Space of modular forms of level 1734, weight 2, and trivial character

• Label: 1734.2.a
• Level: $1734$
• Weight: $2$
• Character orbit: 1734.a
• Rep. character: $\chi_{1734}(1,\cdot)$
• Character field: $\Q$
• Dimension: $45$
• Newform subspaces: $23$
• Sturm bound: $612$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1734 = 2 \cdot 3 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1734.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(612\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1734))\).

	Total	New	Old
Modular forms	342	45	297
Cusp forms	271	45	226
Eisenstein series	71	0	71

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

\(2\)	\(3\)	\(17\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(36\)	\(3\)	\(33\)		\(28\)	\(3\)	\(25\)		\(8\)	\(0\)	\(8\)
\(+\)	\(+\)	\(-\)	\(-\)		\(49\)	\(8\)	\(41\)		\(40\)	\(8\)	\(32\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(+\)	\(-\)		\(45\)	\(6\)	\(39\)		\(36\)	\(6\)	\(30\)		\(9\)	\(0\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)		\(40\)	\(5\)	\(35\)		\(31\)	\(5\)	\(26\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(+\)	\(-\)		\(45\)	\(8\)	\(37\)		\(36\)	\(8\)	\(28\)		\(9\)	\(0\)	\(9\)
\(-\)	\(+\)	\(-\)	\(+\)		\(41\)	\(4\)	\(37\)		\(32\)	\(4\)	\(28\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(+\)	\(+\)		\(45\)	\(2\)	\(43\)		\(36\)	\(2\)	\(34\)		\(9\)	\(0\)	\(9\)
\(-\)	\(-\)	\(-\)	\(-\)		\(41\)	\(9\)	\(32\)		\(32\)	\(9\)	\(23\)		\(9\)	\(0\)	\(9\)
Plus space			\(+\)		\(162\)	\(14\)	\(148\)		\(127\)	\(14\)	\(113\)		\(35\)	\(0\)	\(35\)
Minus space			\(-\)		\(180\)	\(31\)	\(149\)		\(144\)	\(31\)	\(113\)		\(36\)	\(0\)	\(36\)

