Space of modular forms of level 931, weight 2, and trivial character

• Label: 931.2.a
• Level: $931$
• Weight: $2$
• Character orbit: 931.a
• Rep. character: $\chi_{931}(1,\cdot)$
• Character field: $\Q$
• Dimension: $62$
• Newform subspaces: $17$
• Sturm bound: $186$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 931 = 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	931.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(186\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(3\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(931))\).

	Total	New	Old
Modular forms	100	62	38
Cusp forms	85	62	23
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.

\(7\)	\(19\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(21\)	\(14\)	\(7\)		\(18\)	\(14\)	\(4\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(29\)	\(18\)	\(11\)		\(25\)	\(18\)	\(7\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(27\)	\(18\)	\(9\)		\(23\)	\(18\)	\(5\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(23\)	\(12\)	\(11\)		\(19\)	\(12\)	\(7\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(44\)	\(26\)	\(18\)		\(37\)	\(26\)	\(11\)		\(7\)	\(0\)	\(7\)
Minus space		\(-\)		\(56\)	\(36\)	\(20\)		\(48\)	\(36\)	\(12\)		\(8\)	\(0\)	\(8\)

Trace form

62 * q - q^2 + 2 * q^3 + 61 * q^4 + 3 * q^5 + 3 * q^8 + 58 * q^9 + 10 * q^10 - 3 * q^11 + 12 * q^12 + 10 * q^13 + 2 * q^15 + 51 * q^16 + 5 * q^17 - 17 * q^18 - 2 * q^19 - 20 * q^22 - 4 * q^23 - 20 * q^24 + 69 * q^25 - 2 * q^26 + 8 * q^27 - 20 * q^29 - 32 * q^30 + 12 * q^31 - 25 * q^32 - 30 * q^33 - 18 * q^34 + 33 * q^36 + 12 * q^37 - 3 * q^38 - 8 * q^39 - 6 * q^40 - 4 * q^41 + 9 * q^43 - 26 * q^44 + 15 * q^45 - 56 * q^46 - 9 * q^47 + 44 * q^48 + 13 * q^50 - 70 * q^51 + 6 * q^52 + 30 * q^53 - 16 * q^54 - 21 * q^55 + 2 * q^57 + 10 * q^58 + 6 * q^59 - 12 * q^60 + 7 * q^61 - 4 * q^62 + 39 * q^64 + 24 * q^65 + 24 * q^66 - 32 * q^67 + 36 * q^68 - 28 * q^69 - 46 * q^71 + 7 * q^72 + 13 * q^73 - 22 * q^74 + 12 * q^75 - 5 * q^76 - 12 * q^78 - 24 * q^79 - 38 * q^80 + 10 * q^81 - 18 * q^82 - 32 * q^83 + 37 * q^85 + 20 * q^86 - 40 * q^87 + 2 * q^89 + 102 * q^90 + 60 * q^92 + 16 * q^93 + 12 * q^94 + 7 * q^95 - 16 * q^96 + 2 * q^97 - 3 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(931))\) 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	19
931.2.a.a	$931$	$2$	931.a	1.a	$1$	$1$	$1$	$7.434$	\(\Q\)	None	✓				19.2.a.a	$1$	$0$	\(0\)	\(2\)	\(-3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{4}-3q^{5}+q^{9}+3q^{11}+\cdots\)
931.2.a.b	$931$	$2$	931.a	1.a	$1$	$1$	$1$	$7.434$	\(\Q\)	None	✓				133.2.f.a	$1$	$0$	\(2\)	\(-2\)	\(-3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-2q^{3}+2q^{4}-3q^{5}-4q^{6}+\cdots\)
931.2.a.c	$931$	$2$	931.a	1.a	$1$	$1$	$1$	$7.434$	\(\Q\)	None	✓				133.2.f.a	$1$	$0$	\(2\)	\(2\)	\(3\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{3}+2q^{4}+3q^{5}+4q^{6}+\cdots\)
