Properties

Label 50.2.d
Level 50
Weight 2
Character orbit d
Rep. character \(\chi_{50}(11,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 12
Newforms 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: \(\Q(\zeta_{5})\)
Newforms: \( 2 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(50, [\chi])\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
50.2.d.a \(4\) \(0.399\) \(\Q(\zeta_{10})\) None \(1\) \(-1\) \(5\) \(-12\) \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-1+\cdots)q^{3}+\cdots\)
50.2.d.b \(8\) \(0.399\) 8.0.58140625.2 None \(-2\) \(-3\) \(0\) \(4\) \(q+(-1+\beta _{2}+\beta _{3}+\beta _{6})q^{2}+\beta _{4}q^{3}+\cdots\)

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

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