Properties

Label 147.4.a.g
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: no (minimal twist has level 21)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{2} + 3q^{3} + 8q^{4} + 4q^{5} + 12q^{6} + 9q^{9} + O(q^{10}) \) \( q + 4q^{2} + 3q^{3} + 8q^{4} + 4q^{5} + 12q^{6} + 9q^{9} + 16q^{10} + 62q^{11} + 24q^{12} + 62q^{13} + 12q^{15} - 64q^{16} - 84q^{17} + 36q^{18} - 100q^{19} + 32q^{20} + 248q^{22} - 42q^{23} - 109q^{25} + 248q^{26} + 27q^{27} - 10q^{29} + 48q^{30} + 48q^{31} - 256q^{32} + 186q^{33} - 336q^{34} + 72q^{36} - 246q^{37} - 400q^{38} + 186q^{39} + 248q^{41} + 68q^{43} + 496q^{44} + 36q^{45} - 168q^{46} - 324q^{47} - 192q^{48} - 436q^{50} - 252q^{51} + 496q^{52} + 258q^{53} + 108q^{54} + 248q^{55} - 300q^{57} - 40q^{58} - 120q^{59} + 96q^{60} - 622q^{61} + 192q^{62} - 512q^{64} + 248q^{65} + 744q^{66} + 904q^{67} - 672q^{68} - 126q^{69} - 678q^{71} + 642q^{73} - 984q^{74} - 327q^{75} - 800q^{76} + 744q^{78} + 740q^{79} - 256q^{80} + 81q^{81} + 992q^{82} - 468q^{83} - 336q^{85} + 272q^{86} - 30q^{87} - 200q^{89} + 144q^{90} - 336q^{92} + 144q^{93} - 1296q^{94} - 400q^{95} - 768q^{96} + 1266q^{97} + 558q^{99} + O(q^{100}) \)

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 4.00000 12.0000 0 0 9.00000 16.0000
\(n\): 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.g 1
3.b odd 2 1 441.4.a.b 1
4.b odd 2 1 2352.4.a.l 1
7.b odd 2 1 21.4.a.b 1
7.c even 3 2 147.4.e.b 2
7.d odd 6 2 147.4.e.c 2
21.c even 2 1 63.4.a.a 1
21.g even 6 2 441.4.e.m 2
21.h odd 6 2 441.4.e.n 2
28.d even 2 1 336.4.a.h 1
35.c odd 2 1 525.4.a.b 1
35.f even 4 2 525.4.d.b 2
56.e even 2 1 1344.4.a.i 1
56.h odd 2 1 1344.4.a.w 1
84.h odd 2 1 1008.4.a.m 1
105.g even 2 1 1575.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
21.4.a.b 1 7.b odd 2 1
63.4.a.a 1 21.c even 2 1
147.4.a.g 1 1.a even 1 1 trivial
147.4.e.b 2 7.c even 3 2
147.4.e.c 2 7.d odd 6 2
336.4.a.h 1 28.d even 2 1
441.4.a.b 1 3.b odd 2 1
441.4.e.m 2 21.g even 6 2
441.4.e.n 2 21.h odd 6 2
525.4.a.b 1 35.c odd 2 1
525.4.d.b 2 35.f even 4 2
1008.4.a.m 1 84.h odd 2 1
1344.4.a.i 1 56.e even 2 1
1344.4.a.w 1 56.h odd 2 1
1575.4.a.k 1 105.g even 2 1
2352.4.a.l 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(147))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( -3 + T \)
$5$ \( -4 + T \)
$7$ \( T \)
$11$ \( -62 + T \)
$13$ \( -62 + T \)
$17$ \( 84 + T \)
$19$ \( 100 + T \)
$23$ \( 42 + T \)
$29$ \( 10 + T \)
$31$ \( -48 + T \)
$37$ \( 246 + T \)
$41$ \( -248 + T \)
$43$ \( -68 + T \)
$47$ \( 324 + T \)
$53$ \( -258 + T \)
$59$ \( 120 + T \)
$61$ \( 622 + T \)
$67$ \( -904 + T \)
$71$ \( 678 + T \)
$73$ \( -642 + T \)
$79$ \( -740 + T \)
$83$ \( 468 + T \)
$89$ \( 200 + T \)
$97$ \( -1266 + T \)
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