Space of modular forms of level 65, weight 6, and trivial character

• Label: 65.6.a
• Level: $65$
• Weight: $6$
• Character orbit: 65.a
• Rep. character: $\chi_{65}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $5$
• Sturm bound: $42$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	65.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(42\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(65))\).

	Total	New	Old
Modular forms	36	20	16
Cusp forms	32	20	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.

\(5\)	\(13\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(8\)	\(5\)	\(3\)		\(7\)	\(5\)	\(2\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(10\)	\(6\)	\(4\)		\(9\)	\(6\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(10\)	\(6\)	\(4\)		\(9\)	\(6\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(8\)	\(3\)	\(5\)		\(7\)	\(3\)	\(4\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(16\)	\(8\)	\(8\)		\(14\)	\(8\)	\(6\)		\(2\)	\(0\)	\(2\)
Minus space		\(-\)		\(20\)	\(12\)	\(8\)		\(18\)	\(12\)	\(6\)		\(2\)	\(0\)	\(2\)

Trace form

20 * q - 4 * q^2 + 44 * q^3 + 344 * q^4 - 50 * q^5 + 244 * q^6 - 196 * q^7 - 612 * q^8 + 2132 * q^9 + 100 * q^10 + 292 * q^11 - 68 * q^12 - 338 * q^13 - 232 * q^14 - 900 * q^15 + 488 * q^16 + 3140 * q^17 + 3840 * q^18 + 856 * q^19 - 2400 * q^20 - 9028 * q^21 + 10200 * q^22 + 744 * q^23 + 14632 * q^24 + 12500 * q^25 + 8228 * q^27 + 12348 * q^28 + 7384 * q^29 - 7600 * q^30 - 612 * q^31 - 34008 * q^32 - 28732 * q^33 - 27612 * q^34 + 3100 * q^35 + 36648 * q^36 + 28728 * q^37 - 41244 * q^38 + 15600 * q^40 + 44252 * q^41 - 56460 * q^42 - 5968 * q^43 - 81764 * q^44 + 5550 * q^45 - 5012 * q^46 - 57124 * q^47 + 85632 * q^48 + 76548 * q^49 - 2500 * q^50 - 82240 * q^51 - 16224 * q^52 + 40664 * q^53 - 10564 * q^54 + 1900 * q^55 - 50832 * q^56 - 71116 * q^57 - 154172 * q^58 - 28320 * q^59 - 76000 * q^60 + 60520 * q^61 + 172084 * q^62 + 123668 * q^63 - 97912 * q^64 - 16900 * q^65 + 53000 * q^66 - 120036 * q^67 - 2284 * q^68 + 34744 * q^69 + 47000 * q^70 - 67896 * q^71 + 163584 * q^72 - 91872 * q^73 + 85840 * q^74 + 27500 * q^75 + 224920 * q^76 + 71940 * q^77 + 73684 * q^78 + 74568 * q^79 - 89600 * q^80 + 242196 * q^81 - 178452 * q^82 + 117844 * q^83 - 352500 * q^84 + 101600 * q^85 + 7328 * q^86 + 50148 * q^87 - 133444 * q^88 - 210568 * q^89 + 103600 * q^90 + 37180 * q^91 + 58188 * q^92 + 165988 * q^93 + 207984 * q^94 + 75800 * q^95 + 155676 * q^96 + 388540 * q^97 + 30052 * q^98 + 325856 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(65))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	13
65.6.a.a	$65$	$6$	65.a	1.a	$1$	$1$	$1$	$10.425$	\(\Q\)	None	✓	✓			65.6.a.a	$1$	$1$	\(5\)	\(6\)	\(-25\)	\(-244\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+5q^{2}+6q^{3}-7q^{4}-5^{2}q^{5}+30q^{6}+\cdots\)
65.6.a.b	$65$	$6$	65.a	1.a	$1$	$3$	$3$	$10.425$	3.3.49857.1	None	✓	✓	✓	✓	65.6.a.b	$1$	$1$	\(-2\)	\(-16\)	\(75\)	\(-208\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-5-\beta _{2})q^{3}+(-4+\cdots)q^{4}+\cdots\)
65.6.a.c	$65$	$6$	65.a	1.a	$1$	$4$	$4$	$10.425$	4.4.1878612.1	None	✓	✓	✓		65.6.a.c	$1$	$1$	\(-9\)	\(-4\)	\(-100\)	\(-136\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1}+\beta _{2})q^{2}+(-1-\beta _{2}-\beta _{3})q^{3}+\cdots\)
65.6.a.d	$65$	$6$	65.a	1.a	$1$	$6$	$6$	$10.425$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	65.6.a.d	$1$	$0$	\(0\)	\(38\)	\(-150\)	\(220\)		$+$	$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(6+\beta _{1}+\beta _{2})q^{3}+(22+\beta _{2}+\cdots)q^{4}+\cdots\)
65.6.a.e	$65$	$6$	65.a	1.a	$1$	$6$	$6$	$10.425$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	65.6.a.e	$1$	$0$	\(2\)	\(20\)	\(150\)	\(172\)		$-$	$+$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3-\beta _{2})q^{3}+(23-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(65))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(65)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)