Properties

Label 87.1.d
Level 87
Weight 1
Character orbit d
Rep. character \(\chi_{87}(86,\cdot)\)
Character field \(\Q\)
Dimension 2
Newform subspaces 2
Sturm bound 10
Trace bound 2

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Defining parameters

Level: \( N \) \(=\) \( 87 = 3 \cdot 29 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 87.d (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 87 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(10\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(87, [\chi])\).

Total New Old
Modular forms 4 4 0
Cusp forms 2 2 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 2 0 0 0

Trace form

\( 2q - 2q^{6} - 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{6} - 2q^{7} + 2q^{9} - 2q^{13} - 2q^{16} + 2q^{22} + 2q^{24} + 2q^{25} - 2q^{33} + 2q^{34} + 2q^{42} - 2q^{51} - 2q^{54} - 2q^{58} - 2q^{63} + 2q^{64} - 2q^{67} + 2q^{78} + 2q^{81} - 4q^{82} + 2q^{87} - 2q^{88} + 2q^{91} + 2q^{94} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(87, [\chi])\) into newform subspaces

Label Dim. \(A\) Field Image CM RM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
87.1.d.a \(1\) \(0.043\) \(\Q\) \(D_{3}\) \(\Q(\sqrt{-87}) \) None \(-1\) \(1\) \(0\) \(-1\) \(q-q^{2}+q^{3}-q^{6}-q^{7}+q^{8}+q^{9}+\cdots\)
87.1.d.b \(1\) \(0.043\) \(\Q\) \(D_{3}\) \(\Q(\sqrt{-87}) \) None \(1\) \(-1\) \(0\) \(-1\) \(q+q^{2}-q^{3}-q^{6}-q^{7}-q^{8}+q^{9}+\cdots\)

Hecke characteristic polynomials

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