Space of modular forms of level 1360, weight 2, and trivial character

• Label: 1360.2.a
• Level: $1360$
• Weight: $2$
• Character orbit: 1360.a
• Rep. character: $\chi_{1360}(1,\cdot)$
• Character field: $\Q$
• Dimension: $32$
• Newform subspaces: $19$
• Sturm bound: $432$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1360 = 2^{4} \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1360.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(432\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1360))\).

	Total	New	Old
Modular forms	228	32	196
Cusp forms	205	32	173
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.

\(2\)	\(5\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(6\)
Plus space			\(+\)	\(14\)
Minus space			\(-\)	\(18\)

Trace form

32 * q - 4 * q^3 - 4 * q^7 + 32 * q^9 + 8 * q^11 - 8 * q^19 + 8 * q^21 + 20 * q^23 + 32 * q^25 + 8 * q^27 - 8 * q^29 - 32 * q^31 - 16 * q^37 + 16 * q^39 + 8 * q^41 - 24 * q^43 + 8 * q^45 + 24 * q^49 - 16 * q^53 - 16 * q^57 + 24 * q^59 - 12 * q^63 + 8 * q^65 + 32 * q^67 + 24 * q^69 + 8 * q^71 - 16 * q^73 - 4 * q^75 + 32 * q^79 + 24 * q^81 + 24 * q^87 + 16 * q^89 - 40 * q^91 - 32 * q^93 + 16 * q^95 + 72 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1360))\) 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	5	17
1360.2.a.a	$1360$	$2$	1360.a	1.a	$1$	$1$	$1$	$10.860$	\(\Q\)	None	✓				170.2.a.d	$1$	$1$	\(0\)	\(-3\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{5}-2q^{7}+6q^{9}+4q^{11}+\cdots\)
1360.2.a.b	$1360$	$2$	1360.a	1.a	$1$	$1$	$1$	$10.860$	\(\Q\)	None	✓				85.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-q^{5}+2q^{7}+q^{9}-2q^{11}+\cdots\)
1360.2.a.c	$1360$	$2$	1360.a	1.a	$1$	$1$	$1$	$10.860$	\(\Q\)	None	✓				680.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-2q^{7}+q^{9}+2q^{11}+\cdots\)
1360.2.a.d	$1360$	$2$	1360.a	1.a	$1$	$1$	$1$	$10.860$	\(\Q\)	None	✓				170.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-2q^{7}-2q^{9}-q^{13}+q^{15}+\cdots\)
1360.2.a.e	$1360$	$2$	1360.a	1.a	$1$	$1$	$1$	$10.860$	\(\Q\)	None	✓				170.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-2q^{7}-2q^{9}+5q^{13}+\cdots\)
1360.2.a.f	$1360$	$2$	1360.a	1.a	$1$	$1$	$1$	$10.860$	\(\Q\)	None	✓				680.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{9}-2q^{13}+q^{17}+4q^{19}+\cdots\)
1360.2.a.g	$1360$	$2$	1360.a	1.a	$1$	$1$	$1$	$10.860$	\(\Q\)	None	✓				340.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+4q^{7}-3q^{9}-2q^{11}-6q^{13}+\cdots\)
1360.2.a.h	$1360$	$2$	1360.a	1.a	$1$	$1$	$1$	$10.860$	\(\Q\)	None	✓				680.2.a.a	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{7}-2q^{9}-4q^{11}+\cdots\)
1360.2.a.i	$1360$	$2$	1360.a	1.a	$1$	$1$	$1$	$10.860$	\(\Q\)	None	✓				170.2.a.a	$1$	$1$	\(0\)	\(2\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}-2q^{7}+q^{9}-6q^{11}+\cdots\)
1360.2.a.j	$1360$	$2$	1360.a	1.a	$1$	$1$	$1$	$10.860$	\(\Q\)	None	✓				170.2.a.b	$1$	$0$	\(0\)	\(2\)	\(1\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+2q^{7}+q^{9}+2q^{11}+\cdots\)
1360.2.a.k	$1360$	$2$	1360.a	1.a	$1$	$2$	$2$	$10.860$	\(\Q(\sqrt{3}) \)	None	✓		✓		85.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+q^{5}+(1-\beta )q^{7}+(1+\cdots)q^{9}+\cdots\)
1360.2.a.l	$1360$	$2$	1360.a	1.a	$1$	$2$	$2$	$10.860$	\(\Q(\sqrt{2}) \)	None	✓				680.2.a.e	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}-\beta q^{7}-q^{9}+(2-3\beta )q^{11}+\cdots\)
1360.2.a.m	$1360$	$2$	1360.a	1.a	$1$	$2$	$2$	$10.860$	\(\Q(\sqrt{17}) \)	None	✓				170.2.a.f	$1$	$0$	\(0\)	\(1\)	\(2\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}-2\beta q^{7}+(1+\beta )q^{9}+4q^{11}+\cdots\)
1360.2.a.n	$1360$	$2$	1360.a	1.a	$1$	$2$	$2$	$10.860$	\(\Q(\sqrt{3}) \)	None	✓				680.2.a.d	$1$	$0$	\(0\)	\(2\)	\(2\)	\(6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+q^{5}+(3-\beta )q^{7}+(1+2\beta )q^{9}+\cdots\)
1360.2.a.o	$1360$	$2$	1360.a	1.a	$1$	$2$	$2$	$10.860$	\(\Q(\sqrt{2}) \)	None	✓		✓		85.2.a.b	$1$	$0$	\(0\)	\(4\)	\(-2\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{3}-q^{5}+(2-\beta )q^{7}+(3+4\beta )q^{9}+\cdots\)
1360.2.a.p	$1360$	$2$	1360.a	1.a	$1$	$3$	$3$	$10.860$	3.3.940.1	None	✓		✓		680.2.a.h	$1$	$0$	\(0\)	\(-3\)	\(3\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+q^{5}+(-1+\beta _{2})q^{7}+\cdots\)
1360.2.a.q	$1360$	$2$	1360.a	1.a	$1$	$3$	$3$	$10.860$	3.3.1016.1	None	✓		✓		680.2.a.g	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-6\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+(-2-\beta _{2})q^{7}+(1+\beta _{1}+\cdots)q^{9}+\cdots\)
1360.2.a.r	$1360$	$2$	1360.a	1.a	$1$	$3$	$3$	$10.860$	3.3.229.1	None	✓		✓		680.2.a.f	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-q^{5}+(-\beta _{1}+\beta _{2})q^{7}+(1+\cdots)q^{9}+\cdots\)
1360.2.a.s	$1360$	$2$	1360.a	1.a	$1$	$3$	$3$	$10.860$	3.3.404.1	None	✓		✓		340.2.a.b	$1$	$0$	\(0\)	\(0\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+q^{5}+\beta _{2}q^{7}+(2-\beta _{1}+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1360))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1360)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(85))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(170))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(272))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(340))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(680))\)\(^{\oplus 2}\)