Properties

Label 111.1
Level 111
Weight 1
Dimension 7
Nonzero newspaces 3
Newform subspaces 4
Sturm bound 912
Trace bound 4

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Defining parameters

Level: \( N \) = \( 111\( 111 = 3 \cdot 37 \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 4 \)
Sturm bound: \(912\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(111))\).

Total New Old
Modular forms 79 41 38
Cusp forms 7 7 0
Eisenstein series 72 34 38

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 7 0 0 0

Trace form

\( 7q - 3q^{3} + q^{4} - 2q^{7} + q^{9} + O(q^{10}) \) \( 7q - 3q^{3} + q^{4} - 2q^{7} + q^{9} - 4q^{10} - 3q^{12} - 2q^{13} - 3q^{16} - 2q^{19} - 2q^{21} + q^{25} + 3q^{27} + 4q^{28} + 4q^{30} - 2q^{31} + 4q^{34} + q^{36} + 3q^{37} + 4q^{39} - 2q^{43} - 4q^{46} + 7q^{48} + q^{49} - 2q^{52} - 2q^{57} + 4q^{58} - 2q^{61} - 2q^{63} - 3q^{64} - 2q^{67} - 2q^{73} - 3q^{75} - 2q^{76} - 2q^{79} + q^{81} - 2q^{84} - 4q^{85} - 4q^{90} + 2q^{91} - 2q^{93} - 2q^{97} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(111))\)

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
111.1.b \(\chi_{111}(38, \cdot)\) None 0 1
111.1.d \(\chi_{111}(110, \cdot)\) 111.1.d.a 1 1
111.1.d.b 2
111.1.f \(\chi_{111}(31, \cdot)\) None 0 2
111.1.h \(\chi_{111}(11, \cdot)\) 111.1.h.a 2 2
111.1.i \(\chi_{111}(26, \cdot)\) 111.1.i.a 2 2
111.1.l \(\chi_{111}(82, \cdot)\) None 0 4
111.1.n \(\chi_{111}(41, \cdot)\) None 0 6
111.1.p \(\chi_{111}(44, \cdot)\) None 0 6
111.1.r \(\chi_{111}(13, \cdot)\) None 0 12

Hecke characteristic polynomials

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