Space of modular forms of level 65, weight 2, and trivial character

• Label: 65.2.a
• Level: $65$
• Weight: $2$
• Character orbit: 65.a
• Rep. character: $\chi_{65}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $3$
• Sturm bound: $14$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 65 = 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	65.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(14\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(65))\).

	Total	New	Old
Modular forms	8	5	3
Cusp forms	5	5	0
Eisenstein series	3	0	3

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	Dim
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(2\)
Plus space		\(+\)	\(1\)
Minus space		\(-\)	\(4\)

Trace form

5 * q - 3 * q^2 + 3 * q^4 - q^5 + 4 * q^7 - 3 * q^8 + q^9 - q^10 - 4 * q^12 - q^13 - 8 * q^14 - 5 * q^16 - 2 * q^17 - 11 * q^18 - 4 * q^19 + q^20 + 4 * q^21 - 4 * q^22 + 4 * q^24 + 5 * q^25 + 3 * q^26 + 12 * q^27 + 28 * q^28 - 10 * q^29 + 8 * q^30 + 12 * q^31 + q^32 - 20 * q^33 + 6 * q^34 + 4 * q^35 - q^36 - 10 * q^37 + 24 * q^38 + 4 * q^39 - 9 * q^40 - 18 * q^41 - 4 * q^42 + 12 * q^43 + 4 * q^44 - 5 * q^45 + 8 * q^46 + 12 * q^47 - 8 * q^48 + 13 * q^49 - 3 * q^50 - 24 * q^51 + q^52 - 10 * q^53 - 20 * q^54 + 8 * q^55 - 32 * q^56 - 4 * q^57 + 2 * q^58 + 12 * q^59 - 12 * q^60 - 10 * q^61 - 8 * q^62 - 4 * q^63 - 5 * q^64 - 3 * q^65 + 40 * q^66 - 16 * q^67 + 10 * q^68 + 8 * q^69 - 16 * q^70 + 16 * q^71 + 21 * q^72 - 14 * q^73 + 26 * q^74 - 4 * q^77 - 12 * q^78 - 8 * q^79 + 17 * q^80 - 19 * q^81 - 2 * q^82 - 40 * q^83 - 28 * q^84 - 6 * q^85 + 36 * q^86 + 12 * q^87 - 16 * q^88 + 2 * q^89 + 15 * q^90 + 4 * q^91 + 20 * q^92 + 60 * q^93 + 8 * q^94 + 12 * q^95 + 32 * q^96 - 2 * q^97 - 51 * q^98 - 20 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(65))\) 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
65.2.a.a	$65$	$2$	65.a	1.a	$1$	$1$	$1$	$0.519$	\(\Q\)	None	✓	✓	✓	✓	65.2.a.a	$1$	$1$	\(-1\)	\(-2\)	\(-1\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}-q^{4}-q^{5}+2q^{6}-4q^{7}+\cdots\)
65.2.a.b	$65$	$2$	65.a	1.a	$1$	$2$	$2$	$0.519$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	65.2.a.b	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
65.2.a.c	$65$	$2$	65.a	1.a	$1$	$2$	$2$	$0.519$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	65.2.a.c	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1-\beta )q^{3}+q^{4}-q^{5}+(-3+\cdots)q^{6}+\cdots\)