Properties

Label 171.1.c.a
Level $171$
Weight $1$
Character orbit 171.c
Self dual yes
Analytic conductor $0.085$
Analytic rank $0$
Dimension $1$
Projective image $D_{2}$
CM/RM discs -3, -19, 57
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.0853401171602\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{-19})\)
Artin image: $D_4$
Artin field: Galois closure of 4.0.513.1

$q$-expansion

\(f(q)\) \(=\) \( q + q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{4} - 2 q^{7} + q^{16} - q^{19} - q^{25} - 2 q^{28} + 2 q^{43} + 3 q^{49} - 2 q^{61} + q^{64} + 2 q^{73} - q^{76}+O(q^{100}) \) Copy content Toggle raw display

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 1.00000 0 0 −2.00000 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
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)
57.d even 2 1 RM by \(\Q(\sqrt{57}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.1.c.a 1
3.b odd 2 1 CM 171.1.c.a 1
4.b odd 2 1 2736.1.o.a 1
9.c even 3 2 1539.1.o.b 2
9.d odd 6 2 1539.1.o.b 2
12.b even 2 1 2736.1.o.a 1
19.b odd 2 1 CM 171.1.c.a 1
19.c even 3 2 3249.1.p.a 2
19.d odd 6 2 3249.1.p.a 2
19.e even 9 6 3249.1.ba.c 6
19.f odd 18 6 3249.1.ba.c 6
57.d even 2 1 RM 171.1.c.a 1
57.f even 6 2 3249.1.p.a 2
57.h odd 6 2 3249.1.p.a 2
57.j even 18 6 3249.1.ba.c 6
57.l odd 18 6 3249.1.ba.c 6
76.d even 2 1 2736.1.o.a 1
171.l even 6 2 1539.1.o.b 2
171.o odd 6 2 1539.1.o.b 2
228.b odd 2 1 2736.1.o.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
171.1.c.a 1 1.a even 1 1 trivial
171.1.c.a 1 3.b odd 2 1 CM
171.1.c.a 1 19.b odd 2 1 CM
171.1.c.a 1 57.d even 2 1 RM
1539.1.o.b 2 9.c even 3 2
1539.1.o.b 2 9.d odd 6 2
1539.1.o.b 2 171.l even 6 2
1539.1.o.b 2 171.o odd 6 2
2736.1.o.a 1 4.b odd 2 1
2736.1.o.a 1 12.b even 2 1
2736.1.o.a 1 76.d even 2 1
2736.1.o.a 1 228.b odd 2 1
3249.1.p.a 2 19.c even 3 2
3249.1.p.a 2 19.d odd 6 2
3249.1.p.a 2 57.f even 6 2
3249.1.p.a 2 57.h odd 6 2
3249.1.ba.c 6 19.e even 9 6
3249.1.ba.c 6 19.f odd 18 6
3249.1.ba.c 6 57.j even 18 6
3249.1.ba.c 6 57.l odd 18 6

Hecke kernels

This newform subspace is the entire newspace \(S_{1}^{\mathrm{new}}(171, [\chi])\).

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
$7$ \( T + 2 \) 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
$19$ \( T + 1 \) 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
$41$ \( T \) Copy content Toggle raw display
$43$ \( T - 2 \) Copy content Toggle raw display
$47$ \( T \) Copy content Toggle raw display
$53$ \( T \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 2 \) Copy content Toggle raw display
$67$ \( T \) Copy content Toggle raw display
$71$ \( T \) Copy content Toggle raw display
$73$ \( T - 2 \) Copy content Toggle raw display
$79$ \( T \) Copy content Toggle raw display
$83$ \( T \) Copy content Toggle raw display
$89$ \( T \) Copy content Toggle raw display
$97$ \( T \) Copy content Toggle raw display
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