↑

Properties

Label 52.6.e
Level $52$
Weight $6$
Character orbit 52.e
Rep. character $\chi_{52}(9,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $12$
Newform subspaces $1$
Sturm bound $42$
Trace bound $0$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 52 = 2^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 52.e (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 13 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 1 \)
Sturm bound: \(42\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(52, [\chi])\).

Total New Old
Modular forms 76 12 64
Cusp forms 64 12 52
Eisenstein series 12 0 12

Trace form

\( 12 q - 9 q^{3} - 22 q^{5} + 7 q^{7} - 677 q^{9} + O(q^{10}) \) \( 12 q - 9 q^{3} - 22 q^{5} + 7 q^{7} - 677 q^{9} - 459 q^{11} + 525 q^{13} + 260 q^{15} + 358 q^{17} - 4295 q^{19} - 7990 q^{21} - 725 q^{23} + 5954 q^{25} + 14778 q^{27} - 5996 q^{29} + 1908 q^{31} + 6893 q^{33} - 4102 q^{35} + 1384 q^{37} + 627 q^{39} - 7802 q^{41} - 2813 q^{43} + 30373 q^{45} + 56316 q^{47} + 1959 q^{49} - 52762 q^{51} - 7502 q^{53} + 8042 q^{55} - 4990 q^{57} - 4965 q^{59} - 55892 q^{61} + 98766 q^{63} - 114069 q^{65} - 102273 q^{67} - 90543 q^{69} - 32579 q^{71} + 33342 q^{73} + 59563 q^{75} + 153786 q^{77} + 260592 q^{79} - 99530 q^{81} - 143328 q^{83} + 172409 q^{85} - 304959 q^{87} - 21239 q^{89} - 128849 q^{91} - 247188 q^{93} + 78940 q^{95} - 62543 q^{97} + 470452 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{6}^{\mathrm{new}}(52, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
52.6.e.a 52.e 13.c $12$ $8.340$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(-9\) \(-22\) \(7\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+2\beta _{2})q^{3}+(-2+\beta _{3})q^{5}+(2+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(52, [\chi])\) into lower level spaces

\( S_{6}^{\mathrm{old}}(52, [\chi]) \cong \) \(S_{6}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)