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Properties

Label 56.3.o
Level $56$
Weight $3$
Character orbit 56.o
Rep. character $\chi_{56}(17,\cdot)$
Character field $\Q(\zeta_{6})$
Dimension $8$
Newform subspaces $1$
Sturm bound $24$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 56.o (of order \(6\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 7 \)
Character field: \(\Q(\zeta_{6})\)
Newform subspaces: \( 1 \)
Sturm bound: \(24\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(56, [\chi])\).

Total New Old
Modular forms 40 8 32
Cusp forms 24 8 16
Eisenstein series 16 0 16

Trace form

\( 8 q - 4 q^{7} + 20 q^{9} + O(q^{10}) \) \( 8 q - 4 q^{7} + 20 q^{9} + 4 q^{11} + 8 q^{15} - 24 q^{17} - 84 q^{19} - 52 q^{21} - 48 q^{23} + 24 q^{25} - 104 q^{29} + 156 q^{31} + 204 q^{33} + 156 q^{35} + 68 q^{37} - 52 q^{39} + 160 q^{43} - 276 q^{45} - 108 q^{47} - 80 q^{49} - 180 q^{51} - 28 q^{53} - 8 q^{57} + 120 q^{59} + 252 q^{61} + 76 q^{63} + 220 q^{65} - 56 q^{67} - 208 q^{71} - 156 q^{73} - 576 q^{75} - 240 q^{77} - 160 q^{79} + 96 q^{81} - 88 q^{85} + 348 q^{87} + 204 q^{89} + 120 q^{91} + 268 q^{93} + 216 q^{95} + 728 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{3}^{\mathrm{new}}(56, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
56.3.o.a 56.o 7.d $8$ $1.526$ 8.0.\(\cdots\).2 None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{5}q^{3}+\beta _{4}q^{5}+(1-3\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(56, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(56, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 2}\)