Properties

Label 147.4.a.e
Level $147$
Weight $4$
Character orbit 147.a
Self dual yes
Analytic conductor $8.673$
Analytic rank $0$
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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} + 3q^{3} - 7q^{4} - 12q^{5} - 3q^{6} + 15q^{8} + 9q^{9} + O(q^{10}) \) \( q - q^{2} + 3q^{3} - 7q^{4} - 12q^{5} - 3q^{6} + 15q^{8} + 9q^{9} + 12q^{10} + 20q^{11} - 21q^{12} + 84q^{13} - 36q^{15} + 41q^{16} + 96q^{17} - 9q^{18} - 12q^{19} + 84q^{20} - 20q^{22} - 176q^{23} + 45q^{24} + 19q^{25} - 84q^{26} + 27q^{27} + 58q^{29} + 36q^{30} + 264q^{31} - 161q^{32} + 60q^{33} - 96q^{34} - 63q^{36} + 258q^{37} + 12q^{38} + 252q^{39} - 180q^{40} + 156q^{43} - 140q^{44} - 108q^{45} + 176q^{46} + 408q^{47} + 123q^{48} - 19q^{50} + 288q^{51} - 588q^{52} - 722q^{53} - 27q^{54} - 240q^{55} - 36q^{57} - 58q^{58} - 492q^{59} + 252q^{60} + 492q^{61} - 264q^{62} - 167q^{64} - 1008q^{65} - 60q^{66} + 412q^{67} - 672q^{68} - 528q^{69} + 296q^{71} + 135q^{72} - 240q^{73} - 258q^{74} + 57q^{75} + 84q^{76} - 252q^{78} + 776q^{79} - 492q^{80} + 81q^{81} - 924q^{83} - 1152q^{85} - 156q^{86} + 174q^{87} + 300q^{88} + 744q^{89} + 108q^{90} + 1232q^{92} + 792q^{93} - 408q^{94} + 144q^{95} - 483q^{96} + 168q^{97} + 180q^{99} + O(q^{100}) \)

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
−1.00000 3.00000 −7.00000 −12.0000 −3.00000 0 15.0000 9.00000 12.0000
\(n\): 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.e yes 1
3.b odd 2 1 441.4.a.h 1
4.b odd 2 1 2352.4.a.b 1
7.b odd 2 1 147.4.a.d 1
7.c even 3 2 147.4.e.e 2
7.d odd 6 2 147.4.e.f 2
21.c even 2 1 441.4.a.g 1
21.g even 6 2 441.4.e.g 2
21.h odd 6 2 441.4.e.f 2
28.d even 2 1 2352.4.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.d 1 7.b odd 2 1
147.4.a.e yes 1 1.a even 1 1 trivial
147.4.e.e 2 7.c even 3 2
147.4.e.f 2 7.d odd 6 2
441.4.a.g 1 21.c even 2 1
441.4.a.h 1 3.b odd 2 1
441.4.e.f 2 21.h odd 6 2
441.4.e.g 2 21.g even 6 2
2352.4.a.b 1 4.b odd 2 1
2352.4.a.bi 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2} + 1 \)
\( T_{5} + 12 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T \)
$3$ \( -3 + T \)
$5$ \( 12 + T \)
$7$ \( T \)
$11$ \( -20 + T \)
$13$ \( -84 + T \)
$17$ \( -96 + T \)
$19$ \( 12 + T \)
$23$ \( 176 + T \)
$29$ \( -58 + T \)
$31$ \( -264 + T \)
$37$ \( -258 + T \)
$41$ \( T \)
$43$ \( -156 + T \)
$47$ \( -408 + T \)
$53$ \( 722 + T \)
$59$ \( 492 + T \)
$61$ \( -492 + T \)
$67$ \( -412 + T \)
$71$ \( -296 + T \)
$73$ \( 240 + T \)
$79$ \( -776 + T \)
$83$ \( 924 + T \)
$89$ \( -744 + T \)
$97$ \( -168 + T \)
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