Properties

Label 171.3.c.a
Level $171$
Weight $3$
Character orbit 171.c
Self dual yes
Analytic conductor $4.659$
Analytic rank $0$
Dimension $1$
CM discriminant -19
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 171.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(4.65941252056\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \( q + 4 q^{4} + 9 q^{5} - 5 q^{7} + O(q^{10}) \) \( q + 4 q^{4} + 9 q^{5} - 5 q^{7} - 3 q^{11} + 16 q^{16} - 15 q^{17} - 19 q^{19} + 36 q^{20} + 30 q^{23} + 56 q^{25} - 20 q^{28} - 45 q^{35} - 85 q^{43} - 12 q^{44} - 75 q^{47} - 24 q^{49} - 27 q^{55} + 103 q^{61} + 64 q^{64} - 60 q^{68} - 25 q^{73} - 76 q^{76} + 15 q^{77} + 144 q^{80} - 90 q^{83} - 135 q^{85} + 120 q^{92} - 171 q^{95} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/171\mathbb{Z}\right)^\times\).

\(n\) \(20\) \(154\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0
0 0 4.00000 9.00000 0 −5.00000 0 0 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
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.3.c.a 1
3.b odd 2 1 19.3.b.a 1
4.b odd 2 1 2736.3.o.a 1
12.b even 2 1 304.3.e.a 1
15.d odd 2 1 475.3.c.a 1
15.e even 4 2 475.3.d.a 2
19.b odd 2 1 CM 171.3.c.a 1
24.f even 2 1 1216.3.e.b 1
24.h odd 2 1 1216.3.e.a 1
57.d even 2 1 19.3.b.a 1
57.f even 6 2 361.3.d.a 2
57.h odd 6 2 361.3.d.a 2
57.j even 18 6 361.3.f.a 6
57.l odd 18 6 361.3.f.a 6
76.d even 2 1 2736.3.o.a 1
228.b odd 2 1 304.3.e.a 1
285.b even 2 1 475.3.c.a 1
285.j odd 4 2 475.3.d.a 2
456.l odd 2 1 1216.3.e.b 1
456.p even 2 1 1216.3.e.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.a 1 3.b odd 2 1
19.3.b.a 1 57.d even 2 1
171.3.c.a 1 1.a even 1 1 trivial
171.3.c.a 1 19.b odd 2 1 CM
304.3.e.a 1 12.b even 2 1
304.3.e.a 1 228.b odd 2 1
361.3.d.a 2 57.f even 6 2
361.3.d.a 2 57.h odd 6 2
361.3.f.a 6 57.j even 18 6
361.3.f.a 6 57.l odd 18 6
475.3.c.a 1 15.d odd 2 1
475.3.c.a 1 285.b even 2 1
475.3.d.a 2 15.e even 4 2
475.3.d.a 2 285.j odd 4 2
1216.3.e.a 1 24.h odd 2 1
1216.3.e.a 1 456.p even 2 1
1216.3.e.b 1 24.f even 2 1
1216.3.e.b 1 456.l odd 2 1
2736.3.o.a 1 4.b odd 2 1
2736.3.o.a 1 76.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(171, [\chi])\):

\( T_{2} \)
\( T_{5} - 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \)
$3$ \( T \)
$5$ \( -9 + T \)
$7$ \( 5 + T \)
$11$ \( 3 + T \)
$13$ \( T \)
$17$ \( 15 + T \)
$19$ \( 19 + T \)
$23$ \( -30 + T \)
$29$ \( T \)
$31$ \( T \)
$37$ \( T \)
$41$ \( T \)
$43$ \( 85 + T \)
$47$ \( 75 + T \)
$53$ \( T \)
$59$ \( T \)
$61$ \( -103 + T \)
$67$ \( T \)
$71$ \( T \)
$73$ \( 25 + T \)
$79$ \( T \)
$83$ \( 90 + T \)
$89$ \( T \)
$97$ \( T \)
show more
show less