Space of modular forms of level 165, weight 4, and trivial character

• Label: 165.4.a
• Level: $165$
• Weight: $4$
• Character orbit: 165.a
• Rep. character: $\chi_{165}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $8$
• Sturm bound: $96$
• Trace bound: $4$

Defining parameters

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(96\)
Trace bound:			\(4\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(165))\).

	Total	New	Old
Modular forms	76	20	56
Cusp forms	68	20	48
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.

\(3\)	\(5\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(11\)	\(1\)	\(10\)		\(10\)	\(1\)	\(9\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(8\)	\(3\)	\(5\)		\(7\)	\(3\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(10\)	\(3\)	\(7\)		\(9\)	\(3\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(9\)	\(3\)	\(6\)		\(8\)	\(3\)	\(5\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(10\)	\(2\)	\(8\)		\(9\)	\(2\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(9\)	\(4\)	\(5\)		\(8\)	\(4\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(11\)	\(4\)	\(7\)		\(10\)	\(4\)	\(6\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(8\)	\(0\)	\(8\)		\(7\)	\(0\)	\(7\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(40\)	\(12\)	\(28\)		\(36\)	\(12\)	\(24\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(36\)	\(8\)	\(28\)		\(32\)	\(8\)	\(24\)		\(4\)	\(0\)	\(4\)

Trace form

20 * q + 68 * q^4 + 12 * q^6 + 64 * q^7 + 180 * q^9 + 60 * q^10 + 48 * q^12 - 56 * q^13 - 104 * q^14 - 60 * q^15 + 300 * q^16 - 128 * q^17 + 152 * q^19 + 40 * q^20 + 264 * q^21 - 88 * q^22 - 136 * q^23 + 324 * q^24 + 500 * q^25 - 896 * q^26 - 24 * q^28 - 64 * q^29 + 192 * q^31 - 280 * q^32 - 132 * q^33 + 352 * q^34 - 80 * q^35 + 612 * q^36 + 504 * q^37 + 104 * q^38 + 336 * q^39 + 420 * q^40 - 656 * q^41 - 384 * q^42 - 784 * q^43 + 616 * q^44 + 1264 * q^46 + 472 * q^47 + 576 * q^48 - 28 * q^49 + 504 * q^51 - 32 * q^52 - 80 * q^53 + 108 * q^54 - 440 * q^55 - 408 * q^56 - 168 * q^57 - 2088 * q^58 - 736 * q^59 + 180 * q^60 + 696 * q^61 - 5128 * q^62 + 576 * q^63 - 1292 * q^64 - 320 * q^65 - 1136 * q^68 - 408 * q^69 + 680 * q^70 - 936 * q^71 + 3016 * q^73 - 4624 * q^74 + 544 * q^76 + 352 * q^77 + 432 * q^78 + 88 * q^79 - 1440 * q^80 + 1620 * q^81 + 4216 * q^82 - 2176 * q^83 + 2784 * q^84 + 1360 * q^85 + 2928 * q^86 - 864 * q^87 - 1056 * q^88 - 952 * q^89 + 540 * q^90 + 736 * q^91 - 6728 * q^92 + 3456 * q^93 - 6960 * q^94 - 1040 * q^95 + 2244 * q^96 + 1336 * q^97 + 2264 * q^98

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	11
165.4.a.a	$165$	$4$	165.a	1.a	$1$	$1$	$1$	$9.735$	\(\Q\)	None	✓	✓	✓	✓	165.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(-5\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-8q^{4}-5q^{5}+2q^{7}+9q^{9}+\cdots\)
165.4.a.b	$165$	$4$	165.a	1.a	$1$	$1$	$1$	$9.735$	\(\Q\)	None	✓	✓			165.4.a.b	$1$	$0$	\(1\)	\(3\)	\(-5\)	\(36\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+3q^{3}-7q^{4}-5q^{5}+3q^{6}+\cdots\)
165.4.a.c	$165$	$4$	165.a	1.a	$1$	$2$	$2$	$9.735$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	165.4.a.c	$1$	$1$	\(-1\)	\(6\)	\(-10\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+3q^{3}+(-4+\beta )q^{4}-5q^{5}+\cdots\)
165.4.a.d	$165$	$4$	165.a	1.a	$1$	$3$	$3$	$9.735$	3.3.23612.1	None	✓	✓	✓	✓	165.4.a.d	$1$	$1$	\(-4\)	\(-9\)	\(-15\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}-3q^{3}+(7+\beta _{1}+\beta _{2})q^{4}+\cdots\)
165.4.a.e	$165$	$4$	165.a	1.a	$1$	$3$	$3$	$9.735$	3.3.47528.1	None	✓	✓	✓		165.4.a.e	$1$	$0$	\(-2\)	\(9\)	\(-15\)	\(10\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+3q^{3}+(10+\beta _{2})q^{4}+\cdots\)
165.4.a.f	$165$	$4$	165.a	1.a	$1$	$3$	$3$	$9.735$	3.3.788.1	None	✓	✓	✓	✓	165.4.a.f	$1$	$1$	\(1\)	\(-9\)	\(15\)	\(-16\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-3q^{3}+(-2-\beta _{1})q^{4}+5q^{5}+\cdots\)
165.4.a.g	$165$	$4$	165.a	1.a	$1$	$3$	$3$	$9.735$	3.3.1957.1	None	✓	✓	✓	✓	165.4.a.g	$1$	$0$	\(1\)	\(-9\)	\(15\)	\(6\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-3q^{3}+(6+\beta _{1}+2\beta _{2})q^{4}+\cdots\)
165.4.a.h	$165$	$4$	165.a	1.a	$1$	$4$	$4$	$9.735$	4.4.1540841.1	None	✓	✓	✓	✓	165.4.a.h	$1$	$0$	\(4\)	\(12\)	\(20\)	\(34\)		$-$	$-$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+3q^{3}+(7-\beta _{1}+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(165))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(165)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(33))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)