Properties

Label 56.2.i
Level $56$
Weight $2$
Character orbit 56.i
Rep. character $\chi_{56}(9,\cdot)$
Character field $\Q(\zeta_{3})$
Dimension $4$
Newform subspaces $2$
Sturm bound $16$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 56.i (of order \(3\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{3})\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(56, [\chi])\).

Total New Old
Modular forms 24 4 20
Cusp forms 8 4 4
Eisenstein series 16 0 16

Trace form

\( 4 q - 2 q^{3} + 2 q^{5} - 4 q^{9} + O(q^{10}) \) \( 4 q - 2 q^{3} + 2 q^{5} - 4 q^{9} - 2 q^{11} - 8 q^{13} - 4 q^{15} + 2 q^{17} - 6 q^{19} - 2 q^{21} + 10 q^{23} + 8 q^{25} + 28 q^{27} - 8 q^{29} + 6 q^{31} + 6 q^{33} + 6 q^{35} + 2 q^{37} - 12 q^{39} - 24 q^{41} - 16 q^{43} + 4 q^{45} - 6 q^{47} + 4 q^{49} - 14 q^{51} + 10 q^{53} - 4 q^{55} + 28 q^{57} - 18 q^{59} + 2 q^{61} - 28 q^{63} - 4 q^{65} - 2 q^{67} - 4 q^{69} + 32 q^{71} - 14 q^{73} + 8 q^{75} + 14 q^{77} + 10 q^{79} - 10 q^{81} + 16 q^{83} + 4 q^{85} + 20 q^{87} + 2 q^{89} + 32 q^{91} - 2 q^{93} + 6 q^{95} + 8 q^{97} - 24 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(56, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
56.2.i.a $2$ $0.447$ \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(1\) \(4\) \(q-3\zeta_{6}q^{3}+(1-\zeta_{6})q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
56.2.i.b $2$ $0.447$ \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(1\) \(-4\) \(q+\zeta_{6}q^{3}+(1-\zeta_{6})q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(56, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(56, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)