Space of modular forms of level 1380, weight 2, and trivial character

• Label: 1380.2.a
• Level: $1380$
• Weight: $2$
• Character orbit: 1380.a
• Rep. character: $\chi_{1380}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $10$
• Sturm bound: $576$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1380.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(576\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1380))\).

	Total	New	Old
Modular forms	300	16	284
Cusp forms	277	16	261
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.

\(2\)	\(3\)	\(5\)	\(23\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)	\(1\)
Plus space				\(+\)	\(6\)
Minus space				\(-\)	\(10\)

Trace form

16 * q + 16 * q^9 - 8 * q^13 - 16 * q^17 + 16 * q^25 - 12 * q^29 + 12 * q^31 + 8 * q^33 + 4 * q^35 + 16 * q^37 + 8 * q^39 + 12 * q^41 + 24 * q^43 + 24 * q^47 + 28 * q^49 + 8 * q^51 + 8 * q^53 + 8 * q^57 + 4 * q^59 + 8 * q^67 - 4 * q^71 - 8 * q^73 - 24 * q^77 + 16 * q^81 - 16 * q^83 - 4 * q^85 - 8 * q^87 - 8 * q^89 - 16 * q^91 + 16 * q^93 - 16 * q^95 + 32 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1380))\) 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	23
1380.2.a.a	$1380$	$2$	1380.a	1.a	$1$	$1$	$1$	$11.019$	\(\Q\)	None	✓	✓			1380.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(-5\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}-5q^{7}+q^{9}+4q^{13}+q^{15}+\cdots\)
1380.2.a.b	$1380$	$2$	1380.a	1.a	$1$	$1$	$1$	$11.019$	\(\Q\)	None	✓	✓			1380.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{9}-6q^{13}+q^{15}+2q^{17}+\cdots\)
1380.2.a.c	$1380$	$2$	1380.a	1.a	$1$	$1$	$1$	$11.019$	\(\Q\)	None	✓	✓	✓	✓	1380.2.a.c	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-1\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}+q^{9}-4q^{13}-q^{15}+\cdots\)
1380.2.a.d	$1380$	$2$	1380.a	1.a	$1$	$1$	$1$	$11.019$	\(\Q\)	None	✓	✓			1380.2.a.d	$1$	$0$	\(0\)	\(1\)	\(1\)	\(-4\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-4q^{7}+q^{9}+2q^{13}+q^{15}+\cdots\)
1380.2.a.e	$1380$	$2$	1380.a	1.a	$1$	$1$	$1$	$11.019$	\(\Q\)	None	✓	✓	✓	✓	1380.2.a.e	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-3\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-3q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
1380.2.a.f	$1380$	$2$	1380.a	1.a	$1$	$2$	$2$	$11.019$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓	✓	1380.2.a.f	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)
1380.2.a.g	$1380$	$2$	1380.a	1.a	$1$	$2$	$2$	$11.019$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	1380.2.a.g	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+\beta q^{7}+q^{9}+2\beta q^{11}+\cdots\)
1380.2.a.h	$1380$	$2$	1380.a	1.a	$1$	$2$	$2$	$11.019$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	1380.2.a.h	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(2\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+(1+2\beta )q^{7}+q^{9}+(-1+\cdots)q^{11}+\cdots\)
1380.2.a.i	$1380$	$2$	1380.a	1.a	$1$	$2$	$2$	$11.019$	\(\Q(\sqrt{15}) \)	None	✓	✓	✓		1380.2.a.i	$1$	$0$	\(0\)	\(2\)	\(2\)	\(6\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+3q^{7}+q^{9}+(1+\beta )q^{11}+\cdots\)
1380.2.a.j	$1380$	$2$	1380.a	1.a	$1$	$3$	$3$	$11.019$	3.3.3144.1	None	✓	✓	✓	✓	1380.2.a.j	$1$	$0$	\(0\)	\(3\)	\(-3\)	\(2\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+(1-\beta _{1})q^{7}+q^{9}+(1+\beta _{2})q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1380))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1380)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(230))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(345))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(460))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(690))\)\(^{\oplus 2}\)