Properties

Label 75.3.f
Level $75$
Weight $3$
Character orbit 75.f
Rep. character $\chi_{75}(7,\cdot)$
Character field $\Q(\zeta_{4})$
Dimension $12$
Newform subspaces $3$
Sturm bound $30$
Trace bound $6$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 3 \)
Sturm bound: \(30\)
Trace bound: \(6\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(75, [\chi])\).

Total New Old
Modular forms 52 12 40
Cusp forms 28 12 16
Eisenstein series 24 0 24

Trace form

\( 12q + 4q^{2} + 24q^{6} - 4q^{7} - 12q^{8} + O(q^{10}) \) \( 12q + 4q^{2} + 24q^{6} - 4q^{7} - 12q^{8} - 32q^{11} - 24q^{12} + 32q^{13} - 40q^{16} + 40q^{17} + 12q^{18} - 12q^{21} - 20q^{22} - 56q^{23} - 56q^{26} - 44q^{28} + 172q^{31} + 76q^{32} + 36q^{33} + 96q^{36} - 64q^{37} + 96q^{38} - 368q^{41} - 12q^{42} + 8q^{43} - 88q^{46} - 128q^{47} - 48q^{48} - 144q^{51} + 80q^{52} - 56q^{53} + 840q^{56} + 72q^{57} + 12q^{58} + 60q^{61} - 88q^{62} - 12q^{63} - 48q^{66} + 200q^{67} + 104q^{68} + 544q^{71} + 36q^{72} - 76q^{73} - 464q^{76} - 88q^{77} - 120q^{78} - 108q^{81} - 128q^{82} + 16q^{83} - 1112q^{86} + 84q^{87} - 12q^{88} - 268q^{91} - 104q^{92} + 72q^{93} - 192q^{96} + 20q^{97} + 188q^{98} + O(q^{100}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(75, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
75.3.f.a \(4\) \(2.044\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}-\beta _{3}q^{3}-\beta _{2}q^{4}+3q^{6}+6\beta _{1}q^{7}+\cdots\)
75.3.f.b \(4\) \(2.044\) \(\Q(i, \sqrt{6})\) None \(0\) \(0\) \(0\) \(0\) \(q+2\beta _{1}q^{2}-\beta _{3}q^{3}+8\beta _{2}q^{4}+6q^{6}+\cdots\)
75.3.f.c \(4\) \(2.044\) \(\Q(i, \sqrt{6})\) None \(4\) \(0\) \(0\) \(-4\) \(q+(1+\beta _{1}+\beta _{2})q^{2}+\beta _{3}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(75, [\chi]) \cong \) \(S_{3}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)