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Properties

Label 750.2.a
Level $750$
Weight $2$
Character orbit 750.a
Rep. character $\chi_{750}(1,\cdot)$
Character field $\Q$
Dimension $16$
Newform subspaces $8$
Sturm bound $300$
Trace bound $7$

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Defining parameters

Level: \( N \) \(=\) \( 750 = 2 \cdot 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 750.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(300\)
Trace bound: \(7\)
Distinguishing \(T_p\): \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(750))\).

Total New Old
Modular forms 170 16 154
Cusp forms 131 16 115
Eisenstein series 39 0 39

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

\(2\)\(3\)\(5\)FrickeDim
\(+\)\(+\)\(+\)$+$\(2\)
\(+\)\(+\)\(-\)$-$\(2\)
\(+\)\(-\)\(+\)$-$\(4\)
\(-\)\(+\)\(+\)$-$\(2\)
\(-\)\(+\)\(-\)$+$\(2\)
\(-\)\(-\)\(-\)$-$\(4\)
Plus space\(+\)\(4\)
Minus space\(-\)\(12\)

Trace form

\( 16 q + 16 q^{4} + 16 q^{9} + O(q^{10}) \) \( 16 q + 16 q^{4} + 16 q^{9} - 4 q^{11} - 4 q^{14} + 16 q^{16} + 28 q^{19} + 32 q^{21} - 4 q^{26} - 8 q^{29} + 20 q^{31} - 4 q^{34} + 16 q^{36} + 28 q^{39} - 12 q^{41} - 4 q^{44} - 8 q^{46} + 28 q^{49} - 8 q^{51} - 4 q^{56} - 20 q^{59} + 16 q^{61} + 16 q^{64} - 4 q^{66} - 8 q^{69} - 24 q^{71} - 12 q^{74} + 28 q^{76} + 4 q^{79} + 16 q^{81} + 32 q^{84} - 16 q^{86} - 36 q^{89} + 32 q^{91} - 16 q^{94} - 4 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(750))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3 5
750.2.a.a 750.a 1.a $2$ $5.989$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(0\) \(-6\) $+$ $+$ $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+(-2-2\beta )q^{7}+\cdots\)
750.2.a.b 750.a 1.a $2$ $5.989$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(0\) \(-1\) $+$ $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{6}+(-1+\beta )q^{7}+\cdots\)
750.2.a.c 750.a 1.a $2$ $5.989$ \(\Q(\sqrt{5}) \) None \(-2\) \(2\) \(0\) \(2\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+2\beta q^{7}-q^{8}+\cdots\)
750.2.a.d 750.a 1.a $2$ $5.989$ \(\Q(\sqrt{5}) \) None \(-2\) \(2\) \(0\) \(7\) $+$ $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+(4-\beta )q^{7}+\cdots\)
750.2.a.e 750.a 1.a $2$ $5.989$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(0\) \(-7\) $-$ $+$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}+(-3-\beta )q^{7}+\cdots\)
750.2.a.f 750.a 1.a $2$ $5.989$ \(\Q(\sqrt{5}) \) None \(2\) \(-2\) \(0\) \(-2\) $-$ $+$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2\beta q^{7}+q^{8}+\cdots\)
750.2.a.g 750.a 1.a $2$ $5.989$ \(\Q(\sqrt{5}) \) None \(2\) \(2\) \(0\) \(1\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+(1-\beta )q^{7}+\cdots\)
750.2.a.h 750.a 1.a $2$ $5.989$ \(\Q(\sqrt{5}) \) None \(2\) \(2\) \(0\) \(6\) $-$ $-$ $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{6}+(4-2\beta )q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(750))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(750)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(150))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(250))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(375))\)\(^{\oplus 2}\)