Space of modular forms of level 1740, weight 2, and trivial character

• Label: 1740.2.a
• Level: $1740$
• Weight: $2$
• Character orbit: 1740.a
• Rep. character: $\chi_{1740}(1,\cdot)$
• Character field: $\Q$
• Dimension: $20$
• Newform subspaces: $14$
• Sturm bound: $720$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1740 = 2^{2} \cdot 3 \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1740.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(720\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1740))\).

	Total	New	Old
Modular forms	372	20	352
Cusp forms	349	20	329
Eisenstein series	23	0	23

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

\(2\)	\(3\)	\(5\)	\(29\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(20\)	\(0\)	\(20\)		\(19\)	\(0\)	\(19\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(26\)	\(0\)	\(26\)		\(24\)	\(0\)	\(24\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(24\)	\(0\)	\(24\)		\(22\)	\(0\)	\(22\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(24\)	\(0\)	\(24\)		\(22\)	\(0\)	\(22\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(23\)	\(0\)	\(23\)		\(21\)	\(0\)	\(21\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(23\)	\(0\)	\(23\)		\(21\)	\(0\)	\(21\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(21\)	\(0\)	\(21\)		\(19\)	\(0\)	\(19\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(27\)	\(0\)	\(27\)		\(25\)	\(0\)	\(25\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(25\)	\(2\)	\(23\)		\(24\)	\(2\)	\(22\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(22\)	\(3\)	\(19\)		\(21\)	\(3\)	\(18\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(24\)	\(3\)	\(21\)		\(23\)	\(3\)	\(20\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(21\)	\(2\)	\(19\)		\(20\)	\(2\)	\(18\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(25\)	\(1\)	\(24\)		\(24\)	\(1\)	\(23\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(22\)	\(4\)	\(18\)		\(21\)	\(4\)	\(17\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(24\)	\(4\)	\(20\)		\(23\)	\(4\)	\(19\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(21\)	\(1\)	\(20\)		\(20\)	\(1\)	\(19\)		\(1\)	\(0\)	\(1\)
Plus space				\(+\)		\(180\)	\(8\)	\(172\)		\(169\)	\(8\)	\(161\)		\(11\)	\(0\)	\(11\)
Minus space				\(-\)		\(192\)	\(12\)	\(180\)		\(180\)	\(12\)	\(168\)		\(12\)	\(0\)	\(12\)

Trace form

20 * q + 20 * q^9 + 8 * q^13 + 8 * q^17 - 8 * q^19 + 8 * q^23 + 20 * q^25 + 8 * q^31 - 8 * q^35 + 8 * q^39 - 8 * q^41 + 8 * q^43 - 8 * q^47 + 28 * q^49 + 8 * q^51 + 8 * q^55 + 8 * q^57 - 24 * q^59 + 24 * q^61 + 8 * q^67 + 8 * q^69 - 32 * q^73 - 8 * q^77 + 16 * q^79 + 20 * q^81 + 32 * q^83 - 8 * q^85 - 24 * q^89 + 8 * q^91 + 24 * q^93 - 16 * q^95 + 40 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1740))\) 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	5	29
1740.2.a.a	$1740$	$2$	1740.a	1.a	$1$	$1$	$1$	$13.894$	\(\Q\)	None	✓	✓			1740.2.a.a	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-6q^{11}+6q^{13}+\cdots\)
1740.2.a.b	$1740$	$2$	1740.a	1.a	$1$	$1$	$1$	$13.894$	\(\Q\)	None	✓	✓			1740.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(-2\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{7}+q^{9}+3q^{11}-2q^{13}+\cdots\)
1740.2.a.c	$1740$	$2$	1740.a	1.a	$1$	$1$	$1$	$13.894$	\(\Q\)	None	✓	✓			1740.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+2q^{7}+q^{9}-4q^{11}+2q^{13}+\cdots\)
1740.2.a.d	$1740$	$2$	1740.a	1.a	$1$	$1$	$1$	$13.894$	\(\Q\)	None	✓	✓			1740.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(1\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+3q^{7}+q^{9}+3q^{11}+3q^{13}+\cdots\)
1740.2.a.e	$1740$	$2$	1740.a	1.a	$1$	$1$	$1$	$13.894$	\(\Q\)	None	✓	✓	✓	✓	1740.2.a.e	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}+q^{9}-3q^{11}-q^{13}+\cdots\)
1740.2.a.f	$1740$	$2$	1740.a	1.a	$1$	$1$	$1$	$13.894$	\(\Q\)	None	✓	✓			1740.2.a.f	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-5\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-5q^{7}+q^{9}+5q^{11}-5q^{13}+\cdots\)
1740.2.a.g	$1740$	$2$	1740.a	1.a	$1$	$1$	$1$	$13.894$	\(\Q\)	None	✓	✓	✓	✓	1740.2.a.g	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-3\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-3q^{7}+q^{9}-3q^{11}+q^{13}+\cdots\)
1740.2.a.h	$1740$	$2$	1740.a	1.a	$1$	$1$	$1$	$13.894$	\(\Q\)	None	✓	✓			1740.2.a.h	$1$	$0$	\(0\)	\(1\)	\(1\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+2q^{7}+q^{9}+4q^{11}+2q^{13}+\cdots\)
1740.2.a.i	$1740$	$2$	1740.a	1.a	$1$	$2$	$2$	$13.894$	\(\Q(\sqrt{33}) \)	None	✓	✓	✓		1740.2.a.i	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-1\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-\beta q^{7}+q^{9}+(2+\beta )q^{11}+\cdots\)
1740.2.a.j	$1740$	$2$	1740.a	1.a	$1$	$2$	$2$	$13.894$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	1740.2.a.j	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(3\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+(1+\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
1740.2.a.k	$1740$	$2$	1740.a	1.a	$1$	$2$	$2$	$13.894$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓		1740.2.a.k	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-5\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(-2-\beta )q^{7}+q^{9}-\beta q^{11}+\cdots\)
1740.2.a.l	$1740$	$2$	1740.a	1.a	$1$	$2$	$2$	$13.894$	\(\Q(\sqrt{33}) \)	None	✓	✓			1740.2.a.l	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(1\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+\beta q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
1740.2.a.m	$1740$	$2$	1740.a	1.a	$1$	$2$	$2$	$13.894$	\(\Q(\sqrt{17}) \)	None	✓	✓			1740.2.a.m	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+2\beta q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)
1740.2.a.n	$1740$	$2$	1740.a	1.a	$1$	$2$	$2$	$13.894$	\(\Q(\sqrt{57}) \)	None	✓	✓	✓		1740.2.a.n	$1$	$0$	\(0\)	\(2\)	\(2\)	\(4\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+2q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1740))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1740)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(116))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(145))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(290))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(348))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(435))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(580))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(870))\)\(^{\oplus 2}\)