Space of modular forms of level 230, weight 2, and trivial character

• Label: 230.2.a
• Level: $230$
• Weight: $2$
• Character orbit: 230.a
• Rep. character: $\chi_{230}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $4$
• Sturm bound: $72$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	230.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(72\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(230))\).

	Total	New	Old
Modular forms	40	9	31
Cusp forms	33	9	24
Eisenstein series	7	0	7

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

\(2\)	\(5\)	\(23\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)		\(0\)	\(0\)	\(0\)
\(+\)	\(+\)	\(-\)	\(-\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(7\)	\(2\)	\(5\)		\(6\)	\(2\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(6\)	\(3\)	\(3\)		\(5\)	\(3\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(4\)	\(0\)	\(4\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(15\)	\(0\)	\(15\)		\(12\)	\(0\)	\(12\)		\(3\)	\(0\)	\(3\)
Minus space			\(-\)		\(25\)	\(9\)	\(16\)		\(21\)	\(9\)	\(12\)		\(4\)	\(0\)	\(4\)

Trace form

9 * q + q^2 + 4 * q^3 + 9 * q^4 - q^5 + 8 * q^7 + q^8 + 17 * q^9 - q^10 + 4 * q^12 + 6 * q^13 + 4 * q^15 + 9 * q^16 - 6 * q^17 - 3 * q^18 + 8 * q^19 - q^20 - 16 * q^21 + 8 * q^22 - q^23 + 9 * q^25 - 14 * q^26 - 8 * q^27 + 8 * q^28 - 10 * q^29 - 4 * q^30 + 4 * q^31 + q^32 - 24 * q^33 - 6 * q^34 + 17 * q^36 + 10 * q^37 - 8 * q^38 - 16 * q^39 - q^40 - 26 * q^41 - 32 * q^42 + 24 * q^43 - 13 * q^45 - q^46 - 24 * q^47 + 4 * q^48 + 13 * q^49 + q^50 - 32 * q^51 + 6 * q^52 - 22 * q^53 - 24 * q^54 - 12 * q^55 - 48 * q^57 - 26 * q^58 - 28 * q^59 + 4 * q^60 + 10 * q^61 - 16 * q^63 + 9 * q^64 - 6 * q^65 - 8 * q^66 + 24 * q^67 - 6 * q^68 - 4 * q^69 - 4 * q^70 - 12 * q^71 - 3 * q^72 + 2 * q^73 - 6 * q^74 + 4 * q^75 + 8 * q^76 - 8 * q^77 + 16 * q^78 + 24 * q^79 - q^80 + 33 * q^81 + 10 * q^82 + 8 * q^83 - 16 * q^84 + 14 * q^85 + 24 * q^86 + 16 * q^87 + 8 * q^88 + 18 * q^89 - 13 * q^90 + 16 * q^91 - q^92 + 40 * q^93 + 8 * q^94 - 12 * q^95 + 10 * q^97 + 25 * q^98 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(230))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5	23
230.2.a.a	$230$	$2$	230.a	1.a	$1$	$2$	$2$	$1.837$	\(\Q(\sqrt{21}) \)	None	✓	✓	✓	✓	230.2.a.a	$1$	$0$	\(-2\)	\(-1\)	\(-2\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta q^{3}+q^{4}-q^{5}+\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
230.2.a.b	$230$	$2$	230.a	1.a	$1$	$2$	$2$	$1.837$	\(\Q(\sqrt{13}) \)	None	✓	✓	✓	✓	230.2.a.b	$1$	$0$	\(-2\)	\(3\)	\(2\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta )q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots\)
230.2.a.c	$230$	$2$	230.a	1.a	$1$	$2$	$2$	$1.837$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓	✓	230.2.a.c	$1$	$0$	\(2\)	\(1\)	\(2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+q^{5}+\beta q^{6}+(1+\cdots)q^{7}+\cdots\)
230.2.a.d	$230$	$2$	230.a	1.a	$1$	$3$	$3$	$1.837$	3.3.1101.1	None	✓	✓	✓	✓	230.2.a.d	$1$	$0$	\(3\)	\(1\)	\(-3\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(230))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(230)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)