Space of modular forms of level 44, weight 4, and trivial character

• Label: 44.4.a
• Level: $44$
• Weight: $4$
• Character orbit: 44.a
• Rep. character: $\chi_{44}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $24$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 44 = 2^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	44.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(24\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(44))\).

	Total	New	Old
Modular forms	21	3	18
Cusp forms	15	3	12
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.

\(2\)	\(11\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(4\)	\(0\)	\(4\)		\(2\)	\(0\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(5\)	\(2\)	\(3\)		\(4\)	\(2\)	\(2\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(12\)	\(2\)	\(10\)		\(9\)	\(2\)	\(7\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(9\)	\(1\)	\(8\)		\(6\)	\(1\)	\(5\)		\(3\)	\(0\)	\(3\)

Trace form

3 * q + 4 * q^3 + 4 * q^5 - 16 * q^7 + 33 * q^9 + 11 * q^11 + 30 * q^13 + 36 * q^15 - 114 * q^17 - 84 * q^19 - 116 * q^21 - 92 * q^23 - 217 * q^25 + 496 * q^27 - 122 * q^29 + 108 * q^31 + 154 * q^33 + 528 * q^35 + 312 * q^37 - 456 * q^39 - 790 * q^41 + 340 * q^43 - 230 * q^45 - 536 * q^47 + 1443 * q^49 - 1144 * q^51 + 354 * q^53 + 198 * q^55 + 1504 * q^57 + 804 * q^59 - 1010 * q^61 - 2392 * q^63 - 388 * q^65 + 204 * q^67 - 1786 * q^69 + 1604 * q^71 - 814 * q^73 - 788 * q^75 + 396 * q^77 + 2552 * q^79 + 1467 * q^81 - 1204 * q^83 - 1008 * q^85 + 1336 * q^87 - 496 * q^89 - 880 * q^91 + 3318 * q^93 - 232 * q^95 + 1204 * q^97 + 407 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(44))\) 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	11
44.4.a.a	$44$	$4$	44.a	1.a	$1$	$1$	$1$	$2.596$	\(\Q\)	None	✓	✓	✓	✓	44.4.a.a	$1$	$1$	\(0\)	\(-5\)	\(-7\)	\(-26\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}-7q^{5}-26q^{7}-2q^{9}-11q^{11}+\cdots\)
44.4.a.b	$44$	$4$	44.a	1.a	$1$	$2$	$2$	$2.596$	\(\Q(\sqrt{97}) \)	None	✓	✓	✓	✓	44.4.a.b	$1$	$0$	\(0\)	\(9\)	\(11\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{3}+(5+\beta )q^{5}+(2+6\beta )q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(44))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(44)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 2}\)