Space of modular forms of level 76, weight 4, and trivial character

• Label: 76.4.a
• Level: $76$
• Weight: $4$
• Character orbit: 76.a
• Rep. character: $\chi_{76}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $2$
• Sturm bound: $40$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	76.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(40\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(76))\).

	Total	New	Old
Modular forms	33	5	28
Cusp forms	27	5	22
Eisenstein series	6	0	6

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

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

Trace form

5 * q - 4 * q^3 + 4 * q^5 + 14 * q^7 + 35 * q^9 + 8 * q^11 - 46 * q^13 + 20 * q^15 + 82 * q^17 + 19 * q^19 - 72 * q^21 + 98 * q^23 - 35 * q^25 + 8 * q^27 + 62 * q^29 - 204 * q^31 - 404 * q^33 - 348 * q^35 - 86 * q^37 + 606 * q^39 - 330 * q^41 + 144 * q^43 + 88 * q^45 - 292 * q^47 + 97 * q^49 + 104 * q^51 - 78 * q^53 - 4 * q^55 + 114 * q^57 + 508 * q^59 + 40 * q^61 + 1592 * q^63 + 368 * q^65 + 1588 * q^67 - 2300 * q^69 - 660 * q^71 - 1570 * q^73 - 156 * q^75 + 3250 * q^77 - 2676 * q^79 + 437 * q^81 + 1712 * q^83 - 750 * q^85 - 2474 * q^87 + 1078 * q^89 - 472 * q^91 + 1316 * q^93 + 266 * q^95 + 758 * q^97 + 20 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(76))\) 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	19
76.4.a.a	$76$	$4$	76.a	1.a	$1$	$2$	$2$	$4.484$	\(\Q(\sqrt{33}) \)	None	✓	✓	✓	✓	76.4.a.a	$1$	$1$	\(0\)	\(-5\)	\(-5\)	\(-30\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta )q^{3}+(-5+5\beta )q^{5}+(-13+\cdots)q^{7}+\cdots\)
76.4.a.b	$76$	$4$	76.a	1.a	$1$	$3$	$3$	$4.484$	3.3.35529.1	None	✓	✓	✓	✓	76.4.a.b	$1$	$0$	\(0\)	\(1\)	\(9\)	\(44\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{2})q^{3}+(3+\beta _{2})q^{5}+(15-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(76))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(76)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 2}\)