Properties

Label 633.1
Level 633
Weight 1
Dimension 42
Nonzero newspaces 3
Newform subspaces 4
Sturm bound 29680
Trace bound 1

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Defining parameters

Level: \( N \) = \( 633\( 633 = 3 \cdot 211 \) \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 4 \)
Sturm bound: \(29680\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(633))\).

Total New Old
Modular forms 464 250 214
Cusp forms 44 42 2
Eisenstein series 420 208 212

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 34 0 0 8

Trace form

\( 42q + q^{3} - 5q^{4} - 4q^{7} - 3q^{9} + O(q^{10}) \) \( 42q + q^{3} - 5q^{4} - 4q^{7} - 3q^{9} + 4q^{10} + 3q^{12} - q^{16} - 4q^{19} - 2q^{22} - q^{25} + q^{27} + 4q^{28} + 6q^{30} - 6q^{31} - 8q^{34} - 5q^{36} - 2q^{37} - 4q^{39} - 8q^{40} - 10q^{43} + 4q^{46} - q^{48} - 3q^{49} - 8q^{52} + 4q^{55} + 12q^{58} - 8q^{61} + 6q^{63} - 7q^{64} + 2q^{66} - 2q^{67} + 4q^{70} - 10q^{73} - q^{75} + 4q^{76} - 2q^{79} - 3q^{81} + 2q^{82} - 8q^{84} - 4q^{85} + 4q^{88} - 6q^{90} - 2q^{91} + 2q^{93} - 12q^{94} - 2q^{97} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(633))\)

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
633.1.b \(\chi_{633}(212, \cdot)\) None 0 1
633.1.c \(\chi_{633}(421, \cdot)\) None 0 1
633.1.h \(\chi_{633}(14, \cdot)\) None 0 2
633.1.i \(\chi_{633}(226, \cdot)\) None 0 2
633.1.l \(\chi_{633}(367, \cdot)\) None 0 4
633.1.m \(\chi_{633}(71, \cdot)\) 633.1.m.a 4 4
633.1.m.b 8
633.1.o \(\chi_{633}(40, \cdot)\) None 0 6
633.1.p \(\chi_{633}(269, \cdot)\) 633.1.p.a 6 6
633.1.s \(\chi_{633}(10, \cdot)\) None 0 8
633.1.t \(\chi_{633}(83, \cdot)\) None 0 8
633.1.w \(\chi_{633}(31, \cdot)\) None 0 12
633.1.x \(\chi_{633}(101, \cdot)\) None 0 12
633.1.z \(\chi_{633}(5, \cdot)\) 633.1.z.a 24 24
633.1.ba \(\chi_{633}(28, \cdot)\) None 0 24
633.1.be \(\chi_{633}(20, \cdot)\) None 0 48
633.1.bf \(\chi_{633}(7, \cdot)\) None 0 48

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(633))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(633)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(211))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} ) \))
$3$ (\( 1 + T + T^{2} + T^{3} + T^{4} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \))(\( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} \))
$5$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} ) \))
$7$ (\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4}( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} ) \))
$11$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} ) \))
$13$ (\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )^{2} \))
$17$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} ) \))
$19$ (\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T - T^{3} - T^{4} - T^{5} + T^{7} + T^{8} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4}( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} ) \))
$23$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \))(\( ( 1 - T )^{6}( 1 + T )^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} )^{6} \))
$29$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} ) \))
$31$ (\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )^{2} \))
$37$ (\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4}( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} ) \))
$41$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} ) \))
$43$ (\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} )^{8} \))(\( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{6}( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} ) \))
$47$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 + T^{2} )^{4}( 1 - T^{2} + T^{4} - T^{6} + T^{8} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} ) \))
$53$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} ) \))
$59$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2}( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} ) \))
$61$ (\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T )^{8}( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )^{2} \))
$67$ (\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T^{2} )^{8} \))(\( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{6}( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} ) \))
$71$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T )^{6}( 1 + T )^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} )^{6} \))
$73$ (\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 + T + T^{2} )^{8} \))(\( ( 1 - T )^{6}( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{6}( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} ) \))
$79$ (\( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )^{2} \))
$83$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 - T )^{6}( 1 + T )^{6} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} )^{6}( 1 + T + T^{2} + T^{3} + T^{4} )^{6} \))
$89$ (\( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( 1 + T^{2} - T^{6} - T^{8} - T^{10} + T^{14} + T^{16} \))(\( ( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} )( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} ) \))(\( ( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} )( 1 + T - T^{5} - T^{6} - T^{7} - T^{8} + T^{10} + T^{11} + T^{12} + T^{13} + T^{14} - T^{16} - T^{17} - T^{18} - T^{19} + T^{23} + T^{24} ) \))
$97$ (\( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \))(\( ( 1 - T^{2} + T^{4} - T^{6} + T^{8} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{2} \))(\( ( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} )^{4}( 1 - T + T^{5} - T^{6} + T^{7} - T^{8} + T^{10} - T^{11} + T^{12} - T^{13} + T^{14} - T^{16} + T^{17} - T^{18} + T^{19} - T^{23} + T^{24} ) \))
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