Space of modular forms of level 441, weight 6, and trivial character

• Label: 441.6.a
• Level: $441$
• Weight: $6$
• Character orbit: 441.a
• Rep. character: $\chi_{441}(1,\cdot)$
• Character field: $\Q$
• Dimension: $83$
• Newform subspaces: $31$
• Sturm bound: $336$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 31 \)
Sturm bound:			\(336\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(2\), \(5\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(441))\).

	Total	New	Old
Modular forms	296	88	208
Cusp forms	264	83	181
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(72\)	\(15\)	\(57\)		\(64\)	\(15\)	\(49\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)		\(76\)	\(19\)	\(57\)		\(68\)	\(19\)	\(49\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)		\(76\)	\(27\)	\(49\)		\(68\)	\(25\)	\(43\)		\(8\)	\(2\)	\(6\)
\(-\)	\(-\)	\(+\)		\(72\)	\(27\)	\(45\)		\(64\)	\(24\)	\(40\)		\(8\)	\(3\)	\(5\)
Plus space		\(+\)		\(144\)	\(42\)	\(102\)		\(128\)	\(39\)	\(89\)		\(16\)	\(3\)	\(13\)
Minus space		\(-\)		\(152\)	\(46\)	\(106\)		\(136\)	\(44\)	\(92\)		\(16\)	\(2\)	\(14\)

