Properties

Label 75.2.g
Level 75
Weight 2
Character orbit g
Rep. character \(\chi_{75}(16,\cdot)\)
Character field \(\Q(\zeta_{5})\)
Dimension 24
Newforms 3
Sturm bound 20
Trace bound 1

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Defining parameters

Level: \( N \) = \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 75.g (of order \(5\) and degree \(4\))
Character conductor: \(\operatorname{cond}(\chi)\) = \( 25 \)
Character field: \(\Q(\zeta_{5})\)
Newforms: \( 3 \)
Sturm bound: \(20\)
Trace bound: \(1\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(75, [\chi])\).

Total New Old
Modular forms 48 24 24
Cusp forms 32 24 8
Eisenstein series 16 0 16

Trace form

\(24q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut -\mathstrut 6q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(24q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut -\mathstrut 8q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut -\mathstrut 2q^{6} \) \(\mathstrut -\mathstrut 8q^{7} \) \(\mathstrut +\mathstrut 12q^{8} \) \(\mathstrut -\mathstrut 6q^{9} \) \(\mathstrut -\mathstrut 14q^{10} \) \(\mathstrut +\mathstrut 6q^{11} \) \(\mathstrut -\mathstrut 8q^{12} \) \(\mathstrut -\mathstrut 12q^{13} \) \(\mathstrut -\mathstrut 12q^{14} \) \(\mathstrut -\mathstrut 4q^{15} \) \(\mathstrut -\mathstrut 10q^{17} \) \(\mathstrut +\mathstrut 8q^{18} \) \(\mathstrut -\mathstrut 2q^{19} \) \(\mathstrut +\mathstrut 16q^{20} \) \(\mathstrut -\mathstrut 4q^{21} \) \(\mathstrut +\mathstrut 30q^{22} \) \(\mathstrut +\mathstrut 36q^{23} \) \(\mathstrut +\mathstrut 24q^{24} \) \(\mathstrut -\mathstrut 16q^{25} \) \(\mathstrut -\mathstrut 32q^{26} \) \(\mathstrut -\mathstrut 26q^{28} \) \(\mathstrut -\mathstrut 4q^{29} \) \(\mathstrut +\mathstrut 44q^{30} \) \(\mathstrut -\mathstrut 6q^{31} \) \(\mathstrut +\mathstrut 12q^{32} \) \(\mathstrut -\mathstrut 6q^{33} \) \(\mathstrut -\mathstrut 6q^{34} \) \(\mathstrut -\mathstrut 30q^{35} \) \(\mathstrut -\mathstrut 8q^{36} \) \(\mathstrut -\mathstrut 8q^{37} \) \(\mathstrut -\mathstrut 26q^{38} \) \(\mathstrut -\mathstrut 8q^{39} \) \(\mathstrut +\mathstrut 12q^{40} \) \(\mathstrut -\mathstrut 6q^{41} \) \(\mathstrut -\mathstrut 14q^{42} \) \(\mathstrut +\mathstrut 32q^{43} \) \(\mathstrut +\mathstrut 26q^{44} \) \(\mathstrut -\mathstrut 6q^{45} \) \(\mathstrut -\mathstrut 16q^{46} \) \(\mathstrut -\mathstrut 16q^{47} \) \(\mathstrut -\mathstrut 32q^{48} \) \(\mathstrut +\mathstrut 40q^{49} \) \(\mathstrut +\mathstrut 86q^{50} \) \(\mathstrut +\mathstrut 32q^{51} \) \(\mathstrut +\mathstrut 40q^{52} \) \(\mathstrut +\mathstrut 36q^{53} \) \(\mathstrut -\mathstrut 2q^{54} \) \(\mathstrut +\mathstrut 14q^{55} \) \(\mathstrut -\mathstrut 16q^{57} \) \(\mathstrut +\mathstrut 38q^{58} \) \(\mathstrut +\mathstrut 12q^{59} \) \(\mathstrut -\mathstrut 6q^{60} \) \(\mathstrut -\mathstrut 20q^{61} \) \(\mathstrut +\mathstrut 14q^{62} \) \(\mathstrut +\mathstrut 2q^{63} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut +\mathstrut 12q^{65} \) \(\mathstrut -\mathstrut 16q^{66} \) \(\mathstrut -\mathstrut 124q^{68} \) \(\mathstrut -\mathstrut 12q^{69} \) \(\mathstrut -\mathstrut 70q^{70} \) \(\mathstrut +\mathstrut 8q^{71} \) \(\mathstrut +\mathstrut 12q^{72} \) \(\mathstrut -\mathstrut 24q^{73} \) \(\mathstrut -\mathstrut 72q^{74} \) \(\mathstrut -\mathstrut 24q^{75} \) \(\mathstrut +\mathstrut 32q^{76} \) \(\mathstrut -\mathstrut 56q^{77} \) \(\mathstrut -\mathstrut 40q^{78} \) \(\mathstrut -\mathstrut 20q^{79} \) \(\mathstrut -\mathstrut 154q^{80} \) \(\mathstrut -\mathstrut 6q^{81} \) \(\mathstrut +\mathstrut 4q^{82} \) \(\mathstrut -\mathstrut 26q^{83} \) \(\mathstrut +\mathstrut 12q^{84} \) \(\mathstrut -\mathstrut 2q^{85} \) \(\mathstrut +\mathstrut 36q^{86} \) \(\mathstrut +\mathstrut 8q^{87} \) \(\mathstrut +\mathstrut 72q^{88} \) \(\mathstrut +\mathstrut 48q^{89} \) \(\mathstrut +\mathstrut 16q^{90} \) \(\mathstrut -\mathstrut 26q^{91} \) \(\mathstrut +\mathstrut 22q^{92} \) \(\mathstrut +\mathstrut 88q^{93} \) \(\mathstrut -\mathstrut 58q^{94} \) \(\mathstrut +\mathstrut 96q^{95} \) \(\mathstrut +\mathstrut 6q^{96} \) \(\mathstrut +\mathstrut 42q^{98} \) \(\mathstrut -\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(75, [\chi])\) into irreducible Hecke orbits

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
75.2.g.a \(4\) \(0.599\) \(\Q(\zeta_{10})\) None \(-1\) \(-1\) \(5\) \(0\) \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
75.2.g.b \(8\) \(0.599\) 8.0.26265625.1 None \(-1\) \(-2\) \(-5\) \(4\) \(q+(-\beta _{3}-\beta _{7})q^{2}+(-1+\beta _{1}+\beta _{3}+\cdots)q^{3}+\cdots\)
75.2.g.c \(12\) \(0.599\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(3\) \(-6\) \(-12\) \(q-\beta _{2}q^{2}+\beta _{8}q^{3}+(-1+\beta _{1}-\beta _{5}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

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