Space of modular forms of level 161, weight 4, and trivial character

• Label: 161.4.a
• Level: $161$
• Weight: $4$
• Character orbit: 161.a
• Rep. character: $\chi_{161}(1,\cdot)$
• Character field: $\Q$
• Dimension: $34$
• Newform subspaces: $4$
• Sturm bound: $64$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 161 = 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	161.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(64\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(161))\).

	Total	New	Old
Modular forms	50	34	16
Cusp forms	46	34	12
Eisenstein series	4	0	4

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

\(7\)	\(23\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(15\)	\(9\)	\(6\)		\(14\)	\(9\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(11\)	\(8\)	\(3\)		\(10\)	\(8\)	\(2\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(10\)	\(5\)	\(5\)		\(9\)	\(5\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(14\)	\(12\)	\(2\)		\(13\)	\(12\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(29\)	\(21\)	\(8\)		\(27\)	\(21\)	\(6\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(21\)	\(13\)	\(8\)		\(19\)	\(13\)	\(6\)		\(2\)	\(0\)	\(2\)

Trace form

34 * q - 4 * q^3 + 144 * q^4 - 16 * q^5 - 10 * q^6 - 6 * q^8 + 434 * q^9 + 152 * q^10 - 92 * q^11 - 78 * q^12 - 168 * q^13 - 40 * q^15 + 496 * q^16 - 140 * q^17 + 34 * q^18 + 180 * q^19 + 100 * q^20 - 112 * q^21 - 360 * q^22 + 138 * q^23 + 468 * q^24 + 406 * q^25 + 286 * q^26 - 268 * q^27 - 156 * q^29 + 16 * q^30 + 240 * q^31 + 1032 * q^32 + 608 * q^33 - 540 * q^34 + 280 * q^35 + 2094 * q^36 + 872 * q^37 - 284 * q^38 + 364 * q^39 + 820 * q^40 - 756 * q^41 - 140 * q^42 + 268 * q^43 - 1824 * q^44 - 296 * q^45 + 184 * q^46 + 1192 * q^47 - 3458 * q^48 + 1666 * q^49 - 320 * q^50 + 336 * q^51 - 3182 * q^52 - 144 * q^53 - 2454 * q^54 + 1280 * q^55 - 1944 * q^57 + 214 * q^58 + 472 * q^59 - 148 * q^60 - 1984 * q^61 + 1374 * q^62 + 3126 * q^64 - 2536 * q^65 - 4084 * q^66 - 548 * q^67 - 3180 * q^68 + 784 * q^70 + 1292 * q^71 - 1474 * q^72 + 900 * q^73 - 6856 * q^74 - 620 * q^75 + 5160 * q^76 + 840 * q^77 + 398 * q^78 + 2480 * q^79 - 6712 * q^80 + 4802 * q^81 - 1094 * q^82 + 268 * q^83 - 1792 * q^84 - 768 * q^85 - 2116 * q^86 + 44 * q^87 + 1004 * q^88 - 1284 * q^89 - 2752 * q^90 + 504 * q^91 + 1104 * q^92 + 2968 * q^93 + 1846 * q^94 + 3024 * q^95 - 4062 * q^96 - 2532 * q^97 - 884 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(161))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7	23
161.4.a.a	$161$	$4$	161.a	1.a	$1$	$5$	$5$	$9.499$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	161.4.a.a	$1$	$1$	\(-4\)	\(-11\)	\(-4\)	\(35\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-2-\beta _{4})q^{3}+\beta _{2}q^{4}+\cdots\)
161.4.a.b	$161$	$4$	161.a	1.a	$1$	$8$	$8$	$9.499$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	161.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(-24\)	\(-56\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(3+\beta _{1}+\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots\)
161.4.a.c	$161$	$4$	161.a	1.a	$1$	$9$	$9$	$9.499$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓	✓	161.4.a.c	$1$	$0$	\(0\)	\(9\)	\(-4\)	\(-63\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(5+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
161.4.a.d	$161$	$4$	161.a	1.a	$1$	$12$	$12$	$9.499$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓	✓	161.4.a.d	$1$	$0$	\(4\)	\(1\)	\(16\)	\(84\)		$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{4}q^{3}+(6+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(161))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(161)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 2}\)