Properties

Label 51.2.e
Level $51$
Weight $2$
Character orbit 51.e
Rep. character $\chi_{51}(4,\cdot)$
Character field $\Q(\zeta_{4})$
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.e (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 17 \)
Character field: \(\Q(i)\)
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 16 8 8
Cusp forms 8 8 0
Eisenstein series 8 0 8

Trace form

\( 8 q - 12 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} + O(q^{10}) \) \( 8 q - 12 q^{4} - 4 q^{5} + 4 q^{6} - 4 q^{7} - 4 q^{13} + 24 q^{14} + 12 q^{16} - 4 q^{17} - 4 q^{18} - 12 q^{20} - 8 q^{21} + 12 q^{22} - 16 q^{23} - 16 q^{24} - 8 q^{28} + 4 q^{29} + 24 q^{30} - 8 q^{31} + 12 q^{33} - 12 q^{34} + 8 q^{35} + 8 q^{37} + 24 q^{38} + 8 q^{39} + 36 q^{40} + 28 q^{41} - 32 q^{44} + 4 q^{45} - 20 q^{46} - 8 q^{47} - 32 q^{48} - 60 q^{50} - 4 q^{51} - 16 q^{52} + 4 q^{54} - 4 q^{55} - 24 q^{56} - 4 q^{58} + 16 q^{61} + 16 q^{62} - 4 q^{63} + 4 q^{64} + 16 q^{65} + 8 q^{67} + 60 q^{68} - 12 q^{69} + 24 q^{71} + 12 q^{72} + 20 q^{73} - 28 q^{74} - 16 q^{75} + 20 q^{78} + 20 q^{79} + 60 q^{80} - 8 q^{81} - 40 q^{82} + 48 q^{84} - 32 q^{85} - 8 q^{86} - 8 q^{88} - 8 q^{89} - 36 q^{91} + 56 q^{92} + 8 q^{95} + 8 q^{96} - 12 q^{97} - 76 q^{98} + O(q^{100}) \)

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.e.a 51.e 17.c $8$ $0.407$ 8.0.836829184.2 None 51.2.e.a \(0\) \(0\) \(-4\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(-1-\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)