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		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(17\)	\(4\)	\(13\)		\(16\)	\(4\)	\(12\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(12\)	\(2\)	\(10\)		\(11\)	\(2\)	\(9\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(12\)	\(2\)	\(10\)		\(11\)	\(2\)	\(9\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(15\)	\(4\)	\(11\)		\(14\)	\(4\)	\(10\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(13\)	\(1\)	\(12\)		\(12\)	\(1\)	\(11\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(15\)	\(4\)	\(11\)		\(14\)	\(4\)	\(10\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(14\)	\(4\)	\(10\)		\(13\)	\(4\)	\(9\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(14\)	\(1\)	\(13\)		\(13\)	\(1\)	\(12\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(61\)	\(16\)	\(45\)		\(57\)	\(16\)	\(41\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(51\)	\(6\)	\(45\)		\(47\)	\(6\)	\(41\)		\(4\)	\(0\)	\(4\)

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 + 84 * q^11 + 64 * q^12 + 148 * q^13 - 80 * q^14 + 80 * q^15 + 352 * q^16 + 68 * q^17 + 76 * q^18 + 212 * q^19 + 392 * q^21 + 40 * q^22 + 160 * q^24 + 550 * q^25 + 832 * q^27 + 32 * q^28 + 584 * q^29 - 440 * q^31 - 64 * q^32 - 232 * q^33 + 360 * q^34 - 180 * q^35 + 456 * q^36 + 200 * q^37 + 920 * q^38 + 304 * q^39 - 308 * q^41 + 732 * q^43 + 336 * q^44 - 912 * q^47 + 256 * q^48 + 2678 * q^49 - 100 * q^50 + 1920 * q^51 + 592 * q^52 - 872 * q^53 + 928 * q^54 - 320 * q^56 + 1568 * q^57 - 232 * q^58 - 444 * q^59 + 320 * q^60 - 512 * q^61 - 288 * q^62 - 1168 * q^63 + 1408 * q^64 + 1280 * q^65 - 848 * q^66 + 1580 * q^67 + 272 * q^68 - 280 * q^70 - 280 * q^71 + 304 * q^72 - 60 * q^73 - 1664 * q^74 + 400 * q^75 + 848 * q^76 - 1712 * q^77 - 3440 * q^78 - 1344 * q^79 - 3002 * q^81 - 360 * q^82 - 3236 * q^83 + 1568 * q^84 - 1220 * q^85 - 3320 * q^86 - 3616 * q^87 + 160 * q^88 - 2924 * q^89 - 1032 * q^91 - 5632 * q^93 - 144 * q^94 + 40 * q^95 + 640 * q^96 - 36 * q^97 - 804 * q^98 - 2548 * q^99

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

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

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