Space of modular forms of level 230, weight 4, and trivial character

• Label: 230.4.a
• Level: $230$
• Weight: $4$
• Character orbit: 230.a
• Rep. character: $\chi_{230}(1,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $10$
• Sturm bound: $144$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	230.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(144\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(230))\).

	Total	New	Old
Modular forms	112	22	90
Cusp forms	104	22	82
Eisenstein series	8	0	8

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	Dim
\(+\)	\(+\)	\(+\)	$+$	\(4\)
\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(-\)	$+$	\(4\)
\(-\)	\(+\)	\(+\)	$-$	\(1\)
\(-\)	\(+\)	\(-\)	$+$	\(4\)
\(-\)	\(-\)	\(+\)	$+$	\(4\)
\(-\)	\(-\)	\(-\)	$-$	\(1\)
Plus space			\(+\)	\(16\)
Minus space			\(-\)	\(6\)

Trace form

\( 22 q - 4 q^{2} + 16 q^{3} + 88 q^{4} + 40 q^{6} + 8 q^{7} - 16 q^{8} + 114 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(230))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5	23
230.4.a.a	$230$	$4$	230.a	1.a	$1$	$1$	$1$	$13.570$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(-5\)	\(-5\)	\(12\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-5q^{3}+4q^{4}-5q^{5}+10q^{6}+\cdots\)
230.4.a.b	$230$	$4$	230.a	1.a	$1$	$1$	$1$	$13.570$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(4\)	\(-5\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{3}+4q^{4}-5q^{5}-8q^{6}+\cdots\)
230.4.a.c	$230$	$4$	230.a	1.a	$1$	$1$	$1$	$13.570$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(7\)	\(5\)	\(20\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+7q^{3}+4q^{4}+5q^{5}-14q^{6}+\cdots\)
230.4.a.d	$230$	$4$	230.a	1.a	$1$	$1$	$1$	$13.570$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(-1\)	\(5\)	\(-32\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+4q^{4}+5q^{5}-2q^{6}+\cdots\)
230.4.a.e	$230$	$4$	230.a	1.a	$1$	$1$	$1$	$13.570$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(1\)	\(-5\)	\(-18\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+4q^{4}-5q^{5}+2q^{6}+\cdots\)
230.4.a.f	$230$	$4$	230.a	1.a	$1$	$2$	$2$	$13.570$	\(\Q(\sqrt{73}) \)	None	✓	$1$	$1$	\(-4\)	\(-3\)	\(10\)	\(-17\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1-\beta )q^{3}+4q^{4}+5q^{5}+\cdots\)
230.4.a.g	$230$	$4$	230.a	1.a	$1$	$3$	$3$	$13.570$	3.3.318165.1	None	✓	$1$	$0$	\(-6\)	\(-1\)	\(15\)	\(7\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-\beta _{1}q^{3}+4q^{4}+5q^{5}+2\beta _{1}q^{6}+\cdots\)
230.4.a.h	$230$	$4$	230.a	1.a	$1$	$4$	$4$	$13.570$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-8\)	\(-4\)	\(-20\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
230.4.a.i	$230$	$4$	230.a	1.a	$1$	$4$	$4$	$13.570$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(4\)	\(-20\)	\(26\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots\)
230.4.a.j	$230$	$4$	230.a	1.a	$1$	$4$	$4$	$13.570$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(8\)	\(14\)	\(20\)	\(8\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+2q^{2}+(4-\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(230))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(230)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)