Properties

Label 75.2.a
Level 75
Weight 2
Character orbit a
Rep. character \(\chi_{75}(1,\cdot)\)
Character field \(\Q\)
Dimension 3
Newforms 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: \(\Q\)
Newforms: \( 3 \)
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

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(75))\) into irreducible Hecke orbits

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

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