Properties

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

Related objects

Downloads

Learn more about

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

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

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + T + 2 T^{2} \))(\( 1 + T + 2 T^{2} \))(\( 1 - T + 2 T^{2} \))(\( 1 - 2 T + 2 T^{2} \))
$3$ (\( \))(\( \))(\( \))(\( \))
$5$ (\( 1 + 4 T + 5 T^{2} \))(\( 1 - 2 T + 5 T^{2} \))(\( 1 - 4 T + 5 T^{2} \))(\( 1 + T + 5 T^{2} \))
$7$ (\( 1 + 2 T + 7 T^{2} \))(\( 1 - 4 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))(\( 1 + 2 T + 7 T^{2} \))
$11$ (\( 1 + T \))(\( 1 + T \))(\( 1 - T \))(\( 1 + T \))
$13$ (\( 1 + 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 + 2 T + 13 T^{2} \))(\( 1 - 4 T + 13 T^{2} \))
$17$ (\( 1 - 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))(\( 1 + 2 T + 17 T^{2} \))(\( 1 - 2 T + 17 T^{2} \))
$19$ (\( 1 + 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))(\( 1 + 6 T + 19 T^{2} \))(\( 1 + 19 T^{2} \))
$23$ (\( 1 - 4 T + 23 T^{2} \))(\( 1 + 8 T + 23 T^{2} \))(\( 1 + 4 T + 23 T^{2} \))(\( 1 - T + 23 T^{2} \))
$29$ (\( 1 + 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 - 6 T + 29 T^{2} \))(\( 1 + 29 T^{2} \))
$31$ (\( 1 - 4 T + 31 T^{2} \))(\( 1 + 8 T + 31 T^{2} \))(\( 1 - 4 T + 31 T^{2} \))(\( 1 - 7 T + 31 T^{2} \))
$37$ (\( 1 + 6 T + 37 T^{2} \))(\( 1 - 6 T + 37 T^{2} \))(\( 1 + 6 T + 37 T^{2} \))(\( 1 - 3 T + 37 T^{2} \))
$41$ (\( 1 + 10 T + 41 T^{2} \))(\( 1 - 2 T + 41 T^{2} \))(\( 1 - 10 T + 41 T^{2} \))(\( 1 - 8 T + 41 T^{2} \))
$43$ (\( 1 - 6 T + 43 T^{2} \))(\( 1 + 43 T^{2} \))(\( 1 - 6 T + 43 T^{2} \))(\( 1 + 6 T + 43 T^{2} \))
$47$ (\( 1 + 8 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))(\( 1 - 8 T + 47 T^{2} \))(\( 1 + 8 T + 47 T^{2} \))
$53$ (\( 1 + 53 T^{2} \))(\( 1 + 6 T + 53 T^{2} \))(\( 1 + 53 T^{2} \))(\( 1 - 6 T + 53 T^{2} \))
$59$ (\( 1 - 4 T + 59 T^{2} \))(\( 1 - 4 T + 59 T^{2} \))(\( 1 + 4 T + 59 T^{2} \))(\( 1 + 5 T + 59 T^{2} \))
$61$ (\( 1 + 6 T + 61 T^{2} \))(\( 1 - 6 T + 61 T^{2} \))(\( 1 + 6 T + 61 T^{2} \))(\( 1 - 12 T + 61 T^{2} \))
$67$ (\( 1 - 8 T + 67 T^{2} \))(\( 1 + 4 T + 67 T^{2} \))(\( 1 - 8 T + 67 T^{2} \))(\( 1 + 7 T + 67 T^{2} \))
$71$ (\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 + 71 T^{2} \))(\( 1 - 3 T + 71 T^{2} \))
$73$ (\( 1 + 2 T + 73 T^{2} \))(\( 1 + 14 T + 73 T^{2} \))(\( 1 + 2 T + 73 T^{2} \))(\( 1 - 4 T + 73 T^{2} \))
$79$ (\( 1 + 10 T + 79 T^{2} \))(\( 1 + 4 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))(\( 1 + 10 T + 79 T^{2} \))
$83$ (\( 1 - 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 + 12 T + 83 T^{2} \))(\( 1 - 6 T + 83 T^{2} \))
$89$ (\( 1 + 89 T^{2} \))(\( 1 - 6 T + 89 T^{2} \))(\( 1 + 89 T^{2} \))(\( 1 + 15 T + 89 T^{2} \))
$97$ (\( 1 - 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 - 2 T + 97 T^{2} \))(\( 1 + 7 T + 97 T^{2} \))
show more
show less