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Properties

Label 539.2.j
Level $539$
Weight $2$
Character orbit 539.j
Rep. character $\chi_{539}(78,\cdot)$
Character field $\Q(\zeta_{7})$
Dimension $288$
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.j (of order \(7\) and degree \(6\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 49 \)
Character field: \(\Q(\zeta_{7})\)
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 348 288 60
Cusp forms 324 288 36
Eisenstein series 24 0 24

Trace form

\( 288 q - 4 q^{3} - 52 q^{4} - 4 q^{5} + 20 q^{6} - 6 q^{7} - 12 q^{8} - 32 q^{9} + O(q^{10}) \) \( 288 q - 4 q^{3} - 52 q^{4} - 4 q^{5} + 20 q^{6} - 6 q^{7} - 12 q^{8} - 32 q^{9} - 12 q^{10} + 38 q^{12} - 20 q^{13} + 46 q^{14} - 12 q^{15} - 68 q^{16} - 12 q^{17} - 20 q^{18} - 20 q^{19} - 32 q^{20} - 34 q^{21} - 44 q^{24} - 56 q^{25} - 40 q^{26} - 52 q^{27} - 52 q^{28} + 52 q^{29} - 64 q^{30} + 32 q^{31} + 20 q^{32} - 4 q^{33} + 84 q^{34} - 42 q^{35} - 28 q^{36} + 62 q^{37} - 80 q^{38} + 36 q^{39} + 16 q^{40} + 12 q^{41} + 44 q^{42} - 48 q^{43} + 20 q^{44} - 48 q^{45} + 26 q^{47} + 76 q^{48} - 44 q^{49} + 188 q^{50} - 64 q^{51} + 64 q^{52} + 10 q^{53} + 8 q^{54} - 8 q^{55} - 38 q^{56} - 10 q^{57} - 40 q^{58} - 40 q^{59} - 88 q^{60} + 76 q^{61} - 88 q^{62} + 62 q^{63} - 48 q^{64} - 28 q^{65} + 42 q^{66} - 76 q^{67} - 116 q^{68} - 84 q^{69} + 8 q^{70} + 62 q^{71} + 110 q^{72} - 80 q^{73} - 24 q^{74} + 108 q^{75} + 50 q^{76} - 2 q^{77} + 54 q^{78} - 52 q^{79} + 88 q^{80} + 52 q^{81} + 58 q^{82} + 54 q^{83} - 32 q^{84} + 48 q^{85} - 34 q^{86} - 60 q^{87} - 100 q^{89} + 70 q^{90} - 32 q^{91} - 22 q^{92} + 166 q^{93} + 34 q^{94} - 88 q^{95} + 214 q^{96} - 20 q^{97} - 126 q^{98} + O(q^{100}) \)

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.j.a 539.j 49.e $144$ $4.304$ None \(0\) \(-4\) \(-6\) \(-4\) $\mathrm{SU}(2)[C_{7}]$
539.2.j.b 539.j 49.e $144$ $4.304$ None \(0\) \(0\) \(2\) \(-2\) $\mathrm{SU}(2)[C_{7}]$

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

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