Properties

Label 51.2.h
Level $51$
Weight $2$
Character orbit 51.h
Rep. character $\chi_{51}(19,\cdot)$
Character field $\Q(\zeta_{8})$
Dimension $8$
Newform subspaces $1$
Sturm bound $12$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 51 = 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 51.h (of order \(8\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q(\zeta_{8})\)
Newform subspaces: \( 1 \)
Sturm bound: \(12\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(51, [\chi])\).

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

Trace form

\( 8 q - 8 q^{5} - 8 q^{6} + O(q^{10}) \) \( 8 q - 8 q^{5} - 8 q^{6} + 8 q^{11} - 16 q^{14} + 16 q^{16} - 8 q^{17} - 8 q^{19} + 16 q^{20} - 8 q^{22} + 8 q^{23} + 8 q^{24} - 16 q^{25} + 16 q^{26} - 8 q^{28} + 8 q^{31} + 8 q^{33} - 8 q^{34} + 32 q^{35} + 8 q^{36} - 8 q^{37} + 16 q^{39} - 8 q^{40} - 24 q^{41} + 8 q^{42} - 8 q^{43} - 8 q^{45} - 8 q^{49} - 32 q^{50} - 16 q^{52} - 32 q^{53} - 8 q^{54} - 16 q^{56} - 16 q^{57} + 24 q^{58} + 16 q^{59} - 8 q^{60} + 16 q^{61} + 16 q^{62} + 24 q^{65} - 16 q^{66} - 16 q^{67} - 24 q^{69} + 40 q^{70} - 16 q^{71} + 48 q^{73} + 64 q^{74} + 16 q^{75} + 24 q^{76} + 8 q^{78} - 16 q^{80} - 8 q^{82} + 32 q^{83} + 40 q^{85} - 32 q^{86} + 24 q^{87} - 8 q^{88} - 16 q^{91} - 32 q^{92} - 32 q^{93} - 40 q^{95} - 16 q^{96} + 8 q^{97} + 8 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(51, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
51.2.h.a 51.h 17.d $8$ $0.407$ \(\Q(\zeta_{16})\) None 51.2.h.a \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q+(\zeta_{16}+\zeta_{16}^{3})q^{2}+\zeta_{16}^{7}q^{3}+(\zeta_{16}^{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(51, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(51, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(17, [\chi])\)\(^{\oplus 2}\)