Properties

Label 171.4.a.d
Level $171$
Weight $4$
Character orbit 171.a
Self dual yes
Analytic conductor $10.089$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 171 = 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 171.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.0893266110\)
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)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 3q^{2} + q^{4} + 12q^{5} + 11q^{7} - 21q^{8} + O(q^{10}) \) \( q + 3q^{2} + q^{4} + 12q^{5} + 11q^{7} - 21q^{8} + 36q^{10} + 54q^{11} + 11q^{13} + 33q^{14} - 71q^{16} + 93q^{17} + 19q^{19} + 12q^{20} + 162q^{22} - 183q^{23} + 19q^{25} + 33q^{26} + 11q^{28} + 249q^{29} + 56q^{31} - 45q^{32} + 279q^{34} + 132q^{35} - 250q^{37} + 57q^{38} - 252q^{40} - 240q^{41} - 196q^{43} + 54q^{44} - 549q^{46} + 168q^{47} - 222q^{49} + 57q^{50} + 11q^{52} - 435q^{53} + 648q^{55} - 231q^{56} + 747q^{58} - 195q^{59} - 358q^{61} + 168q^{62} + 433q^{64} + 132q^{65} - 961q^{67} + 93q^{68} + 396q^{70} + 246q^{71} + 353q^{73} - 750q^{74} + 19q^{76} + 594q^{77} - 34q^{79} - 852q^{80} - 720q^{82} - 234q^{83} + 1116q^{85} - 588q^{86} - 1134q^{88} + 168q^{89} + 121q^{91} - 183q^{92} + 504q^{94} + 228q^{95} + 758q^{97} - 666q^{98} + O(q^{100}) \)

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
3.00000 0 1.00000 12.0000 0 11.0000 −21.0000 0 36.0000
\(n\): 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 171.4.a.d 1
3.b odd 2 1 19.4.a.a 1
12.b even 2 1 304.4.a.b 1
15.d odd 2 1 475.4.a.e 1
15.e even 4 2 475.4.b.c 2
21.c even 2 1 931.4.a.a 1
24.f even 2 1 1216.4.a.a 1
24.h odd 2 1 1216.4.a.f 1
33.d even 2 1 2299.4.a.b 1
57.d even 2 1 361.4.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.4.a.a 1 3.b odd 2 1
171.4.a.d 1 1.a even 1 1 trivial
304.4.a.b 1 12.b even 2 1
361.4.a.b 1 57.d even 2 1
475.4.a.e 1 15.d odd 2 1
475.4.b.c 2 15.e even 4 2
931.4.a.a 1 21.c even 2 1
1216.4.a.a 1 24.f even 2 1
1216.4.a.f 1 24.h odd 2 1
2299.4.a.b 1 33.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(171))\):

\( T_{2} - 3 \)
\( T_{5} - 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -3 + T \)
$3$ \( T \)
$5$ \( -12 + T \)
$7$ \( -11 + T \)
$11$ \( -54 + T \)
$13$ \( -11 + T \)
$17$ \( -93 + T \)
$19$ \( -19 + T \)
$23$ \( 183 + T \)
$29$ \( -249 + T \)
$31$ \( -56 + T \)
$37$ \( 250 + T \)
$41$ \( 240 + T \)
$43$ \( 196 + T \)
$47$ \( -168 + T \)
$53$ \( 435 + T \)
$59$ \( 195 + T \)
$61$ \( 358 + T \)
$67$ \( 961 + T \)
$71$ \( -246 + T \)
$73$ \( -353 + T \)
$79$ \( 34 + T \)
$83$ \( 234 + T \)
$89$ \( -168 + T \)
$97$ \( -758 + T \)
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