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Properties

Label 78.6.e
Level $78$
Weight $6$
Character orbit 78.e
Rep. character $\chi_{78}(55,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $20$
Newform subspaces $4$
Sturm bound $84$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 78 = 2 \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 78.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 4 \)
Sturm bound: \(84\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(78, [\chi])\).

Total New Old
Modular forms 148 20 128
Cusp forms 132 20 112
Eisenstein series 16 0 16

Trace form

\( 20 q - 8 q^{2} - 160 q^{4} + 44 q^{5} - 40 q^{7} + 256 q^{8} - 810 q^{9} + O(q^{10}) \) \( 20 q - 8 q^{2} - 160 q^{4} + 44 q^{5} - 40 q^{7} + 256 q^{8} - 810 q^{9} + 88 q^{10} - 376 q^{11} - 890 q^{13} - 2112 q^{14} + 396 q^{15} - 2560 q^{16} - 862 q^{17} + 1296 q^{18} - 2256 q^{19} - 352 q^{20} - 6192 q^{21} - 448 q^{22} + 4416 q^{23} - 4904 q^{25} + 5272 q^{26} - 640 q^{28} - 8862 q^{29} + 3600 q^{30} + 49952 q^{31} - 2048 q^{32} + 1476 q^{33} - 6768 q^{34} - 28504 q^{35} - 12960 q^{36} - 3474 q^{37} - 14400 q^{38} - 16740 q^{39} - 2816 q^{40} + 41474 q^{41} + 7056 q^{42} - 1312 q^{43} + 12032 q^{44} - 1782 q^{45} + 33664 q^{46} - 73648 q^{47} + 47514 q^{49} - 53760 q^{50} - 54576 q^{51} - 9344 q^{52} + 32252 q^{53} + 63040 q^{55} + 16896 q^{56} - 12888 q^{57} - 18824 q^{58} + 129352 q^{59} - 12672 q^{60} + 18530 q^{61} - 14176 q^{62} - 3240 q^{63} + 81920 q^{64} - 135934 q^{65} - 97056 q^{66} + 72824 q^{67} - 13792 q^{68} - 40716 q^{69} + 159424 q^{70} - 29912 q^{71} - 10368 q^{72} + 242860 q^{73} + 1896 q^{74} + 9432 q^{75} - 36096 q^{76} - 14080 q^{77} + 46656 q^{78} + 3328 q^{79} - 5632 q^{80} - 65610 q^{81} + 67608 q^{82} + 175488 q^{83} + 49536 q^{84} + 49674 q^{85} - 259392 q^{86} - 28620 q^{87} - 7168 q^{88} - 14524 q^{89} - 14256 q^{90} - 270704 q^{91} - 141312 q^{92} - 102060 q^{93} + 258560 q^{94} - 128392 q^{95} + 193912 q^{97} - 214440 q^{98} + 60912 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(78, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
78.6.e.a 78.e 13.c $4$ $12.510$ \(\Q(\sqrt{-3}, \sqrt{649})\) None \(8\) \(-18\) \(50\) \(-39\) $\mathrm{SU}(2)[C_{3}]$ \(q+(4-4\beta _{2})q^{2}+(-9+9\beta _{2})q^{3}-2^{4}\beta _{2}q^{4}+\cdots\)
78.6.e.b 78.e 13.c $4$ $12.510$ \(\Q(\sqrt{-3}, \sqrt{313})\) None \(8\) \(18\) \(-6\) \(-113\) $\mathrm{SU}(2)[C_{3}]$ \(q+(4-4\beta _{2})q^{2}+(9-9\beta _{2})q^{3}-2^{4}\beta _{2}q^{4}+\cdots\)
78.6.e.c 78.e 13.c $6$ $12.510$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(-27\) \(-72\) \(191\) $\mathrm{SU}(2)[C_{3}]$ \(q+4\beta _{2}q^{2}+9\beta _{2}q^{3}+(-2^{4}-2^{4}\beta _{2}+\cdots)q^{4}+\cdots\)
78.6.e.d 78.e 13.c $6$ $12.510$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-12\) \(27\) \(72\) \(-79\) $\mathrm{SU}(2)[C_{3}]$ \(q-4\beta _{3}q^{2}+9\beta _{3}q^{3}+(-2^{4}+2^{4}\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(78, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(78, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(39, [\chi])\)\(^{\oplus 2}\)