Properties

Label 75.2.a
Level $75$
Weight $2$
Character orbit 75.a
Rep. character $\chi_{75}(1,\cdot)$
Character field $\Q$
Dimension $3$
Newform subspaces $3$
Sturm bound $20$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 3 \)
Sturm bound: \(20\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(75))\).

Total New Old
Modular forms 16 3 13
Cusp forms 5 3 2
Eisenstein series 11 0 11

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)\(5\)FrickeDim.
\(+\)\(-\)\(-\)\(1\)
\(-\)\(+\)\(-\)\(2\)
Plus space\(+\)\(0\)
Minus space\(-\)\(3\)

Trace form

\( 3q + q^{2} + q^{3} + 3q^{4} - 3q^{6} - 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + q^{2} + q^{3} + 3q^{4} - 3q^{6} - 3q^{8} + 3q^{9} - q^{12} + 2q^{13} - 12q^{14} - 9q^{16} - 2q^{17} + q^{18} - 6q^{19} + 6q^{21} - 4q^{22} - 3q^{24} + 6q^{26} + q^{27} + 18q^{29} - 6q^{31} + 5q^{32} - 4q^{33} + 6q^{34} + 3q^{36} + 10q^{37} + 4q^{38} - 6q^{41} - 4q^{43} + 12q^{44} + 24q^{46} - 8q^{47} - q^{48} - 3q^{49} - 6q^{51} - 2q^{52} + 10q^{53} - 3q^{54} + 4q^{57} - 2q^{58} - 24q^{59} + 12q^{61} - 9q^{64} - 12q^{66} - 12q^{67} + 2q^{68} - 12q^{69} - 24q^{71} - 3q^{72} - 10q^{73} + 18q^{74} - 24q^{76} + 2q^{78} + 3q^{81} + 10q^{82} - 12q^{83} + 12q^{84} - 2q^{87} + 12q^{88} - 6q^{89} - 6q^{91} + 21q^{96} - 2q^{97} - 7q^{98} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(75))\) into newform subspaces

Label Dim. \(A\) Field CM Traces A-L signs $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\) 3 5
75.2.a.a \(1\) \(0.599\) \(\Q\) None \(-2\) \(1\) \(0\) \(3\) \(-\) \(+\) \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+3q^{7}+\cdots\)
75.2.a.b \(1\) \(0.599\) \(\Q\) None \(1\) \(1\) \(0\) \(0\) \(-\) \(+\) \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
75.2.a.c \(1\) \(0.599\) \(\Q\) None \(2\) \(-1\) \(0\) \(-3\) \(+\) \(-\) \(q+2q^{2}-q^{3}+2q^{4}-2q^{6}-3q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(75))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(75)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)