Properties

Label 539.2.q
Level $539$
Weight $2$
Character orbit 539.q
Rep. character $\chi_{539}(214,\cdot)$
Character field $\Q(\zeta_{15})$
Dimension $288$
Newform subspaces $9$
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.q (of order \(15\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 77 \)
Character field: \(\Q(\zeta_{15})\)
Newform subspaces: \( 9 \)
Sturm bound: \(112\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(3\)

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

\( 288 q + 7 q^{2} + 3 q^{3} + 39 q^{4} + q^{5} + 12 q^{6} + 19 q^{9} + O(q^{10}) \) \( 288 q + 7 q^{2} + 3 q^{3} + 39 q^{4} + q^{5} + 12 q^{6} + 19 q^{9} - 4 q^{10} + 9 q^{11} + 16 q^{12} + 4 q^{13} + 27 q^{16} + 11 q^{17} - 57 q^{18} + 7 q^{19} + 40 q^{20} - 108 q^{22} + 2 q^{23} + 7 q^{24} - 15 q^{25} + 15 q^{26} + 6 q^{27} - 84 q^{29} + 41 q^{30} + 9 q^{31} - 8 q^{32} + 26 q^{33} - 72 q^{34} - 134 q^{36} - 15 q^{37} - 3 q^{38} - 53 q^{39} - 5 q^{40} - 62 q^{41} - 60 q^{43} - 45 q^{44} + 16 q^{45} + 17 q^{46} - 19 q^{47} - 146 q^{48} - 70 q^{50} - 55 q^{51} + 3 q^{52} - 33 q^{53} - 44 q^{54} - 4 q^{55} - 72 q^{57} + 57 q^{58} - 3 q^{59} + 137 q^{60} - 18 q^{61} + 80 q^{62} - 204 q^{64} + 22 q^{65} + 47 q^{66} + 104 q^{67} + 21 q^{68} + 126 q^{69} - 36 q^{71} - 31 q^{72} - 26 q^{73} + 47 q^{74} + 11 q^{75} + 144 q^{76} + 348 q^{78} - 48 q^{79} + 3 q^{80} + 65 q^{81} - 13 q^{82} + 28 q^{83} + 118 q^{85} + 145 q^{86} - 66 q^{87} - 101 q^{88} + 34 q^{89} + 18 q^{90} - 168 q^{92} - 68 q^{93} - 45 q^{94} - 98 q^{95} + 55 q^{96} + 68 q^{97} + 104 q^{99} + O(q^{100}) \)

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.q.a 539.q 77.m $8$ $4.304$ \(\Q(\zeta_{15})\) None \(-2\) \(-1\) \(5\) \(0\) $\mathrm{SU}(2)[C_{15}]$ \(q+(-1+\zeta_{15}-\zeta_{15}^{5}+\zeta_{15}^{7})q^{2}+\cdots\)
539.2.q.b 539.q 77.m $16$ $4.304$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(1\) \(-4\) \(3\) \(0\) $\mathrm{SU}(2)[C_{15}]$ \(q+(\beta _{2}+\beta _{4}+\beta _{11}-\beta _{12})q^{2}+(\beta _{5}-\beta _{7}+\cdots)q^{3}+\cdots\)
539.2.q.c 539.q 77.m $16$ $4.304$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(1\) \(4\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{15}]$ \(q+(\beta _{2}+\beta _{4}+\beta _{11}-\beta _{12})q^{2}+(-\beta _{5}+\cdots)q^{3}+\cdots\)
539.2.q.d 539.q 77.m $16$ $4.304$ 16.0.\(\cdots\).1 \(\Q(\sqrt{-7}) \) \(2\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{15}]$ \(q+(\beta _{1}-\beta _{4}-\beta _{5}+\beta _{6}+\beta _{7}-\beta _{9}-\beta _{10}+\cdots)q^{2}+\cdots\)
539.2.q.e 539.q 77.m $32$ $4.304$ None \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{15}]$
539.2.q.f 539.q 77.m $32$ $4.304$ None \(3\) \(-2\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{15}]$
539.2.q.g 539.q 77.m $32$ $4.304$ None \(3\) \(2\) \(5\) \(0\) $\mathrm{SU}(2)[C_{15}]$
539.2.q.h 539.q 77.m $40$ $4.304$ None \(-3\) \(4\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{15}]$
539.2.q.i 539.q 77.m $96$ $4.304$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{15}]$

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

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