Space of modular forms of level 680, weight 2, and trivial character

• Label: 680.2.a
• Level: $680$
• Weight: $2$
• Character orbit: 680.a
• Rep. character: $\chi_{680}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $8$
• Sturm bound: $216$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 680 = 2^{3} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	680.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(216\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(680))\).

	Total	New	Old
Modular forms	116	16	100
Cusp forms	101	16	85
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.

\(2\)	\(5\)	\(17\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(12\)	\(2\)	\(10\)		\(11\)	\(2\)	\(9\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(17\)	\(3\)	\(14\)		\(15\)	\(3\)	\(12\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(15\)	\(3\)	\(12\)		\(13\)	\(3\)	\(10\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(14\)	\(0\)	\(14\)		\(12\)	\(0\)	\(12\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(17\)	\(3\)	\(14\)		\(15\)	\(3\)	\(12\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(12\)	\(1\)	\(11\)		\(10\)	\(1\)	\(9\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(14\)	\(2\)	\(12\)		\(12\)	\(2\)	\(10\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(15\)	\(2\)	\(13\)		\(13\)	\(2\)	\(11\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(52\)	\(5\)	\(47\)		\(45\)	\(5\)	\(40\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(64\)	\(11\)	\(53\)		\(56\)	\(11\)	\(45\)		\(8\)	\(0\)	\(8\)

Trace form

16 * q + 4 * q^3 - 2 * q^5 + 8 * q^7 + 12 * q^9 - 4 * q^11 + 8 * q^13 - 4 * q^17 + 4 * q^19 + 8 * q^21 - 8 * q^23 + 16 * q^25 + 16 * q^27 + 12 * q^29 + 24 * q^31 - 8 * q^33 - 4 * q^35 + 4 * q^37 + 8 * q^39 + 32 * q^43 + 6 * q^45 - 8 * q^47 + 24 * q^49 - 16 * q^53 - 12 * q^59 - 20 * q^61 + 4 * q^65 + 16 * q^69 - 40 * q^71 - 24 * q^73 + 4 * q^75 + 8 * q^77 - 8 * q^79 - 8 * q^81 - 24 * q^83 - 2 * q^85 - 48 * q^87 + 28 * q^89 + 16 * q^93 + 24 * q^97 - 20 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(680))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5	17
680.2.a.a	$680$	$2$	680.a	1.a	$1$	$1$	$1$	$5.430$	\(\Q\)	None	✓	✓			680.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+2q^{7}-2q^{9}+4q^{11}+\cdots\)
680.2.a.b	$680$	$2$	680.a	1.a	$1$	$1$	$1$	$5.430$	\(\Q\)	None	✓	✓	✓	✓	680.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{9}-2q^{13}+q^{17}-4q^{19}+\cdots\)
680.2.a.c	$680$	$2$	680.a	1.a	$1$	$1$	$1$	$5.430$	\(\Q\)	None	✓	✓			680.2.a.c	$1$	$0$	\(0\)	\(2\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+2q^{7}+q^{9}-2q^{11}+\cdots\)
680.2.a.d	$680$	$2$	680.a	1.a	$1$	$2$	$2$	$5.430$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	680.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-6\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+q^{5}+(-3-\beta )q^{7}+\cdots\)
680.2.a.e	$680$	$2$	680.a	1.a	$1$	$2$	$2$	$5.430$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	680.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}-\beta q^{7}-q^{9}+(-2-3\beta )q^{11}+\cdots\)
680.2.a.f	$680$	$2$	680.a	1.a	$1$	$3$	$3$	$5.430$	3.3.229.1	None	✓	✓	✓	✓	680.2.a.f	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-q^{5}+(\beta _{1}-\beta _{2})q^{7}+(1-\beta _{1}+\cdots)q^{9}+\cdots\)
680.2.a.g	$680$	$2$	680.a	1.a	$1$	$3$	$3$	$5.430$	3.3.1016.1	None	✓	✓	✓	✓	680.2.a.g	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(2+\beta _{2})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
680.2.a.h	$680$	$2$	680.a	1.a	$1$	$3$	$3$	$5.430$	3.3.940.1	None	✓	✓	✓	✓	680.2.a.h	$1$	$0$	\(0\)	\(3\)	\(3\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+q^{5}+(1-\beta _{2})q^{7}+(3+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(680))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(680)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(340))\)\(^{\oplus 2}\)