Properties

Label 612.1
Level 612
Weight 1
Dimension 39
Nonzero newspaces 6
Newform subspaces 10
Sturm bound 20736
Trace bound 4

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Defining parameters

Level: \( N \) = \( 612 = 2^{2} \cdot 3^{2} \cdot 17 \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 10 \)
Sturm bound: \(20736\)
Trace bound: \(4\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(612))\).

Total New Old
Modular forms 692 173 519
Cusp forms 52 39 13
Eisenstein series 640 134 506

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 35 0 4 0

Trace form

\( 39q + q^{2} - 5q^{4} + 2q^{5} + q^{8} + O(q^{10}) \) \( 39q + q^{2} - 5q^{4} + 2q^{5} + q^{8} - 6q^{10} + 2q^{13} - 9q^{16} + 9q^{17} - 4q^{18} - 2q^{20} + 8q^{21} - 11q^{25} - 10q^{26} - 2q^{29} - 4q^{31} + q^{32} - 4q^{33} - q^{34} - 2q^{37} - 2q^{40} - 2q^{41} - 4q^{42} - 5q^{49} + 3q^{50} - 2q^{52} - 10q^{53} - 4q^{55} - 2q^{58} - 2q^{61} + 7q^{64} + 8q^{66} - 7q^{68} + 8q^{69} - 4q^{72} - 10q^{73} - 2q^{74} + 4q^{77} - 4q^{79} + 2q^{80} - 4q^{81} - 6q^{82} - 4q^{84} - 10q^{85} + 2q^{89} - 4q^{93} - 2q^{97} + 13q^{98} + O(q^{100}) \)

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(612))\)

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
612.1.d \(\chi_{612}(341, \cdot)\) None 0 1
612.1.e \(\chi_{612}(271, \cdot)\) 612.1.e.a 1 1
612.1.f \(\chi_{612}(307, \cdot)\) None 0 1
612.1.g \(\chi_{612}(305, \cdot)\) None 0 1
612.1.j \(\chi_{612}(89, \cdot)\) 612.1.j.a 4 2
612.1.l \(\chi_{612}(55, \cdot)\) 612.1.l.a 2 2
612.1.o \(\chi_{612}(101, \cdot)\) None 0 2
612.1.p \(\chi_{612}(103, \cdot)\) None 0 2
612.1.q \(\chi_{612}(67, \cdot)\) 612.1.q.a 2 2
612.1.q.b 2
612.1.q.c 4
612.1.r \(\chi_{612}(137, \cdot)\) None 0 2
612.1.u \(\chi_{612}(53, \cdot)\) None 0 4
612.1.v \(\chi_{612}(19, \cdot)\) 612.1.v.a 4 4
612.1.v.b 4
612.1.y \(\chi_{612}(149, \cdot)\) None 0 4
612.1.ba \(\chi_{612}(115, \cdot)\) None 0 4
612.1.be \(\chi_{612}(37, \cdot)\) None 0 8
612.1.bf \(\chi_{612}(71, \cdot)\) 612.1.bf.a 8 8
612.1.bf.b 8
612.1.bg \(\chi_{612}(77, \cdot)\) None 0 8
612.1.bh \(\chi_{612}(43, \cdot)\) None 0 8
612.1.bm \(\chi_{612}(11, \cdot)\) None 0 16
612.1.bn \(\chi_{612}(61, \cdot)\) None 0 16

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(612))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(612)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(68))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(204))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

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