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 $16$

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

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

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