Properties

Label 50.3.c
Level $50$
Weight $3$
Character orbit 50.c
Rep. character $\chi_{50}(7,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $6$
Newform subspaces $3$
Sturm bound $22$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 50.c (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
Sturm bound: \(22\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(50, [\chi])\).

Total New Old
Modular forms 42 6 36
Cusp forms 18 6 12
Eisenstein series 24 0 24

Trace form

\( 6q + 2q^{2} + 4q^{3} - 16q^{6} - 4q^{7} - 4q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 4q^{3} - 16q^{6} - 4q^{7} - 4q^{8} + 32q^{11} + 8q^{12} - 6q^{13} - 24q^{16} - 14q^{17} - 2q^{18} - 88q^{21} - 16q^{22} + 4q^{23} + 84q^{26} + 40q^{27} + 8q^{28} + 72q^{31} - 8q^{32} - 32q^{33} + 68q^{36} + 6q^{37} + 40q^{38} - 208q^{41} - 16q^{42} + 84q^{43} - 16q^{46} + 36q^{47} - 16q^{48} + 232q^{51} - 12q^{52} - 106q^{53} - 32q^{56} - 80q^{57} - 80q^{58} + 32q^{61} + 104q^{62} + 4q^{63} - 352q^{66} - 124q^{67} + 28q^{68} - 248q^{71} - 4q^{72} + 94q^{73} - 80q^{76} + 32q^{77} - 24q^{78} + 466q^{81} - 16q^{82} - 36q^{83} + 384q^{86} + 160q^{87} + 32q^{88} + 312q^{91} + 8q^{92} + 208q^{93} + 64q^{96} + 126q^{97} + 82q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(50, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
50.3.c.a \(2\) \(1.362\) \(\Q(\sqrt{-1}) \) None \(-2\) \(6\) \(0\) \(-6\) \(q+(-1-i)q^{2}+(3-3i)q^{3}+2iq^{4}+\cdots\)
50.3.c.b \(2\) \(1.362\) \(\Q(\sqrt{-1}) \) None \(2\) \(-6\) \(0\) \(6\) \(q+(1+i)q^{2}+(-3+3i)q^{3}+2iq^{4}+\cdots\)
50.3.c.c \(2\) \(1.362\) \(\Q(\sqrt{-1}) \) None \(2\) \(4\) \(0\) \(-4\) \(q+(1+i)q^{2}+(2-2i)q^{3}+2iq^{4}+4q^{6}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(50, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)