Properties

Label 75.3.c
Level $75$
Weight $3$
Character orbit 75.c
Rep. character $\chi_{75}(26,\cdot)$
Character field $\Q$
Dimension $10$
Newform subspaces $6$
Sturm bound $30$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(30\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\), \(7\), \(13\)

Dimensions

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

Total New Old
Modular forms 26 16 10
Cusp forms 14 10 4
Eisenstein series 12 6 6

Trace form

\( 10q + 4q^{3} - 16q^{4} - 18q^{6} + 12q^{7} + 12q^{9} + O(q^{10}) \) \( 10q + 4q^{3} - 16q^{4} - 18q^{6} + 12q^{7} + 12q^{9} - 4q^{12} - 32q^{13} + 24q^{16} + 40q^{18} - 6q^{19} - 42q^{21} - 20q^{22} + 54q^{24} - 44q^{27} - 12q^{28} + 110q^{31} + 20q^{33} + 4q^{34} - 78q^{36} + 32q^{37} + 42q^{39} + 60q^{42} - 32q^{43} - 136q^{46} - 76q^{48} - 176q^{49} + 42q^{51} + 32q^{52} - 228q^{54} - 8q^{57} + 140q^{58} - 10q^{61} - 12q^{63} + 408q^{64} + 510q^{66} - 48q^{67} + 12q^{69} + 120q^{72} + 148q^{73} - 356q^{76} - 160q^{78} - 156q^{79} - 60q^{81} - 280q^{82} - 288q^{84} - 140q^{87} - 60q^{88} - 214q^{91} - 72q^{93} + 424q^{94} - 114q^{96} + 332q^{97} + 630q^{99} + O(q^{100}) \)

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.c.a \(1\) \(2.044\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(-3\) \(0\) \(11\) \(q-3q^{3}+4q^{4}+11q^{7}+9q^{9}-12q^{12}+\cdots\)
75.3.c.b \(1\) \(2.044\) \(\Q\) \(\Q(\sqrt{-3}) \) \(0\) \(3\) \(0\) \(-11\) \(q+3q^{3}+4q^{4}-11q^{7}+9q^{9}+12q^{12}+\cdots\)
75.3.c.c \(2\) \(2.044\) \(\Q(\sqrt{-11}) \) None \(0\) \(-5\) \(0\) \(0\) \(q+(1-2\beta )q^{2}+(-2-\beta )q^{3}-7q^{4}+\cdots\)
75.3.c.d \(2\) \(2.044\) \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-15}) \) \(0\) \(0\) \(0\) \(0\) \(q+iq^{2}+3iq^{3}+3q^{4}-3q^{6}+7iq^{8}+\cdots\)
75.3.c.e \(2\) \(2.044\) \(\Q(\sqrt{-5}) \) None \(0\) \(4\) \(0\) \(12\) \(q+\beta q^{2}+(2-\beta )q^{3}-q^{4}+(5+2\beta )q^{6}+\cdots\)
75.3.c.f \(2\) \(2.044\) \(\Q(\sqrt{-11}) \) None \(0\) \(5\) \(0\) \(0\) \(q+(1-2\beta )q^{2}+(3-\beta )q^{3}-7q^{4}+(-3+\cdots)q^{6}+\cdots\)

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

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