Properties

Label 5625.2.a
Level $5625$
Weight $2$
Character orbit 5625.a
Rep. character $\chi_{5625}(1,\cdot)$
Character field $\Q$
Dimension $192$
Newform subspaces $32$
Sturm bound $1500$
Trace bound $16$

Related objects

Downloads

Learn more

Defining parameters

Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 32 \)
Sturm bound: \(1500\)
Trace bound: \(16\)
Distinguishing \(T_p\): \(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5625))\).

Total New Old
Modular forms 810 208 602
Cusp forms 691 192 499
Eisenstein series 119 16 103

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
\(+\)\(+\)\(+\)\(36\)
\(+\)\(-\)\(-\)\(44\)
\(-\)\(+\)\(-\)\(58\)
\(-\)\(-\)\(+\)\(54\)
Plus space\(+\)\(90\)
Minus space\(-\)\(102\)

Trace form

\( 192 q + 184 q^{4} + O(q^{10}) \) \( 192 q + 184 q^{4} - 4 q^{11} - 8 q^{14} + 172 q^{16} + 10 q^{19} + 26 q^{26} + 10 q^{29} + 14 q^{31} + 18 q^{34} + 6 q^{41} - 38 q^{44} - 26 q^{46} + 154 q^{49} - 50 q^{56} + 30 q^{59} + 24 q^{61} + 114 q^{64} - 24 q^{71} - 28 q^{74} + 40 q^{76} + 40 q^{79} - 4 q^{86} - 10 q^{89} + 54 q^{91} - 32 q^{94} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5625))\) into newform subspaces

