Properties

Label 375.2.a
Level $375$
Weight $2$
Character orbit 375.a
Rep. character $\chi_{375}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $6$
Sturm bound $100$
Trace bound $2$

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Defining parameters

Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 6 \)
Sturm bound: \(100\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(375))\).

Total New Old
Modular forms 60 16 44
Cusp forms 41 16 25
Eisenstein series 19 0 19

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
\(+\)\(+\)$+$\(2\)
\(+\)\(-\)$-$\(6\)
\(-\)\(+\)$-$\(6\)
\(-\)\(-\)$+$\(2\)
Plus space\(+\)\(4\)
Minus space\(-\)\(12\)

Trace form

\( 16 q + 18 q^{4} + 2 q^{6} + 16 q^{9} + O(q^{10}) \) \( 16 q + 18 q^{4} + 2 q^{6} + 16 q^{9} + 4 q^{11} + 12 q^{14} + 30 q^{16} - 8 q^{19} - 16 q^{21} + 6 q^{24} + 12 q^{26} + 4 q^{29} - 4 q^{31} + 16 q^{34} + 18 q^{36} - 12 q^{39} + 16 q^{41} - 76 q^{44} - 84 q^{46} + 20 q^{49} + 8 q^{51} + 2 q^{54} - 60 q^{56} + 28 q^{59} - 38 q^{64} + 16 q^{66} + 12 q^{69} + 32 q^{71} - 68 q^{74} - 12 q^{76} - 40 q^{79} + 16 q^{81} - 52 q^{84} - 56 q^{86} + 32 q^{89} - 44 q^{91} - 2 q^{94} - 36 q^{96} + 4 q^{99} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(375))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
375.2.a.a 375.a 1.a $2$ $2.994$ \(\Q(\sqrt{5}) \) None \(-3\) \(2\) \(0\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}+(-1+\cdots)q^{6}+\cdots\)
375.2.a.b 375.a 1.a $2$ $2.994$ \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(-1\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
375.2.a.c 375.a 1.a $2$ $2.994$ \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(0\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
375.2.a.d 375.a 1.a $2$ $2.994$ \(\Q(\sqrt{5}) \) None \(3\) \(-2\) \(0\) \(5\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-q^{3}+3\beta q^{4}+(-1-\beta )q^{6}+\cdots\)
375.2.a.e 375.a 1.a $4$ $2.994$ 4.4.2525.1 None \(-3\) \(-4\) \(0\) \(4\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{3})q^{2}-q^{3}+(1+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
375.2.a.f 375.a 1.a $4$ $2.994$ 4.4.2525.1 None \(3\) \(4\) \(0\) \(-4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2}-\beta _{3})q^{2}+q^{3}+(2-\beta _{1}-\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(375))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(375)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 2}\)