Properties

Label 171.2.a.b
Level 171
Weight 2
Character orbit 171.a
Self dual yes
Analytic conductor 1.365
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 171.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.36544187456\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 2q^{4} - 3q^{5} - q^{7} + O(q^{10}) \) \( q - 2q^{4} - 3q^{5} - q^{7} - 3q^{11} - 4q^{13} + 4q^{16} + 3q^{17} + q^{19} + 6q^{20} + 4q^{25} + 2q^{28} - 6q^{29} - 4q^{31} + 3q^{35} + 2q^{37} + 6q^{41} - q^{43} + 6q^{44} + 3q^{47} - 6q^{49} + 8q^{52} - 12q^{53} + 9q^{55} + 6q^{59} - q^{61} - 8q^{64} + 12q^{65} - 4q^{67} - 6q^{68} - 6q^{71} - 7q^{73} - 2q^{76} + 3q^{77} + 8q^{79} - 12q^{80} - 12q^{83} - 9q^{85} - 12q^{89} + 4q^{91} - 3q^{95} + 8q^{97} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
0 0 −2.00000 −3.00000 0 −1.00000 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 171.2.a.b 1
3.b odd 2 1 19.2.a.a 1
4.b odd 2 1 2736.2.a.c 1
5.b even 2 1 4275.2.a.i 1
7.b odd 2 1 8379.2.a.j 1
12.b even 2 1 304.2.a.f 1
15.d odd 2 1 475.2.a.b 1
15.e even 4 2 475.2.b.a 2
19.b odd 2 1 3249.2.a.d 1
21.c even 2 1 931.2.a.a 1
21.g even 6 2 931.2.f.b 2
21.h odd 6 2 931.2.f.c 2
24.f even 2 1 1216.2.a.b 1
24.h odd 2 1 1216.2.a.o 1
33.d even 2 1 2299.2.a.b 1
39.d odd 2 1 3211.2.a.a 1
51.c odd 2 1 5491.2.a.b 1
57.d even 2 1 361.2.a.b 1
57.f even 6 2 361.2.c.a 2
57.h odd 6 2 361.2.c.c 2
57.j even 18 6 361.2.e.e 6
57.l odd 18 6 361.2.e.d 6
60.h even 2 1 7600.2.a.c 1
228.b odd 2 1 5776.2.a.c 1
285.b even 2 1 9025.2.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.a.a 1 3.b odd 2 1
171.2.a.b 1 1.a even 1 1 trivial
304.2.a.f 1 12.b even 2 1
361.2.a.b 1 57.d even 2 1
361.2.c.a 2 57.f even 6 2
361.2.c.c 2 57.h odd 6 2
361.2.e.d 6 57.l odd 18 6
361.2.e.e 6 57.j even 18 6
475.2.a.b 1 15.d odd 2 1
475.2.b.a 2 15.e even 4 2
931.2.a.a 1 21.c even 2 1
931.2.f.b 2 21.g even 6 2
931.2.f.c 2 21.h odd 6 2
1216.2.a.b 1 24.f even 2 1
1216.2.a.o 1 24.h odd 2 1
2299.2.a.b 1 33.d even 2 1
2736.2.a.c 1 4.b odd 2 1
3211.2.a.a 1 39.d odd 2 1
3249.2.a.d 1 19.b odd 2 1
4275.2.a.i 1 5.b even 2 1
5491.2.a.b 1 51.c odd 2 1
5776.2.a.c 1 228.b odd 2 1
7600.2.a.c 1 60.h even 2 1
8379.2.a.j 1 7.b odd 2 1
9025.2.a.d 1 285.b even 2 1

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(171))\):

\( T_{2} \)
\( T_{5} + 3 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T^{2} \)
$3$ 1
$5$ \( 1 + 3 T + 5 T^{2} \)
$7$ \( 1 + T + 7 T^{2} \)
$11$ \( 1 + 3 T + 11 T^{2} \)
$13$ \( 1 + 4 T + 13 T^{2} \)
$17$ \( 1 - 3 T + 17 T^{2} \)
$19$ \( 1 - T \)
$23$ \( 1 + 23 T^{2} \)
$29$ \( 1 + 6 T + 29 T^{2} \)
$31$ \( 1 + 4 T + 31 T^{2} \)
$37$ \( 1 - 2 T + 37 T^{2} \)
$41$ \( 1 - 6 T + 41 T^{2} \)
$43$ \( 1 + T + 43 T^{2} \)
$47$ \( 1 - 3 T + 47 T^{2} \)
$53$ \( 1 + 12 T + 53 T^{2} \)
$59$ \( 1 - 6 T + 59 T^{2} \)
$61$ \( 1 + T + 61 T^{2} \)
$67$ \( 1 + 4 T + 67 T^{2} \)
$71$ \( 1 + 6 T + 71 T^{2} \)
$73$ \( 1 + 7 T + 73 T^{2} \)
$79$ \( 1 - 8 T + 79 T^{2} \)
$83$ \( 1 + 12 T + 83 T^{2} \)
$89$ \( 1 + 12 T + 89 T^{2} \)
$97$ \( 1 - 8 T + 97 T^{2} \)
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