Properties

Label 1875.2.a
Level $1875$
Weight $2$
Character orbit 1875.a
Rep. character $\chi_{1875}(1,\cdot)$
Character field $\Q$
Dimension $80$
Newform subspaces $16$
Sturm bound $500$
Trace bound $3$

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Defining parameters

Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 16 \)
Sturm bound: \(500\)
Trace bound: \(3\)
Distinguishing \(T_p\): \(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1875))\).

Total New Old
Modular forms 280 80 200
Cusp forms 221 80 141
Eisenstein series 59 0 59

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.
\(+\)\(+\)\(+\)\(18\)
\(+\)\(-\)\(-\)\(22\)
\(-\)\(+\)\(-\)\(26\)
\(-\)\(-\)\(+\)\(14\)
Plus space\(+\)\(32\)
Minus space\(-\)\(48\)

Trace form

\( 80 q + 80 q^{4} + 80 q^{9} + O(q^{10}) \) \( 80 q + 80 q^{4} + 80 q^{9} + 80 q^{16} + 10 q^{19} + 10 q^{21} - 20 q^{26} - 20 q^{29} + 10 q^{31} - 20 q^{34} + 80 q^{36} + 10 q^{39} - 20 q^{41} + 60 q^{44} + 60 q^{46} + 90 q^{49} + 60 q^{56} - 10 q^{61} + 140 q^{64} + 40 q^{74} + 40 q^{76} + 40 q^{79} + 80 q^{81} + 40 q^{84} + 60 q^{86} - 20 q^{89} + 50 q^{91} - 40 q^{94} - 40 q^{96} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\) 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
1875.2.a.a 1875.a 1.a $2$ $14.972$ \(\Q(\sqrt{5}) \) None \(-2\) \(-2\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+q^{6}+(2-4\beta )q^{7}+\cdots\)
1875.2.a.b 1875.a 1.a $2$ $14.972$ \(\Q(\sqrt{5}) \) None \(-1\) \(-2\) \(0\) \(4\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
1875.2.a.c 1875.a 1.a $2$ $14.972$ \(\Q(\sqrt{5}) \) None \(1\) \(2\) \(0\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
1875.2.a.d 1875.a 1.a $2$ $14.972$ \(\Q(\sqrt{5}) \) None \(2\) \(2\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{6}+(-2+4\beta )q^{7}+\cdots\)
1875.2.a.e 1875.a 1.a $4$ $14.972$ 4.4.5125.1 None \(-2\) \(-4\) \(0\) \(-2\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.f 1875.a 1.a $4$ $14.972$ \(\Q(\zeta_{15})^+\) None \(-1\) \(4\) \(0\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(\beta _{2}+\beta _{3})q^{2}+q^{3}+\beta _{1}q^{4}+(\beta _{2}+\beta _{3})q^{6}+\cdots\)
1875.2.a.g 1875.a 1.a $4$ $14.972$ \(\Q(\zeta_{15})^+\) None \(1\) \(-4\) \(0\) \(5\) $+$ $+$ $\mathrm{SU}(2)$ \(q+(-\beta _{2}-\beta _{3})q^{2}-q^{3}+\beta _{1}q^{4}+(\beta _{2}+\cdots)q^{6}+\cdots\)
1875.2.a.h 1875.a 1.a $4$ $14.972$ 4.4.5125.1 None \(2\) \(4\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.i 1875.a 1.a $6$ $14.972$ 6.6.46840000.1 None \(-1\) \(6\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
1875.2.a.j 1875.a 1.a $6$ $14.972$ 6.6.44400625.1 None \(0\) \(-6\) \(0\) \(-6\) $+$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1-\beta _{3}+\beta _{4})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.k 1875.a 1.a $6$ $14.972$ 6.6.44400625.1 None \(0\) \(6\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1-\beta _{3}+\beta _{4})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.l 1875.a 1.a $6$ $14.972$ 6.6.46840000.1 None \(1\) \(-6\) \(0\) \(-2\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
1875.2.a.m 1875.a 1.a $8$ $14.972$ 8.8.5444000000.1 None \(-4\) \(8\) \(0\) \(-8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
1875.2.a.n 1875.a 1.a $8$ $14.972$ 8.8.\(\cdots\).1 None \(-1\) \(-8\) \(0\) \(-12\) $+$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{4}+\beta _{5})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.o 1875.a 1.a $8$ $14.972$ 8.8.\(\cdots\).1 None \(1\) \(8\) \(0\) \(12\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{4}+\beta _{5})q^{4}+\beta _{1}q^{6}+\cdots\)
1875.2.a.p 1875.a 1.a $8$ $14.972$ 8.8.5444000000.1 None \(4\) \(-8\) \(0\) \(8\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1875))\) into lower level spaces

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