931.2.a.d	$931$	$2$	931.a	1.a	$1$	$2$	$2$	$7.434$	\(\Q(\sqrt{5}) \)	None	✓				133.2.a.a	$1$	$1$	\(-3\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(1+\beta )q^{3}+3\beta q^{4}+\cdots\)
931.2.a.e	$931$	$2$	931.a	1.a	$1$	$2$	$2$	$7.434$	\(\Q(\sqrt{2}) \)	None	✓				133.2.f.c	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
931.2.a.f	$931$	$2$	931.a	1.a	$1$	$2$	$2$	$7.434$	\(\Q(\sqrt{2}) \)	None	✓				133.2.f.c	$1$	$1$	\(-2\)	\(0\)	\(2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-\beta q^{3}+(1-2\beta )q^{4}+\cdots\)
931.2.a.g	$931$	$2$	931.a	1.a	$1$	$2$	$2$	$7.434$	\(\Q(\sqrt{13}) \)	None	✓				133.2.a.b	$1$	$0$	\(-1\)	\(3\)	\(6\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2-\beta )q^{3}+(1+\beta )q^{4}+3q^{5}+\cdots\)
931.2.a.h	$931$	$2$	931.a	1.a	$1$	$2$	$2$	$7.434$	\(\Q(\sqrt{2}) \)	None	✓				133.2.f.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-\beta q^{3}+2\beta q^{5}-2q^{6}-2\beta q^{8}+\cdots\)
931.2.a.i	$931$	$2$	931.a	1.a	$1$	$2$	$2$	$7.434$	\(\Q(\sqrt{2}) \)	None	✓				133.2.f.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+\beta q^{3}-2\beta q^{5}+2q^{6}-2\beta q^{8}+\cdots\)
931.2.a.j	$931$	$2$	931.a	1.a	$1$	$2$	$2$	$7.434$	\(\Q(\sqrt{5}) \)	None	✓				133.2.a.c	$1$	$1$	\(1\)	\(-3\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-2+\beta )q^{3}+(-1+\beta )q^{4}+\cdots\)
931.2.a.k	$931$	$2$	931.a	1.a	$1$	$3$	$3$	$7.434$	3.3.229.1	None	✓				133.2.a.d	$1$	$0$	\(2\)	\(-3\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{2}+(-1+\beta _{1})q^{3}+(2+\beta _{1}+\cdots)q^{4}+\cdots\)
931.2.a.l	$931$	$2$	931.a	1.a	$1$	$4$	$4$	$7.434$	4.4.5744.1	None	✓	✓	✓		931.2.a.l	$1$	$1$	\(0\)	\(-2\)	\(-8\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
931.2.a.m	$931$	$2$	931.a	1.a	$1$	$4$	$4$	$7.434$	4.4.5744.1	None	✓	✓			931.2.a.l	$1$	$0$	\(0\)	\(2\)	\(8\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
931.2.a.n	$931$	$2$	931.a	1.a	$1$	$7$	$7$	$7.434$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				133.2.f.d	$1$	$0$	\(2\)	\(-2\)	\(-2\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}-\beta _{2}q^{3}+(1+\beta _{3}-\beta _{5})q^{4}+\cdots\)
931.2.a.o	$931$	$2$	931.a	1.a	$1$	$7$	$7$	$7.434$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		133.2.f.d	$1$	$0$	\(2\)	\(2\)	\(2\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+\beta _{2}q^{3}+(1+\beta _{3}-\beta _{5})q^{4}+\cdots\)
931.2.a.p	$931$	$2$	931.a	1.a	$1$	$10$	$10$	$7.434$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		931.2.a.p	$1$	$1$	\(-2\)	\(-4\)	\(-16\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{4}q^{3}+(1+\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
931.2.a.q	$931$	$2$	931.a	1.a	$1$	$10$	$10$	$7.434$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		931.2.a.p	$1$	$0$	\(-2\)	\(4\)	\(16\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(1+\beta _{2})q^{4}+(2+\beta _{9})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(931))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(931)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(133))\)\(^{\oplus 2}\)