Properties

Label 50.2.e
Level 50
Weight 2
Character orbit e
Rep. character \(\chi_{50}(9,\cdot)\)
Character field \(\Q(\zeta_{10})\)
Dimension 8
Newforms 1
Sturm bound 15
Trace bound 0

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Defining parameters

Level: \( N \) = \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 50.e (of order \(10\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 25 \)
Character field: \(\Q(\zeta_{10})\)
Newforms: \( 1 \)
Sturm bound: \(15\)
Trace bound: \(0\)

Dimensions

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

Total New Old
Modular forms 40 8 32
Cusp forms 24 8 16
Eisenstein series 16 0 16

Trace form

\(8q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(8q \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 10q^{5} \) \(\mathstrut +\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 4q^{11} \) \(\mathstrut -\mathstrut 10q^{12} \) \(\mathstrut -\mathstrut 4q^{14} \) \(\mathstrut -\mathstrut 20q^{15} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 10q^{17} \) \(\mathstrut +\mathstrut 10q^{19} \) \(\mathstrut +\mathstrut 16q^{21} \) \(\mathstrut +\mathstrut 20q^{22} \) \(\mathstrut +\mathstrut 10q^{23} \) \(\mathstrut +\mathstrut 8q^{24} \) \(\mathstrut +\mathstrut 10q^{25} \) \(\mathstrut -\mathstrut 28q^{26} \) \(\mathstrut +\mathstrut 10q^{28} \) \(\mathstrut +\mathstrut 10q^{29} \) \(\mathstrut +\mathstrut 20q^{30} \) \(\mathstrut +\mathstrut 6q^{31} \) \(\mathstrut -\mathstrut 30q^{33} \) \(\mathstrut -\mathstrut 4q^{34} \) \(\mathstrut +\mathstrut 10q^{35} \) \(\mathstrut +\mathstrut 4q^{36} \) \(\mathstrut -\mathstrut 10q^{37} \) \(\mathstrut -\mathstrut 8q^{39} \) \(\mathstrut -\mathstrut 14q^{41} \) \(\mathstrut -\mathstrut 10q^{42} \) \(\mathstrut -\mathstrut 6q^{44} \) \(\mathstrut +\mathstrut 10q^{45} \) \(\mathstrut -\mathstrut 8q^{46} \) \(\mathstrut -\mathstrut 30q^{47} \) \(\mathstrut -\mathstrut 10q^{48} \) \(\mathstrut -\mathstrut 16q^{49} \) \(\mathstrut +\mathstrut 16q^{51} \) \(\mathstrut -\mathstrut 20q^{54} \) \(\mathstrut -\mathstrut 10q^{55} \) \(\mathstrut +\mathstrut 4q^{56} \) \(\mathstrut -\mathstrut 14q^{61} \) \(\mathstrut +\mathstrut 20q^{63} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut 50q^{65} \) \(\mathstrut +\mathstrut 24q^{66} \) \(\mathstrut +\mathstrut 10q^{67} \) \(\mathstrut -\mathstrut 8q^{69} \) \(\mathstrut -\mathstrut 34q^{71} \) \(\mathstrut +\mathstrut 36q^{74} \) \(\mathstrut +\mathstrut 10q^{75} \) \(\mathstrut +\mathstrut 40q^{77} \) \(\mathstrut +\mathstrut 20q^{78} \) \(\mathstrut -\mathstrut 12q^{81} \) \(\mathstrut +\mathstrut 50q^{83} \) \(\mathstrut +\mathstrut 14q^{84} \) \(\mathstrut -\mathstrut 20q^{85} \) \(\mathstrut +\mathstrut 22q^{86} \) \(\mathstrut +\mathstrut 20q^{87} \) \(\mathstrut -\mathstrut 10q^{88} \) \(\mathstrut +\mathstrut 20q^{89} \) \(\mathstrut -\mathstrut 30q^{90} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 10q^{92} \) \(\mathstrut -\mathstrut 24q^{94} \) \(\mathstrut +\mathstrut 2q^{96} \) \(\mathstrut -\mathstrut 20q^{97} \) \(\mathstrut -\mathstrut 40q^{98} \) \(\mathstrut -\mathstrut 28q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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.e.a \(8\) \(0.399\) \(\Q(\zeta_{20})\) None \(0\) \(0\) \(-10\) \(0\) \(q+\zeta_{20}q^{2}+(-1+\zeta_{20}^{2}-\zeta_{20}^{4}+2\zeta_{20}^{6}+\cdots)q^{3}+\cdots\)

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

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