Space of modular forms of level 768, weight 2, and Character 768.d

• Label: 768.2.d
• Level: $768$
• Weight: $2$
• Character orbit: 768.d
• Rep. character: $\chi_{768}(385,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $8$
• Sturm bound: $256$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	768.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(256\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(5\), \(7\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(768, [\chi])\).

	Total	New	Old
Modular forms	152	16	136
Cusp forms	104	16	88
Eisenstein series	48	0	48

Trace form

\( 16 q - 16 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(768, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
768.2.d.a	$768$	$2$	768.d	8.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-1}) \)	None					96.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 i q^{5}-4 q^{7}-q^{9}-4 i q^{11}+\cdots\)
768.2.d.b	$768$	$2$	768.d	8.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-1}) \)	None					384.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}-2 q^{7}-q^{9}-4 i q^{11}-6 i q^{13}+\cdots\)
768.2.d.c	$768$	$2$	768.d	8.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-1}) \)	None					384.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+4 i q^{5}-2 q^{7}-q^{9}-4 i q^{11}+\cdots\)
768.2.d.d	$768$	$2$	768.d	8.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-1}) \)	None					24.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 i q^{5}-q^{9}+4 i q^{11}+\cdots\)
768.2.d.e	$768$	$2$	768.d	8.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-1}) \)	None					24.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+2 i q^{5}-q^{9}-4 i q^{11}+\cdots\)
768.2.d.f	$768$	$2$	768.d	8.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-1}) \)	None					384.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+4 i q^{5}+2 q^{7}-q^{9}+4 i q^{11}+\cdots\)
768.2.d.g	$768$	$2$	768.d	8.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-1}) \)	None					384.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+2 q^{7}-q^{9}+4 i q^{11}-6 i q^{13}+\cdots\)
768.2.d.h	$768$	$2$	768.d	8.b	$2$	$2$	$2$	$6.133$	\(\Q(\sqrt{-1}) \)	None					96.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(8\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{3}+2 i q^{5}+4 q^{7}-q^{9}+4 i q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(768, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(768, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(96, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(192, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(256, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(384, [\chi])\)\(^{\oplus 2}\)