Properties

Label 99.2.a
Level $99$
Weight $2$
Character orbit 99.a
Rep. character $\chi_{99}(1,\cdot)$
Character field $\Q$
Dimension $4$
Newform subspaces $4$
Sturm bound $24$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 4 \)
Sturm bound: \(24\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(99))\).

Total New Old
Modular forms 16 4 12
Cusp forms 9 4 5
Eisenstein series 7 0 7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(11\)FrickeDim.
\(+\)\(+\)\(+\)\(1\)
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(1\)
Minus space\(-\)\(3\)

Trace form

\( 4 q + q^{2} - q^{4} + q^{5} - 2 q^{7} + 3 q^{8} + O(q^{10}) \) \( 4 q + q^{2} - q^{4} + q^{5} - 2 q^{7} + 3 q^{8} + 4 q^{10} - 2 q^{11} - 2 q^{13} - 8 q^{14} - 7 q^{16} + 4 q^{17} - 12 q^{19} - 4 q^{20} + q^{22} - 7 q^{23} + 17 q^{25} + 10 q^{26} - 4 q^{28} + 6 q^{29} + 7 q^{31} - 13 q^{32} - 2 q^{34} + 10 q^{35} - 3 q^{37} - 18 q^{40} + 10 q^{41} + 6 q^{43} - q^{44} + 2 q^{46} - 16 q^{47} - 7 q^{50} + 14 q^{52} + 7 q^{55} + 12 q^{56} + 6 q^{58} - q^{59} + 6 q^{61} + 22 q^{62} + 13 q^{64} - 8 q^{65} + 5 q^{67} + 2 q^{68} - 20 q^{70} + 3 q^{71} - 14 q^{73} + 12 q^{76} - 2 q^{77} - 34 q^{79} + 2 q^{80} + 34 q^{82} - 6 q^{83} - 14 q^{85} - 12 q^{86} - 9 q^{88} - 9 q^{89} - 8 q^{91} + 10 q^{92} + 8 q^{94} - q^{97} - 15 q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(99))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 11
99.2.a.a \(1\) \(0.791\) \(\Q\) None \(-1\) \(0\) \(-4\) \(-2\) \(+\) \(+\) \(q-q^{2}-q^{4}-4q^{5}-2q^{7}+3q^{8}+4q^{10}+\cdots\)
99.2.a.b \(1\) \(0.791\) \(\Q\) None \(-1\) \(0\) \(2\) \(4\) \(-\) \(+\) \(q-q^{2}-q^{4}+2q^{5}+4q^{7}+3q^{8}-2q^{10}+\cdots\)
99.2.a.c \(1\) \(0.791\) \(\Q\) None \(1\) \(0\) \(4\) \(-2\) \(+\) \(-\) \(q+q^{2}-q^{4}+4q^{5}-2q^{7}-3q^{8}+4q^{10}+\cdots\)
99.2.a.d \(1\) \(0.791\) \(\Q\) None \(2\) \(0\) \(-1\) \(-2\) \(-\) \(+\) \(q+2q^{2}+2q^{4}-q^{5}-2q^{7}-2q^{10}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(99))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(99)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)