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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.

\(2\)\(3\)\(23\)FrickeDim
\(+\)\(+\)\(-\)$-$\(1\)
\(+\)\(-\)\(+\)$-$\(1\)
\(+\)\(-\)\(-\)$+$\(2\)
\(-\)\(+\)\(+\)$-$\(1\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(+\)$+$\(3\)
Plus space\(+\)\(7\)
Minus space\(-\)\(3\)

Trace form

\( 10 q + 4 q^{2} + 6 q^{3} + 40 q^{4} + 20 q^{5} - 12 q^{6} + 16 q^{7} + 16 q^{8} + 90 q^{9} + O(q^{10}) \) \( 10 q + 4 q^{2} + 6 q^{3} + 40 q^{4} + 20 q^{5} - 12 q^{6} + 16 q^{7} + 16 q^{8} + 90 q^{9} + 64 q^{10} + 24 q^{12} + 52 q^{13} - 80 q^{14} + 84 q^{15} + 160 q^{16} + 60 q^{17} + 36 q^{18} - 68 q^{19} + 80 q^{20} - 12 q^{21} + 280 q^{22} - 48 q^{24} + 166 q^{25} - 72 q^{26} + 54 q^{27} + 64 q^{28} + 68 q^{29} + 128 q^{31} + 64 q^{32} - 180 q^{33} + 152 q^{34} - 896 q^{35} + 360 q^{36} - 760 q^{37} + 496 q^{38} + 252 q^{39} + 256 q^{40} - 292 q^{41} + 408 q^{42} - 196 q^{43} + 180 q^{45} - 184 q^{46} - 1208 q^{47} + 96 q^{48} + 1450 q^{49} - 308 q^{50} + 240 q^{51} + 208 q^{52} - 2500 q^{53} - 108 q^{54} - 1136 q^{55} - 320 q^{56} - 348 q^{57} - 520 q^{58} - 920 q^{59} + 336 q^{60} + 1880 q^{61} + 720 q^{62} + 144 q^{63} + 640 q^{64} - 584 q^{65} - 1468 q^{67} + 240 q^{68} - 276 q^{69} - 368 q^{70} - 1128 q^{71} + 144 q^{72} + 996 q^{73} - 440 q^{74} + 1146 q^{75} - 272 q^{76} + 832 q^{77} - 648 q^{78} - 2352 q^{79} + 320 q^{80} + 810 q^{81} - 1096 q^{82} - 2888 q^{83} - 48 q^{84} + 520 q^{85} - 2784 q^{86} + 1644 q^{87} + 1120 q^{88} - 1092 q^{89} + 576 q^{90} + 2992 q^{91} - 288 q^{93} - 1104 q^{94} - 520 q^{95} - 192 q^{96} + 396 q^{97} - 1372 q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

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

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

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