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 $2$

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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

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

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 \(S_{3}^{\mathrm{new}}(163, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T \) Copy content Toggle raw display
$13$ \( T \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T + 81 \) Copy content Toggle raw display
$43$ \( T + 77 \) Copy content Toggle raw display
$47$ \( T + 69 \) Copy content Toggle raw display
$53$ \( T + 57 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 41 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T + 21 \) Copy content Toggle raw display
$73$ \( T \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T - 3 \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T - 31 \) Copy content Toggle raw display
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Additional information

The Fourier coefficient $a_{n^2}$ for this newform is $n^2$ for all $n < 41$, and $a_p = 0$ for $p < 41$, yielding a striking beginning to the $q$-expansion.