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Properties

Label 539.2.s
Level $539$
Weight $2$
Character orbit 539.s
Rep. character $\chi_{539}(19,\cdot)$
Character field $\Q(\zeta_{30})$
Dimension $288$
Newform subspaces $5$
Sturm bound $112$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 539 = 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 539.s (of order \(30\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 77 \)
Character field: \(\Q(\zeta_{30})\)
Newform subspaces: \( 5 \)
Sturm bound: \(112\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 512 352 160
Cusp forms 384 288 96
Eisenstein series 128 64 64

Trace form

\( 288 q + 5 q^{2} + 9 q^{3} - 25 q^{4} + 15 q^{5} - 80 q^{8} - 13 q^{9} + O(q^{10}) \) \( 288 q + 5 q^{2} + 9 q^{3} - 25 q^{4} + 15 q^{5} - 80 q^{8} - 13 q^{9} + q^{11} + 12 q^{12} - 48 q^{15} + 67 q^{16} - 15 q^{17} - 45 q^{18} + 15 q^{19} + 44 q^{22} - 14 q^{23} - 75 q^{24} - 31 q^{25} - 27 q^{26} - 20 q^{29} + 65 q^{30} - 9 q^{31} - 42 q^{33} - 34 q^{36} - 3 q^{37} - 33 q^{38} + 85 q^{39} - 75 q^{40} - 77 q^{44} + 84 q^{45} + 115 q^{46} - 3 q^{47} - 90 q^{50} - 95 q^{51} + 15 q^{52} + 15 q^{53} - 120 q^{57} - 71 q^{58} + 3 q^{59} - 23 q^{60} + 30 q^{61} - 36 q^{64} + 93 q^{66} - 72 q^{67} + 75 q^{68} - 12 q^{71} + 85 q^{72} + 60 q^{73} - 45 q^{74} + 57 q^{75} - 76 q^{78} + 120 q^{79} + 75 q^{80} + 21 q^{81} + 129 q^{82} - 150 q^{85} - 55 q^{86} + 9 q^{88} - 6 q^{89} + 208 q^{92} + 188 q^{93} - 105 q^{94} - 70 q^{95} - 75 q^{96} - 212 q^{99} + O(q^{100}) \)

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.s.a 539.s 77.n $16$ $4.304$ 16.0.\(\cdots\).1 \(\Q(\sqrt{-7}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{30}]$ \(q+(1+\beta _{1}-\beta _{3}+\beta _{5}-\beta _{6}-\beta _{7}-\beta _{9}+\cdots)q^{2}+\cdots\)
539.2.s.b 539.s 77.n $16$ $4.304$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{30}]$ \(q+(\beta _{4}-\beta _{9}-\beta _{11}-\beta _{14})q^{2}+\beta _{2}q^{3}+\cdots\)
539.2.s.c 539.s 77.n $16$ $4.304$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{30}]$ \(q+(-\beta _{4}-\beta _{8}+\beta _{12}+\beta _{14})q^{2}+(-\beta _{1}+\cdots)q^{3}+\cdots\)
539.2.s.d 539.s 77.n $48$ $4.304$ None \(-5\) \(9\) \(15\) \(0\) $\mathrm{SU}(2)[C_{30}]$
539.2.s.e 539.s 77.n $192$ $4.304$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{30}]$

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

\( S_{2}^{\mathrm{old}}(539, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 2}\)