Properties

Label 111.1.d
Level 111
Weight 1
Character orbit d
Rep. character \(\chi_{111}(110,\cdot)\)
Character field \(\Q\)
Dimension 3
Newform subspaces 2
Sturm bound 12
Trace bound 1

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Defining parameters

Level: \( N \) \(=\) \( 111 = 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 111.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 111 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(12\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(111, [\chi])\).

Total New Old
Modular forms 5 5 0
Cusp forms 3 3 0
Eisenstein series 2 2 0

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 3 0 0 0

Trace form

\( 3q - q^{3} + q^{4} - 2q^{7} + 3q^{9} + O(q^{10}) \) \( 3q - q^{3} + q^{4} - 2q^{7} + 3q^{9} - 4q^{10} - 3q^{12} - q^{16} - 2q^{21} + q^{25} - q^{27} + 2q^{28} + 4q^{30} + 4q^{34} + q^{36} - q^{37} - 4q^{46} + 3q^{48} + q^{49} + 4q^{58} - 2q^{63} - 3q^{64} - 2q^{67} - 2q^{73} - 3q^{75} + 3q^{81} + 2q^{84} - 4q^{85} - 4q^{90} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(111, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
111.1.d.a \(1\) \(0.055\) \(\Q\) \(D_{2}\) \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-111}) \) \(\Q(\sqrt{37}) \) \(0\) \(1\) \(0\) \(-2\) \(q+q^{3}-q^{4}-2q^{7}+q^{9}-q^{12}+q^{16}+\cdots\)
111.1.d.b \(2\) \(0.055\) \(\Q(\sqrt{2}) \) \(D_{4}\) \(\Q(\sqrt{-111}) \) None \(0\) \(-2\) \(0\) \(0\) \(q-\beta q^{2}-q^{3}+q^{4}+\beta q^{5}+\beta q^{6}+q^{9}+\cdots\)

Hecke characteristic polynomials

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