Space of modular forms of level 707, weight 2, and trivial character

• Label: 707.2.a
• Level: $707$
• Weight: $2$
• Character orbit: 707.a
• Rep. character: $\chi_{707}(1,\cdot)$
• Character field: $\Q$
• Dimension: $51$
• Newform subspaces: $7$
• Sturm bound: $136$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 707 = 7 \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	707.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(136\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(707))\).

	Total	New	Old
Modular forms	70	51	19
Cusp forms	67	51	16
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.

\(7\)	\(101\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(11\)	\(10\)	\(1\)		\(11\)	\(10\)	\(1\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(24\)	\(16\)	\(8\)		\(23\)	\(16\)	\(7\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(17\)	\(15\)	\(2\)		\(16\)	\(15\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(18\)	\(10\)	\(8\)		\(17\)	\(10\)	\(7\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(29\)	\(20\)	\(9\)		\(28\)	\(20\)	\(8\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(41\)	\(31\)	\(10\)		\(39\)	\(31\)	\(8\)		\(2\)	\(0\)	\(2\)

Trace form

51 * q - q^2 - 4 * q^3 + 47 * q^4 + 2 * q^5 - q^7 + 3 * q^8 + 43 * q^9 - 6 * q^10 - 8 * q^11 - 16 * q^12 + 6 * q^13 + q^14 - 16 * q^15 + 27 * q^16 + 2 * q^17 + 19 * q^18 - 20 * q^19 + 26 * q^20 - 4 * q^22 + 8 * q^23 + 24 * q^24 + 57 * q^25 + 42 * q^26 - 4 * q^27 - 7 * q^28 - 2 * q^29 + 12 * q^30 + 4 * q^31 - 5 * q^32 - 28 * q^33 - 18 * q^34 - 6 * q^35 + 19 * q^36 + 14 * q^37 - 16 * q^38 + 20 * q^39 - 34 * q^40 + 10 * q^41 - 12 * q^42 - 28 * q^43 - 36 * q^44 - 2 * q^45 - 8 * q^46 + 8 * q^47 - 28 * q^48 + 51 * q^49 - 51 * q^50 - 20 * q^51 + 30 * q^52 - 14 * q^53 - 52 * q^54 - 28 * q^55 - 3 * q^56 + 8 * q^57 + 2 * q^58 + 24 * q^59 - 56 * q^60 + 10 * q^61 + 36 * q^62 + 3 * q^63 - q^64 + 44 * q^65 + 28 * q^66 - 32 * q^67 - 50 * q^68 - 28 * q^69 - 6 * q^70 - 12 * q^71 + 63 * q^72 + 2 * q^73 + 26 * q^74 - 32 * q^75 - 68 * q^76 - 4 * q^77 - 76 * q^78 + 12 * q^79 + 86 * q^80 + 59 * q^81 + 46 * q^82 + 8 * q^83 + 12 * q^84 + 20 * q^85 - 36 * q^86 + 28 * q^87 - 52 * q^88 - 2 * q^89 + 6 * q^90 - 14 * q^91 + 108 * q^92 - 52 * q^94 + 24 * q^95 + 56 * q^96 + 30 * q^97 - q^98 - 44 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(707))\) 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	101
707.2.a.a	$707$	$2$	707.a	1.a	$1$	$1$	$1$	$5.645$	\(\Q\)	None	✓	✓			707.2.a.a	$1$	$2$	\(-2\)	\(-2\)	\(-3\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-2q^{3}+2q^{4}-3q^{5}+4q^{6}+\cdots\)
707.2.a.b	$707$	$2$	707.a	1.a	$1$	$2$	$2$	$5.645$	\(\Q(\sqrt{13}) \)	None	✓	✓			707.2.a.b	$1$	$1$	\(-1\)	\(2\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(1+\beta )q^{4}+(-1+\beta )q^{5}+\cdots\)
707.2.a.c	$707$	$2$	707.a	1.a	$1$	$4$	$4$	$5.645$	4.4.16609.1	None	✓	✓			707.2.a.c	$1$	$0$	\(0\)	\(4\)	\(1\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}-2q^{4}+\beta _{3}q^{5}+q^{7}+\cdots\)
707.2.a.d	$707$	$2$	707.a	1.a	$1$	$8$	$8$	$5.645$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		707.2.a.d	$1$	$1$	\(-3\)	\(-3\)	\(-2\)	\(-8\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{2}-\beta _{3}-\beta _{7})q^{3}+\cdots\)
707.2.a.e	$707$	$2$	707.a	1.a	$1$	$10$	$10$	$5.645$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓	✓	707.2.a.e	$1$	$1$	\(-2\)	\(-9\)	\(-7\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{3})q^{3}+(-\beta _{7}-\beta _{8}+\cdots)q^{4}+\cdots\)
707.2.a.f	$707$	$2$	707.a	1.a	$1$	$11$	$11$	$5.645$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓	✓	✓		707.2.a.f	$1$	$0$	\(2\)	\(3\)	\(4\)	\(11\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+\beta _{5}q^{3}+(2+\beta _{1})q^{4}-\beta _{7}q^{5}+\cdots\)
707.2.a.g	$707$	$2$	707.a	1.a	$1$	$15$	$15$	$5.645$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓	✓		707.2.a.g	$1$	$0$	\(5\)	\(1\)	\(10\)	\(-15\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(1+\beta _{5}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(707))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(707)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(101))\)\(^{\oplus 2}\)