Space of modular forms of level 825, weight 4, and trivial character

• Label: 825.4.a
• Level: $825$
• Weight: $4$
• Character orbit: 825.a
• Rep. character: $\chi_{825}(1,\cdot)$
• Character field: $\Q$
• Dimension: $94$
• Newform subspaces: $30$
• Sturm bound: $480$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	825.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(480\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(825))\).

	Total	New	Old
Modular forms	372	94	278
Cusp forms	348	94	254
Eisenstein series	24	0	24

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(3\)	\(5\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(12\)
\(+\)	\(+\)	\(-\)	$-$	\(9\)
\(+\)	\(-\)	\(+\)	$-$	\(12\)
\(+\)	\(-\)	\(-\)	$+$	\(14\)
\(-\)	\(+\)	\(+\)	$-$	\(11\)
\(-\)	\(+\)	\(-\)	$+$	\(14\)
\(-\)	\(-\)	\(+\)	$+$	\(12\)
\(-\)	\(-\)	\(-\)	$-$	\(10\)
Plus space			\(+\)	\(52\)
Minus space			\(-\)	\(42\)

Trace form

\( 94 q + 4 q^{2} + 356 q^{4} + 12 q^{6} - 32 q^{7} - 36 q^{8} + 846 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(825))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	11
825.4.a.a	$825$	$4$	825.a	1.a	$1$	$1$	$1$	$48.677$	\(\Q\)	None	✓	$1$	$0$	\(-5\)	\(-3\)	\(0\)	\(-3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-3q^{3}+17q^{4}+15q^{6}-3q^{7}+\cdots\)
825.4.a.b	$825$	$4$	825.a	1.a	$1$	$1$	$1$	$48.677$	\(\Q\)	None	✓	$1$	$0$	\(-4\)	\(3\)	\(0\)	\(21\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+3q^{3}+8q^{4}-12q^{6}+21q^{7}+\cdots\)
825.4.a.c	$825$	$4$	825.a	1.a	$1$	$1$	$1$	$48.677$	\(\Q\)	None	✓	$1$	$0$	\(-3\)	\(-3\)	\(0\)	\(-7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{2}-3q^{3}+q^{4}+9q^{6}-7q^{7}+\cdots\)
825.4.a.d	$825$	$4$	825.a	1.a	$1$	$1$	$1$	$48.677$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(-3\)	\(0\)	\(-36\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-3q^{3}-7q^{4}+3q^{6}-6^{2}q^{7}+\cdots\)
825.4.a.e	$825$	$4$	825.a	1.a	$1$	$1$	$1$	$48.677$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(3\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-8q^{4}-2q^{7}+9q^{9}-11q^{11}+\cdots\)
825.4.a.f	$825$	$4$	825.a	1.a	$1$	$1$	$1$	$48.677$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(3\)	\(0\)	\(26\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}-7q^{4}+3q^{6}+26q^{7}+\cdots\)
825.4.a.g	$825$	$4$	825.a	1.a	$1$	$1$	$1$	$48.677$	\(\Q\)	None	✓	$1$	$0$	\(3\)	\(3\)	\(0\)	\(7\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+3q^{3}+q^{4}+9q^{6}+7q^{7}+\cdots\)
825.4.a.h	$825$	$4$	825.a	1.a	$1$	$1$	$1$	$48.677$	\(\Q\)	None	✓	$1$	$0$	\(4\)	\(-3\)	\(0\)	\(-21\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}-3q^{3}+8q^{4}-12q^{6}-21q^{7}+\cdots\)
825.4.a.i	$825$	$4$	825.a	1.a	$1$	$1$	$1$	$48.677$	\(\Q\)	None	✓	$1$	$0$	\(5\)	\(-3\)	\(0\)	\(32\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}-3q^{3}+17q^{4}-15q^{6}+2^{5}q^{7}+\cdots\)
825.4.a.j	$825$	$4$	825.a	1.a	$1$	$1$	$1$	$48.677$	\(\Q\)	None	✓	$1$	$0$	\(5\)	\(3\)	\(0\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+3q^{3}+17q^{4}+15q^{6}+3q^{7}+\cdots\)
825.4.a.k	$825$	$4$	825.a	1.a	$1$	$2$	$2$	$48.677$	\(\Q(\sqrt{33}) \)	None	✓	$1$	$1$	\(-1\)	\(-6\)	\(0\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-3q^{3}+\beta q^{4}+3\beta q^{6}+(-2+\cdots)q^{7}+\cdots\)
825.4.a.l	$825$	$4$	825.a	1.a	$1$	$2$	$2$	$48.677$	\(\Q(\sqrt{97}) \)	None	✓	$1$	$1$	\(-1\)	\(6\)	\(0\)	\(-24\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+3q^{3}+(2^{4}+\beta )q^{4}-3\beta q^{6}+\cdots\)
825.4.a.m	$825$	$4$	825.a	1.a	$1$	$2$	$2$	$48.677$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(1\)	\(-6\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-3q^{3}+(-4+\beta )q^{4}-3\beta q^{6}+\cdots\)
825.4.a.n	$825$	$4$	825.a	1.a	$1$	$3$	$3$	$48.677$	3.3.788.1	None	✓	$1$	$1$	\(-1\)	\(9\)	\(0\)	\(16\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+3q^{3}+(-2-\beta _{1})q^{4}-3\beta _{2}q^{6}+\cdots\)
