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Properties

Label 756.2.i
Level $756$
Weight $2$
Character orbit 756.i
Rep. character $\chi_{756}(37,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $16$
Newform subspaces $2$
Sturm bound $288$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 63 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(288\)
Trace bound: \(1\)
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 16 308
Cusp forms 252 16 236
Eisenstein series 72 0 72

Trace form

\( 16 q + 4 q^{5} + q^{7} + O(q^{10}) \) \( 16 q + 4 q^{5} + q^{7} + 2 q^{11} - q^{13} + 5 q^{17} + 2 q^{19} - 7 q^{23} - 8 q^{25} - 2 q^{29} - 4 q^{31} + 11 q^{35} - q^{37} + 24 q^{41} + 2 q^{43} - 12 q^{47} + 7 q^{49} + 18 q^{53} - 12 q^{55} - 14 q^{59} + 26 q^{61} + 18 q^{65} + 14 q^{67} + 14 q^{71} + 14 q^{73} + 43 q^{77} + 2 q^{79} + 26 q^{83} - 6 q^{85} + 21 q^{89} + 5 q^{91} - 76 q^{95} - q^{97} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
756.2.i.a 756.i 63.h $2$ $6.037$ \(\Q(\sqrt{-3}) \) None 252.2.i.a \(0\) \(0\) \(2\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{5}+(-2-\zeta_{6})q^{7}+4\zeta_{6}q^{11}+\cdots\)
756.2.i.b 756.i 63.h $14$ $6.037$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None 252.2.i.b \(0\) \(0\) \(2\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{3}q^{5}+(\beta _{5}+\beta _{12})q^{7}+\beta _{13}q^{11}+\cdots\)

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

\( S_{2}^{\mathrm{old}}(756, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(63, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(126, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(189, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(252, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(378, [\chi])\)\(^{\oplus 2}\)