Space of modular forms of level 1225, weight 2, and trivial character

• Label: 1225.2.a
• Level: $1225$
• Weight: $2$
• Character orbit: 1225.a
• Rep. character: $\chi_{1225}(1,\cdot)$
• Character field: $\Q$
• Dimension: $58$
• Newform subspaces: $29$
• Sturm bound: $280$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 29 \)
Sturm bound:			\(280\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1225))\).

	Total	New	Old
Modular forms	164	73	91
Cusp forms	117	58	59
Eisenstein series	47	15	32

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

\(5\)	\(7\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(38\)	\(15\)	\(23\)		\(27\)	\(13\)	\(14\)		\(11\)	\(2\)	\(9\)
\(+\)	\(-\)	\(-\)		\(44\)	\(18\)	\(26\)		\(32\)	\(15\)	\(17\)		\(12\)	\(3\)	\(9\)
\(-\)	\(+\)	\(-\)		\(42\)	\(21\)	\(21\)		\(30\)	\(17\)	\(13\)		\(12\)	\(4\)	\(8\)
\(-\)	\(-\)	\(+\)		\(40\)	\(19\)	\(21\)		\(28\)	\(13\)	\(15\)		\(12\)	\(6\)	\(6\)
Plus space		\(+\)		\(78\)	\(34\)	\(44\)		\(55\)	\(26\)	\(29\)		\(23\)	\(8\)	\(15\)
Minus space		\(-\)		\(86\)	\(39\)	\(47\)		\(62\)	\(32\)	\(30\)		\(24\)	\(7\)	\(17\)