Label Char Prim Dim $A$ Field CM Minimal twist Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 3 5
5625.2.a.a 5625.a 1.a $2$ $44.916$ \(\Q(\sqrt{5}) \) None 75.2.g.a \(-2\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+(-2+4\beta )q^{7}+3q^{8}+\cdots\)
5625.2.a.b 5625.a 1.a $2$ $44.916$ \(\Q(\sqrt{5}) \) None 1875.2.a.b \(-1\) \(0\) \(0\) \(-4\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}-2q^{7}+(-1+\cdots)q^{8}+\cdots\)
5625.2.a.c 5625.a 1.a $2$ $44.916$ \(\Q(\sqrt{5}) \) None 625.2.a.a \(-1\) \(0\) \(0\) \(1\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+(-1+3\beta )q^{7}+\cdots\)
5625.2.a.d 5625.a 1.a $2$ $44.916$ \(\Q(\sqrt{5}) \) None 25.2.d.a \(-1\) \(0\) \(0\) \(1\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1+\beta )q^{4}+(1-\beta )q^{7}+(-1+\cdots)q^{8}+\cdots\)
5625.2.a.e 5625.a 1.a $2$ $44.916$ \(\Q(\sqrt{5}) \) None 625.2.a.a \(1\) \(0\) \(0\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+(1-3\beta )q^{7}+\cdots\)
5625.2.a.f 5625.a 1.a $2$ $44.916$ \(\Q(\sqrt{5}) \) None 25.2.d.a \(1\) \(0\) \(0\) \(-1\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+(-1+\beta )q^{7}+\cdots\)
5625.2.a.g 5625.a 1.a $2$ $44.916$ \(\Q(\sqrt{5}) \) None 1875.2.a.b \(1\) \(0\) \(0\) \(4\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(-1+\beta )q^{4}+2q^{7}+(1-2\beta )q^{8}+\cdots\)
5625.2.a.h 5625.a 1.a $2$ $44.916$ \(\Q(\sqrt{5}) \) None 75.2.g.a \(2\) \(0\) \(0\) \(0\) $-$ $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+(2-4\beta )q^{7}-3q^{8}-2\beta q^{11}+\cdots\)
5625.2.a.i 5625.a 1.a $4$ $44.916$ 4.4.5125.1 None 75.2.g.b \(-2\) \(0\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
5625.2.a.j 5625.a 1.a $4$ $44.916$ \(\Q(\zeta_{15})^+\) None 1875.2.a.f \(-1\) \(0\) \(0\) \(5\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(\beta _{2}+\beta _{3})q^{2}+\beta _{1}q^{4}+(2+\beta _{2}+2\beta _{3})q^{7}+\cdots\)
5625.2.a.k 5625.a 1.a $4$ $44.916$ \(\Q(\zeta_{15})^+\) \(\Q(\sqrt{-3}) \) 5625.2.a.k \(0\) \(0\) \(0\) \(-5\) $+$ $+$ $N(\mathrm{U}(1))$ \(q-2q^{4}+(-\beta _{2}+2\beta _{3})q^{7}+(-1-3\beta _{1}+\cdots)q^{13}+\cdots\)
5625.2.a.l 5625.a 1.a $4$ $44.916$ \(\Q(\zeta_{15})^+\) \(\Q(\sqrt{-3}) \) 5625.2.a.k \(0\) \(0\) \(0\) \(5\) $+$ $-$ $N(\mathrm{U}(1))$ \(q-2q^{4}+(\beta _{2}-2\beta _{3})q^{7}+(1+3\beta _{1}+\beta _{3})q^{13}+\cdots\)
5625.2.a.m 5625.a 1.a $4$ $44.916$ \(\Q(\zeta_{15})^+\) None 1875.2.a.f \(1\) \(0\) \(0\) \(-5\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-\beta _{2}-\beta _{3})q^{2}+\beta _{1}q^{4}+(-2-\beta _{2}+\cdots)q^{7}+\cdots\)
5625.2.a.n 5625.a 1.a $4$ $44.916$ 4.4.5125.1 None 75.2.g.b \(2\) \(0\) \(0\) \(-2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
5625.2.a.o 5625.a 1.a $6$ $44.916$ 6.6.46840000.1 None 1875.2.a.i \(-1\) \(0\) \(0\) \(-2\) $-$ $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
5625.2.a.p 5625.a 1.a $6$ $44.916$ 6.6.44400625.1 None 75.2.g.c \(0\) \(0\) \(0\) \(-6\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{3}+\beta _{4})q^{4}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
5625.2.a.q 5625.a 1.a $6$ $44.916$ 6.6.44400625.1 None 75.2.g.c \(0\) \(0\) \(0\) \(6\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1-\beta _{3}+\beta _{4})q^{4}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
5625.2.a.r 5625.a 1.a $6$ $44.916$ 6.6.46840000.1 None 1875.2.a.i \(1\) \(0\) \(0\) \(2\) $-$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{2}-\beta _{3})q^{7}+\cdots\)
5625.2.a.s 5625.a 1.a $8$ $44.916$ 8.8.6152203125.1 None 625.2.a.e \(-5\) \(0\) \(0\) \(10\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
5625.2.a.t 5625.a 1.a $8$ $44.916$ 8.8.5444000000.1 None 75.2.i.a \(-4\) \(0\) \(0\) \(8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(1-\beta _{1}+\beta _{2})q^{4}+\cdots\)
5625.2.a.u 5625.a 1.a $8$ $44.916$ 8.8.\(\cdots\).1 None 1875.2.a.n \(-1\) \(0\) \(0\) \(12\) $-$ $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{4}+\beta _{5})q^{4}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)
5625.2.a.v 5625.a 1.a $8$ $44.916$ 8.8.\(\cdots\).1 None 225.2.h.e \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
5625.2.a.w 5625.a 1.a $8$ $44.916$ 8.8.\(\cdots\).1 None 225.2.h.e \(0\) \(0\) \(0\) \(0\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
5625.2.a.x 5625.a 1.a $8$ $44.916$ 8.8.\(\cdots\).2 None 25.2.e.a \(0\) \(0\) \(0\) \(0\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}-\beta _{6}+\cdots)q^{7}+\cdots\)
5625.2.a.y 5625.a 1.a $8$ $44.916$ 8.8.\(\cdots\).2 None 5625.2.a.y \(0\) \(0\) \(0\) \(-10\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{3}+2\beta _{5}+\cdots)q^{7}+\cdots\)
5625.2.a.z 5625.a 1.a $8$ $44.916$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 5625.2.a.z \(0\) \(0\) \(0\) \(-10\) $+$ $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1-\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
5625.2.a.ba 5625.a 1.a $8$ $44.916$ 8.8.\(\cdots\).2 None 5625.2.a.y \(0\) \(0\) \(0\) \(10\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{3}-2\beta _{5})q^{7}+\cdots\)
5625.2.a.bb 5625.a 1.a $8$ $44.916$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None 5625.2.a.z \(0\) \(0\) \(0\) \(10\) $+$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(1+\beta _{2}-\beta _{4}+\cdots)q^{7}+\cdots\)
5625.2.a.bc 5625.a 1.a $8$ $44.916$ 8.8.\(\cdots\).1 None 1875.2.a.n \(1\) \(0\) \(0\) \(-12\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{4}+\beta _{5})q^{4}+(-1-\beta _{2}+\cdots)q^{7}+\cdots\)
5625.2.a.bd 5625.a 1.a $8$ $44.916$ 8.8.5444000000.1 None 75.2.i.a \(4\) \(0\) \(0\) \(-8\) $-$ $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{3}+\cdots)q^{7}+\cdots\)
5625.2.a.be 5625.a 1.a $8$ $44.916$ 8.8.6152203125.1 None 625.2.a.e \(5\) \(0\) \(0\) \(-10\) $-$ $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(2-\beta _{1}+\beta _{2})q^{4}+(-1+\cdots)q^{7}+\cdots\)
5625.2.a.bf 5625.a 1.a $24$ $44.916$ None 225.2.m.c \(0\) \(0\) \(0\) \(0\) $+$ $-$ $\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5625))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5625)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(125))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(225))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(375))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(625))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1125))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1875))\)\(^{\oplus 2}\)