Properties

Label 138.4.a
Level $138$
Weight $4$
Character orbit 138.a
Rep. character $\chi_{138}(1,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $6$
Sturm bound $96$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(96\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(138))\).

Total New Old
Modular forms 76 10 66
Cusp forms 68 10 58
Eisenstein series 8 0 8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)\(3\)\(23\)FrickeTotalCuspEisenstein
AllNewOldAllNewOldAllNewOld
\(+\)\(+\)\(+\)\(+\)\(12\)\(0\)\(12\)\(11\)\(0\)\(11\)\(1\)\(0\)\(1\)
\(+\)\(+\)\(-\)\(-\)\(7\)\(1\)\(6\)\(6\)\(1\)\(5\)\(1\)\(0\)\(1\)
\(+\)\(-\)\(+\)\(-\)\(10\)\(1\)\(9\)\(9\)\(1\)\(8\)\(1\)\(0\)\(1\)
\(+\)\(-\)\(-\)\(+\)\(9\)\(2\)\(7\)\(8\)\(2\)\(6\)\(1\)\(0\)\(1\)
\(-\)\(+\)\(+\)\(-\)\(11\)\(1\)\(10\)\(10\)\(1\)\(9\)\(1\)\(0\)\(1\)
\(-\)\(+\)\(-\)\(+\)\(8\)\(2\)\(6\)\(7\)\(2\)\(5\)\(1\)\(0\)\(1\)
\(-\)\(-\)\(+\)\(+\)\(11\)\(3\)\(8\)\(10\)\(3\)\(7\)\(1\)\(0\)\(1\)
\(-\)\(-\)\(-\)\(-\)\(8\)\(0\)\(8\)\(7\)\(0\)\(7\)\(1\)\(0\)\(1\)
Plus space\(+\)\(40\)\(7\)\(33\)\(36\)\(7\)\(29\)\(4\)\(0\)\(4\)
Minus space\(-\)\(36\)\(3\)\(33\)\(32\)\(3\)\(29\)\(4\)\(0\)\(4\)

Trace form

\( 10 q + 4 q^{2} + 6 q^{3} + 40 q^{4} + 20 q^{5} - 12 q^{6} + 16 q^{7} + 16 q^{8} + 90 q^{9} + 64 q^{10} + 24 q^{12} + 52 q^{13} - 80 q^{14} + 84 q^{15} + 160 q^{16} + 60 q^{17} + 36 q^{18} - 68 q^{19} + 80 q^{20}+ \cdots - 1372 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(138))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 23
138.4.a.a 138.a 1.a $1$ $8.142$ \(\Q\) None 138.4.a.a \(-2\) \(-3\) \(-10\) \(32\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-3q^{3}+4q^{4}-10q^{5}+6q^{6}+\cdots\)
138.4.a.b 138.a 1.a $1$ $8.142$ \(\Q\) None 138.4.a.b \(-2\) \(3\) \(2\) \(-32\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+2q^{5}-6q^{6}+\cdots\)
138.4.a.c 138.a 1.a $1$ $8.142$ \(\Q\) None 138.4.a.c \(2\) \(-3\) \(-2\) \(-34\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}-2q^{5}-6q^{6}+\cdots\)
138.4.a.d 138.a 1.a $2$ $8.142$ \(\Q(\sqrt{277}) \) None 138.4.a.d \(-4\) \(6\) \(2\) \(28\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+3q^{3}+4q^{4}+(1-\beta )q^{5}-6q^{6}+\cdots\)
138.4.a.e 138.a 1.a $2$ $8.142$ \(\Q(\sqrt{2}) \) None 138.4.a.e \(4\) \(-6\) \(8\) \(12\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-3q^{3}+4q^{4}+(4+\beta )q^{5}-6q^{6}+\cdots\)
138.4.a.f 138.a 1.a $3$ $8.142$ 3.3.16372.1 None 138.4.a.f \(6\) \(9\) \(20\) \(10\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+3q^{3}+4q^{4}+(7+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(138))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(138)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)