Space of modular forms of level 253, weight 2, and trivial character

• Label: 253.2.a
• Level: $253$
• Weight: $2$
• Character orbit: 253.a
• Rep. character: $\chi_{253}(1,\cdot)$
• Character field: $\Q$
• Dimension: $17$
• Newform subspaces: $4$
• Sturm bound: $48$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 253 = 11 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	253.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(48\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(253))\).

	Total	New	Old
Modular forms	26	17	9
Cusp forms	23	17	6
Eisenstein series	3	0	3

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

\(11\)	\(23\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(5\)	\(5\)	\(0\)		\(5\)	\(5\)	\(0\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(6\)	\(3\)	\(3\)		\(5\)	\(3\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(8\)	\(6\)	\(2\)		\(7\)	\(6\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(7\)	\(3\)	\(4\)		\(6\)	\(3\)	\(3\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(12\)	\(8\)	\(4\)		\(11\)	\(8\)	\(3\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(14\)	\(9\)	\(5\)		\(12\)	\(9\)	\(3\)		\(2\)	\(0\)	\(2\)

Trace form

17 * q + q^2 + 17 * q^4 - 2 * q^5 + 6 * q^6 - 4 * q^7 - 9 * q^8 + 21 * q^9 - 6 * q^10 + q^11 - 10 * q^12 - 22 * q^13 - 16 * q^14 + 12 * q^15 + 21 * q^16 - 10 * q^17 - 25 * q^18 - 26 * q^20 - 4 * q^21 + 3 * q^22 - 5 * q^23 + 11 * q^25 + 4 * q^26 - 24 * q^27 + 4 * q^28 + 10 * q^29 - 20 * q^30 - 16 * q^31 - 47 * q^32 + 4 * q^33 + 26 * q^34 - 16 * q^35 + 27 * q^36 - 22 * q^37 + 20 * q^38 + 20 * q^39 + 30 * q^40 + 10 * q^41 + 12 * q^42 + 8 * q^43 - q^44 + 6 * q^45 + 3 * q^46 - 24 * q^47 + 10 * q^48 + 5 * q^49 + 11 * q^50 - 12 * q^51 - 20 * q^52 + 22 * q^53 + 82 * q^54 - 2 * q^55 + 4 * q^56 + 8 * q^57 - 12 * q^58 - 8 * q^59 + 28 * q^60 - 22 * q^61 + 30 * q^62 - 28 * q^63 + 31 * q^64 + 12 * q^66 - 6 * q^68 - 4 * q^69 + 28 * q^70 - 8 * q^71 - 61 * q^72 - 30 * q^73 + 34 * q^74 - 52 * q^75 - 20 * q^76 - 4 * q^77 + 22 * q^78 - 4 * q^79 - 42 * q^80 + 73 * q^81 - 12 * q^82 - 12 * q^83 - 64 * q^84 - 12 * q^85 - 64 * q^86 - 12 * q^87 - 9 * q^88 + 34 * q^89 - 70 * q^90 - 8 * q^91 - 5 * q^92 + 40 * q^93 + 70 * q^94 + 20 * q^95 + 54 * q^96 - 6 * q^97 + 21 * q^98 + 13 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(253))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		11	23
253.2.a.a	$253$	$2$	253.a	1.a	$1$	$3$	$3$	$2.020$	3.3.169.1	None	✓	✓	✓	✓	253.2.a.a	$1$	$1$	\(-1\)	\(-5\)	\(-5\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-2-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
253.2.a.b	$253$	$2$	253.a	1.a	$1$	$3$	$3$	$2.020$	\(\Q(\zeta_{18})^+\)	None	✓	✓	✓	✓	253.2.a.b	$1$	$0$	\(3\)	\(3\)	\(3\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{1}-\beta _{2})q^{3}+(1+\cdots)q^{4}+\cdots\)
253.2.a.c	$253$	$2$	253.a	1.a	$1$	$5$	$5$	$2.020$	5.5.170701.1	None	✓	✓	✓	✓	253.2.a.c	$1$	$1$	\(-4\)	\(-5\)	\(-3\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1-\beta _{1}-\beta _{2})q^{3}+\cdots\)
253.2.a.d	$253$	$2$	253.a	1.a	$1$	$6$	$6$	$2.020$	6.6.8639957.1	None	✓	✓	✓	✓	253.2.a.d	$1$	$0$	\(3\)	\(7\)	\(3\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{2})q^{3}+(1-\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(253))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(253)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)