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 discriminant -163
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

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

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: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

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

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 Parity Ord Mult Type
1.a even 1 1 trivial
163.b odd 2 1 CM by \(\Q(\sqrt{-163}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 163.3.b.a 1
163.b odd 2 1 CM 163.3.b.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
163.3.b.a 1 1.a even 1 1 trivial
163.3.b.a 1 163.b odd 2 1 CM

Hecke kernels

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - 2 T )( 1 + 2 T ) \)
$3$ \( ( 1 - 3 T )( 1 + 3 T ) \)
$5$ \( ( 1 - 5 T )( 1 + 5 T ) \)
$7$ \( ( 1 - 7 T )( 1 + 7 T ) \)
$11$ \( ( 1 - 11 T )( 1 + 11 T ) \)
$13$ \( ( 1 - 13 T )( 1 + 13 T ) \)
$17$ \( ( 1 - 17 T )( 1 + 17 T ) \)
$19$ \( ( 1 - 19 T )( 1 + 19 T ) \)
$23$ \( ( 1 - 23 T )( 1 + 23 T ) \)
$29$ \( ( 1 - 29 T )( 1 + 29 T ) \)
$31$ \( ( 1 - 31 T )( 1 + 31 T ) \)
$37$ \( ( 1 - 37 T )( 1 + 37 T ) \)
$41$ \( 1 + 81 T + 1681 T^{2} \)
$43$ \( 1 + 77 T + 1849 T^{2} \)
$47$ \( 1 + 69 T + 2209 T^{2} \)
$53$ \( 1 + 57 T + 2809 T^{2} \)
$59$ \( ( 1 - 59 T )( 1 + 59 T ) \)
$61$ \( 1 + 41 T + 3721 T^{2} \)
$67$ \( ( 1 - 67 T )( 1 + 67 T ) \)
$71$ \( 1 + 21 T + 5041 T^{2} \)
$73$ \( ( 1 - 73 T )( 1 + 73 T ) \)
$79$ \( ( 1 - 79 T )( 1 + 79 T ) \)
$83$ \( 1 - 3 T + 6889 T^{2} \)
$89$ \( ( 1 - 89 T )( 1 + 89 T ) \)
$97$ \( 1 - 31 T + 9409 T^{2} \)
show more
show less