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Properties

Label 539.2.r
Level $539$
Weight $2$
Character orbit 539.r
Rep. character $\chi_{539}(23,\cdot)$
Character field $\Q(\zeta_{21})$
Dimension $552$
Newform subspaces $2$
Sturm bound $112$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.r (of order \(21\) and degree \(12\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 49 \)
Character field: \(\Q(\zeta_{21})\)
Newform subspaces: \( 2 \)
Sturm bound: \(112\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(539, [\chi])\).

Total New Old
Modular forms 696 552 144
Cusp forms 648 552 96
Eisenstein series 48 0 48

Trace form

\( 552 q + 2 q^{3} + 44 q^{4} + 4 q^{5} - 20 q^{6} - 2 q^{7} + 12 q^{8} + 12 q^{9} + O(q^{10}) \) \( 552 q + 2 q^{3} + 44 q^{4} + 4 q^{5} - 20 q^{6} - 2 q^{7} + 12 q^{8} + 12 q^{9} - 6 q^{10} - 76 q^{12} + 16 q^{13} - 70 q^{14} + 12 q^{15} + 36 q^{16} - 6 q^{17} - 10 q^{18} - 2 q^{19} - 40 q^{20} + 2 q^{21} + 8 q^{24} + 42 q^{25} + 10 q^{26} - 16 q^{27} - 16 q^{28} - 52 q^{29} - 32 q^{30} - 78 q^{31} - 116 q^{32} + 4 q^{33} - 108 q^{34} + 6 q^{35} - 112 q^{36} - 46 q^{37} + 8 q^{38} + 4 q^{39} - 112 q^{40} + 48 q^{41} - 8 q^{42} + 20 q^{43} - 32 q^{44} - 6 q^{45} + 126 q^{46} - 62 q^{47} + 128 q^{48} + 24 q^{49} - 188 q^{50} - 32 q^{51} + 114 q^{52} + 26 q^{53} + 142 q^{54} - 16 q^{55} - 76 q^{56} - 18 q^{57} + 22 q^{58} + 64 q^{59} - 44 q^{60} - 54 q^{61} - 20 q^{62} - 116 q^{63} - 92 q^{64} - 14 q^{65} - 72 q^{66} - 24 q^{67} - 16 q^{68} - 36 q^{69} - 158 q^{70} - 62 q^{71} - 260 q^{72} - 14 q^{73} - 12 q^{74} - 218 q^{75} - 114 q^{76} + 2 q^{77} - 54 q^{78} - 12 q^{79} - 172 q^{80} - 198 q^{81} + 38 q^{82} + 66 q^{83} - 180 q^{84} - 48 q^{85} + 130 q^{86} + 6 q^{87} - 14 q^{89} + 500 q^{90} + 118 q^{91} + 22 q^{92} + 202 q^{93} + 242 q^{94} - 44 q^{95} + 254 q^{96} + 4 q^{97} - 12 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(539, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
539.2.r.a 539.r 49.g $276$ $4.304$ None \(0\) \(-1\) \(-2\) \(-2\) $\mathrm{SU}(2)[C_{21}]$
539.2.r.b 539.r 49.g $276$ $4.304$ None \(0\) \(3\) \(6\) \(0\) $\mathrm{SU}(2)[C_{21}]$

Decomposition of \(S_{2}^{\mathrm{old}}(539, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(539, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(49, [\chi])\)\(^{\oplus 2}\)