Properties

Label 19.7.b.a
Level 19
Weight 7
Character orbit 19.b
Self dual Yes
Analytic conductor 4.371
Analytic rank 0
Dimension 1
CM disc. -19
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 19 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 19.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(4.37102758878\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 54q^{5} \) \(\mathstrut +\mathstrut 610q^{7} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 54q^{5} \) \(\mathstrut +\mathstrut 610q^{7} \) \(\mathstrut +\mathstrut 729q^{9} \) \(\mathstrut -\mathstrut 1062q^{11} \) \(\mathstrut +\mathstrut 4096q^{16} \) \(\mathstrut -\mathstrut 9630q^{17} \) \(\mathstrut -\mathstrut 6859q^{19} \) \(\mathstrut -\mathstrut 3456q^{20} \) \(\mathstrut +\mathstrut 20610q^{23} \) \(\mathstrut -\mathstrut 12709q^{25} \) \(\mathstrut +\mathstrut 39040q^{28} \) \(\mathstrut -\mathstrut 32940q^{35} \) \(\mathstrut +\mathstrut 46656q^{36} \) \(\mathstrut -\mathstrut 142630q^{43} \) \(\mathstrut -\mathstrut 67968q^{44} \) \(\mathstrut -\mathstrut 39366q^{45} \) \(\mathstrut -\mathstrut 75150q^{47} \) \(\mathstrut +\mathstrut 254451q^{49} \) \(\mathstrut +\mathstrut 57348q^{55} \) \(\mathstrut -\mathstrut 57062q^{61} \) \(\mathstrut +\mathstrut 444690q^{63} \) \(\mathstrut +\mathstrut 262144q^{64} \) \(\mathstrut -\mathstrut 616320q^{68} \) \(\mathstrut +\mathstrut 384050q^{73} \) \(\mathstrut -\mathstrut 438976q^{76} \) \(\mathstrut -\mathstrut 647820q^{77} \) \(\mathstrut -\mathstrut 221184q^{80} \) \(\mathstrut +\mathstrut 531441q^{81} \) \(\mathstrut -\mathstrut 1131030q^{83} \) \(\mathstrut +\mathstrut 520020q^{85} \) \(\mathstrut +\mathstrut 1319040q^{92} \) \(\mathstrut +\mathstrut 370386q^{95} \) \(\mathstrut -\mathstrut 774198q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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 64.0000 −54.0000 0 610.000 0 729.000 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 \(S_{7}^{\mathrm{new}}(19, [\chi])\).