Space of modular forms of level 762, weight 2, and trivial character

• Label: 762.2.a
• Level: $762$
• Weight: $2$
• Character orbit: 762.a
• Rep. character: $\chi_{762}(1,\cdot)$
• Character field: $\Q$
• Dimension: $21$
• Newform subspaces: $12$
• Sturm bound: $256$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 762 = 2 \cdot 3 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	762.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(256\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(762))\).

	Total	New	Old
Modular forms	132	21	111
Cusp forms	125	21	104
Eisenstein series	7	0	7

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(127\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(2\)
\(+\)	\(-\)	\(+\)	$-$	\(2\)
\(+\)	\(-\)	\(-\)	$+$	\(3\)
\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(4\)
Plus space			\(+\)	\(8\)
Minus space			\(-\)	\(13\)

Trace form

\( 21 q + q^{2} - q^{3} + 21 q^{4} + 6 q^{5} - q^{6} + q^{8} + 21 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(762))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	3	127
762.2.a.a	$762$	$2$	762.a	1.a	$1$	$1$	$1$	$6.085$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-3\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-3q^{5}-q^{6}+3q^{7}+\cdots\)
762.2.a.b	$762$	$2$	762.a	1.a	$1$	$1$	$1$	$6.085$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+q^{7}-q^{8}+\cdots\)
762.2.a.c	$762$	$2$	762.a	1.a	$1$	$1$	$1$	$6.085$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(3\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+3q^{5}-q^{6}+q^{7}+\cdots\)
762.2.a.d	$762$	$2$	762.a	1.a	$1$	$1$	$1$	$6.085$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(-1\)	\(-1\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-3q^{7}+\cdots\)
762.2.a.e	$762$	$2$	762.a	1.a	$1$	$1$	$1$	$6.085$	\(\Q\)	None	✓	$1$	$1$	\(1\)	\(1\)	\(-3\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-3q^{5}+q^{6}-5q^{7}+\cdots\)
762.2.a.f	$762$	$2$	762.a	1.a	$1$	$1$	$1$	$6.085$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{7}+\cdots\)
762.2.a.g	$762$	$2$	762.a	1.a	$1$	$1$	$1$	$6.085$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}-q^{7}+q^{8}+\cdots\)
762.2.a.h	$762$	$2$	762.a	1.a	$1$	$2$	$2$	$6.085$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(-2\)	\(-2\)	\(1\)	\(6\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+\beta q^{5}+q^{6}+3q^{7}+\cdots\)
762.2.a.i	$762$	$2$	762.a	1.a	$1$	$2$	$2$	$6.085$	\(\Q(\sqrt{13}) \)	None	✓	$1$	$1$	\(-2\)	\(2\)	\(-1\)	\(-5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-\beta q^{5}-q^{6}+(-3+\cdots)q^{7}+\cdots\)
762.2.a.j	$762$	$2$	762.a	1.a	$1$	$2$	$2$	$6.085$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(2\)	\(5\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(3-\beta )q^{5}+q^{6}+\cdots\)
762.2.a.k	$762$	$2$	762.a	1.a	$1$	$3$	$3$	$6.085$	3.3.229.1	None	✓	$1$	$1$	\(-3\)	\(-3\)	\(4\)	\(-6\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(1-\beta _{2})q^{5}+q^{6}+\cdots\)
762.2.a.l	$762$	$2$	762.a	1.a	$1$	$5$	$5$	$6.085$	5.5.18925901.1	None	✓	$1$	$0$	\(5\)	\(-5\)	\(2\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+\beta _{1}q^{5}-q^{6}+(1+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(762))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(762)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(127))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(254))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(381))\)\(^{\oplus 2}\)