Properties

Label 138.8.a
Level $138$
Weight $8$
Character orbit 138.a
Rep. character $\chi_{138}(1,\cdot)$
Character field $\Q$
Dimension $26$
Newform subspaces $8$
Sturm bound $192$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 138 = 2 \cdot 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 138.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(192\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(138))\).

Total New Old
Modular forms 172 26 146
Cusp forms 164 26 138
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.
\(+\)\(+\)\(+\)\(+\)\(3\)
\(+\)\(+\)\(-\)\(-\)\(4\)
\(+\)\(-\)\(+\)\(-\)\(3\)
\(+\)\(-\)\(-\)\(+\)\(4\)
\(-\)\(+\)\(+\)\(-\)\(3\)
\(-\)\(+\)\(-\)\(+\)\(4\)
\(-\)\(-\)\(+\)\(+\)\(4\)
\(-\)\(-\)\(-\)\(-\)\(1\)
Plus space\(+\)\(15\)
Minus space\(-\)\(11\)

Trace form

\( 26 q - 16 q^{2} - 54 q^{3} + 1664 q^{4} + 220 q^{5} - 432 q^{6} + 1576 q^{7} - 1024 q^{8} + 18954 q^{9} + O(q^{10}) \) \( 26 q - 16 q^{2} - 54 q^{3} + 1664 q^{4} + 220 q^{5} - 432 q^{6} + 1576 q^{7} - 1024 q^{8} + 18954 q^{9} + 5824 q^{10} - 3456 q^{12} - 1564 q^{13} + 7232 q^{14} - 21276 q^{15} + 106496 q^{16} + 82884 q^{17} - 11664 q^{18} - 53188 q^{19} + 14080 q^{20} + 122148 q^{21} - 68896 q^{22} - 27648 q^{24} + 415606 q^{25} + 142368 q^{26} - 39366 q^{27} + 100864 q^{28} - 128012 q^{29} + 122208 q^{31} - 65536 q^{32} + 210276 q^{33} + 275872 q^{34} - 479776 q^{35} + 1213056 q^{36} + 681376 q^{37} - 707776 q^{38} - 299916 q^{39} + 372736 q^{40} + 1400908 q^{41} + 325728 q^{42} + 1949804 q^{43} + 160380 q^{45} - 389344 q^{46} + 1258232 q^{47} - 221184 q^{48} - 540870 q^{49} - 569008 q^{50} - 81432 q^{51} - 100096 q^{52} + 1400788 q^{53} - 314928 q^{54} + 227344 q^{55} + 462848 q^{56} + 1812996 q^{57} + 3391904 q^{58} + 4373144 q^{59} - 1361664 q^{60} - 3612976 q^{61} + 1507008 q^{62} + 1148904 q^{63} + 6815744 q^{64} + 2195096 q^{65} + 8781028 q^{67} + 5304576 q^{68} - 1314036 q^{69} + 969152 q^{70} - 2266104 q^{71} - 746496 q^{72} + 4215780 q^{73} - 3695008 q^{74} + 220806 q^{75} - 3404032 q^{76} - 5894464 q^{77} + 1148256 q^{78} - 3578136 q^{79} + 901120 q^{80} + 13817466 q^{81} + 6326688 q^{82} - 12723448 q^{83} + 7817472 q^{84} + 10049960 q^{85} + 3319680 q^{86} + 4714740 q^{87} - 4409344 q^{88} + 16050948 q^{89} + 4245696 q^{90} - 22649456 q^{91} - 20024064 q^{93} + 11826240 q^{94} + 8718280 q^{95} - 1769472 q^{96} + 12783468 q^{97} + 9329776 q^{98} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(138))\) 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
138.8.a.a 138.a 1.a $1$ $43.109$ \(\Q\) None \(8\) \(27\) \(-230\) \(106\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}-230q^{5}+\cdots\)
138.8.a.b 138.a 1.a $3$ $43.109$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(-24\) \(-81\) \(160\) \(-364\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(55+5\beta _{1}+\cdots)q^{5}+\cdots\)
138.8.a.c 138.a 1.a $3$ $43.109$ 3.3.3351293.1 None \(-24\) \(81\) \(-162\) \(-296\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(-55-5\beta _{1}+\cdots)q^{5}+\cdots\)
138.8.a.d 138.a 1.a $3$ $43.109$ \(\mathbb{Q}[x]/(x^{3} - \cdots)\) None \(24\) \(-81\) \(92\) \(-858\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(31+\beta _{2})q^{5}+\cdots\)
138.8.a.e 138.a 1.a $4$ $43.109$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-32\) \(-108\) \(-90\) \(-222\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(-23+\beta _{1}+\cdots)q^{5}+\cdots\)
138.8.a.f 138.a 1.a $4$ $43.109$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-32\) \(108\) \(-162\) \(1218\) $+$ $-$ $-$ $\mathrm{SU}(2)$ \(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(-40-\beta _{1}+\cdots)q^{5}+\cdots\)
138.8.a.g 138.a 1.a $4$ $43.109$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(32\) \(-108\) \(342\) \(-30\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+(85+\beta _{1}+\cdots)q^{5}+\cdots\)
138.8.a.h 138.a 1.a $4$ $43.109$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(32\) \(108\) \(270\) \(2022\) $-$ $-$ $+$ $\mathrm{SU}(2)$ \(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+(68-\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(138))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(138)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)