Properties

Label 147.2.a.a
Level 147
Weight 2
Character orbit 147.a
Self dual Yes
Analytic conductor 1.174
Analytic rank 0
Dimension 1
CM No
Inner twists 1

Related objects

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Newspace parameters

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

Newform invariants

Self dual: Yes
Analytic conductor: \(1.17380090971\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} + 3q^{8} + q^{9} + O(q^{10}) \) \( q - q^{2} - q^{3} - q^{4} + 2q^{5} + q^{6} + 3q^{8} + q^{9} - 2q^{10} + 4q^{11} + q^{12} + 2q^{13} - 2q^{15} - q^{16} + 6q^{17} - q^{18} - 4q^{19} - 2q^{20} - 4q^{22} - 3q^{24} - q^{25} - 2q^{26} - q^{27} - 2q^{29} + 2q^{30} - 5q^{32} - 4q^{33} - 6q^{34} - q^{36} + 6q^{37} + 4q^{38} - 2q^{39} + 6q^{40} - 2q^{41} - 4q^{43} - 4q^{44} + 2q^{45} + q^{48} + q^{50} - 6q^{51} - 2q^{52} + 6q^{53} + q^{54} + 8q^{55} + 4q^{57} + 2q^{58} - 12q^{59} + 2q^{60} + 2q^{61} + 7q^{64} + 4q^{65} + 4q^{66} + 4q^{67} - 6q^{68} + 3q^{72} + 6q^{73} - 6q^{74} + q^{75} + 4q^{76} + 2q^{78} - 16q^{79} - 2q^{80} + q^{81} + 2q^{82} + 12q^{83} + 12q^{85} + 4q^{86} + 2q^{87} + 12q^{88} + 14q^{89} - 2q^{90} - 8q^{95} + 5q^{96} - 18q^{97} + 4q^{99} + O(q^{100}) \)

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 −1.00000 −1.00000 2.00000 1.00000 0 3.00000 1.00000 −2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

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

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(147))\):

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