Properties

Label 1875.4.a
Level $1875$
Weight $4$
Character orbit 1875.a
Rep. character $\chi_{1875}(1,\cdot)$
Character field $\Q$
Dimension $240$
Newform subspaces $14$
Sturm bound $1000$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 14 \)
Sturm bound: \(1000\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1875))\).

Total New Old
Modular forms 780 240 540
Cusp forms 720 240 480
Eisenstein series 60 0 60

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.
\(+\)\(+\)\(+\)\(62\)
\(+\)\(-\)\(-\)\(58\)
\(-\)\(+\)\(-\)\(54\)
\(-\)\(-\)\(+\)\(66\)
Plus space\(+\)\(128\)
Minus space\(-\)\(112\)

Trace form

\( 240 q + 960 q^{4} + 2160 q^{9} + O(q^{10}) \) \( 240 q + 960 q^{4} + 2160 q^{9} + 3840 q^{16} - 180 q^{19} - 120 q^{21} + 440 q^{26} + 260 q^{29} + 180 q^{31} - 40 q^{34} + 8640 q^{36} + 420 q^{39} - 460 q^{41} - 2580 q^{44} - 2220 q^{46} + 12480 q^{49} - 420 q^{56} - 580 q^{61} + 16380 q^{64} + 380 q^{74} - 2880 q^{76} - 2520 q^{79} + 19440 q^{81} - 1920 q^{84} + 4980 q^{86} - 3340 q^{89} + 900 q^{91} + 3720 q^{94} + 3480 q^{96} + O(q^{100}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1875))\) 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
1875.4.a.a 1875.a 1.a $10$ $110.629$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-1\) \(-30\) \(0\) \(51\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
1875.4.a.b 1875.a 1.a $10$ $110.629$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(-1\) \(30\) \(0\) \(21\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(2+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1875.4.a.c 1875.a 1.a $10$ $110.629$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(1\) \(-30\) \(0\) \(-21\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(2+\beta _{2}-\beta _{3})q^{4}+\cdots\)
1875.4.a.d 1875.a 1.a $10$ $110.629$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None \(1\) \(30\) \(0\) \(-51\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
1875.4.a.e 1875.a 1.a $14$ $110.629$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(-8\) \(42\) \(0\) \(-29\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1875.4.a.f 1875.a 1.a $14$ $110.629$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(-42\) \(0\) \(27\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
1875.4.a.g 1875.a 1.a $14$ $110.629$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(0\) \(42\) \(0\) \(-27\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+3q^{3}+(4+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
1875.4.a.h 1875.a 1.a $14$ $110.629$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None \(8\) \(-42\) \(0\) \(29\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-3q^{3}+(4-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1875.4.a.i 1875.a 1.a $16$ $110.629$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(-1\) \(48\) \(0\) \(-52\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+3q^{3}+(2+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
1875.4.a.j 1875.a 1.a $16$ $110.629$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(1\) \(-48\) \(0\) \(52\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(2+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
1875.4.a.k 1875.a 1.a $24$ $110.629$ None \(-1\) \(-72\) \(0\) \(-62\) $+$ $+$ $\mathrm{SU}(2)$
1875.4.a.l 1875.a 1.a $24$ $110.629$ None \(1\) \(72\) \(0\) \(62\) $-$ $-$ $\mathrm{SU}(2)$
1875.4.a.m 1875.a 1.a $32$ $110.629$ None \(-8\) \(-96\) \(0\) \(-56\) $+$ $-$ $\mathrm{SU}(2)$
1875.4.a.n 1875.a 1.a $32$ $110.629$ None \(8\) \(96\) \(0\) \(56\) $-$ $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1875))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1875)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(375))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(625))\)\(^{\oplus 2}\)