Properties

Label 147.2.a
Level $147$
Weight $2$
Character orbit 147.a
Rep. character $\chi_{147}(1,\cdot)$
Character field $\Q$
Dimension $7$
Newform subspaces $5$
Sturm bound $37$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(37\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(147))\).

Total New Old
Modular forms 26 7 19
Cusp forms 11 7 4
Eisenstein series 15 0 15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(7\)FrickeDim.
\(+\)\(+\)\(+\)\(2\)
\(+\)\(-\)\(-\)\(2\)
\(-\)\(+\)\(-\)\(3\)
Plus space\(+\)\(2\)
Minus space\(-\)\(5\)

Trace form

\( 7 q - q^{2} - q^{3} + 7 q^{4} + 2 q^{5} + q^{6} - 9 q^{8} + 7 q^{9} + O(q^{10}) \) \( 7 q - q^{2} - q^{3} + 7 q^{4} + 2 q^{5} + q^{6} - 9 q^{8} + 7 q^{9} - 2 q^{10} - 8 q^{11} + q^{12} + 2 q^{13} + 2 q^{15} + 3 q^{16} + 6 q^{17} - q^{18} - 4 q^{19} - 2 q^{20} - 4 q^{22} - 8 q^{23} - 3 q^{24} + q^{25} - 2 q^{26} - q^{27} - 10 q^{29} - 6 q^{30} - 9 q^{32} - 4 q^{33} - 6 q^{34} + 7 q^{36} - 4 q^{37} + 4 q^{38} + 16 q^{39} + 6 q^{40} - 2 q^{41} + 6 q^{43} - 20 q^{44} + 2 q^{45} - 24 q^{46} + q^{48} + 25 q^{50} + 2 q^{51} - 2 q^{52} + 22 q^{53} + q^{54} + 8 q^{55} + 6 q^{57} + 18 q^{58} - 12 q^{59} - 14 q^{60} + 2 q^{61} - 37 q^{64} + 24 q^{65} + 4 q^{66} - 6 q^{67} - 6 q^{68} - 20 q^{71} - 9 q^{72} + 6 q^{73} + 22 q^{74} + q^{75} + 4 q^{76} - 18 q^{78} + 14 q^{79} - 2 q^{80} + 7 q^{81} + 2 q^{82} + 12 q^{83} + 4 q^{85} + 56 q^{86} + 2 q^{87} + 36 q^{88} + 14 q^{89} - 2 q^{90} + 56 q^{92} + 2 q^{93} - 28 q^{95} + 5 q^{96} - 18 q^{97} - 8 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(147))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 7
147.2.a.a 147.a 1.a $1$ $1.174$ \(\Q\) None \(-1\) \(-1\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+2q^{5}+q^{6}+3q^{8}+\cdots\)
147.2.a.b 147.a 1.a $1$ $1.174$ \(\Q\) None \(2\) \(-1\) \(2\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+2q^{4}+2q^{5}-2q^{6}+\cdots\)
147.2.a.c 147.a 1.a $1$ $1.174$ \(\Q\) None \(2\) \(1\) \(-2\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+q^{3}+2q^{4}-2q^{5}+2q^{6}+\cdots\)
147.2.a.d 147.a 1.a $2$ $1.174$ \(\Q(\sqrt{2}) \) None \(-2\) \(-2\) \(-4\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(-2+\cdots)q^{5}+\cdots\)
147.2.a.e 147.a 1.a $2$ $1.174$ \(\Q(\sqrt{2}) \) None \(-2\) \(2\) \(4\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(2+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(147))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(147)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)