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Properties

Label 38.8.c
Level $38$
Weight $8$
Character orbit 38.c
Rep. character $\chi_{38}(7,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $26$
Newform subspaces $2$
Sturm bound $40$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 38 = 2 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 38.c (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 19 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(40\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(38, [\chi])\).

Total New Old
Modular forms 74 26 48
Cusp forms 66 26 40
Eisenstein series 8 0 8

Trace form

\( 26 q + 8 q^{2} + 67 q^{3} - 832 q^{4} - 2 q^{5} - 344 q^{6} + 892 q^{7} - 1024 q^{8} - 11834 q^{9} + O(q^{10}) \) \( 26 q + 8 q^{2} + 67 q^{3} - 832 q^{4} - 2 q^{5} - 344 q^{6} + 892 q^{7} - 1024 q^{8} - 11834 q^{9} + 2000 q^{10} - 11114 q^{11} - 8576 q^{12} + 16082 q^{13} + 11856 q^{14} + 4678 q^{15} - 53248 q^{16} + 55782 q^{17} + 42080 q^{18} - 113727 q^{19} + 256 q^{20} + 15450 q^{21} + 49624 q^{22} - 37268 q^{23} - 22016 q^{24} - 283611 q^{25} - 299168 q^{26} + 226510 q^{27} - 28544 q^{28} + 277514 q^{29} + 204832 q^{30} - 165224 q^{31} + 32768 q^{32} + 326993 q^{33} + 277808 q^{34} - 564612 q^{35} - 757376 q^{36} + 1303064 q^{37} + 133696 q^{38} - 1296876 q^{39} + 128000 q^{40} + 555443 q^{41} + 612944 q^{42} + 2507656 q^{43} + 355648 q^{44} - 3398656 q^{45} - 3046016 q^{46} - 2194838 q^{47} + 274432 q^{48} + 5995178 q^{49} + 2933200 q^{50} + 4784254 q^{51} + 1029248 q^{52} + 1101694 q^{53} - 3465224 q^{54} - 2775962 q^{55} - 1517568 q^{56} - 2714088 q^{57} + 926368 q^{58} + 28797 q^{59} + 299392 q^{60} - 3392246 q^{61} + 2275136 q^{62} - 851888 q^{63} + 6815744 q^{64} + 9127808 q^{65} - 3423624 q^{66} - 3547953 q^{67} - 7140096 q^{68} - 24724432 q^{69} - 3986208 q^{70} + 4497358 q^{71} - 1346560 q^{72} + 5721781 q^{73} - 2164864 q^{74} - 12783786 q^{75} + 63936 q^{76} - 11604420 q^{77} + 16224016 q^{78} + 799284 q^{79} - 8192 q^{80} - 10123481 q^{81} + 3382728 q^{82} + 17700078 q^{83} - 1977600 q^{84} + 25388772 q^{85} + 8436224 q^{86} + 40339436 q^{87} - 6351872 q^{88} + 16795458 q^{89} + 17026112 q^{90} - 33000876 q^{91} - 2385152 q^{92} - 31032284 q^{93} - 40843808 q^{94} + 22424876 q^{95} + 2818048 q^{96} - 18311673 q^{97} + 23210056 q^{98} + 26293582 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{8}^{\mathrm{new}}(38, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
38.8.c.a 38.c 19.c $12$ $11.871$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(-48\) \(12\) \(124\) \(-1036\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\beta _{2}q^{2}+(\beta _{1}-2\beta _{2})q^{3}+(-2^{6}-2^{6}\beta _{2}+\cdots)q^{4}+\cdots\)
38.8.c.b 38.c 19.c $14$ $11.871$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(56\) \(55\) \(-126\) \(1928\) $\mathrm{SU}(2)[C_{3}]$ \(q+8\beta _{3}q^{2}+(\beta _{1}+\beta _{2}+8\beta _{3})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(38, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(38, [\chi]) \cong \) \(S_{8}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 2}\)