Space of modular forms of level 380, weight 2, and trivial character

• Label: 380.2.a
• Level: $380$
• Weight: $2$
• Character orbit: 380.a
• Rep. character: $\chi_{380}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $4$
• Sturm bound: $120$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 380 = 2^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	380.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(120\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(380))\).

	Total	New	Old
Modular forms	66	6	60
Cusp forms	55	6	49
Eisenstein series	11	0	11

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

\(2\)	\(5\)	\(19\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(2\)
Plus space			\(+\)	\(3\)
Minus space			\(-\)	\(3\)

Trace form

6 * q + 2 * q^5 + 6 * q^9 - 8 * q^11 - 4 * q^13 - 4 * q^15 + 4 * q^17 + 8 * q^21 - 16 * q^23 + 6 * q^25 - 12 * q^27 - 4 * q^29 - 8 * q^31 - 4 * q^33 - 8 * q^37 + 24 * q^39 - 20 * q^41 + 8 * q^43 + 10 * q^45 + 8 * q^47 - 2 * q^49 - 8 * q^51 + 4 * q^57 - 4 * q^61 + 32 * q^63 - 8 * q^65 + 4 * q^67 + 8 * q^69 + 16 * q^71 - 12 * q^73 + 16 * q^77 - 16 * q^79 + 46 * q^81 + 8 * q^83 - 12 * q^85 + 24 * q^87 - 4 * q^89 + 24 * q^93 - 16 * q^97 - 24 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(380))\) 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	19
380.2.a.a	$380$	$2$	380.a	1.a	$1$	$1$	$1$	$3.034$	\(\Q\)	None	✓	✓	✓	✓	380.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-2q^{7}-3q^{9}-4q^{11}-4q^{13}+\cdots\)
380.2.a.b	$380$	$2$	380.a	1.a	$1$	$1$	$1$	$3.034$	\(\Q\)	None	✓	✓	✓	✓	380.2.a.b	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+2q^{7}+q^{9}+6q^{13}+\cdots\)
380.2.a.c	$380$	$2$	380.a	1.a	$1$	$2$	$2$	$3.034$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	380.2.a.c	$1$	$1$	\(0\)	\(-4\)	\(2\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{3}+q^{5}+(-2-2\beta )q^{7}+\cdots\)
380.2.a.d	$380$	$2$	380.a	1.a	$1$	$2$	$2$	$3.034$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	380.2.a.d	$1$	$0$	\(0\)	\(2\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+q^{5}+2q^{7}+(1+2\beta )q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(380))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(380)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(95))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(190))\)\(^{\oplus 2}\)