Space of modular forms of level 2, weight 78, and trivial character

• Label: 2.78.a
• Level: $2$
• Weight: $78$
• Character orbit: 2.a
• Rep. character: $\chi_{2}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $2$
• Sturm bound: $19$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 2 \)
Weight:	\( k \)	\(=\)	\( 78 \)
Character orbit:	\([\chi]\)	\(=\)	2.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(19\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{78}(\Gamma_0(2))\).

	Total	New	Old
Modular forms	20	6	14
Cusp forms	18	6	12
Eisenstein series	2	0	2

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

\(2\)	Dim
\(+\)	\(3\)
\(-\)	\(3\)

Trace form

\( 6 q - 2638706479849430280 q^{3} + 453347182355485940514816 q^{4} + 951948703777156427890806660 q^{5} + 664069599363895225939000295424 q^{6} - 404664187371826723533400217750160 q^{7} + 27635744030038056171543919721032526958 q^{9} + O(q^{10}) \)

Decomposition of \(S_{78}^{\mathrm{new}}(\Gamma_0(2))\) 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
2.78.a.a	$2$	$78$	2.a	1.a	$1$	$3$	$3$	$75.096$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(-824633720832\)	\(-25\!\cdots\!88\)	\(30\!\cdots\!10\)	\(-23\!\cdots\!16\)		$+$	$+$	$2^{24}\cdot 3^{11}\cdot 5^{2}\cdot 7\cdot 11\cdot 13\cdot 19$	$\mathrm{SU}(2)$	\(q-2^{38}q^{2}+(-842429601461527596+\cdots)q^{3}+\cdots\)
2.78.a.b	$2$	$78$	2.a	1.a	$1$	$3$	$3$	$75.096$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(824633720832\)	\(-11\!\cdots\!92\)	\(64\!\cdots\!50\)	\(-17\!\cdots\!44\)		$-$	$-$	$2^{24}\cdot 3^{8}\cdot 5^{4}\cdot 7\cdot 11$	$\mathrm{SU}(2)$	\(q+2^{38}q^{2}+(-37139225154949164+\cdots)q^{3}+\cdots\)

Decomposition of \(S_{78}^{\mathrm{old}}(\Gamma_0(2))\) into lower level spaces

\( S_{78}^{\mathrm{old}}(\Gamma_0(2)) \cong \) \(S_{78}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 2}\)