Properties

Label 147.3.b
Level $147$
Weight $3$
Character orbit 147.b
Rep. character $\chi_{147}(50,\cdot)$
Character field $\Q$
Dimension $22$
Newform subspaces $7$
Sturm bound $56$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 147.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 7 \)
Sturm bound: \(56\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(147, [\chi])\).

Total New Old
Modular forms 46 32 14
Cusp forms 30 22 8
Eisenstein series 16 10 6

Trace form

\( 22 q + 2 q^{3} - 28 q^{4} - 14 q^{6} + 22 q^{9} + O(q^{10}) \) \( 22 q + 2 q^{3} - 28 q^{4} - 14 q^{6} + 22 q^{9} - 28 q^{10} + 22 q^{12} + 36 q^{13} - 52 q^{15} - 12 q^{16} - 16 q^{18} - 12 q^{19} + 92 q^{22} + 126 q^{24} - 14 q^{25} - 10 q^{27} - 48 q^{30} - 136 q^{31} - 28 q^{33} - 188 q^{36} + 30 q^{37} + 46 q^{39} - 84 q^{40} + 86 q^{43} + 140 q^{45} + 248 q^{46} - 38 q^{48} + 104 q^{51} - 164 q^{52} + 154 q^{54} - 56 q^{55} - 114 q^{57} - 196 q^{58} + 52 q^{60} + 156 q^{61} + 204 q^{64} + 28 q^{66} - 234 q^{67} - 168 q^{69} - 48 q^{72} + 32 q^{73} - 146 q^{75} + 316 q^{76} - 48 q^{78} + 170 q^{79} + 142 q^{81} - 392 q^{82} - 160 q^{85} - 28 q^{87} - 60 q^{88} + 112 q^{90} + 38 q^{93} + 336 q^{94} + 98 q^{96} + 8 q^{97} + 88 q^{99} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(147, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
147.3.b.a 147.b 3.b $1$ $4.005$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-3q^{3}+4q^{4}+9q^{9}-12q^{12}+23q^{13}+\cdots\)
147.3.b.b 147.b 3.b $1$ $4.005$ \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+3q^{3}+4q^{4}+9q^{9}+12q^{12}-23q^{13}+\cdots\)
147.3.b.c 147.b 3.b $2$ $4.005$ \(\Q(\sqrt{-5}) \) None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}+(-2-\beta )q^{3}-q^{4}+\beta q^{5}+\cdots\)
147.3.b.d 147.b 3.b $2$ $4.005$ \(\Q(\sqrt{-5}) \) None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}+(2+\beta )q^{3}-q^{4}-\beta q^{5}+(-5+\cdots)q^{6}+\cdots\)
147.3.b.e 147.b 3.b $4$ $4.005$ \(\Q(\sqrt{-5}, \sqrt{-6})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+\beta _{1}q^{3}-q^{4}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)
147.3.b.f 147.b 3.b $4$ $4.005$ 4.0.65856.1 None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2}-\beta _{3})q^{3}+(-3+\cdots)q^{4}+\cdots\)
147.3.b.g 147.b 3.b $8$ $4.005$ 8.0.\(\cdots\).7 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+(-\beta _{5}-\beta _{7})q^{3}+(-2+\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(147, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(147, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(21, [\chi])\)\(^{\oplus 2}\)