Space of modular forms of level 1734, weight 2, and Character 1734.f

• Label: 1734.2.f
• Level: $1734$
• Weight: $2$
• Character orbit: 1734.f
• Rep. character: $\chi_{1734}(829,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $92$
• Newform subspaces: $15$
• Sturm bound: $612$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1734.f (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 17 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(612\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1734, [\chi])\).

	Total	New	Old
Modular forms	684	92	592
Cusp forms	540	92	448
Eisenstein series	144	0	144

Trace form

\( 92 q - 92 q^{4} + 4 q^{5} - 8 q^{7} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1734, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1734.2.f.a	$1734$	$2$	1734.f	17.c	$4$	$4$	$2$	$13.846$	\(\Q(\zeta_{8})\)	None		✓			1734.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{2}q^{2}-\zeta_{8}q^{3}-q^{4}+\zeta_{8}q^{5}-\zeta_{8}^{3}q^{6}+\cdots\)
1734.2.f.b	$1734$	$2$	1734.f	17.c	$4$	$4$	$2$	$13.846$	\(\Q(\zeta_{8})\)	None					102.2.a.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}^{2}q^{2}+\zeta_{8}^{3}q^{3}-q^{4}+\zeta_{8}q^{6}+\cdots\)
1734.2.f.c	$1734$	$2$	1734.f	17.c	$4$	$4$	$2$	$13.846$	\(\Q(\zeta_{8})\)	None		✓			1734.2.a.k	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+3\zeta_{8}q^{5}+\cdots\)
1734.2.f.d	$1734$	$2$	1734.f	17.c	$4$	$4$	$2$	$13.846$	\(\Q(\zeta_{8})\)	None		✓			1734.2.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}^{2}q^{2}-\zeta_{8}^{3}q^{3}-q^{4}-\zeta_{8}q^{6}+\cdots\)
1734.2.f.e	$1734$	$2$	1734.f	17.c	$4$	$4$	$2$	$13.846$	\(\Q(\zeta_{8})\)	None					102.2.a.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+2\zeta_{8}q^{5}+\cdots\)
1734.2.f.f	$1734$	$2$	1734.f	17.c	$4$	$4$	$2$	$13.846$	\(\Q(\zeta_{8})\)	None		✓			1734.2.a.i	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}^{2}q^{2}-\zeta_{8}q^{3}-q^{4}+4\zeta_{8}q^{5}+\cdots\)
1734.2.f.g	$1734$	$2$	1734.f	17.c	$4$	$4$	$2$	$13.846$	\(\Q(\zeta_{8})\)	None					102.2.a.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{2}q^{2}-\zeta_{8}q^{3}-q^{4}+4\zeta_{8}q^{5}+\cdots\)
1734.2.f.h	$1734$	$2$	1734.f	17.c	$4$	$4$	$2$	$13.846$	\(\Q(\zeta_{8})\)	None					102.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+2\zeta_{8}q^{5}+\cdots\)
1734.2.f.i	$1734$	$2$	1734.f	17.c	$4$	$4$	$2$	$13.846$	\(\Q(\zeta_{8})\)	None					102.2.f.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(-8\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+(1+2\zeta_{8}+\cdots)q^{5}+\cdots\)
1734.2.f.j	$1734$	$2$	1734.f	17.c	$4$	$8$	$4$	$13.846$	\(\Q(\zeta_{16})\)	None					102.2.h.a	$2$	$0$	\(0\)	\(0\)	\(-8\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{16}^{3}q^{2}+\zeta_{16}^{5}q^{3}-q^{4}+(-1+\cdots)q^{5}+\cdots\)
1734.2.f.k	$1734$	$2$	1734.f	17.c	$4$	$8$	$4$	$13.846$	\(\Q(\zeta_{16})\)	None					102.2.h.b	$2$	$0$	\(0\)	\(0\)	\(-8\)	\(8\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{16}^{3}q^{2}+\zeta_{16}q^{3}-q^{4}+(-1+2\zeta_{16}+\cdots)q^{5}+\cdots\)
1734.2.f.l	$1734$	$2$	1734.f	17.c	$4$	$8$	$4$	$13.846$	\(\Q(\zeta_{16})\)	None					102.2.h.b	$2$	$0$	\(0\)	\(0\)	\(8\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\zeta_{16}^{3}q^{2}+\zeta_{16}q^{3}-q^{4}+(1+2\zeta_{16}+\cdots)q^{5}+\cdots\)
1734.2.f.m	$1734$	$2$	1734.f	17.c	$4$	$8$	$4$	$13.846$	\(\Q(\zeta_{16})\)	None					102.2.h.a	$2$	$0$	\(0\)	\(0\)	\(8\)	\(-8\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q-\zeta_{16}^{3}q^{2}-\zeta_{16}q^{3}-q^{4}+(1+\zeta_{16}^{3}+\cdots)q^{5}+\cdots\)
1734.2.f.n	$1734$	$2$	1734.f	17.c	$4$	$12$	$6$	$13.846$	12.0.\(\cdots\).1	None		✓			1734.2.a.p	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{7}q^{2}-\beta _{10}q^{3}-q^{4}+(2\beta _{10}-\beta _{11})q^{5}+\cdots\)
1734.2.f.o	$1734$	$2$	1734.f	17.c	$4$	$12$	$6$	$13.846$	12.0.\(\cdots\).1	None		✓			1734.2.a.r	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{7}q^{2}-\beta _{10}q^{3}-q^{4}+(2\beta _{10}-\beta _{11})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1734, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1734, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(51, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(102, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(289, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(578, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(867, [\chi])\)\(^{\oplus 2}\)