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Properties

Label 693.2.bz
Level $693$
Weight $2$
Character orbit 693.bz
Rep. character $\chi_{693}(148,\cdot)$
Character field $\Q(\zeta_{15})$
Dimension $576$
Newform subspaces $2$
Sturm bound $192$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.bz (of order \(15\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 99 \)
Character field: \(\Q(\zeta_{15})\)
Newform subspaces: \( 2 \)
Sturm bound: \(192\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(693, [\chi])\).

Total New Old
Modular forms 800 576 224
Cusp forms 736 576 160
Eisenstein series 64 0 64

Trace form

\( 576 q + 4 q^{2} + 6 q^{3} + 72 q^{4} - 2 q^{5} - 6 q^{6} - 24 q^{8} + 26 q^{9} + O(q^{10}) \) \( 576 q + 4 q^{2} + 6 q^{3} + 72 q^{4} - 2 q^{5} - 6 q^{6} - 24 q^{8} + 26 q^{9} - 4 q^{11} - 56 q^{12} - 2 q^{15} + 72 q^{16} + 24 q^{17} + 12 q^{18} - 36 q^{19} - 8 q^{20} + 4 q^{21} - 6 q^{22} - 16 q^{23} + 14 q^{24} + 66 q^{25} + 64 q^{26} - 48 q^{27} + 50 q^{29} - 100 q^{30} - 6 q^{31} - 112 q^{32} - 34 q^{33} - 12 q^{34} - 32 q^{35} - 90 q^{36} + 12 q^{37} + 32 q^{38} + 34 q^{39} - 28 q^{41} - 12 q^{43} - 96 q^{44} - 136 q^{45} + 16 q^{47} - 12 q^{48} + 72 q^{49} + 28 q^{50} - 82 q^{51} + 40 q^{53} - 148 q^{54} + 76 q^{57} + 40 q^{59} + 104 q^{60} - 180 q^{62} - 144 q^{64} - 156 q^{65} - 164 q^{66} + 12 q^{67} + 50 q^{68} - 6 q^{69} + 52 q^{71} - 26 q^{72} + 72 q^{73} + 8 q^{74} - 12 q^{75} - 36 q^{76} + 24 q^{78} - 12 q^{79} - 296 q^{80} - 6 q^{81} - 36 q^{82} - 34 q^{83} + 48 q^{84} - 12 q^{85} + 138 q^{86} + 20 q^{87} - 114 q^{88} + 152 q^{89} - 36 q^{90} - 36 q^{91} - 60 q^{92} + 10 q^{93} + 48 q^{94} + 88 q^{95} + 140 q^{96} + 48 q^{97} + 32 q^{98} + 218 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(693, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
693.2.bz.a 693.bz 99.m $288$ $5.534$ None \(2\) \(-3\) \(-9\) \(-36\) $\mathrm{SU}(2)[C_{15}]$
693.2.bz.b 693.bz 99.m $288$ $5.534$ None \(2\) \(9\) \(7\) \(36\) $\mathrm{SU}(2)[C_{15}]$

Decomposition of \(S_{2}^{\mathrm{old}}(693, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(693, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(99, [\chi])\)\(^{\oplus 2}\)