Properties

Label 2009.1
Level 2009
Weight 1
Dimension 12
Nonzero newspaces 1
Newform subspaces 2
Sturm bound 329280
Trace bound 0

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Defining parameters

Level: \( N \) = \( 2009\( 2009 = 7^{2} \cdot 41 \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 1 \)
Newform subspaces: \( 2 \)
Sturm bound: \(329280\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(2009))\).

Total New Old
Modular forms 2424 1904 520
Cusp forms 24 12 12
Eisenstein series 2400 1892 508

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 12 0 0 0

Trace form

\( 12q + 2q^{2} - 4q^{4} - 8q^{8} - 4q^{9} + O(q^{10}) \) \( 12q + 2q^{2} - 4q^{4} - 8q^{8} - 4q^{9} - 2q^{16} + 6q^{18} + 2q^{23} - 6q^{25} + 6q^{32} - 4q^{36} + 2q^{37} + 4q^{39} - 4q^{43} + 4q^{46} - 4q^{50} + 4q^{51} - 8q^{57} - 2q^{72} - 10q^{74} + 12q^{78} - 2q^{81} + 4q^{86} + 16q^{92} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(2009))\)

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
2009.1.b \(\chi_{2009}(1518, \cdot)\) None 0 1
2009.1.d \(\chi_{2009}(2008, \cdot)\) None 0 1
2009.1.g \(\chi_{2009}(1567, \cdot)\) None 0 2
2009.1.i \(\chi_{2009}(901, \cdot)\) 2009.1.i.a 6 2
2009.1.i.b 6
2009.1.k \(\chi_{2009}(411, \cdot)\) None 0 2
2009.1.n \(\chi_{2009}(834, \cdot)\) None 0 4
2009.1.p \(\chi_{2009}(734, \cdot)\) None 0 4
2009.1.q \(\chi_{2009}(146, \cdot)\) None 0 4
2009.1.r \(\chi_{2009}(460, \cdot)\) None 0 4
2009.1.t \(\chi_{2009}(286, \cdot)\) None 0 6
2009.1.v \(\chi_{2009}(83, \cdot)\) None 0 6
2009.1.x \(\chi_{2009}(244, \cdot)\) None 0 8
2009.1.ba \(\chi_{2009}(79, \cdot)\) None 0 8
2009.1.bc \(\chi_{2009}(132, \cdot)\) None 0 12
2009.1.be \(\chi_{2009}(31, \cdot)\) None 0 8
2009.1.bf \(\chi_{2009}(215, \cdot)\) None 0 8
2009.1.bi \(\chi_{2009}(99, \cdot)\) None 0 16
2009.1.bk \(\chi_{2009}(124, \cdot)\) None 0 12
2009.1.bm \(\chi_{2009}(40, \cdot)\) None 0 12
2009.1.bn \(\chi_{2009}(85, \cdot)\) None 0 24
2009.1.bq \(\chi_{2009}(80, \cdot)\) None 0 16
2009.1.br \(\chi_{2009}(209, \cdot)\) None 0 24
2009.1.bs \(\chi_{2009}(139, \cdot)\) None 0 24
2009.1.bv \(\chi_{2009}(73, \cdot)\) None 0 24
2009.1.by \(\chi_{2009}(30, \cdot)\) None 0 32
2009.1.ca \(\chi_{2009}(20, \cdot)\) None 0 48
2009.1.cc \(\chi_{2009}(44, \cdot)\) None 0 48
2009.1.ce \(\chi_{2009}(10, \cdot)\) None 0 48
2009.1.cf \(\chi_{2009}(45, \cdot)\) None 0 48
2009.1.ch \(\chi_{2009}(15, \cdot)\) None 0 96
2009.1.ci \(\chi_{2009}(5, \cdot)\) None 0 96
2009.1.ck \(\chi_{2009}(11, \cdot)\) None 0 192

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(2009))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(2009)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(287))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \))(\( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \))
$3$ (\( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \))(\( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \))
$5$ (\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))
$7$ 1
$11$ (\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))
$13$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))
$17$ (\( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \))(\( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \))
$19$ (\( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \))(\( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \))
$23$ (\( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \))(\( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \))
$29$ (\( ( 1 - T )^{6}( 1 + T )^{6} \))(\( ( 1 - T )^{6}( 1 + T )^{6} \))
$31$ (\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))
$37$ (\( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \))(\( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \))
$41$ (\( ( 1 + T )^{6} \))(\( ( 1 - T )^{6} \))
$43$ (\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))
$47$ (\( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \))(\( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \))
$53$ (\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))
$59$ (\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))
$61$ (\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))
$67$ (\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))
$71$ (\( ( 1 - T )^{6}( 1 + T )^{6} \))(\( ( 1 - T )^{6}( 1 + T )^{6} \))
$73$ (\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))
$79$ (\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))(\( ( 1 - T + T^{2} )^{3}( 1 + T + T^{2} )^{3} \))
$83$ (\( ( 1 - T )^{6}( 1 + T )^{6} \))(\( ( 1 - T )^{6}( 1 + T )^{6} \))
$89$ (\( 1 + T - T^{3} - T^{4} + T^{6} - T^{8} - T^{9} + T^{11} + T^{12} \))(\( 1 - T + T^{3} - T^{4} + T^{6} - T^{8} + T^{9} - T^{11} + T^{12} \))
$97$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))
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