Properties

Label 147.4.a.f
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{2} - 3q^{3} + 8q^{4} - 18q^{5} - 12q^{6} + 9q^{9} + O(q^{10}) \) \( q + 4q^{2} - 3q^{3} + 8q^{4} - 18q^{5} - 12q^{6} + 9q^{9} - 72q^{10} - 50q^{11} - 24q^{12} + 36q^{13} + 54q^{15} - 64q^{16} - 126q^{17} + 36q^{18} + 72q^{19} - 144q^{20} - 200q^{22} + 14q^{23} + 199q^{25} + 144q^{26} - 27q^{27} + 158q^{29} + 216q^{30} + 36q^{31} - 256q^{32} + 150q^{33} - 504q^{34} + 72q^{36} - 162q^{37} + 288q^{38} - 108q^{39} + 270q^{41} - 324q^{43} - 400q^{44} - 162q^{45} + 56q^{46} + 72q^{47} + 192q^{48} + 796q^{50} + 378q^{51} + 288q^{52} - 22q^{53} - 108q^{54} + 900q^{55} - 216q^{57} + 632q^{58} - 468q^{59} + 432q^{60} - 792q^{61} + 144q^{62} - 512q^{64} - 648q^{65} + 600q^{66} + 232q^{67} - 1008q^{68} - 42q^{69} - 734q^{71} - 180q^{73} - 648q^{74} - 597q^{75} + 576q^{76} - 432q^{78} + 236q^{79} + 1152q^{80} + 81q^{81} + 1080q^{82} - 36q^{83} + 2268q^{85} - 1296q^{86} - 474q^{87} - 234q^{89} - 648q^{90} + 112q^{92} - 108q^{93} + 288q^{94} - 1296q^{95} + 768q^{96} - 468q^{97} - 450q^{99} + O(q^{100}) \)

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
4.00000 −3.00000 8.00000 −18.0000 −12.0000 0 0 9.00000 −72.0000
\(n\): 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.f 1
3.b odd 2 1 441.4.a.c 1
4.b odd 2 1 2352.4.a.t 1
7.b odd 2 1 147.4.a.h yes 1
7.c even 3 2 147.4.e.d 2
7.d odd 6 2 147.4.e.a 2
21.c even 2 1 441.4.a.a 1
21.g even 6 2 441.4.e.o 2
21.h odd 6 2 441.4.e.l 2
28.d even 2 1 2352.4.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
147.4.a.f 1 1.a even 1 1 trivial
147.4.a.h yes 1 7.b odd 2 1
147.4.e.a 2 7.d odd 6 2
147.4.e.d 2 7.c even 3 2
441.4.a.a 1 21.c even 2 1
441.4.a.c 1 3.b odd 2 1
441.4.e.l 2 21.h odd 6 2
441.4.e.o 2 21.g even 6 2
2352.4.a.s 1 28.d even 2 1
2352.4.a.t 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2} - 4 \)
\( T_{5} + 18 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( 3 + T \)
$5$ \( 18 + T \)
$7$ \( T \)
$11$ \( 50 + T \)
$13$ \( -36 + T \)
$17$ \( 126 + T \)
$19$ \( -72 + T \)
$23$ \( -14 + T \)
$29$ \( -158 + T \)
$31$ \( -36 + T \)
$37$ \( 162 + T \)
$41$ \( -270 + T \)
$43$ \( 324 + T \)
$47$ \( -72 + T \)
$53$ \( 22 + T \)
$59$ \( 468 + T \)
$61$ \( 792 + T \)
$67$ \( -232 + T \)
$71$ \( 734 + T \)
$73$ \( 180 + T \)
$79$ \( -236 + T \)
$83$ \( 36 + T \)
$89$ \( 234 + T \)
$97$ \( 468 + T \)
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