Space of modular forms of level 775, weight 2, and trivial character

• Label: 775.2.a
• Level: $775$
• Weight: $2$
• Character orbit: 775.a
• Rep. character: $\chi_{775}(1,\cdot)$
• Character field: $\Q$
• Dimension: $47$
• Newform subspaces: $12$
• Sturm bound: $160$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 775 = 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	775.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(160\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(775))\).

	Total	New	Old
Modular forms	86	47	39
Cusp forms	75	47	28
Eisenstein series	11	0	11

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

\(5\)	\(31\)	Fricke	Dim.
\(+\)	\(+\)	\(+\)	\(10\)
\(+\)	\(-\)	\(-\)	\(13\)
\(-\)	\(+\)	\(-\)	\(15\)
\(-\)	\(-\)	\(+\)	\(9\)
Plus space		\(+\)	\(19\)
Minus space		\(-\)	\(28\)

Trace form

\( 47 q + 2 q^{2} + 2 q^{3} + 48 q^{4} - 2 q^{6} + 3 q^{8} + 43 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(775))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	31
775.2.a.a	$775$	$2$	775.a	1.a	$1$	$1$	$1$	$6.188$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-2q^{9}-4q^{11}-2q^{12}+\cdots\)
775.2.a.b	$775$	$2$	775.a	1.a	$1$	$1$	$1$	$6.188$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-2\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}-q^{4}-2q^{6}-4q^{7}-3q^{8}+\cdots\)
775.2.a.c	$775$	$2$	775.a	1.a	$1$	$1$	$1$	$6.188$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(1\)	\(0\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+2q^{7}+\cdots\)
775.2.a.d	$775$	$2$	775.a	1.a	$1$	$2$	$2$	$6.188$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(-1\)	\(2\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+2\beta q^{3}+(-1+\beta )q^{4}+(-2+\cdots)q^{6}+\cdots\)
775.2.a.e	$775$	$2$	775.a	1.a	$1$	$4$	$4$	$6.188$	4.4.8468.1	None	✓	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2}+\beta _{3})q^{4}+\cdots\)
775.2.a.f	$775$	$2$	775.a	1.a	$1$	$4$	$4$	$6.188$	\(\Q(\zeta_{24})^+\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{3})q^{2}-\beta _{1}q^{3}+(-1-\beta _{2}+\cdots)q^{6}+\cdots\)
775.2.a.g	$775$	$2$	775.a	1.a	$1$	$4$	$4$	$6.188$	4.4.20308.1	None	✓	$1$	$0$	\(1\)	\(1\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(2+\beta _{1}+\beta _{2})q^{4}+\cdots\)
775.2.a.h	$775$	$2$	775.a	1.a	$1$	$5$	$5$	$6.188$	5.5.144209.1	None	✓	$1$	$1$	\(-4\)	\(-3\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{3})q^{2}+(-1+\beta _{2})q^{3}+(1+\cdots)q^{4}+\cdots\)
775.2.a.i	$775$	$2$	775.a	1.a	$1$	$5$	$5$	$6.188$	5.5.205225.1	None	✓	$1$	$1$	\(-4\)	\(-1\)	\(0\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+\beta _{3}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
775.2.a.j	$775$	$2$	775.a	1.a	$1$	$5$	$5$	$6.188$	5.5.205225.1	None	✓	$1$	$0$	\(4\)	\(1\)	\(0\)	\(6\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-\beta _{3}q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
775.2.a.k	$775$	$2$	775.a	1.a	$1$	$5$	$5$	$6.188$	5.5.144209.1	None	✓	$1$	$0$	\(4\)	\(3\)	\(0\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{2}+(1-\beta _{2})q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
775.2.a.l	$775$	$2$	775.a	1.a	$1$	$10$	$10$	$6.188$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{8}q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{4}+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(775))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(775)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(31))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(155))\)\(^{\oplus 2}\)