Space of modular forms of level 441, weight 2, and trivial character

• Label: 441.2.a
• Level: $441$
• Weight: $2$
• Character orbit: 441.a
• Rep. character: $\chi_{441}(1,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $10$
• Sturm bound: $112$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(112\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(441))\).

	Total	New	Old
Modular forms	72	19	53
Cusp forms	41	14	27
Eisenstein series	31	5	26

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

\(3\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(1\)
\(+\)	\(-\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(5\)
\(-\)	\(-\)	\(+\)	\(3\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(10\)

Trace form

14 * q + 14 * q^4 - 2 * q^5 + 12 * q^8 + 10 * q^10 + 4 * q^11 - 2 * q^13 + 22 * q^16 - 6 * q^17 + 4 * q^19 + 2 * q^20 - 16 * q^22 - 10 * q^25 + 2 * q^26 + 8 * q^29 + 8 * q^31 + 4 * q^32 - 18 * q^34 + 4 * q^37 - 4 * q^38 - 6 * q^40 + 2 * q^41 + 20 * q^43 + 24 * q^44 - 20 * q^50 - 6 * q^52 - 12 * q^53 + 32 * q^55 - 36 * q^58 + 12 * q^59 + 22 * q^61 - 18 * q^64 - 24 * q^65 + 20 * q^67 + 6 * q^68 + 4 * q^71 - 22 * q^73 - 16 * q^74 + 12 * q^76 + 28 * q^79 + 2 * q^80 - 34 * q^82 - 12 * q^83 - 20 * q^85 - 44 * q^86 - 72 * q^88 - 14 * q^89 - 48 * q^92 - 24 * q^94 + 28 * q^95 - 46 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(441))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	7
441.2.a.a	$441$	$2$	441.a	1.a	$1$	$1$	$1$	$3.521$	\(\Q\)	None	✓				21.2.e.a	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-2q^{5}+4q^{10}+2q^{11}+\cdots\)
441.2.a.b	$441$	$2$	441.a	1.a	$1$	$1$	$1$	$3.521$	\(\Q\)	None	✓				21.2.e.a	$1$	$0$	\(-2\)	\(0\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+2q^{5}-4q^{10}+2q^{11}+\cdots\)
441.2.a.c	$441$	$2$	441.a	1.a	$1$	$1$	$1$	$3.521$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				49.2.a.a	$2$	$1$	\(-1\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-q^{2}-q^{4}+3q^{8}-4q^{11}-q^{16}+\cdots\)
441.2.a.d	$441$	$2$	441.a	1.a	$1$	$1$	$1$	$3.521$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓		✓	✓	63.2.e.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}-7q^{13}+4q^{16}-7q^{19}-5q^{25}+\cdots\)
441.2.a.e	$441$	$2$	441.a	1.a	$1$	$1$	$1$	$3.521$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				63.2.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{4}+7q^{13}+4q^{16}+7q^{19}-5q^{25}+\cdots\)
441.2.a.f	$441$	$2$	441.a	1.a	$1$	$1$	$1$	$3.521$	\(\Q\)	None	✓				21.2.a.a	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-2q^{5}-3q^{8}-2q^{10}+\cdots\)
441.2.a.g	$441$	$2$	441.a	1.a	$1$	$2$	$2$	$3.521$	\(\Q(\sqrt{3}) \)	None	✓				63.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{4}+2\beta q^{5}-\beta q^{8}+6q^{10}+\cdots\)
441.2.a.h	$441$	$2$	441.a	1.a	$1$	$2$	$2$	$3.521$	\(\Q(\sqrt{7}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			441.2.a.h	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+\beta q^{2}+5q^{4}+3\beta q^{8}-2\beta q^{11}+11q^{16}+\cdots\)
441.2.a.i	$441$	$2$	441.a	1.a	$1$	$2$	$2$	$3.521$	\(\Q(\sqrt{2}) \)	None	✓				147.2.a.d	$1$	$0$	\(2\)	\(0\)	\(-4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(-2+\beta )q^{5}+\cdots\)
441.2.a.j	$441$	$2$	441.a	1.a	$1$	$2$	$2$	$3.521$	\(\Q(\sqrt{2}) \)	None	✓				147.2.a.d	$1$	$0$	\(2\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+(2-\beta )q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(441))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(441)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)