Trace form

45 * q + q^2 - q^3 + 45 * q^4 + 6 * q^5 - q^6 + q^8 + 45 * q^9 - 2 * q^10 + 4 * q^11 - q^12 + 10 * q^13 + 2 * q^15 + 45 * q^16 + q^18 + 6 * q^20 - 4 * q^21 + 4 * q^22 - q^24 + 47 * q^25 - 2 * q^26 - q^27 + 14 * q^29 + 6 * q^30 + 8 * q^31 + q^32 + 8 * q^33 + 45 * q^36 - 2 * q^37 + 4 * q^38 - 6 * q^39 - 2 * q^40 - 6 * q^41 + 4 * q^42 + 8 * q^43 + 4 * q^44 + 6 * q^45 - 8 * q^47 - q^48 + 69 * q^49 + 15 * q^50 + 10 * q^52 - 2 * q^53 - q^54 + 4 * q^57 + 6 * q^58 + 4 * q^59 + 2 * q^60 + 6 * q^61 - 24 * q^62 + 45 * q^64 - 28 * q^65 + 4 * q^66 + 40 * q^67 + 16 * q^69 + 16 * q^70 + q^72 - 14 * q^73 + 6 * q^74 + 17 * q^75 + 8 * q^77 + 10 * q^78 + 8 * q^79 + 6 * q^80 + 45 * q^81 - 14 * q^82 + 4 * q^83 - 4 * q^84 - 12 * q^86 + 10 * q^87 + 4 * q^88 + 42 * q^89 - 2 * q^90 - 16 * q^91 - 4 * q^93 + 24 * q^94 + 24 * q^95 - q^96 - 6 * q^97 + 9 * q^98 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1734))\) 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					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	17
1734.2.a.a	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓	✓			1734.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(-1\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+4q^{7}+\cdots\)
1734.2.a.b	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓				102.2.a.b	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}-2q^{7}-q^{8}+\cdots\)
1734.2.a.c	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓	✓			1734.2.a.c	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}+q^{7}-q^{8}+\cdots\)
1734.2.a.d	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓				102.2.b.a	$1$	$1$	\(-1\)	\(-1\)	\(2\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+2q^{7}+\cdots\)
1734.2.a.e	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓				102.2.b.a	$1$	$0$	\(-1\)	\(1\)	\(-2\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-2q^{5}-q^{6}-2q^{7}+\cdots\)
1734.2.a.f	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓	✓			1734.2.a.c	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
1734.2.a.g	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓	✓			1734.2.a.a	$1$	$0$	\(-1\)	\(1\)	\(1\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-4q^{7}+\cdots\)
1734.2.a.h	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓				102.2.a.a	$1$	$0$	\(-1\)	\(1\)	\(4\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+4q^{5}-q^{6}+2q^{7}+\cdots\)
1734.2.a.i	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓	✓			1734.2.a.i	$1$	$0$	\(1\)	\(-1\)	\(-4\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-4q^{5}-q^{6}-3q^{7}+\cdots\)
1734.2.a.j	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓				102.2.a.c	$1$	$0$	\(1\)	\(-1\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+2q^{5}-q^{6}+q^{8}+\cdots\)
1734.2.a.k	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓	✓			1734.2.a.k	$1$	$0$	\(1\)	\(-1\)	\(3\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+3q^{5}-q^{6}+4q^{7}+\cdots\)
1734.2.a.l	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓	✓			1734.2.a.k	$1$	$0$	\(1\)	\(1\)	\(-3\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-3q^{5}+q^{6}-4q^{7}+\cdots\)
1734.2.a.m	$1734$	$2$	1734.a	1.a	$1$	$1$	$1$	$13.846$	\(\Q\)	None	✓	✓			1734.2.a.i	$1$	$0$	\(1\)	\(1\)	\(4\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+4q^{5}+q^{6}+3q^{7}+\cdots\)
1734.2.a.n	$1734$	$2$	1734.a	1.a	$1$	$2$	$2$	$13.846$	\(\Q(\sqrt{2}) \)	None	✓				102.2.f.a	$1$	$0$	\(2\)	\(-2\)	\(4\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(2+\beta )q^{5}-q^{6}+\cdots\)
1734.2.a.o	$1734$	$2$	1734.a	1.a	$1$	$2$	$2$	$13.846$	\(\Q(\sqrt{2}) \)	None	✓		✓	✓	102.2.f.a	$1$	$1$	\(2\)	\(2\)	\(-4\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(-2+\beta )q^{5}+q^{6}+\cdots\)
1734.2.a.p	$1734$	$2$	1734.a	1.a	$1$	$3$	$3$	$13.846$	\(\Q(\zeta_{18})^+\)	None	✓	✓			1734.2.a.p	$1$	$0$	\(-3\)	\(-3\)	\(-6\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(-2+\beta _{1})q^{5}+q^{6}+\cdots\)
1734.2.a.q	$1734$	$2$	1734.a	1.a	$1$	$3$	$3$	$13.846$	\(\Q(\zeta_{18})^+\)	None	✓	✓	✓		1734.2.a.p	$1$	$0$	\(-3\)	\(3\)	\(6\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+(2-\beta _{1})q^{5}-q^{6}+\cdots\)
1734.2.a.r	$1734$	$2$	1734.a	1.a	$1$	$3$	$3$	$13.846$	\(\Q(\zeta_{18})^+\)	None	✓	✓	✓		1734.2.a.r	$1$	$0$	\(3\)	\(-3\)	\(-6\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(-2+\beta _{1})q^{5}-q^{6}+\cdots\)
1734.2.a.s	$1734$	$2$	1734.a	1.a	$1$	$3$	$3$	$13.846$	\(\Q(\zeta_{18})^+\)	None	✓	✓			1734.2.a.r	$1$	$0$	\(3\)	\(3\)	\(6\)	\(-3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(2-\beta _{1})q^{5}+q^{6}+\cdots\)
1734.2.a.t	$1734$	$2$	1734.a	1.a	$1$	$4$	$4$	$13.846$	\(\Q(\zeta_{16})^+\)	None	✓		✓		102.2.h.b	$1$	$0$	\(-4\)	\(-4\)	\(8\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(2+\beta _{2}+\beta _{3})q^{5}+\cdots\)
1734.2.a.u	$1734$	$2$	1734.a	1.a	$1$	$4$	$4$	$13.846$	\(\Q(\zeta_{16})^+\)	None	✓		✓		102.2.h.b	$1$	$1$	\(-4\)	\(4\)	\(-8\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+(-2-\beta _{2}+\beta _{3})q^{5}+\cdots\)
1734.2.a.v	$1734$	$2$	1734.a	1.a	$1$	$4$	$4$	$13.846$	\(\Q(\zeta_{16})^+\)	None	✓		✓	✓	102.2.h.a	$1$	$1$	\(4\)	\(-4\)	\(0\)	\(-8\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(-\beta _{2}+\beta _{3})q^{5}+\cdots\)
1734.2.a.w	$1734$	$2$	1734.a	1.a	$1$	$4$	$4$	$13.846$	\(\Q(\zeta_{16})^+\)	None	✓		✓		102.2.h.a	$1$	$0$	\(4\)	\(4\)	\(0\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(\beta _{2}+\beta _{3})q^{5}+q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1734))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1734)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(578))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(867))\)\(^{\oplus 2}\)