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Properties

Label 1988.1
Level 1988
Weight 1
Dimension 90
Nonzero newspaces 6
Newform subspaces 17
Sturm bound 241920
Trace bound 2

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Defining parameters

Level: \( N \) = \( 1988 = 2^{2} \cdot 7 \cdot 71 \)
Weight: \( k \) = \( 1 \)
Nonzero newspaces: \( 6 \)
Newform subspaces: \( 17 \)
Sturm bound: \(241920\)
Trace bound: \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{1}(\Gamma_1(1988))\).

Total New Old
Modular forms 2322 782 1540
Cusp forms 222 90 132
Eisenstein series 2100 692 1408

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

\(D_n\) \(A_4\) \(S_4\) \(A_5\)
Dimension 82 0 0 8

Trace form

\( 90 q + 2 q^{2} + 6 q^{4} + 6 q^{5} - 4 q^{8} + 8 q^{9} + O(q^{10}) \) \( 90 q + 2 q^{2} + 6 q^{4} + 6 q^{5} - 4 q^{8} + 8 q^{9} + 6 q^{10} - 4 q^{11} + 6 q^{16} - 10 q^{18} - 8 q^{23} + 12 q^{25} - 12 q^{29} + 2 q^{32} + 2 q^{35} + 4 q^{36} - 10 q^{37} - 6 q^{40} + 6 q^{43} + 6 q^{45} + 8 q^{49} + 8 q^{50} + 4 q^{53} - 8 q^{57} - 2 q^{58} + 12 q^{64} - 2 q^{65} - 2 q^{67} - 2 q^{71} - 10 q^{72} - 6 q^{73} - 2 q^{74} + 4 q^{77} - 6 q^{79} - 6 q^{80} - 4 q^{81} - 2 q^{85} + 8 q^{91} - 4 q^{95} + 2 q^{98} + 6 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{1}^{\mathrm{new}}(\Gamma_1(1988))\)

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 available newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
1988.1.c \(\chi_{1988}(853, \cdot)\) None 0 1
1988.1.d \(\chi_{1988}(995, \cdot)\) None 0 1
1988.1.g \(\chi_{1988}(1987, \cdot)\) 1988.1.g.a 1 1
1988.1.g.b 1
1988.1.g.c 1
1988.1.g.d 1
1988.1.g.e 1
1988.1.g.f 1
1988.1.g.g 2
1988.1.g.h 2
1988.1.h \(\chi_{1988}(141, \cdot)\) None 0 1
1988.1.k \(\chi_{1988}(709, \cdot)\) None 0 2
1988.1.l \(\chi_{1988}(283, \cdot)\) 1988.1.l.a 2 2
1988.1.l.b 2
1988.1.o \(\chi_{1988}(711, \cdot)\) None 0 2
1988.1.p \(\chi_{1988}(285, \cdot)\) None 0 2
1988.1.s \(\chi_{1988}(85, \cdot)\) None 0 4
1988.1.t \(\chi_{1988}(279, \cdot)\) 1988.1.t.a 4 4
1988.1.t.b 4
1988.1.w \(\chi_{1988}(267, \cdot)\) None 0 4
1988.1.x \(\chi_{1988}(125, \cdot)\) 1988.1.x.a 8 4
1988.1.z \(\chi_{1988}(307, \cdot)\) 1988.1.z.a 6 6
1988.1.z.b 6
1988.1.bb \(\chi_{1988}(449, \cdot)\) None 0 6
1988.1.bc \(\chi_{1988}(321, \cdot)\) None 0 6
1988.1.bf \(\chi_{1988}(463, \cdot)\) None 0 6
1988.1.bj \(\chi_{1988}(5, \cdot)\) None 0 8
1988.1.bk \(\chi_{1988}(431, \cdot)\) None 0 8
1988.1.bn \(\chi_{1988}(159, \cdot)\) None 0 8
1988.1.bo \(\chi_{1988}(137, \cdot)\) None 0 8
1988.1.bq \(\chi_{1988}(179, \cdot)\) None 0 12
1988.1.bt \(\chi_{1988}(45, \cdot)\) None 0 12
1988.1.bu \(\chi_{1988}(165, \cdot)\) None 0 12
1988.1.bw \(\chi_{1988}(467, \cdot)\) None 0 12
1988.1.bx \(\chi_{1988}(15, \cdot)\) None 0 24
1988.1.ca \(\chi_{1988}(237, \cdot)\) None 0 24
1988.1.cb \(\chi_{1988}(113, \cdot)\) None 0 24
1988.1.cd \(\chi_{1988}(55, \cdot)\) 1988.1.cd.a 24 24
1988.1.cd.b 24
1988.1.cf \(\chi_{1988}(31, \cdot)\) None 0 48
1988.1.ch \(\chi_{1988}(53, \cdot)\) None 0 48
1988.1.ci \(\chi_{1988}(73, \cdot)\) None 0 48
1988.1.cl \(\chi_{1988}(79, \cdot)\) None 0 48

Decomposition of \(S_{1}^{\mathrm{old}}(\Gamma_1(1988))\) into lower level spaces

\( S_{1}^{\mathrm{old}}(\Gamma_1(1988)) \cong \) \(S_{1}^{\mathrm{new}}(\Gamma_1(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(2))\)\(^{\oplus 8}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(14))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(28))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(71))\)\(^{\oplus 6}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(142))\)\(^{\oplus 4}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(284))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(497))\)\(^{\oplus 3}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(994))\)\(^{\oplus 2}\)\(\oplus\)\(S_{1}^{\mathrm{new}}(\Gamma_1(1988))\)\(^{\oplus 1}\)