Newform orbit 163.3.b.a

• 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: $2$

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

$f(q)$

$=$

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

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

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

$p$	$F_p(T)$
$2$	$T$
$3$	$T$
$5$	$T$
$7$	$T$
$11$	$T$

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.