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

\( 4 q + q^{2} - q^{4} + q^{5} - 2 q^{7} + 3 q^{8} + 4 q^{10} - 2 q^{11} - 2 q^{13} - 8 q^{14} - 7 q^{16} + 4 q^{17} - 12 q^{19} - 4 q^{20} + q^{22} - 7 q^{23} + 17 q^{25} + 10 q^{26} - 4 q^{28} + 6 q^{29}+ \cdots - 15 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 11
99.2.a.a 99.a 1.a $1$ $0.791$ \(\Q\) None 99.2.a.a \(-1\) \(0\) \(-4\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-4q^{5}-2q^{7}+3q^{8}+4q^{10}+\cdots\)
99.2.a.b 99.a 1.a $1$ $0.791$ \(\Q\) None 33.2.a.a \(-1\) \(0\) \(2\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+2q^{5}+4q^{7}+3q^{8}-2q^{10}+\cdots\)
99.2.a.c 99.a 1.a $1$ $0.791$ \(\Q\) None 99.2.a.a \(1\) \(0\) \(4\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+4q^{5}-2q^{7}-3q^{8}+4q^{10}+\cdots\)
99.2.a.d 99.a 1.a $1$ $0.791$ \(\Q\) None 11.2.a.a \(2\) \(0\) \(-1\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{4}-q^{5}-2q^{7}-2q^{10}+\cdots\)

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

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