Properties

Label 171.2.a.d
Level $171$
Weight $2$
Character orbit 171.a
Self dual yes
Analytic conductor $1.365$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 57)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 2 q^{2} + 2 q^{4} + 3 q^{5} - 5 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + 2 q^{4} + 3 q^{5} - 5 q^{7} + 6 q^{10} - q^{11} + 2 q^{13} - 10 q^{14} - 4 q^{16} + q^{17} - q^{19} + 6 q^{20} - 2 q^{22} + 4 q^{23} + 4 q^{25} + 4 q^{26} - 10 q^{28} + 2 q^{29} - 6 q^{31} - 8 q^{32} + 2 q^{34} - 15 q^{35} - 2 q^{38} - q^{43} - 2 q^{44} + 8 q^{46} + 9 q^{47} + 18 q^{49} + 8 q^{50} + 4 q^{52} - 10 q^{53} - 3 q^{55} + 4 q^{58} + 8 q^{59} - q^{61} - 12 q^{62} - 8 q^{64} + 6 q^{65} + 8 q^{67} + 2 q^{68} - 30 q^{70} + 12 q^{71} - 11 q^{73} - 2 q^{76} + 5 q^{77} + 16 q^{79} - 12 q^{80} - 12 q^{83} + 3 q^{85} - 2 q^{86} + 6 q^{89} - 10 q^{91} + 8 q^{92} + 18 q^{94} - 3 q^{95} - 10 q^{97} + 36 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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
2.00000 0 2.00000 3.00000 0 −5.00000 0 0 6.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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.d 1
3.b odd 2 1 57.2.a.a 1
4.b odd 2 1 2736.2.a.v 1
5.b even 2 1 4275.2.a.b 1
7.b odd 2 1 8379.2.a.p 1
12.b even 2 1 912.2.a.g 1
15.d odd 2 1 1425.2.a.j 1
15.e even 4 2 1425.2.c.b 2
19.b odd 2 1 3249.2.a.b 1
21.c even 2 1 2793.2.a.b 1
24.f even 2 1 3648.2.a.r 1
24.h odd 2 1 3648.2.a.bh 1
33.d even 2 1 6897.2.a.f 1
39.d odd 2 1 9633.2.a.o 1
57.d even 2 1 1083.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.2.a.a 1 3.b odd 2 1
171.2.a.d 1 1.a even 1 1 trivial
912.2.a.g 1 12.b even 2 1
1083.2.a.e 1 57.d even 2 1
1425.2.a.j 1 15.d odd 2 1
1425.2.c.b 2 15.e even 4 2
2736.2.a.v 1 4.b odd 2 1
2793.2.a.b 1 21.c even 2 1
3249.2.a.b 1 19.b odd 2 1
3648.2.a.r 1 24.f even 2 1
3648.2.a.bh 1 24.h odd 2 1
4275.2.a.b 1 5.b even 2 1
6897.2.a.f 1 33.d even 2 1
8379.2.a.p 1 7.b odd 2 1
9633.2.a.o 1 39.d odd 2 1

Hecke kernels

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

\( T_{2} - 2 \) Copy content Toggle raw display
\( T_{5} - 3 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 2 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 3 \) Copy content Toggle raw display
$7$ \( T + 5 \) Copy content Toggle raw display
$11$ \( T + 1 \) Copy content Toggle raw display
$13$ \( T - 2 \) Copy content Toggle raw display
$17$ \( T - 1 \) Copy content Toggle raw display
$19$ \( T + 1 \) Copy content Toggle raw display
$23$ \( T - 4 \) Copy content Toggle raw display
$29$ \( T - 2 \) Copy content Toggle raw display
$31$ \( T + 6 \) Copy content Toggle raw display
$37$ \( T \) Copy content Toggle raw display
$41$ \( T \) Copy content Toggle raw display
$43$ \( T + 1 \) Copy content Toggle raw display
$47$ \( T - 9 \) Copy content Toggle raw display
$53$ \( T + 10 \) Copy content Toggle raw display
$59$ \( T - 8 \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T - 8 \) Copy content Toggle raw display
$71$ \( T - 12 \) Copy content Toggle raw display
$73$ \( T + 11 \) Copy content Toggle raw display
$79$ \( T - 16 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 6 \) Copy content Toggle raw display
$97$ \( T + 10 \) Copy content Toggle raw display
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