↑

Properties

Label 375.4.a
Level $375$
Weight $4$
Character orbit 375.a
Rep. character $\chi_{375}(1,\cdot)$
Character field $\Q$
Dimension $48$
Newform subspaces $8$
Sturm bound $200$
Trace bound $2$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 375.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 8 \)
Sturm bound: \(200\)
Trace bound: \(2\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(375))\).

Total New Old
Modular forms 160 48 112
Cusp forms 140 48 92
Eisenstein series 20 0 20

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
\(+\)\(+\)$+$\(14\)
\(+\)\(-\)$-$\(10\)
\(-\)\(+\)$-$\(10\)
\(-\)\(-\)$+$\(14\)
Plus space\(+\)\(28\)
Minus space\(-\)\(20\)

Trace form

\( 48 q + 186 q^{4} + 6 q^{6} + 432 q^{9} + O(q^{10}) \) \( 48 q + 186 q^{4} + 6 q^{6} + 432 q^{9} - 28 q^{11} - 96 q^{14} + 730 q^{16} + 344 q^{19} + 192 q^{21} + 162 q^{24} + 336 q^{26} - 112 q^{29} - 352 q^{31} + 412 q^{34} + 1674 q^{36} - 684 q^{39} - 412 q^{41} + 4408 q^{44} + 4268 q^{46} + 980 q^{49} + 204 q^{51} + 54 q^{54} + 840 q^{56} - 124 q^{59} - 2440 q^{61} + 494 q^{64} - 372 q^{66} - 216 q^{69} - 344 q^{71} - 5416 q^{74} + 4544 q^{76} + 5040 q^{79} + 3888 q^{81} + 3036 q^{84} - 7768 q^{86} + 124 q^{89} - 352 q^{91} + 4986 q^{94} + 5472 q^{96} - 252 q^{99} + O(q^{100}) \)

Display 100 coefficients 10 coefficients

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(375))\) 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
375.4.a.a 375.a 1.a $4$ $22.126$ \(\Q(\sqrt{5}, \sqrt{21})\) None \(-4\) \(12\) \(0\) \(-3\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1}+\beta _{3})q^{2}+3q^{3}+(-1+\cdots)q^{4}+\cdots\)
375.4.a.b 375.a 1.a $4$ $22.126$ \(\Q(\sqrt{5}, \sqrt{21})\) None \(4\) \(-12\) \(0\) \(3\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1}+\beta _{3})q^{2}-3q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
375.4.a.c 375.a 1.a $6$ $22.126$ 6.6.920588125.1 None \(-5\) \(18\) \(0\) \(-28\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{2})q^{2}+3q^{3}+(4-\beta _{1}-2\beta _{2}+\cdots)q^{4}+\cdots\)
375.4.a.d 375.a 1.a $6$ $22.126$ 6.6.3126413125.1 None \(-3\) \(-18\) \(0\) \(-44\) $+$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3q^{3}+(5-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
375.4.a.e 375.a 1.a $6$ $22.126$ 6.6.3126413125.1 None \(3\) \(18\) \(0\) \(44\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(5-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
375.4.a.f 375.a 1.a $6$ $22.126$ 6.6.920588125.1 None \(5\) \(-18\) \(0\) \(28\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{2})q^{2}-3q^{3}+(4-\beta _{1}-2\beta _{2}+\cdots)q^{4}+\cdots\)
375.4.a.g 375.a 1.a $8$ $22.126$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(-7\) \(-24\) \(0\) \(-19\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-3q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
375.4.a.h 375.a 1.a $8$ $22.126$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(7\) \(24\) \(0\) \(19\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(5-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(375))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(375)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 2}\)