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Properties

Label 693.2.by
Level $693$
Weight $2$
Character orbit 693.by
Rep. character $\chi_{693}(37,\cdot)$
Character field $\Q(\zeta_{15})$
Dimension $304$
Newform subspaces $5$
Sturm bound $192$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 693.by (of order \(15\) and degree \(8\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 77 \)
Character field: \(\Q(\zeta_{15})\)
Newform subspaces: \( 5 \)
Sturm bound: \(192\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

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

Total New Old
Modular forms 832 336 496
Cusp forms 704 304 400
Eisenstein series 128 32 96

Trace form

\( 304 q + q^{2} + 31 q^{4} + 5 q^{5} + q^{7} + 12 q^{8} + O(q^{10}) \) \( 304 q + q^{2} + 31 q^{4} + 5 q^{5} + q^{7} + 12 q^{8} - 4 q^{10} + 7 q^{11} - 4 q^{13} + 20 q^{14} + 41 q^{16} + 11 q^{17} - 7 q^{19} - 8 q^{20} - 72 q^{22} + 6 q^{23} + 5 q^{25} + 7 q^{26} - 8 q^{28} + 32 q^{29} + 3 q^{31} + 4 q^{32} - 88 q^{34} - 17 q^{35} + 19 q^{37} + 5 q^{38} + 41 q^{40} - 14 q^{41} - 28 q^{43} + 20 q^{44} + 40 q^{46} + 25 q^{47} - 57 q^{49} + 50 q^{50} - 27 q^{52} + 31 q^{53} - 4 q^{55} - 2 q^{58} - 11 q^{59} - 30 q^{61} - 16 q^{62} - 140 q^{64} - 6 q^{65} + 4 q^{67} + 77 q^{68} - 67 q^{70} - 132 q^{71} - 54 q^{73} + 51 q^{74} - 336 q^{76} - 36 q^{77} - 18 q^{79} + 91 q^{80} - 35 q^{82} - 100 q^{83} - 10 q^{85} + 34 q^{86} - 37 q^{88} - 38 q^{89} - 68 q^{91} - 206 q^{92} - 147 q^{94} - 12 q^{95} + 92 q^{97} + 120 q^{98} + O(q^{100}) \)

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.by.a 693.by 77.m $8$ $5.534$ \(\Q(\zeta_{15})\) None \(2\) \(0\) \(5\) \(-5\) $\mathrm{SU}(2)[C_{15}]$ \(q+(1-\zeta_{15}^{4}+\zeta_{15}^{5}-\zeta_{15}^{7})q^{2}+(-\zeta_{15}+\cdots)q^{4}+\cdots\)
693.2.by.b 693.by 77.m $40$ $5.534$ None \(3\) \(0\) \(-4\) \(-2\) $\mathrm{SU}(2)[C_{15}]$
693.2.by.c 693.by 77.m $64$ $5.534$ None \(-4\) \(0\) \(4\) \(-1\) $\mathrm{SU}(2)[C_{15}]$
693.2.by.d 693.by 77.m $64$ $5.534$ None \(0\) \(0\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{15}]$
693.2.by.e 693.by 77.m $128$ $5.534$ None \(0\) \(0\) \(0\) \(10\) $\mathrm{SU}(2)[C_{15}]$

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

\( S_{2}^{\mathrm{old}}(693, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(77, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(231, [\chi])\)\(^{\oplus 2}\)