Properties

Label 147.4.a
Level $147$
Weight $4$
Character orbit 147.a
Rep. character $\chi_{147}(1,\cdot)$
Character field $\Q$
Dimension $20$
Newform subspaces $13$
Sturm bound $74$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 13 \)
Sturm bound: \(74\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(147))\).

Total New Old
Modular forms 64 20 44
Cusp forms 48 20 28
Eisenstein series 16 0 16

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.
\(+\)\(+\)\(+\)\(5\)
\(+\)\(-\)\(-\)\(5\)
\(-\)\(+\)\(-\)\(3\)
\(-\)\(-\)\(+\)\(7\)
Plus space\(+\)\(12\)
Minus space\(-\)\(8\)

Trace form

\( 20 q + 4 q^{2} + 60 q^{4} + 16 q^{5} + 12 q^{6} + 48 q^{8} + 180 q^{9} + O(q^{10}) \) \( 20 q + 4 q^{2} + 60 q^{4} + 16 q^{5} + 12 q^{6} + 48 q^{8} + 180 q^{9} + 28 q^{10} - 40 q^{11} - 24 q^{12} + 80 q^{13} - 84 q^{15} + 332 q^{16} - 120 q^{17} + 36 q^{18} - 40 q^{19} - 172 q^{20} - 132 q^{22} - 40 q^{23} + 324 q^{24} + 728 q^{25} - 172 q^{26} + 376 q^{29} + 288 q^{30} + 168 q^{31} + 168 q^{32} + 96 q^{33} - 276 q^{34} + 540 q^{36} - 508 q^{37} - 1360 q^{38} - 756 q^{39} + 924 q^{40} + 1016 q^{41} - 828 q^{43} - 1072 q^{44} + 144 q^{45} - 520 q^{46} - 552 q^{47} - 816 q^{48} - 292 q^{50} + 456 q^{51} + 1420 q^{52} - 784 q^{53} + 108 q^{54} - 952 q^{55} - 276 q^{57} - 124 q^{58} - 792 q^{59} - 204 q^{60} - 592 q^{61} + 1824 q^{62} + 156 q^{64} - 168 q^{65} + 1896 q^{66} + 148 q^{67} - 492 q^{68} - 144 q^{69} + 608 q^{71} + 432 q^{72} + 72 q^{73} + 1196 q^{74} + 624 q^{75} + 832 q^{76} - 1692 q^{78} + 1936 q^{79} - 3484 q^{80} + 1620 q^{81} - 1492 q^{82} - 2208 q^{83} + 3904 q^{85} - 9836 q^{86} + 1032 q^{87} - 5484 q^{88} - 920 q^{89} + 252 q^{90} - 2744 q^{92} + 1788 q^{93} + 1344 q^{94} - 416 q^{95} + 204 q^{96} + 744 q^{97} - 360 q^{99} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(147))\) 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
147.4.a.a 147.a 1.a $1$ $8.673$ \(\Q\) None \(-3\) \(-3\) \(3\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}-3q^{3}+q^{4}+3q^{5}+9q^{6}+\cdots\)
147.4.a.b 147.a 1.a $1$ $8.673$ \(\Q\) None \(-3\) \(3\) \(-3\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-3q^{2}+3q^{3}+q^{4}-3q^{5}-9q^{6}+\cdots\)
147.4.a.c 147.a 1.a $1$ $8.673$ \(\Q\) None \(-3\) \(3\) \(18\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-3q^{2}+3q^{3}+q^{4}+18q^{5}-9q^{6}+\cdots\)
147.4.a.d 147.a 1.a $1$ $8.673$ \(\Q\) None \(-1\) \(-3\) \(12\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-3q^{3}-7q^{4}+12q^{5}+3q^{6}+\cdots\)
147.4.a.e 147.a 1.a $1$ $8.673$ \(\Q\) None \(-1\) \(3\) \(-12\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}+3q^{3}-7q^{4}-12q^{5}-3q^{6}+\cdots\)
147.4.a.f 147.a 1.a $1$ $8.673$ \(\Q\) None \(4\) \(-3\) \(-18\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-3q^{3}+8q^{4}-18q^{5}-12q^{6}+\cdots\)
147.4.a.g 147.a 1.a $1$ $8.673$ \(\Q\) None \(4\) \(3\) \(4\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+3q^{3}+8q^{4}+4q^{5}+12q^{6}+\cdots\)
147.4.a.h 147.a 1.a $1$ $8.673$ \(\Q\) None \(4\) \(3\) \(18\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+3q^{3}+8q^{4}+18q^{5}+12q^{6}+\cdots\)
147.4.a.i 147.a 1.a $2$ $8.673$ \(\Q(\sqrt{57}) \) None \(-3\) \(-6\) \(-6\) \(0\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}-3q^{3}+(7+3\beta )q^{4}+\cdots\)
147.4.a.j 147.a 1.a $2$ $8.673$ \(\Q(\sqrt{2}) \) None \(2\) \(-6\) \(20\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-3q^{3}+(-5+2\beta )q^{4}+\cdots\)
147.4.a.k 147.a 1.a $2$ $8.673$ \(\Q(\sqrt{2}) \) None \(2\) \(6\) \(-20\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+3q^{3}+(-5+2\beta )q^{4}+\cdots\)
147.4.a.l 147.a 1.a $3$ $8.673$ 3.3.57516.1 None \(1\) \(-9\) \(11\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(8+\beta _{1}+\beta _{2})q^{4}+\cdots\)
147.4.a.m 147.a 1.a $3$ $8.673$ 3.3.57516.1 None \(1\) \(9\) \(-11\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(8+\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(147))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(147)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)