Trace form

58 * q + 54 * q^4 - 4 * q^6 + 38 * q^9 + 4 * q^12 + 10 * q^13 + 54 * q^16 - 2 * q^17 + 20 * q^18 + 4 * q^19 + 16 * q^22 - 12 * q^23 + 24 * q^24 + 2 * q^26 - 12 * q^27 - 8 * q^29 + 12 * q^31 - 12 * q^33 + 2 * q^34 + 22 * q^36 + 20 * q^38 - 8 * q^39 - 22 * q^41 - 4 * q^43 + 40 * q^44 + 24 * q^46 + 4 * q^47 + 28 * q^48 + 20 * q^51 - 6 * q^52 + 36 * q^53 + 8 * q^54 - 32 * q^57 - 8 * q^58 - 32 * q^59 + 14 * q^61 + 34 * q^64 - 4 * q^67 - 10 * q^68 + 12 * q^69 - 36 * q^71 + 44 * q^72 - 6 * q^73 + 16 * q^76 + 32 * q^78 + 4 * q^79 - 46 * q^81 - 18 * q^82 + 20 * q^83 - 48 * q^86 + 28 * q^87 - 32 * q^88 - 54 * q^89 - 12 * q^92 - 8 * q^93 - 12 * q^94 - 8 * q^96 - 10 * q^97 - 60 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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	7
1225.2.a.a	$1225$	$2$	1225.a	1.a	$1$	$1$	$1$	$9.782$	\(\Q\)	None	✓				35.2.b.a	$1$	$1$	\(-2\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-2q^{6}-2q^{9}+\cdots\)
1225.2.a.b	$1225$	$2$	1225.a	1.a	$1$	$1$	$1$	$9.782$	\(\Q\)	None	✓				35.2.j.a	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{8}-2q^{9}+\cdots\)
1225.2.a.c	$1225$	$2$	1225.a	1.a	$1$	$1$	$1$	$9.782$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				49.2.a.a	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-q^{2}-q^{4}+3q^{8}-3q^{9}+4q^{11}+\cdots\)
1225.2.a.d	$1225$	$2$	1225.a	1.a	$1$	$1$	$1$	$9.782$	\(\Q\)	None	✓				35.2.j.a	$1$	$0$	\(-1\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}-2q^{9}+\cdots\)
1225.2.a.e	$1225$	$2$	1225.a	1.a	$1$	$1$	$1$	$9.782$	\(\Q\)	None	✓				35.2.a.a	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-2q^{9}-3q^{11}-2q^{12}+\cdots\)
1225.2.a.f	$1225$	$2$	1225.a	1.a	$1$	$1$	$1$	$9.782$	\(\Q\)	None	✓				35.2.j.a	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}-2q^{9}+\cdots\)
1225.2.a.g	$1225$	$2$	1225.a	1.a	$1$	$1$	$1$	$9.782$	\(\Q\)	None	✓				35.2.j.a	$1$	$1$	\(1\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}-2q^{9}+\cdots\)
1225.2.a.h	$1225$	$2$	1225.a	1.a	$1$	$1$	$1$	$9.782$	\(\Q\)	None	✓				245.2.a.a	$1$	$0$	\(2\)	\(-3\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+2q^{4}-6q^{6}+6q^{9}+\cdots\)
1225.2.a.i	$1225$	$2$	1225.a	1.a	$1$	$1$	$1$	$9.782$	\(\Q\)	None	✓				35.2.b.a	$1$	$1$	\(2\)	\(-1\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}-2q^{9}+\cdots\)
1225.2.a.j	$1225$	$2$	1225.a	1.a	$1$	$1$	$1$	$9.782$	\(\Q\)	None	✓				245.2.a.a	$1$	$0$	\(2\)	\(3\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+3q^{3}+2q^{4}+6q^{6}+6q^{9}+\cdots\)
1225.2.a.k	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{2}) \)	None	✓				35.2.e.a	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(-1-\beta )q^{3}+(1-2\beta )q^{4}+\cdots\)
1225.2.a.l	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{2}) \)	None	✓				245.2.b.f	$2$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2\beta q^{3}-q^{4}-2\beta q^{6}+3q^{8}+\cdots\)
1225.2.a.m	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{2}) \)	None	✓				35.2.e.a	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(1+\beta )q^{3}+(1-2\beta )q^{4}+\cdots\)
1225.2.a.n	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{5}) \)	None	✓				175.2.a.d	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-2+2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
1225.2.a.o	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{21}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			1225.2.a.o	$2$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-\beta q^{2}+(3+\beta )q^{4}+(-5-2\beta )q^{8}+\cdots\)
1225.2.a.p	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{2}) \)	None	✓				245.2.a.e	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{3}+(2-\beta )q^{6}-2\beta q^{8}+\cdots\)
1225.2.a.q	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{5}) \)	\(\Q(\sqrt{-35}) \)	✓				245.2.b.d	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2$	$N(\mathrm{U}(1))$	\(q-\beta q^{3}-2q^{4}+2q^{9}-3q^{11}+2\beta q^{12}+\cdots\)
1225.2.a.r	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{2}) \)	None	✓				245.2.a.e	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1-\beta )q^{3}+(-2+\beta )q^{6}-2\beta q^{8}+\cdots\)
1225.2.a.s	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{17}) \)	None	✓				35.2.a.b	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{3}+(2+\beta )q^{4}+4q^{6}+\cdots\)
1225.2.a.t	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{21}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			1225.2.a.o	$2$	$0$	\(1\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+\beta q^{2}+(3+\beta )q^{4}+(5+2\beta )q^{8}-3q^{9}+\cdots\)
1225.2.a.u	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{5}) \)	None	✓				175.2.a.d	$1$	$0$	\(1\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(2-2\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
1225.2.a.v	$1225$	$2$	1225.a	1.a	$1$	$2$	$2$	$9.782$	\(\Q(\sqrt{2}) \)	None	✓				245.2.b.f	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2\beta q^{3}-q^{4}-2\beta q^{6}-3q^{8}+\cdots\)
1225.2.a.w	$1225$	$2$	1225.a	1.a	$1$	$3$	$3$	$9.782$	3.3.257.1	None	✓		✓		175.2.e.d	$1$	$1$	\(-1\)	\(-3\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.x	$1225$	$2$	1225.a	1.a	$1$	$3$	$3$	$9.782$	3.3.257.1	None	✓				175.2.e.d	$1$	$0$	\(-1\)	\(3\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.y	$1225$	$2$	1225.a	1.a	$1$	$3$	$3$	$9.782$	3.3.257.1	None	✓				175.2.e.d	$1$	$1$	\(1\)	\(-3\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.z	$1225$	$2$	1225.a	1.a	$1$	$3$	$3$	$9.782$	3.3.257.1	None	✓		✓		175.2.e.d	$1$	$0$	\(1\)	\(3\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1225.2.a.ba	$1225$	$2$	1225.a	1.a	$1$	$4$	$4$	$9.782$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓	✓		1225.2.a.ba	$2$	$1$	\(-6\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}+\beta _{1}q^{3}+3\beta _{2}q^{4}+\cdots\)
1225.2.a.bb	$1225$	$2$	1225.a	1.a	$1$	$4$	$4$	$9.782$	\(\Q(\zeta_{24})^+\)	None	✓				245.2.b.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}-\beta _{1}q^{3}+4q^{4}+3\beta _{2}q^{6}+\cdots\)
1225.2.a.bc	$1225$	$2$	1225.a	1.a	$1$	$4$	$4$	$9.782$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓	✓			1225.2.a.ba	$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+\beta _{1}q^{3}+3\beta _{2}q^{4}+(2\beta _{1}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1225))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1225)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 2}\)