Properties

Label 147.4.a.b
Level $147$
Weight $4$
Character orbit 147.a
Self dual yes
Analytic conductor $8.673$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.67328077084\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 3q^{2} + 3q^{3} + q^{4} - 3q^{5} - 9q^{6} + 21q^{8} + 9q^{9} + O(q^{10}) \) \( q - 3q^{2} + 3q^{3} + q^{4} - 3q^{5} - 9q^{6} + 21q^{8} + 9q^{9} + 9q^{10} - 15q^{11} + 3q^{12} - 64q^{13} - 9q^{15} - 71q^{16} + 84q^{17} - 27q^{18} - 16q^{19} - 3q^{20} + 45q^{22} - 84q^{23} + 63q^{24} - 116q^{25} + 192q^{26} + 27q^{27} - 297q^{29} + 27q^{30} - 253q^{31} + 45q^{32} - 45q^{33} - 252q^{34} + 9q^{36} - 316q^{37} + 48q^{38} - 192q^{39} - 63q^{40} + 360q^{41} + 26q^{43} - 15q^{44} - 27q^{45} + 252q^{46} - 30q^{47} - 213q^{48} + 348q^{50} + 252q^{51} - 64q^{52} + 363q^{53} - 81q^{54} + 45q^{55} - 48q^{57} + 891q^{58} - 15q^{59} - 9q^{60} - 118q^{61} + 759q^{62} + 433q^{64} + 192q^{65} + 135q^{66} - 370q^{67} + 84q^{68} - 252q^{69} - 342q^{71} + 189q^{72} + 362q^{73} + 948q^{74} - 348q^{75} - 16q^{76} + 576q^{78} + 467q^{79} + 213q^{80} + 81q^{81} - 1080q^{82} + 477q^{83} - 252q^{85} - 78q^{86} - 891q^{87} - 315q^{88} + 906q^{89} + 81q^{90} - 84q^{92} - 759q^{93} + 90q^{94} + 48q^{95} + 135q^{96} + 503q^{97} - 135q^{99} + O(q^{100}) \)

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 3.00000 1.00000 −3.00000 −9.00000 0 21.0000 9.00000 9.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)

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 147.4.a.b 1
3.b odd 2 1 441.4.a.l 1
4.b odd 2 1 2352.4.a.i 1
7.b odd 2 1 147.4.a.a 1
7.c even 3 2 21.4.e.a 2
7.d odd 6 2 147.4.e.h 2
21.c even 2 1 441.4.a.k 1
21.g even 6 2 441.4.e.c 2
21.h odd 6 2 63.4.e.a 2
28.d even 2 1 2352.4.a.bd 1
28.g odd 6 2 336.4.q.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.e.a 2 7.c even 3 2
63.4.e.a 2 21.h odd 6 2
147.4.a.a 1 7.b odd 2 1
147.4.a.b 1 1.a even 1 1 trivial
147.4.e.h 2 7.d odd 6 2
336.4.q.e 2 28.g odd 6 2
441.4.a.k 1 21.c even 2 1
441.4.a.l 1 3.b odd 2 1
441.4.e.c 2 21.g even 6 2
2352.4.a.i 1 4.b odd 2 1
2352.4.a.bd 1 28.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 \(S_{4}^{\mathrm{new}}(\Gamma_0(147))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 3 + T \)
$3$ \( -3 + T \)
$5$ \( 3 + T \)
$7$ \( T \)
$11$ \( 15 + T \)
$13$ \( 64 + T \)
$17$ \( -84 + T \)
$19$ \( 16 + T \)
$23$ \( 84 + T \)
$29$ \( 297 + T \)
$31$ \( 253 + T \)
$37$ \( 316 + T \)
$41$ \( -360 + T \)
$43$ \( -26 + T \)
$47$ \( 30 + T \)
$53$ \( -363 + T \)
$59$ \( 15 + T \)
$61$ \( 118 + T \)
$67$ \( 370 + T \)
$71$ \( 342 + T \)
$73$ \( -362 + T \)
$79$ \( -467 + T \)
$83$ \( -477 + T \)
$89$ \( -906 + T \)
$97$ \( -503 + T \)
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