Space of modular forms of level 760, weight 2, and trivial character

• Label: 760.2.a
• Level: $760$
• Weight: $2$
• Character orbit: 760.a
• Rep. character: $\chi_{760}(1,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $10$
• Sturm bound: $240$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	760.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(240\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(760))\).

	Total	New	Old
Modular forms	128	18	110
Cusp forms	113	18	95
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.

\(2\)	\(5\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(3\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(10\)

Trace form

18 * q - 2 * q^5 + 14 * q^9 - 8 * q^17 - 8 * q^21 + 18 * q^25 + 12 * q^29 - 8 * q^31 - 16 * q^33 + 12 * q^35 + 4 * q^39 - 4 * q^41 - 4 * q^43 + 6 * q^45 + 12 * q^47 + 6 * q^49 + 8 * q^51 - 16 * q^53 - 8 * q^55 - 4 * q^57 + 16 * q^59 - 20 * q^61 - 44 * q^63 - 24 * q^67 + 56 * q^69 - 8 * q^71 - 16 * q^73 + 16 * q^77 + 16 * q^79 - 30 * q^81 + 12 * q^83 - 4 * q^85 + 28 * q^87 - 12 * q^89 - 32 * q^91 - 16 * q^97 + 96 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(760))\) 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	19
760.2.a.a	$760$	$2$	760.a	1.a	$1$	$1$	$1$	$6.069$	\(\Q\)	None	✓	✓	✓	✓	760.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+q^{9}-4q^{11}+4q^{13}+\cdots\)
760.2.a.b	$760$	$2$	760.a	1.a	$1$	$1$	$1$	$6.069$	\(\Q\)	None	✓	✓			760.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}+4q^{7}+q^{9}+4q^{11}+\cdots\)
760.2.a.c	$760$	$2$	760.a	1.a	$1$	$1$	$1$	$6.069$	\(\Q\)	None	✓	✓	✓	✓	760.2.a.c	$1$	$1$	\(0\)	\(0\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{9}-4q^{11}-6q^{13}-6q^{17}+\cdots\)
760.2.a.d	$760$	$2$	760.a	1.a	$1$	$1$	$1$	$6.069$	\(\Q\)	None	✓	✓			760.2.a.d	$1$	$0$	\(0\)	\(2\)	\(1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
760.2.a.e	$760$	$2$	760.a	1.a	$1$	$1$	$1$	$6.069$	\(\Q\)	None	✓	✓			760.2.a.e	$1$	$0$	\(0\)	\(3\)	\(1\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+q^{5}-q^{7}+6q^{9}+4q^{11}+\cdots\)
760.2.a.f	$760$	$2$	760.a	1.a	$1$	$2$	$2$	$6.069$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	760.2.a.f	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}+2\beta q^{7}-q^{9}+(2+2\beta )q^{11}+\cdots\)
760.2.a.g	$760$	$2$	760.a	1.a	$1$	$2$	$2$	$6.069$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	760.2.a.g	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}+(1+2\beta )q^{9}+2q^{11}+\cdots\)
760.2.a.h	$760$	$2$	760.a	1.a	$1$	$3$	$3$	$6.069$	3.3.568.1	None	✓	✓	✓	✓	760.2.a.h	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+(-2-\beta _{2})q^{7}+(1+2\beta _{1}+\cdots)q^{9}+\cdots\)
760.2.a.i	$760$	$2$	760.a	1.a	$1$	$3$	$3$	$6.069$	3.3.316.1	None	✓	✓	✓	✓	760.2.a.i	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+(-1+2\beta _{1}-\beta _{2})q^{7}+\cdots\)
760.2.a.j	$760$	$2$	760.a	1.a	$1$	$3$	$3$	$6.069$	3.3.229.1	None	✓	✓	✓	✓	760.2.a.j	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(-1\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+q^{5}+\beta _{1}q^{7}+(1-\beta _{1})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(760))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(760)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(380))\)\(^{\oplus 2}\)