Space of modular forms of level 77, weight 2, and trivial character

• Label: 77.2.a
• Level: $77$
• Weight: $2$
• Character orbit: 77.a
• Rep. character: $\chi_{77}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $4$
• Sturm bound: $16$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	77.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(16\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(77))\).

	Total	New	Old
Modular forms	10	5	5
Cusp forms	7	5	2
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.

\(7\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(3\)
Plus space		\(+\)	\(1\)
Minus space		\(-\)	\(4\)

Trace form

5 * q + q^2 + 2 * q^3 + q^4 - 4 * q^5 - 8 * q^6 + q^7 - 3 * q^8 + 11 * q^9 - 2 * q^10 - 3 * q^11 + 8 * q^12 - 2 * q^13 - q^14 - 2 * q^15 + 5 * q^16 - 2 * q^17 - 19 * q^18 - 14 * q^20 + 4 * q^21 + q^22 - 10 * q^23 - 16 * q^24 - 3 * q^25 + 14 * q^26 + 2 * q^27 + 7 * q^28 + 6 * q^29 + 16 * q^30 + 6 * q^31 + 5 * q^32 + 2 * q^33 - 6 * q^34 + 2 * q^35 + 9 * q^36 - 8 * q^37 + 20 * q^38 + 8 * q^39 + 6 * q^40 - 10 * q^41 - 12 * q^42 + 28 * q^43 - 3 * q^44 - 26 * q^45 + 16 * q^46 + 8 * q^47 - 12 * q^48 + 5 * q^49 - q^50 + 4 * q^51 + 18 * q^52 - 10 * q^53 - 24 * q^54 + 3 * q^56 + 4 * q^57 + 14 * q^58 - 2 * q^59 - 20 * q^60 - 22 * q^61 + 20 * q^62 - 3 * q^63 - 35 * q^64 - 20 * q^65 + 12 * q^66 + 30 * q^67 - 2 * q^68 - 14 * q^69 + 2 * q^70 - 14 * q^71 - 23 * q^72 - 2 * q^73 - 26 * q^74 + 12 * q^75 + 20 * q^76 - 3 * q^77 + 8 * q^78 + 4 * q^79 + 14 * q^80 + 21 * q^81 - 6 * q^82 + 28 * q^83 - 24 * q^85 + 12 * q^86 - 60 * q^87 - 3 * q^88 - 20 * q^89 + 38 * q^90 - 2 * q^91 - 4 * q^92 + 2 * q^93 - 20 * q^94 + 4 * q^95 + 40 * q^96 - 8 * q^97 + q^98 - 9 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(77))\) 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	11
77.2.a.a	$77$	$2$	77.a	1.a	$1$	$1$	$1$	$0.615$	\(\Q\)	None	✓	✓	✓	✓	77.2.a.a	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(-1\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{4}-q^{5}-q^{7}+6q^{9}-q^{11}+\cdots\)
77.2.a.b	$77$	$2$	77.a	1.a	$1$	$1$	$1$	$0.615$	\(\Q\)	None	✓	✓			77.2.a.b	$1$	$0$	\(0\)	\(1\)	\(3\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+3q^{5}+q^{7}-2q^{9}-q^{11}+\cdots\)
77.2.a.c	$77$	$2$	77.a	1.a	$1$	$1$	$1$	$0.615$	\(\Q\)	None	✓	✓	✓	✓	77.2.a.c	$1$	$0$	\(1\)	\(2\)	\(-2\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}-q^{4}-2q^{5}+2q^{6}-q^{7}+\cdots\)
77.2.a.d	$77$	$2$	77.a	1.a	$1$	$2$	$2$	$0.615$	\(\Q(\sqrt{5}) \)	None	✓	✓	✓		77.2.a.d	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(2\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(1+\beta )q^{3}+3q^{4}-2q^{5}+(-5+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(77))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(77)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 2}\)