -\mathstrut 42433q^{17} \) \(\mathstrut +\mathstrut 130321q^{19} \) \(\mathstrut -\mathstrut 73984q^{20} \) \(\mathstrut -\mathstrut 534718q^{23} \) \(\mathstrut -\mathstrut 307104q^{25} \) \(\mathstrut +\mathstrut 134912q^{28} \) \(\mathstrut -\mathstrut 152303q^{35} \) \(\mathstrut +\mathstrut 1679616q^{36} \) \(\mathstrut +\mathstrut 5602127q^{43} \) \(\mathstrut +\mathstrut 6401792q^{44} \) \(\mathstrut -\mathstrut 1896129q^{45} \) \(\mathstrut -\mathstrut 8302513q^{47} \) \(\mathstrut -\mathstrut 5487072q^{49} \) \(\mathstrut -\mathstrut 7227023q^{55} \) \(\mathstrut -\mathstrut 17661793q^{61} \) \(\mathstrut +\mathstrut 3457647q^{63} \) \(\mathstrut +\mathstrut 16777216q^{64} \) \(\mathstrut -\mathstrut 10862848q^{68} \) \(\mathstrut +\mathstrut 43864607q^{73} \) \(\mathstrut +\mathstrut 33362176q^{76} \) \(\mathstrut +\mathstrut 13178689q^{77} \) \(\mathstrut -\mathstrut 18939904q^{80} \) \(\mathstrut +\mathstrut 43046721q^{81} \) \(\mathstrut -\mathstrut 62676958q^{83} \) \(\mathstrut +\mathstrut 12263137q^{85} \) \(\mathstrut -\mathstrut 136887808q^{92} \) \(\mathstrut -\mathstrut 37662769q^{95} \) \(\mathstrut +\mathstrut 164070927q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 0 256.000 −289.000 0 527.000 0 6561.00 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 \(S_{9}^{\mathrm{new}}(19, [\chi])\).