Space of modular forms of level 230, weight 6, and trivial character

• Label: 230.6.a
• Level: $230$
• Weight: $6$
• Character orbit: 230.a
• Rep. character: $\chi_{230}(1,\cdot)$
• Character field: $\Q$
• Dimension: $34$
• Newform subspaces: $9$
• Sturm bound: $216$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 230 = 2 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	230.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(216\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(230))\).

	Total	New	Old
Modular forms	184	34	150
Cusp forms	176	34	142
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
\(+\)	\(+\)	\(+\)	\(+\)		\(20\)	\(3\)	\(17\)		\(19\)	\(3\)	\(16\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(25\)	\(5\)	\(20\)		\(24\)	\(5\)	\(19\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(25\)	\(5\)	\(20\)		\(24\)	\(5\)	\(19\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(22\)	\(3\)	\(19\)		\(21\)	\(3\)	\(18\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(24\)	\(6\)	\(18\)		\(23\)	\(6\)	\(17\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(22\)	\(3\)	\(19\)		\(21\)	\(3\)	\(18\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(23\)	\(3\)	\(20\)		\(22\)	\(3\)	\(19\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(23\)	\(6\)	\(17\)		\(22\)	\(6\)	\(16\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(87\)	\(12\)	\(75\)		\(83\)	\(12\)	\(71\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(97\)	\(22\)	\(75\)		\(93\)	\(22\)	\(71\)		\(4\)	\(0\)	\(4\)

Trace form

34 * q + 8 * q^2 - 8 * q^3 + 544 * q^4 + 16 * q^6 + 624 * q^7 + 128 * q^8 + 2502 * q^9 + 156 * q^11 - 128 * q^12 + 580 * q^13 + 352 * q^14 - 2200 * q^15 + 8704 * q^16 - 3172 * q^17 + 5800 * q^18 + 4428 * q^19 + 1568 * q^21 - 8560 * q^22 + 256 * q^24 + 21250 * q^25 + 11488 * q^27 + 9984 * q^28 + 812 * q^29 + 20244 * q^31 + 2048 * q^32 + 12056 * q^33 - 15312 * q^34 + 1800 * q^35 + 40032 * q^36 + 39232 * q^37 + 23408 * q^38 + 36232 * q^39 - 1112 * q^41 - 55308 * q^43 + 2496 * q^44 + 27576 * q^47 - 2048 * q^48 + 133894 * q^49 + 5000 * q^50 + 18120 * q^51 + 9280 * q^52 - 11792 * q^53 + 6208 * q^54 + 5632 * q^56 - 21472 * q^57 + 13456 * q^58 - 41904 * q^59 - 35200 * q^60 + 33624 * q^61 + 116928 * q^62 + 286712 * q^63 + 139264 * q^64 + 87800 * q^65 + 106144 * q^66 - 97492 * q^67 - 50752 * q^68 + 19600 * q^70 - 106420 * q^71 + 92800 * q^72 - 7076 * q^73 + 54016 * q^74 - 5000 * q^75 + 70848 * q^76 + 194632 * q^77 + 60448 * q^78 + 457704 * q^79 + 553522 * q^81 + 200528 * q^82 + 284428 * q^83 + 25088 * q^84 + 66200 * q^85 + 12112 * q^86 + 55784 * q^87 - 136960 * q^88 + 75100 * q^89 + 27968 * q^91 + 125912 * q^93 + 132832 * q^94 - 75800 * q^95 + 4096 * q^96 + 178892 * q^97 - 449592 * q^98 + 776372 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(230))\) 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
230.6.a.a	$230$	$6$	230.a	1.a	$1$	$1$	$1$	$36.888$	\(\Q\)	None	✓	✓			230.6.a.a	$1$	$1$	\(-4\)	\(8\)	\(-25\)	\(199\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+8q^{3}+2^{4}q^{4}-5^{2}q^{5}-2^{5}q^{6}+\cdots\)
230.6.a.b	$230$	$6$	230.a	1.a	$1$	$2$	$2$	$36.888$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓		230.6.a.b	$1$	$1$	\(-8\)	\(6\)	\(-50\)	\(-164\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-4q^{2}+(3+6\beta )q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
230.6.a.c	$230$	$6$	230.a	1.a	$1$	$3$	$3$	$36.888$	3.3.27980.1	None	✓	✓	✓	✓	230.6.a.c	$1$	$1$	\(-12\)	\(-26\)	\(75\)	\(1\)		$+$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-4q^{2}+(-9+\beta _{2})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
230.6.a.d	$230$	$6$	230.a	1.a	$1$	$3$	$3$	$36.888$	3.3.27980.1	None	✓	✓	✓	✓	230.6.a.d	$1$	$1$	\(12\)	\(-34\)	\(75\)	\(-121\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+4q^{2}+(-11-\beta _{2})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
230.6.a.e	$230$	$6$	230.a	1.a	$1$	$3$	$3$	$36.888$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	✓	✓	✓	230.6.a.e	$1$	$1$	\(12\)	\(6\)	\(-75\)	\(5\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+4q^{2}+(2-\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
230.6.a.f	$230$	$6$	230.a	1.a	$1$	$5$	$5$	$36.888$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	230.6.a.f	$1$	$0$	\(-20\)	\(1\)	\(125\)	\(102\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+5^{2}q^{5}-4\beta _{1}q^{6}+\cdots\)
230.6.a.g	$230$	$6$	230.a	1.a	$1$	$5$	$5$	$36.888$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	230.6.a.g	$1$	$0$	\(-20\)	\(5\)	\(-125\)	\(130\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-4q^{2}+(1+\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)
230.6.a.h	$230$	$6$	230.a	1.a	$1$	$6$	$6$	$36.888$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	230.6.a.h	$1$	$0$	\(24\)	\(11\)	\(150\)	\(366\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+4q^{2}+(2-\beta _{1})q^{3}+2^{4}q^{4}+5^{2}q^{5}+\cdots\)
230.6.a.i	$230$	$6$	230.a	1.a	$1$	$6$	$6$	$36.888$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	230.6.a.i	$1$	$0$	\(24\)	\(15\)	\(-150\)	\(106\)		$-$	$+$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+4q^{2}+(2+\beta _{1})q^{3}+2^{4}q^{4}-5^{2}q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(230))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(230)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)