Space of modular forms of level 234, weight 4, and trivial character

• Label: 234.4.a
• Level: $234$
• Weight: $4$
• Character orbit: 234.a
• Rep. character: $\chi_{234}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $13$
• Sturm bound: $168$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 234 = 2 \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	234.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(168\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(234))\).

	Total	New	Old
Modular forms	134	15	119
Cusp forms	118	15	103
Eisenstein series	16	0	16

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

\(2\)	\(3\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(1\)
Plus space			\(+\)	\(9\)
Minus space			\(-\)	\(6\)

Trace form

15 * q - 2 * q^2 + 60 * q^4 + 22 * q^5 - 20 * q^7 - 8 * q^8 + 24 * q^10 - 28 * q^11 + 13 * q^13 + 108 * q^14 + 240 * q^16 - 120 * q^17 + 168 * q^19 + 88 * q^20 + 32 * q^22 + 172 * q^23 + 347 * q^25 - 78 * q^26 - 80 * q^28 + 118 * q^29 + 284 * q^31 - 32 * q^32 - 284 * q^34 + 370 * q^35 - 294 * q^37 + 400 * q^38 + 96 * q^40 + 750 * q^41 - 86 * q^43 - 112 * q^44 + 1160 * q^46 + 404 * q^47 + 1041 * q^49 - 414 * q^50 + 52 * q^52 - 750 * q^53 - 1104 * q^55 + 432 * q^56 + 356 * q^58 - 1976 * q^59 - 1274 * q^61 + 240 * q^62 + 960 * q^64 - 676 * q^65 - 2668 * q^67 - 480 * q^68 + 1128 * q^70 + 308 * q^71 - 726 * q^73 + 760 * q^74 + 672 * q^76 + 1928 * q^77 - 636 * q^79 + 352 * q^80 - 380 * q^82 - 1452 * q^83 + 712 * q^85 + 1296 * q^86 + 128 * q^88 - 378 * q^89 + 806 * q^91 + 688 * q^92 - 2260 * q^94 - 4260 * q^95 + 398 * q^97 - 2898 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(234))\) 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	3	13
234.4.a.a	$234$	$4$	234.a	1.a	$1$	$1$	$1$	$13.806$	\(\Q\)	None	✓				26.4.a.b	$1$	$0$	\(-2\)	\(0\)	\(-17\)	\(-35\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-17q^{5}-35q^{7}-8q^{8}+\cdots\)
234.4.a.b	$234$	$4$	234.a	1.a	$1$	$1$	$1$	$13.806$	\(\Q\)	None	✓				78.4.a.e	$1$	$0$	\(-2\)	\(0\)	\(-6\)	\(20\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-6q^{5}+20q^{7}-8q^{8}+\cdots\)
234.4.a.c	$234$	$4$	234.a	1.a	$1$	$1$	$1$	$13.806$	\(\Q\)	None	✓				78.4.a.f	$1$	$1$	\(-2\)	\(0\)	\(-4\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}-4q^{5}+4q^{7}-8q^{8}+\cdots\)
234.4.a.d	$234$	$4$	234.a	1.a	$1$	$1$	$1$	$13.806$	\(\Q\)	None	✓	✓	✓	✓	234.4.a.d	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(-26\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+2q^{5}-26q^{7}-8q^{8}+\cdots\)
234.4.a.e	$234$	$4$	234.a	1.a	$1$	$1$	$1$	$13.806$	\(\Q\)	None	✓				26.4.a.c	$1$	$0$	\(-2\)	\(0\)	\(18\)	\(20\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+18q^{5}+20q^{7}-8q^{8}+\cdots\)
234.4.a.f	$234$	$4$	234.a	1.a	$1$	$1$	$1$	$13.806$	\(\Q\)	None	✓				78.4.a.d	$1$	$1$	\(-2\)	\(0\)	\(20\)	\(-32\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+20q^{5}-2^{5}q^{7}-8q^{8}+\cdots\)
234.4.a.g	$234$	$4$	234.a	1.a	$1$	$1$	$1$	$13.806$	\(\Q\)	None	✓				26.4.a.a	$1$	$0$	\(2\)	\(0\)	\(-11\)	\(19\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-11q^{5}+19q^{7}+8q^{8}+\cdots\)
234.4.a.h	$234$	$4$	234.a	1.a	$1$	$1$	$1$	$13.806$	\(\Q\)	None	✓		✓	✓	78.4.a.c	$1$	$1$	\(2\)	\(0\)	\(-10\)	\(-8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-10q^{5}-8q^{7}+8q^{8}+\cdots\)
234.4.a.i	$234$	$4$	234.a	1.a	$1$	$1$	$1$	$13.806$	\(\Q\)	None	✓	✓	✓	✓	234.4.a.d	$1$	$1$	\(2\)	\(0\)	\(-2\)	\(-26\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}-2q^{5}-26q^{7}+8q^{8}+\cdots\)
234.4.a.j	$234$	$4$	234.a	1.a	$1$	$1$	$1$	$13.806$	\(\Q\)	None	✓				78.4.a.b	$1$	$0$	\(2\)	\(0\)	\(16\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+2^{4}q^{5}-8q^{7}+8q^{8}+\cdots\)
234.4.a.k	$234$	$4$	234.a	1.a	$1$	$1$	$1$	$13.806$	\(\Q\)	None	✓				78.4.a.a	$1$	$0$	\(2\)	\(0\)	\(16\)	\(28\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+2^{4}q^{5}+28q^{7}+8q^{8}+\cdots\)
234.4.a.l	$234$	$4$	234.a	1.a	$1$	$2$	$2$	$13.806$	\(\Q(\sqrt{22}) \)	None	✓	✓	✓	✓	234.4.a.l	$1$	$1$	\(-4\)	\(0\)	\(-8\)	\(12\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-2q^{2}+4q^{4}+(-4+\beta )q^{5}+(6-\beta )q^{7}+\cdots\)
234.4.a.m	$234$	$4$	234.a	1.a	$1$	$2$	$2$	$13.806$	\(\Q(\sqrt{22}) \)	None	✓	✓	✓	✓	234.4.a.l	$1$	$0$	\(4\)	\(0\)	\(8\)	\(12\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+2q^{2}+4q^{4}+(4+\beta )q^{5}+(6+\beta )q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(234))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(234)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 2}\)