825.4.a.o	$825$	$4$	825.a	1.a	$1$	$3$	$3$	$48.677$	3.3.3368.1	None	✓	$1$	$1$	\(-1\)	\(9\)	\(0\)	\(-16\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(3+2\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
825.4.a.p	$825$	$4$	825.a	1.a	$1$	$3$	$3$	$48.677$	3.3.1957.1	None	✓	$1$	$0$	\(-1\)	\(9\)	\(0\)	\(-6\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{1}+2\beta _{2})q^{4}+\cdots\)
825.4.a.q	$825$	$4$	825.a	1.a	$1$	$3$	$3$	$48.677$	3.3.3368.1	None	✓	$1$	$1$	\(1\)	\(-9\)	\(0\)	\(16\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(3+2\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
825.4.a.r	$825$	$4$	825.a	1.a	$1$	$3$	$3$	$48.677$	3.3.47528.1	None	✓	$1$	$1$	\(2\)	\(-9\)	\(0\)	\(-10\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3q^{3}+(10+\beta _{2})q^{4}+\cdots\)
825.4.a.s	$825$	$4$	825.a	1.a	$1$	$3$	$3$	$48.677$	3.3.23612.1	None	✓	$1$	$0$	\(4\)	\(9\)	\(0\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+3q^{3}+(7+\beta _{1}+\beta _{2})q^{4}+\cdots\)
825.4.a.t	$825$	$4$	825.a	1.a	$1$	$4$	$4$	$48.677$	4.4.1540841.1	None	✓	$1$	$0$	\(-4\)	\(-12\)	\(0\)	\(-34\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3q^{3}+(7-\beta _{1}+\beta _{3})q^{4}+\cdots\)
825.4.a.u	$825$	$4$	825.a	1.a	$1$	$4$	$4$	$48.677$	4.4.91035289.2	None	✓	$1$	$0$	\(-2\)	\(12\)	\(0\)	\(-11\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+3q^{3}+(-4-\beta _{3})q^{4}+3\beta _{3}q^{6}+\cdots\)
825.4.a.v	$825$	$4$	825.a	1.a	$1$	$4$	$4$	$48.677$	4.4.91035289.2	None	✓	$1$	$0$	\(2\)	\(-12\)	\(0\)	\(11\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}-3q^{3}+(-4-\beta _{3})q^{4}+3\beta _{3}q^{6}+\cdots\)
825.4.a.w	$825$	$4$	825.a	1.a	$1$	$5$	$5$	$48.677$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-4\)	\(-15\)	\(0\)	\(38\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-3q^{3}+(9-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
825.4.a.x	$825$	$4$	825.a	1.a	$1$	$5$	$5$	$48.677$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-1\)	\(15\)	\(0\)	\(-18\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+3q^{3}+(1+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
825.4.a.y	$825$	$4$	825.a	1.a	$1$	$5$	$5$	$48.677$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(-15\)	\(0\)	\(18\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-3q^{3}+(1+\beta _{2})q^{4}-3\beta _{1}q^{6}+\cdots\)
825.4.a.z	$825$	$4$	825.a	1.a	$1$	$5$	$5$	$48.677$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(15\)	\(0\)	\(-38\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+3q^{3}+(9-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
825.4.a.ba	$825$	$4$	825.a	1.a	$1$	$7$	$7$	$48.677$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-5\)	\(21\)	\(0\)	\(-34\)		$-$	$-$	$-$	$-$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3q^{3}+(2-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
825.4.a.bb	$825$	$4$	825.a	1.a	$1$	$7$	$7$	$48.677$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(-21\)	\(0\)	\(-50\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(6+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
825.4.a.bc	$825$	$4$	825.a	1.a	$1$	$7$	$7$	$48.677$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(3\)	\(21\)	\(0\)	\(50\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+3q^{3}+(6+\beta _{2})q^{4}+3\beta _{1}q^{6}+\cdots\)
825.4.a.bd	$825$	$4$	825.a	1.a	$1$	$7$	$7$	$48.677$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(-21\)	\(0\)	\(34\)		$+$	$-$	$-$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-3q^{3}+(2-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(825))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(825)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(165))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 2}\)