Properties

Label 163.3.b.a
Level 163
Weight 3
Character orbit 163.b
Self dual Yes
Analytic conductor 4.441
Analytic rank 0
Dimension 1
CM disc. -163
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 163 \)
Weight: \( k \) = \( 3 \)
Character orbit: \([\chi]\) = 163.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(4.44142830907\)
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

\(f(q)\)  \(=\)  \(q \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 4q^{4} \) \(\mathstrut +\mathstrut 9q^{9} \) \(\mathstrut +\mathstrut 16q^{16} \) \(\mathstrut +\mathstrut 25q^{25} \) \(\mathstrut +\mathstrut 36q^{36} \) \(\mathstrut -\mathstrut 81q^{41} \) \(\mathstrut -\mathstrut 77q^{43} \) \(\mathstrut -\mathstrut 69q^{47} \) \(\mathstrut +\mathstrut 49q^{49} \) \(\mathstrut -\mathstrut 57q^{53} \) \(\mathstrut -\mathstrut 41q^{61} \) \(\mathstrut +\mathstrut 64q^{64} \) \(\mathstrut -\mathstrut 21q^{71} \) \(\mathstrut +\mathstrut 81q^{81} \) \(\mathstrut +\mathstrut 3q^{83} \) \(\mathstrut +\mathstrut 31q^{97} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/163\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
162.1
0
0 0 4.00000 0 0 0 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
163.b Odd 1 CM by \(\Q(\sqrt{-163}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{2} \) acting on \(S_{3}^{\mathrm{new}}(163, \chi)\).