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

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

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

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
75.2.a.a 75.a 1.a $1$ $0.599$ \(\Q\) None 75.2.a.a \(-2\) \(1\) \(0\) \(3\) $-$ $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+3q^{7}+\cdots\)
75.2.a.b 75.a 1.a $1$ $0.599$ \(\Q\) None 15.2.a.a \(1\) \(1\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
75.2.a.c 75.a 1.a $1$ $0.599$ \(\Q\) None 75.2.a.a \(2\) \(-1\) \(0\) \(-3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+2q^{2}-q^{3}+2q^{4}-2q^{6}-3q^{7}+\cdots\)

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

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