Space of modular forms of level 178, weight 2, and trivial character

• Label: 178.2.a
• Level: $178$
• Weight: $2$
• Character orbit: 178.a
• Rep. character: $\chi_{178}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $4$
• Sturm bound: $45$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 178 = 2 \cdot 89 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	178.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(45\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(178))\).

	Total	New	Old
Modular forms	24	7	17
Cusp forms	21	7	14
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.

\(2\)	\(89\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(1\)
\(-\)	\(+\)	$-$	\(4\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(5\)

Trace form

\( 7 q + q^{2} + 2 q^{3} + 7 q^{4} + 2 q^{5} + 2 q^{6} - 8 q^{7} + q^{8} + 7 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(178))\) 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	89
178.2.a.a	$178$	$2$	178.a	1.a	$1$	$1$	$1$	$1.421$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(2\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}+q^{4}+2q^{5}-2q^{6}-q^{8}+\cdots\)
178.2.a.b	$178$	$2$	178.a	1.a	$1$	$1$	$1$	$1.421$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+3q^{5}+q^{6}-4q^{7}+\cdots\)
178.2.a.c	$178$	$2$	178.a	1.a	$1$	$2$	$2$	$1.421$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(-2\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta )q^{3}+q^{4}+(-1-2\beta )q^{5}+\cdots\)
178.2.a.d	$178$	$2$	178.a	1.a	$1$	$3$	$3$	$1.421$	3.3.568.1	None	✓	$1$	$0$	\(3\)	\(1\)	\(-1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{2}q^{5}-\beta _{2}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(178))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(178)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(89))\)\(^{\oplus 2}\)