Space of modular forms of level 5225, weight 2, and trivial character

• Label: 5225.2.a
• Level: $5225$
• Weight: $2$
• Character orbit: 5225.a
• Rep. character: $\chi_{5225}(1,\cdot)$
• Character field: $\Q$
• Dimension: $286$
• Newform subspaces: $29$
• Sturm bound: $1200$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(1200\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(5225))\).

	Total	New	Old
Modular forms	612	286	326
Cusp forms	589	286	303
Eisenstein series	23	0	23

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

\(5\)	\(11\)	\(19\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(+\)	\(32\)
\(+\)	\(+\)	\(-\)	\(-\)	\(37\)
\(+\)	\(-\)	\(+\)	\(-\)	\(35\)
\(+\)	\(-\)	\(-\)	\(+\)	\(30\)
\(-\)	\(+\)	\(+\)	\(-\)	\(41\)
\(-\)	\(+\)	\(-\)	\(+\)	\(31\)
\(-\)	\(-\)	\(+\)	\(+\)	\(35\)
\(-\)	\(-\)	\(-\)	\(-\)	\(45\)
Plus space			\(+\)	\(128\)
Minus space			\(-\)	\(158\)

Trace form

\( 286 q + 2 q^{2} + 2 q^{3} + 288 q^{4} - 4 q^{6} - 6 q^{8} + 300 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(5225))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	11	19
5225.2.a.a	$5225$	$2$	5225.a	1.a	$1$	$1$	$1$	$41.722$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}+3q^{8}-3q^{9}-q^{11}-2q^{13}+\cdots\)
5225.2.a.b	$5225$	$2$	5225.a	1.a	$1$	$1$	$1$	$41.722$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+4q^{7}-2q^{9}+q^{11}+\cdots\)
5225.2.a.c	$5225$	$2$	5225.a	1.a	$1$	$1$	$1$	$41.722$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(2\)	\(0\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}-q^{4}+2q^{6}+2q^{7}-3q^{8}+\cdots\)
5225.2.a.d	$5225$	$2$	5225.a	1.a	$1$	$2$	$2$	$41.722$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$2$	\(-2\)	\(-4\)	\(0\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-2q^{3}+(1-2\beta )q^{4}+\cdots\)
5225.2.a.e	$5225$	$2$	5225.a	1.a	$1$	$2$	$2$	$41.722$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(-4\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-2q^{3}+(1-2\beta )q^{4}+\cdots\)
5225.2.a.f	$5225$	$2$	5225.a	1.a	$1$	$2$	$2$	$41.722$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(2\)	\(0\)	\(4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1-\beta )q^{3}+(-2+\beta )q^{6}+(2+\cdots)q^{7}+\cdots\)
5225.2.a.g	$5225$	$2$	5225.a	1.a	$1$	$2$	$2$	$41.722$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$0$	\(2\)	\(4\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+2q^{3}+(1+2\beta )q^{4}+(2+\cdots)q^{6}+\cdots\)
5225.2.a.h	$5225$	$2$	5225.a	1.a	$1$	$5$	$5$	$41.722$	5.5.246832.1	None	✓	$1$	$0$	\(-2\)	\(-1\)	\(0\)	\(-6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}-\beta _{3}q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
5225.2.a.i	$5225$	$2$	5225.a	1.a	$1$	$5$	$5$	$41.722$	5.5.36497.1	None	✓	$1$	$1$	\(1\)	\(3\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{2}+(1-\beta _{1})q^{3}+\beta _{3}q^{4}+(-1+\cdots)q^{6}+\cdots\)
5225.2.a.j	$5225$	$2$	5225.a	1.a	$1$	$5$	$5$	$41.722$	\(\Q(\zeta_{22})^+\)	None	✓	$1$	$0$	\(3\)	\(7\)	\(0\)	\(11\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{3}-\beta _{4})q^{2}+(2-\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
5225.2.a.k	$5225$	$2$	5225.a	1.a	$1$	$6$	$6$	$41.722$	6.6.131947641.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(0\)	\(-5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(2+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
5225.2.a.l	$5225$	$2$	5225.a	1.a	$1$	$6$	$6$	$41.722$	6.6.7281497.1	None	✓	$1$	$1$	\(2\)	\(1\)	\(0\)	\(-5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}-\beta _{3}q^{3}+(\beta _{4}+\beta _{5})q^{4}+\beta _{2}q^{6}+\cdots\)
