Space of modular forms of level 1225, weight 2, and Character 1225.b

• Label: 1225.2.b
• Level: $1225$
• Weight: $2$
• Character orbit: 1225.b
• Rep. character: $\chi_{1225}(99,\cdot)$
• Character field: $\Q$
• Dimension: $56$
• Newform subspaces: $14$
• Sturm bound: $280$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 1225 = 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1225.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(280\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(19\), \(31\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1225, [\chi])\).

	Total	New	Old
Modular forms	164	66	98
Cusp forms	116	56	60
Eisenstein series	48	10	38

Trace form

56 * q - 56 * q^4 + 8 * q^6 - 36 * q^9 - 8 * q^11 + 40 * q^16 - 8 * q^19 - 28 * q^24 + 40 * q^29 + 20 * q^31 - 36 * q^34 + 48 * q^36 - 16 * q^39 - 8 * q^41 + 64 * q^44 + 24 * q^46 - 52 * q^51 + 24 * q^54 + 4 * q^59 - 16 * q^61 - 36 * q^64 - 8 * q^66 - 12 * q^69 - 68 * q^71 + 60 * q^74 + 52 * q^76 - 4 * q^79 + 8 * q^81 - 24 * q^86 + 48 * q^89 + 52 * q^94 - 20 * q^96 + 44 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(1225, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1225.2.b.a	$1225$	$2$	1225.b	5.b	$2$	$2$	$2$	$9.782$	\(\Q(\sqrt{-1}) \)	None					245.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+3 i q^{3}-2 q^{4}-6 q^{6}+\cdots\)
1225.2.b.b	$1225$	$2$	1225.b	5.b	$2$	$2$	$2$	$9.782$	\(\Q(\sqrt{-1}) \)	None					245.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}-3 i q^{3}-2 q^{4}+6 q^{6}+\cdots\)
1225.2.b.c	$1225$	$2$	1225.b	5.b	$2$	$2$	$2$	$9.782$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-7}) \)					49.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+i q^{2}+q^{4}+3 i q^{8}+3 q^{9}+4 q^{11}+\cdots\)
1225.2.b.d	$1225$	$2$	1225.b	5.b	$2$	$2$	$2$	$9.782$	\(\Q(\sqrt{-1}) \)	None					35.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+2 q^{4}+2 q^{9}-3 q^{11}-2 i q^{12}+\cdots\)
1225.2.b.e	$1225$	$2$	1225.b	5.b	$2$	$4$	$4$	$9.782$	\(\Q(i, \sqrt{21})\)	\(\Q(\sqrt{-7}) \)		✓			1225.2.a.o	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}+(-4+\beta _{2})q^{4}+(-2\beta _{1}+\beta _{3})q^{8}+\cdots\)
1225.2.b.f	$1225$	$2$	1225.b	5.b	$2$	$4$	$4$	$9.782$	\(\Q(i, \sqrt{17})\)	None					35.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{2})q^{3}+(-3+\beta _{3})q^{4}+\cdots\)
1225.2.b.g	$1225$	$2$	1225.b	5.b	$2$	$4$	$4$	$9.782$	\(\Q(\zeta_{8})\)	None					35.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{2}-\beta_{2} q^{3}+(-\beta_{3}-1)q^{4}+\cdots\)
1225.2.b.h	$1225$	$2$	1225.b	5.b	$2$	$4$	$4$	$9.782$	\(\Q(\zeta_{8})\)	None					35.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{2}+\beta_{2} q^{3}+(-\beta_{3}-1)q^{4}+\cdots\)
1225.2.b.i	$1225$	$2$	1225.b	5.b	$2$	$4$	$4$	$9.782$	\(\Q(\zeta_{8})\)	None					245.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{2} q^{2}+(\beta_{2}+\beta_1)q^{3}+(-\beta_{3}-2)q^{6}+\cdots\)
1225.2.b.j	$1225$	$2$	1225.b	5.b	$2$	$4$	$4$	$9.782$	\(\Q(\zeta_{8})\)	None					245.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{2} q^{2}+(-\beta_{2}-\beta_1)q^{3}+(\beta_{3}+2)q^{6}+\cdots\)
1225.2.b.k	$1225$	$2$	1225.b	5.b	$2$	$4$	$4$	$9.782$	\(\Q(i, \sqrt{5})\)	None					175.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(2\beta _{1}+2\beta _{3})q^{3}+(1+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.2.b.l	$1225$	$2$	1225.b	5.b	$2$	$6$	$6$	$9.782$	6.0.4227136.2	None					175.2.e.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-\beta _{3}+\beta _{4})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.2.b.m	$1225$	$2$	1225.b	5.b	$2$	$6$	$6$	$9.782$	6.0.4227136.2	None					175.2.e.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(\beta _{3}-\beta _{4})q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)
1225.2.b.n	$1225$	$2$	1225.b	5.b	$2$	$8$	$8$	$9.782$	8.0.40960000.1	None		✓	✓		1225.2.a.ba	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{2}-\beta _{3})q^{2}+(-\beta _{5}+\beta _{7})q^{3}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1225, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1225, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(245, [\chi])\)\(^{\oplus 2}\)