↑

Properties

Label 56.2.b
Level $56$
Weight $2$
Character orbit 56.b
Rep. character $\chi_{56}(29,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $2$
Sturm bound $16$
Trace bound $1$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 56 = 2^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 56.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 8 \)
Character field: \(\Q\)
Newform subspaces: \( 2 \)
Sturm bound: \(16\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

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

Total New Old
Modular forms 10 6 4
Cusp forms 6 6 0
Eisenstein series 4 0 4

Trace form

\( 6 q - q^{2} - 3 q^{4} + 2 q^{6} - 2 q^{7} - 7 q^{8} - 6 q^{9} + O(q^{10}) \) \( 6 q - q^{2} - 3 q^{4} + 2 q^{6} - 2 q^{7} - 7 q^{8} - 6 q^{9} - 4 q^{10} + 14 q^{12} + q^{14} + 8 q^{15} + q^{16} - 4 q^{17} - 15 q^{18} + 4 q^{20} - 6 q^{22} - 8 q^{23} + 6 q^{24} - 2 q^{25} + 20 q^{26} - 5 q^{28} + 16 q^{30} - 16 q^{31} + 9 q^{32} + 8 q^{33} - 2 q^{34} + 11 q^{36} + 18 q^{38} + 8 q^{39} - 28 q^{40} - 4 q^{41} - 10 q^{42} - 18 q^{44} + 16 q^{46} - 10 q^{48} + 6 q^{49} + 19 q^{50} - 4 q^{52} - 44 q^{54} + 32 q^{55} + 7 q^{56} - 8 q^{57} - 20 q^{58} - 24 q^{60} - 32 q^{62} + 10 q^{63} - 15 q^{64} + 16 q^{65} - 4 q^{66} + 26 q^{68} + 12 q^{70} - 32 q^{71} + 31 q^{72} - 20 q^{73} - 12 q^{74} + 14 q^{76} - 16 q^{78} + 16 q^{79} + 36 q^{80} + 14 q^{81} + 38 q^{82} - 14 q^{84} + 26 q^{86} + 32 q^{87} + 38 q^{88} - 20 q^{89} + 20 q^{90} + 8 q^{92} - 8 q^{95} - 58 q^{96} - 4 q^{97} - q^{98} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

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

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
56.2.b.a 56.b 8.b $2$ $0.447$ \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}+\beta q^{3}-2q^{4}-\beta q^{5}-2q^{6}+\cdots\)
56.2.b.b 56.b 8.b $4$ $0.447$ 4.0.2312.1 None \(-1\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)