Space of modular forms of level 484, weight 4, and trivial character

• Label: 484.4.a
• Level: $484$
• Weight: $4$
• Character orbit: 484.a
• Rep. character: $\chi_{484}(1,\cdot)$
• Character field: $\Q$
• Dimension: $27$
• Newform subspaces: $9$
• Sturm bound: $264$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 484 = 2^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	484.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(264\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(484))\).

	Total	New	Old
Modular forms	216	27	189
Cusp forms	180	27	153
Eisenstein series	36	0	36

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
\(+\)	\(+\)	\(+\)		\(57\)	\(0\)	\(57\)		\(45\)	\(0\)	\(45\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)		\(54\)	\(0\)	\(54\)		\(42\)	\(0\)	\(42\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(51\)	\(12\)	\(39\)		\(45\)	\(12\)	\(33\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(54\)	\(15\)	\(39\)		\(48\)	\(15\)	\(33\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(111\)	\(15\)	\(96\)		\(93\)	\(15\)	\(78\)		\(18\)	\(0\)	\(18\)
Minus space		\(-\)		\(105\)	\(12\)	\(93\)		\(87\)	\(12\)	\(75\)		\(18\)	\(0\)	\(18\)

Trace form

27 * q - 6 * q^3 + 16 * q^7 + 253 * q^9 - 30 * q^13 - 4 * q^15 + 114 * q^17 + 84 * q^19 + 116 * q^21 + 188 * q^23 + 895 * q^25 - 624 * q^27 + 122 * q^29 - 136 * q^31 - 528 * q^35 - 192 * q^37 + 456 * q^39 + 790 * q^41 - 340 * q^43 + 544 * q^45 + 144 * q^47 - 113 * q^49 + 1144 * q^51 - 376 * q^53 - 1504 * q^57 - 658 * q^59 + 1010 * q^61 + 2392 * q^63 + 388 * q^65 + 634 * q^67 + 1084 * q^69 - 2164 * q^71 + 814 * q^73 + 702 * q^75 - 2552 * q^79 + 1755 * q^81 + 1204 * q^83 + 1008 * q^85 - 1336 * q^87 + 786 * q^89 + 4048 * q^91 + 232 * q^95 + 1450 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(484))\) 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
484.4.a.a	$484$	$4$	484.a	1.a	$1$	$1$	$1$	$28.557$	\(\Q\)	None	✓				44.4.a.a	$1$	$0$	\(0\)	\(-5\)	\(-7\)	\(26\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{3}-7q^{5}+26q^{7}-2q^{9}-52q^{13}+\cdots\)
484.4.a.b	$484$	$4$	484.a	1.a	$1$	$2$	$2$	$28.557$	\(\Q(\sqrt{33}) \)	\(\Q(\sqrt{-11}) \)	✓	✓			484.4.a.b	$2$	$1$	\(0\)	\(-8\)	\(-18\)	\(0\)		$-$	$+$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+(-4-\beta )q^{3}+(-9-2\beta )q^{5}+(22+\cdots)q^{9}+\cdots\)
484.4.a.c	$484$	$4$	484.a	1.a	$1$	$2$	$2$	$28.557$	\(\Q(\sqrt{3}) \)	None	✓	✓			484.4.a.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(-18\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+3\beta )q^{3}-3\beta q^{5}+(-9-\beta )q^{7}+\cdots\)
484.4.a.d	$484$	$4$	484.a	1.a	$1$	$2$	$2$	$28.557$	\(\Q(\sqrt{3}) \)	None	✓	✓			484.4.a.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(18\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+3\beta )q^{3}-3\beta q^{5}+(9+\beta )q^{7}+\cdots\)
484.4.a.e	$484$	$4$	484.a	1.a	$1$	$2$	$2$	$28.557$	\(\Q(\sqrt{97}) \)	None	✓				44.4.a.b	$1$	$0$	\(0\)	\(9\)	\(11\)	\(-10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{3}+(5+\beta )q^{5}+(-2-6\beta )q^{7}+\cdots\)
484.4.a.f	$484$	$4$	484.a	1.a	$1$	$3$	$3$	$28.557$	3.3.3124.1	None	✓	✓			484.4.a.f	$1$	$0$	\(0\)	\(-2\)	\(19\)	\(-26\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(6-\beta _{1})q^{5}+(-9+\cdots)q^{7}+\cdots\)
484.4.a.g	$484$	$4$	484.a	1.a	$1$	$3$	$3$	$28.557$	3.3.3124.1	None	✓	✓			484.4.a.f	$1$	$0$	\(0\)	\(-2\)	\(19\)	\(26\)		$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(6-\beta _{1})q^{5}+(9-\beta _{1}+\cdots)q^{7}+\cdots\)
484.4.a.h	$484$	$4$	484.a	1.a	$1$	$6$	$6$	$28.557$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		44.4.e.a	$1$	$0$	\(0\)	\(3\)	\(-12\)	\(-8\)		$-$	$-$	$+$	$2^{2}\cdot 11$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-2+\beta _{1}-\beta _{4})q^{5}+\cdots\)
484.4.a.i	$484$	$4$	484.a	1.a	$1$	$6$	$6$	$28.557$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		44.4.e.a	$1$	$1$	\(0\)	\(3\)	\(-12\)	\(8\)		$-$	$+$	$-$	$2^{2}\cdot 11$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-2+\beta _{1}-\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(484))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(484)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(22))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(242))\)\(^{\oplus 2}\)