Space of modular forms of level 539, weight 2, and trivial character

• Label: 539.2.a
• Level: $539$
• Weight: $2$
• Character orbit: 539.a
• Rep. character: $\chi_{539}(1,\cdot)$
• Character field: $\Q$
• Dimension: $34$
• Newform subspaces: $12$
• Sturm bound: $112$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(112\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(539))\).

	Total	New	Old
Modular forms	64	34	30
Cusp forms	49	34	15
Eisenstein series	15	0	15

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
\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(-\)	\(-\)	\(13\)
\(-\)	\(+\)	\(-\)	\(11\)
\(-\)	\(-\)	\(+\)	\(5\)
Plus space		\(+\)	\(10\)
Minus space		\(-\)	\(24\)

Trace form

34 * q + q^2 - q^3 + 37 * q^4 + 3 * q^5 + 6 * q^6 + 3 * q^8 + 31 * q^9 + 4 * q^10 + 2 * q^11 - 6 * q^12 - 2 * q^13 - 5 * q^15 + 39 * q^16 + 4 * q^17 + 7 * q^18 + 12 * q^20 + q^22 - q^23 + 16 * q^24 + 43 * q^25 - 6 * q^26 - 7 * q^27 - 10 * q^29 - 46 * q^30 - 13 * q^31 - 33 * q^32 - q^33 + 2 * q^34 + 15 * q^36 + 33 * q^37 - 20 * q^38 + 16 * q^39 - 6 * q^40 + 18 * q^41 - 6 * q^43 + q^44 + 28 * q^45 - 26 * q^46 - 16 * q^47 + 8 * q^48 + 5 * q^50 - 22 * q^51 - 26 * q^52 + 44 * q^53 + 34 * q^54 - q^55 - 8 * q^57 + 6 * q^58 - 3 * q^59 - 34 * q^60 + 10 * q^61 - 6 * q^62 + 71 * q^64 + 28 * q^65 - 14 * q^66 - 3 * q^67 + 6 * q^68 + 13 * q^69 + q^71 - 49 * q^72 - 2 * q^73 + 36 * q^74 - 16 * q^75 - 20 * q^76 - 80 * q^78 + 22 * q^79 - 10 * q^80 - 22 * q^81 - 10 * q^82 - 22 * q^83 + 66 * q^85 - 60 * q^86 + 60 * q^87 - 9 * q^88 + 5 * q^89 - 42 * q^90 - 62 * q^92 + 17 * q^93 + 36 * q^94 - 40 * q^95 - 32 * q^96 + 15 * q^97 + 11 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(539))\) 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
539.2.a.a	$539$	$2$	539.a	1.a	$1$	$1$	$1$	$4.304$	\(\Q\)	None	✓				11.2.a.a	$1$	$1$	\(-2\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-q^{5}-2q^{6}-2q^{9}+\cdots\)
539.2.a.b	$539$	$2$	539.a	1.a	$1$	$1$	$1$	$4.304$	\(\Q\)	None	✓				77.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}-3q^{5}-2q^{9}-q^{11}+\cdots\)
539.2.a.c	$539$	$2$	539.a	1.a	$1$	$1$	$1$	$4.304$	\(\Q\)	None	✓				77.2.a.a	$1$	$0$	\(0\)	\(3\)	\(1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-2q^{4}+q^{5}+6q^{9}-q^{11}+\cdots\)
539.2.a.d	$539$	$2$	539.a	1.a	$1$	$1$	$1$	$4.304$	\(\Q\)	None	✓				77.2.a.c	$1$	$1$	\(1\)	\(-2\)	\(2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}-q^{4}+2q^{5}-2q^{6}-3q^{8}+\cdots\)
539.2.a.e	$539$	$2$	539.a	1.a	$1$	$2$	$2$	$4.304$	\(\Q(\sqrt{2}) \)	None	✓	✓			539.2.a.e	$2$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}-q^{4}-\beta q^{5}-\beta q^{6}+3q^{8}+\cdots\)
539.2.a.f	$539$	$2$	539.a	1.a	$1$	$2$	$2$	$4.304$	\(\Q(\sqrt{5}) \)	None	✓				77.2.a.d	$1$	$0$	\(0\)	\(-2\)	\(4\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1-\beta )q^{3}+3q^{4}+2q^{5}+\cdots\)
539.2.a.g	$539$	$2$	539.a	1.a	$1$	$3$	$3$	$4.304$	\(\Q(\zeta_{18})^+\)	None	✓		✓		77.2.e.a	$1$	$1$	\(0\)	\(-3\)	\(-6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
539.2.a.h	$539$	$2$	539.a	1.a	$1$	$3$	$3$	$4.304$	3.3.257.1	None	✓		✓		77.2.e.b	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
539.2.a.i	$539$	$2$	539.a	1.a	$1$	$3$	$3$	$4.304$	3.3.257.1	None	✓				77.2.e.b	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
539.2.a.j	$539$	$2$	539.a	1.a	$1$	$3$	$3$	$4.304$	\(\Q(\zeta_{18})^+\)	None	✓				77.2.e.a	$1$	$0$	\(0\)	\(3\)	\(6\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{1}+\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
539.2.a.k	$539$	$2$	539.a	1.a	$1$	$4$	$4$	$4.304$	4.4.9248.1	None	✓	✓	✓		539.2.a.k	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+\beta _{2}q^{3}+(2-\beta _{3})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
539.2.a.l	$539$	$2$	539.a	1.a	$1$	$10$	$10$	$4.304$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		539.2.a.l	$2$	$0$	\(2\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{5}q^{2}+\beta _{1}q^{3}+(2+\beta _{9})q^{4}+\beta _{7}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(539))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(539)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 2}\)