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Properties

Label 756.2.j
Level $756$
Weight $2$
Character orbit 756.j
Rep. character $\chi_{756}(253,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $12$
Newform subspaces $2$
Sturm bound $288$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.j (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 9 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(288\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(756, [\chi])\).

Total New Old
Modular forms 324 12 312
Cusp forms 252 12 240
Eisenstein series 72 0 72

Trace form

\( 12 q - 2 q^{5} + O(q^{10}) \) \( 12 q - 2 q^{5} - 4 q^{11} - 4 q^{17} + 12 q^{19} + 8 q^{23} - 14 q^{29} + 6 q^{31} + 8 q^{35} - 12 q^{37} - 6 q^{41} - 6 q^{43} + 6 q^{47} - 6 q^{49} + 24 q^{53} - 12 q^{55} + 28 q^{59} + 18 q^{65} - 4 q^{71} + 12 q^{73} - 8 q^{77} + 6 q^{79} + 2 q^{83} + 30 q^{85} - 36 q^{89} - 12 q^{91} - 4 q^{95} + 24 q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(756, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
756.2.j.a 756.j 9.c $6$ $6.037$ 6.0.309123.1 None \(0\) \(0\) \(-3\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1}+\beta _{3})q^{5}-\beta _{3}q^{7}+(-\beta _{2}+\cdots)q^{11}+\cdots\)
756.2.j.b 756.j 9.c $6$ $6.037$ 6.0.309123.1 None \(0\) \(0\) \(1\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}+\beta _{5})q^{5}+(1-\beta _{4})q^{7}+(1-2\beta _{2}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(756, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(756, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(18, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(36, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(54, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(108, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 2}\)