5225.2.a.m	$5225$	$2$	5225.a	1.a	$1$	$7$	$7$	$41.722$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(-1\)	\(-3\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{6}+\cdots\)
5225.2.a.n	$5225$	$2$	5225.a	1.a	$1$	$7$	$7$	$41.722$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(-2\)	\(0\)	\(-10\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+\cdots\)
5225.2.a.o	$5225$	$2$	5225.a	1.a	$1$	$8$	$8$	$41.722$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(6\)	\(7\)	\(0\)	\(11\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{6})q^{3}+(2-\beta _{1}+\cdots)q^{4}+\cdots\)
5225.2.a.p	$5225$	$2$	5225.a	1.a	$1$	$9$	$9$	$41.722$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$1$	\(-3\)	\(-3\)	\(0\)	\(-13\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}-\beta _{8}q^{6}+\cdots\)
5225.2.a.q	$5225$	$2$	5225.a	1.a	$1$	$9$	$9$	$41.722$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	$1$	$0$	\(-3\)	\(-3\)	\(0\)	\(9\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(1-\beta _{1}-\beta _{8})q^{4}+\cdots\)
5225.2.a.r	$5225$	$2$	5225.a	1.a	$1$	$15$	$15$	$41.722$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(-5\)	\(-4\)	\(0\)	\(-21\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(-\beta _{4}+\cdots)q^{6}+\cdots\)
5225.2.a.s	$5225$	$2$	5225.a	1.a	$1$	$15$	$15$	$41.722$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(-5\)	\(-4\)	\(0\)	\(-15\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{6}+\cdots\)
5225.2.a.t	$5225$	$2$	5225.a	1.a	$1$	$15$	$15$	$41.722$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(-1\)	\(4\)	\(0\)	\(11\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{8}q^{3}+(1+\beta _{2})q^{4}+(-\beta _{11}+\cdots)q^{6}+\cdots\)
5225.2.a.u	$5225$	$2$	5225.a	1.a	$1$	$15$	$15$	$41.722$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(-1\)	\(4\)	\(0\)	\(17\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{7}q^{3}+(1+\beta _{2})q^{4}+\beta _{4}q^{6}+\cdots\)
5225.2.a.v	$5225$	$2$	5225.a	1.a	$1$	$15$	$15$	$41.722$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(-4\)	\(0\)	\(-17\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{7}q^{3}+(1+\beta _{2})q^{4}+\beta _{4}q^{6}+\cdots\)
5225.2.a.w	$5225$	$2$	5225.a	1.a	$1$	$15$	$15$	$41.722$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(1\)	\(-4\)	\(0\)	\(-11\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{8}q^{3}+(1+\beta _{2})q^{4}+(-\beta _{11}+\cdots)q^{6}+\cdots\)
5225.2.a.x	$5225$	$2$	5225.a	1.a	$1$	$15$	$15$	$41.722$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(4\)	\(0\)	\(15\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{6}+\cdots\)
5225.2.a.y	$5225$	$2$	5225.a	1.a	$1$	$15$	$15$	$41.722$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(5\)	\(4\)	\(0\)	\(21\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(-\beta _{4}+\cdots)q^{6}+\cdots\)
5225.2.a.z	$5225$	$2$	5225.a	1.a	$1$	$16$	$16$	$41.722$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{11}q^{3}+\beta _{2}q^{4}+(-1+\beta _{6}+\cdots)q^{6}+\cdots\)
5225.2.a.ba	$5225$	$2$	5225.a	1.a	$1$	$20$	$20$	$41.722$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$2^{8}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(-1+\cdots)q^{6}+\cdots\)
5225.2.a.bb	$5225$	$2$	5225.a	1.a	$1$	$22$	$22$	$41.722$		None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$		$\mathrm{SU}(2)$
5225.2.a.bc	$5225$	$2$	5225.a	1.a	$1$	$30$	$30$	$41.722$		None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(5225))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(5225)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(475))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1045))\)\(^{\oplus 2}\)