Space of modular forms of level 441, weight 4, and trivial character

• Label: 441.4.a
• Level: $441$
• Weight: $4$
• Character orbit: 441.a
• Rep. character: $\chi_{441}(1,\cdot)$
• Character field: $\Q$
• Dimension: $49$
• Newform subspaces: $24$
• Sturm bound: $224$
• Trace bound: $10$

Defining parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(224\)
Trace bound:			\(10\)
Distinguishing \(T_p\):			\(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(441))\).

	Total	New	Old
Modular forms	184	54	130
Cusp forms	152	49	103
Eisenstein series	32	5	27

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
\(+\)	\(+\)	\(+\)	\(12\)
\(+\)	\(-\)	\(-\)	\(9\)
\(-\)	\(+\)	\(-\)	\(13\)
\(-\)	\(-\)	\(+\)	\(15\)
Plus space		\(+\)	\(27\)
Minus space		\(-\)	\(22\)

Trace form

49 * q - 4 * q^2 + 188 * q^4 - 36 * q^8 - 32 * q^10 + 26 * q^11 - 42 * q^13 + 748 * q^16 + 174 * q^17 + 54 * q^19 + 60 * q^20 + 420 * q^22 - 2 * q^23 + 1249 * q^25 + 144 * q^26 - 476 * q^29 - 464 * q^31 - 212 * q^32 + 462 * q^34 - 756 * q^37 + 1470 * q^38 + 456 * q^40 - 834 * q^41 + 64 * q^43 + 1672 * q^44 + 68 * q^46 + 876 * q^47 - 1264 * q^50 - 748 * q^52 - 266 * q^53 - 64 * q^55 - 208 * q^58 + 1602 * q^59 - 1866 * q^61 - 1836 * q^62 + 4148 * q^64 + 672 * q^65 - 130 * q^67 + 114 * q^68 - 1912 * q^71 - 668 * q^73 + 216 * q^74 + 950 * q^76 + 1838 * q^79 + 4140 * q^80 + 134 * q^82 + 906 * q^83 + 454 * q^85 + 4740 * q^86 + 7536 * q^88 + 1650 * q^89 - 1024 * q^92 - 2436 * q^94 - 702 * q^95 + 184 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(441))\) 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
441.4.a.a	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	None	✓				147.4.a.f	$1$	$0$	\(-4\)	\(0\)	\(-18\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+8q^{4}-18q^{5}+72q^{10}+50q^{11}+\cdots\)
441.4.a.b	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	None	✓				21.4.a.b	$1$	$0$	\(-4\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+8q^{4}-4q^{5}+2^{4}q^{10}-62q^{11}+\cdots\)
441.4.a.c	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	None	✓				147.4.a.f	$1$	$0$	\(-4\)	\(0\)	\(18\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+8q^{4}+18q^{5}-72q^{10}+50q^{11}+\cdots\)
441.4.a.d	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	None	✓				7.4.c.a	$1$	$1$	\(-2\)	\(0\)	\(-7\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-4q^{4}-7q^{5}+24q^{8}+14q^{10}+\cdots\)
441.4.a.e	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	None	✓				7.4.c.a	$1$	$0$	\(-2\)	\(0\)	\(7\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-4q^{4}+7q^{5}+24q^{8}-14q^{10}+\cdots\)
441.4.a.f	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				9.4.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-8q^{4}+70q^{13}+2^{6}q^{16}-56q^{19}+\cdots\)
441.4.a.g	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	None	✓				147.4.a.d	$1$	$0$	\(1\)	\(0\)	\(-12\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}-12q^{5}-15q^{8}-12q^{10}+\cdots\)
441.4.a.h	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	None	✓				147.4.a.d	$1$	$0$	\(1\)	\(0\)	\(12\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}+12q^{5}-15q^{8}+12q^{10}+\cdots\)
441.4.a.i	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	None	✓				7.4.a.a	$1$	$0$	\(1\)	\(0\)	\(16\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-7q^{4}+2^{4}q^{5}-15q^{8}+2^{4}q^{10}+\cdots\)
