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Properties

Label 50.2.d
Level $50$
Weight $2$
Character orbit 50.d
Rep. character $\chi_{50}(11,\cdot)$
Character field $\Q(\zeta_{5})$
Dimension $12$
Newform subspaces $2$
Sturm bound $15$
Trace bound $1$

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 50.d (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newform subspaces: \( 2 \)
Sturm bound: \(15\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(50, [\chi])\).

Total New Old
Modular forms 36 12 24
Cusp forms 20 12 8
Eisenstein series 16 0 16

Trace form

\( 12 q - q^{2} - 4 q^{3} - 3 q^{4} + 5 q^{5} - 2 q^{6} - 8 q^{7} - q^{8} - 9 q^{9} + O(q^{10}) \) \( 12 q - q^{2} - 4 q^{3} - 3 q^{4} + 5 q^{5} - 2 q^{6} - 8 q^{7} - q^{8} - 9 q^{9} - 5 q^{10} + 4 q^{11} + 6 q^{12} - 14 q^{13} - 4 q^{14} - 3 q^{16} - 8 q^{17} + 12 q^{18} + 10 q^{19} - 16 q^{21} + 8 q^{22} + 6 q^{23} + 8 q^{24} + 25 q^{25} + 18 q^{26} + 20 q^{27} + 2 q^{28} - 6 q^{31} + 4 q^{32} + 22 q^{33} + 11 q^{34} - 30 q^{35} - 9 q^{36} - 23 q^{37} - 20 q^{38} - 8 q^{39} - 5 q^{40} + 4 q^{41} - 22 q^{42} - 4 q^{43} - 6 q^{44} + 5 q^{45} + 8 q^{46} + 22 q^{47} + 6 q^{48} + 4 q^{49} - 25 q^{50} - 16 q^{51} - 14 q^{52} - 9 q^{53} - 20 q^{54} - 10 q^{55} - 4 q^{56} - 30 q^{58} + 4 q^{61} + 28 q^{62} + 16 q^{63} - 3 q^{64} - 15 q^{65} - 24 q^{66} + 2 q^{67} + 42 q^{68} + 52 q^{69} + 40 q^{70} - 6 q^{71} - 13 q^{72} + 6 q^{73} + 26 q^{74} + 50 q^{75} + 24 q^{77} + 24 q^{78} - 5 q^{80} - 33 q^{81} + 38 q^{82} + 26 q^{83} + 14 q^{84} + 35 q^{85} - 22 q^{86} - 2 q^{88} - 5 q^{89} + 25 q^{90} + 4 q^{91} + 6 q^{92} - 68 q^{93} - 24 q^{94} - 60 q^{95} - 2 q^{96} - 58 q^{97} - 17 q^{98} - 68 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(50, [\chi])\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
50.2.d.a 50.d 25.d $4$ $0.399$ \(\Q(\zeta_{10})\) None \(1\) \(-1\) \(5\) \(-12\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-1+\cdots)q^{3}+\cdots\)
50.2.d.b 50.d 25.d $8$ $0.399$ 8.0.58140625.2 None \(-2\) \(-3\) \(0\) \(4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\beta _{2}+\beta _{3}+\beta _{6})q^{2}+\beta _{4}q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(50, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)