Properties

Label 99.2.f
Level $99$
Weight $2$
Character orbit 99.f
Rep. character $\chi_{99}(37,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $16$
Newform subspaces $3$
Sturm bound $24$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.f (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 11 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 3 \)
Sturm bound: \(24\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(99, [\chi])\).

Total New Old
Modular forms 64 24 40
Cusp forms 32 16 16
Eisenstein series 32 8 24

Trace form

\( 16 q + 4 q^{2} - 4 q^{4} + 4 q^{5} - 8 q^{7} - 8 q^{8} + O(q^{10}) \) \( 16 q + 4 q^{2} - 4 q^{4} + 4 q^{5} - 8 q^{7} - 8 q^{8} - 24 q^{10} + 2 q^{11} - 8 q^{13} - 2 q^{14} + 12 q^{16} - 14 q^{17} - 8 q^{19} - 6 q^{20} + 24 q^{22} + 12 q^{23} + 8 q^{25} - 10 q^{26} + 24 q^{28} + 4 q^{29} + 8 q^{31} + 48 q^{32} - 8 q^{34} - 12 q^{37} + 10 q^{38} - 12 q^{40} - 20 q^{41} - 16 q^{43} - 34 q^{44} - 32 q^{46} - 14 q^{47} - 40 q^{49} - 18 q^{50} + 16 q^{52} - 10 q^{53} + 8 q^{55} - 12 q^{56} + 4 q^{58} + 26 q^{59} + 24 q^{61} + 28 q^{62} + 52 q^{64} + 28 q^{65} + 28 q^{68} + 20 q^{70} + 12 q^{71} + 44 q^{73} + 30 q^{74} + 8 q^{76} + 2 q^{77} + 24 q^{79} - 22 q^{80} - 4 q^{82} - 34 q^{83} + 44 q^{85} - 8 q^{86} - 76 q^{88} + 4 q^{89} + 8 q^{91} - 10 q^{92} - 8 q^{94} - 30 q^{95} - 84 q^{97} - 40 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(99, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
99.2.f.a 99.f 11.c $4$ $0.791$ \(\Q(\zeta_{10})\) None \(1\) \(0\) \(3\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+2\zeta_{10}-2\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
99.2.f.b 99.f 11.c $4$ $0.791$ \(\Q(\zeta_{10})\) None \(3\) \(0\) \(1\) \(-3\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\zeta_{10}^{3})q^{2}+(1-\zeta_{10})q^{4}+(1-2\zeta_{10}+\cdots)q^{5}+\cdots\)
99.2.f.c 99.f 11.c $8$ $0.791$ 8.0.484000000.9 None \(0\) \(0\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{1}q^{2}+(1+2\beta _{2}+\beta _{3})q^{4}+(\beta _{4}+\beta _{7})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(99, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(99, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(33, [\chi])\)\(^{\oplus 2}\)