441.4.a.j	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	None	✓				21.4.a.a	$1$	$0$	\(3\)	\(0\)	\(-18\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}-18q^{5}-21q^{8}-54q^{10}+\cdots\)
441.4.a.k	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	None	✓				21.4.e.a	$1$	$0$	\(3\)	\(0\)	\(-3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}-3q^{5}-21q^{8}-9q^{10}+\cdots\)
441.4.a.l	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	None	✓				21.4.e.a	$1$	$1$	\(3\)	\(0\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{2}+q^{4}+3q^{5}-21q^{8}+9q^{10}+\cdots\)
441.4.a.m	$441$	$4$	441.a	1.a	$1$	$1$	$1$	$26.020$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				49.4.a.a	$2$	$0$	\(5\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+5q^{2}+17q^{4}+45q^{8}+68q^{11}+\cdots\)
441.4.a.n	$441$	$4$	441.a	1.a	$1$	$2$	$2$	$26.020$	\(\Q(\sqrt{2}) \)	None	✓				147.4.a.j	$1$	$1$	\(-2\)	\(0\)	\(-20\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(-5-2\beta )q^{4}+(-10+\cdots)q^{5}+\cdots\)
441.4.a.o	$441$	$4$	441.a	1.a	$1$	$2$	$2$	$26.020$	\(\Q(\sqrt{2}) \)	None	✓				147.4.a.j	$1$	$1$	\(-2\)	\(0\)	\(20\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(-5-2\beta )q^{4}+(10+\cdots)q^{5}+\cdots\)
441.4.a.p	$441$	$4$	441.a	1.a	$1$	$2$	$2$	$26.020$	\(\Q(\sqrt{7}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			441.4.a.p	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+\beta q^{2}-q^{4}-9\beta q^{8}+10\beta q^{11}-55q^{16}+\cdots\)
441.4.a.q	$441$	$4$	441.a	1.a	$1$	$2$	$2$	$26.020$	\(\Q(\sqrt{19}) \)	None	✓				63.4.a.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+11q^{4}-2\beta q^{5}+3\beta q^{8}-38q^{10}+\cdots\)
441.4.a.r	$441$	$4$	441.a	1.a	$1$	$2$	$2$	$26.020$	\(\Q(\sqrt{57}) \)	None	✓				21.4.a.c	$1$	$0$	\(3\)	\(0\)	\(6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(7+3\beta )q^{4}+(2+2\beta )q^{5}+\cdots\)
441.4.a.s	$441$	$4$	441.a	1.a	$1$	$3$	$3$	$26.020$	3.3.57516.1	None	✓				21.4.e.b	$1$	$1$	\(-1\)	\(0\)	\(-11\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(8+\beta _{1}+\beta _{2})q^{4}+(-4+\beta _{1}+\cdots)q^{5}+\cdots\)
441.4.a.t	$441$	$4$	441.a	1.a	$1$	$3$	$3$	$26.020$	3.3.57516.1	None	✓		✓		21.4.e.b	$1$	$0$	\(-1\)	\(0\)	\(11\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(8+\beta _{1}+\beta _{2})q^{4}+(4-\beta _{1}+\cdots)q^{5}+\cdots\)
441.4.a.u	$441$	$4$	441.a	1.a	$1$	$4$	$4$	$26.020$	\(\Q(\sqrt{2}, \sqrt{65})\)	None	✓		✓		49.4.a.e	$2$	$1$	\(-2\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$7$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(9+\beta _{1})q^{4}+\beta _{3}q^{5}+\cdots\)
441.4.a.v	$441$	$4$	441.a	1.a	$1$	$4$	$4$	$26.020$	4.4.6257832.1	None	✓		✓		63.4.e.d	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
441.4.a.w	$441$	$4$	441.a	1.a	$1$	$4$	$4$	$26.020$	4.4.6257832.1	None	✓				63.4.e.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(\beta _{1}+\beta _{3})q^{5}+\cdots\)
441.4.a.x	$441$	$4$	441.a	1.a	$1$	$8$	$8$	$26.020$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		441.4.a.x	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(8-\beta _{3})q^{4}+\beta _{1}q^{5}+(-11\beta _{2}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(441))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(441)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 2}\)