Properties

Label 99.1
Level 99
Weight 1
Dimension 2
Nonzero newspaces 1
Newform subspaces 1
Sturm bound 720
Trace bound 0

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Defining parameters

Level: \( N \) = \( 99\( 99 = 3^{2} \cdot 11 \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 1 \)
Sturm bound: \(720\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(99))\).

Total New Old
Modular forms 82 43 39
Cusp forms 2 2 0
Eisenstein series 80 41 39

The following table gives the dimensions of subspaces with specified projective image type.

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 2 0 0 0

Trace form

\( 2q - q^{3} - q^{4} + q^{5} - q^{9} + O(q^{10}) \) \( 2q - q^{3} - q^{4} + q^{5} - q^{9} - q^{11} - q^{12} + q^{15} - q^{16} + q^{20} - 2q^{23} + 2q^{27} + q^{31} + 2q^{33} + 2q^{36} - 2q^{37} + 2q^{44} - 2q^{45} + q^{47} + 2q^{48} - q^{49} - 2q^{53} - 2q^{55} + q^{59} - 2q^{60} + 2q^{64} + q^{67} - 2q^{69} - 2q^{71} - 2q^{80} - q^{81} + 4q^{89} - 2q^{92} + q^{93} + q^{97} - q^{99} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(99))\)

We only show spaces with odd parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
99.1.b \(\chi_{99}(89, \cdot)\) None 0 1
99.1.c \(\chi_{99}(10, \cdot)\) None 0 1
99.1.h \(\chi_{99}(43, \cdot)\) 99.1.h.a 2 2
99.1.i \(\chi_{99}(23, \cdot)\) None 0 2
99.1.k \(\chi_{99}(19, \cdot)\) None 0 4
99.1.l \(\chi_{99}(26, \cdot)\) None 0 4
99.1.n \(\chi_{99}(5, \cdot)\) None 0 8
99.1.o \(\chi_{99}(7, \cdot)\) None 0 8

Hecke characteristic polynomials

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