Trace form

83 * q + 2 * q^2 + 1296 * q^4 - 36 * q^5 + 132 * q^8 + 192 * q^10 + 200 * q^11 + 104 * q^13 + 18876 * q^16 - 2208 * q^17 - 4312 * q^19 - 7764 * q^20 - 4964 * q^22 + 6304 * q^23 + 46469 * q^25 + 5544 * q^26 + 7618 * q^29 + 5576 * q^31 + 5836 * q^32 + 18522 * q^34 + 17366 * q^37 - 26526 * q^38 + 26136 * q^40 - 13008 * q^41 - 26896 * q^43 + 30376 * q^44 + 50612 * q^46 - 8832 * q^47 + 70322 * q^50 + 64532 * q^52 - 29558 * q^53 - 14472 * q^55 + 93092 * q^58 - 58176 * q^59 + 23432 * q^61 + 131004 * q^62 + 238484 * q^64 - 47580 * q^65 - 172056 * q^67 - 105270 * q^68 + 150668 * q^71 + 1964 * q^73 + 280596 * q^74 - 207970 * q^76 + 37540 * q^79 - 498564 * q^80 + 137586 * q^82 + 172560 * q^83 - 159324 * q^85 - 635940 * q^86 - 468000 * q^88 - 77016 * q^89 + 562352 * q^92 + 646404 * q^94 + 321228 * q^95 - 147724 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(441))\) 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
441.6.a.a	$441$	$6$	441.a	1.a	$1$	$1$	$1$	$70.729$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				49.6.a.b	$2$	$1$	\(-11\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-11q^{2}+89q^{4}-627q^{8}+76q^{11}+\cdots\)
441.6.a.b	$441$	$6$	441.a	1.a	$1$	$1$	$1$	$70.729$	\(\Q\)	None	✓				21.6.a.d	$1$	$1$	\(-10\)	\(0\)	\(-106\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-10q^{2}+68q^{4}-106q^{5}-360q^{8}+\cdots\)
441.6.a.c	$441$	$6$	441.a	1.a	$1$	$1$	$1$	$70.729$	\(\Q\)	None	✓				21.6.a.c	$1$	$1$	\(-5\)	\(0\)	\(94\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}-7q^{4}+94q^{5}+195q^{8}-470q^{10}+\cdots\)
441.6.a.d	$441$	$6$	441.a	1.a	$1$	$1$	$1$	$70.729$	\(\Q\)	None	✓				21.6.a.b	$1$	$1$	\(-1\)	\(0\)	\(-34\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-31q^{4}-34q^{5}+63q^{8}+34q^{10}+\cdots\)
441.6.a.e	$441$	$6$	441.a	1.a	$1$	$1$	$1$	$70.729$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				63.6.e.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{5}q^{4}-427q^{13}+2^{10}q^{16}+3143q^{19}+\cdots\)
441.6.a.f	$441$	$6$	441.a	1.a	$1$	$1$	$1$	$70.729$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓				63.6.e.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2^{5}q^{4}+427q^{13}+2^{10}q^{16}-3143q^{19}+\cdots\)
441.6.a.g	$441$	$6$	441.a	1.a	$1$	$1$	$1$	$70.729$	\(\Q\)	None	✓				21.6.e.a	$1$	$0$	\(2\)	\(0\)	\(-11\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-28q^{4}-11q^{5}-120q^{8}+\cdots\)
441.6.a.h	$441$	$6$	441.a	1.a	$1$	$1$	$1$	$70.729$	\(\Q\)	None	✓				21.6.e.a	$1$	$1$	\(2\)	\(0\)	\(11\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-28q^{4}+11q^{5}-120q^{8}+\cdots\)
441.6.a.i	$441$	$6$	441.a	1.a	$1$	$1$	$1$	$70.729$	\(\Q\)	None	✓				3.6.a.a	$1$	$1$	\(6\)	\(0\)	\(6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{2}+4q^{4}+6q^{5}-168q^{8}+6^{2}q^{10}+\cdots\)
441.6.a.j	$441$	$6$	441.a	1.a	$1$	$1$	$1$	$70.729$	\(\Q\)	None	✓				21.6.a.a	$1$	$1$	\(6\)	\(0\)	\(78\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{2}+4q^{4}+78q^{5}-168q^{8}+468q^{10}+\cdots\)
441.6.a.k	$441$	$6$	441.a	1.a	$1$	$1$	$1$	$70.729$	\(\Q\)	None	✓				7.6.a.a	$1$	$1$	\(10\)	\(0\)	\(-56\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{2}+68q^{4}-56q^{5}+360q^{8}+\cdots\)
441.6.a.l	$441$	$6$	441.a	1.a	$1$	$2$	$2$	$70.729$	\(\Q(\sqrt{57}) \)	None	✓				7.6.a.b	$1$	$1$	\(-9\)	\(0\)	\(-18\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{2}+(-2+9\beta )q^{4}+(-14+\cdots)q^{5}+\cdots\)
441.6.a.m	$441$	$6$	441.a	1.a	$1$	$2$	$2$	$70.729$	\(\Q(\sqrt{37}) \)	None	✓				7.6.c.a	$1$	$1$	\(-2\)	\(0\)	\(-38\)	\(0\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(6+2\beta )q^{4}+(-19+\cdots)q^{5}+\cdots\)
441.6.a.n	$441$	$6$	441.a	1.a	$1$	$2$	$2$	$70.729$	\(\Q(\sqrt{37}) \)	None	✓				7.6.c.a	$1$	$0$	\(-2\)	\(0\)	\(38\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+(6+2\beta )q^{4}+(19+10\beta )q^{5}+\cdots\)
441.6.a.o	$441$	$6$	441.a	1.a	$1$	$2$	$2$	$70.729$	\(\Q(\sqrt{7}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			441.6.a.o	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+\beta q^{2}-5^{2}q^{4}-57\beta q^{8}-302\beta q^{11}+\cdots\)
441.6.a.p	$441$	$6$	441.a	1.a	$1$	$2$	$2$	$70.729$	\(\Q(\sqrt{7}) \)	None	✓				63.6.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-4q^{4}+7\beta q^{5}-6^{2}\beta q^{8}+14^{2}q^{10}+\cdots\)
441.6.a.q	$441$	$6$	441.a	1.a	$1$	$2$	$2$	$70.729$	\(\Q(\sqrt{193}) \)	None	✓				147.6.a.h	$1$	$1$	\(3\)	\(0\)	\(-72\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(17+3\beta )q^{4}-6^{2}q^{5}+\cdots\)
441.6.a.r	$441$	$6$	441.a	1.a	$1$	$2$	$2$	$70.729$	\(\Q(\sqrt{193}) \)	None	✓				147.6.a.h	$1$	$1$	\(3\)	\(0\)	\(72\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(17+3\beta )q^{4}+6^{2}q^{5}+\cdots\)
441.6.a.s	$441$	$6$	441.a	1.a	$1$	$2$	$2$	$70.729$	\(\Q(\sqrt{249}) \)	None	✓				21.6.e.b	$1$	$1$	\(3\)	\(0\)	\(-33\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(31+3\beta )q^{4}+(-13-7\beta )q^{5}+\cdots\)
441.6.a.t	$441$	$6$	441.a	1.a	$1$	$2$	$2$	$70.729$	\(\Q(\sqrt{249}) \)	None	✓				21.6.e.b	$1$	$0$	\(3\)	\(0\)	\(33\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(31+3\beta )q^{4}+(13+7\beta )q^{5}+\cdots\)
441.6.a.u	$441$	$6$	441.a	1.a	$1$	$2$	$2$	$70.729$	\(\Q(\sqrt{39}) \)	None	✓				49.6.a.c	$2$	$1$	\(4\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+2q^{2}-28q^{4}-\beta q^{5}-120q^{8}-2\beta q^{10}+\cdots\)
441.6.a.v	$441$	$6$	441.a	1.a	$1$	$4$	$4$	$70.729$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		21.6.e.c	$1$	$1$	\(-3\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$7$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(18-\beta _{1}+\beta _{2})q^{4}+\cdots\)
441.6.a.w	$441$	$6$	441.a	1.a	$1$	$4$	$4$	$70.729$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				21.6.e.c	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$7$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(18-\beta _{1}+\beta _{2})q^{4}+\cdots\)
441.6.a.x	$441$	$6$	441.a	1.a	$1$	$4$	$4$	$70.729$	4.4.358541904.1	None	✓				63.6.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(24+\beta _{3})q^{4}+(-3\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
441.6.a.y	$441$	$6$	441.a	1.a	$1$	$4$	$4$	$70.729$	\(\Q(\sqrt{19}, \sqrt{69})\)	None	✓	✓			441.6.a.y	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+44q^{4}+\beta _{2}q^{5}+12\beta _{1}q^{8}+\cdots\)
441.6.a.z	$441$	$6$	441.a	1.a	$1$	$4$	$4$	$70.729$	\(\Q(\sqrt{2}, \sqrt{113})\)	None	✓				49.6.a.g	$2$	$0$	\(10\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$7$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}-5\beta _{1}q^{4}+(8\beta _{2}-2\beta _{3})q^{5}+\cdots\)
441.6.a.ba	$441$	$6$	441.a	1.a	$1$	$6$	$6$	$70.729$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				147.6.a.n	$1$	$0$	\(-2\)	\(0\)	\(-100\)	\(0\)		$-$	$+$	$-$	$2^{4}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(5^{2}-\beta _{1}+\beta _{4})q^{4}+(-2^{4}+\cdots)q^{5}+\cdots\)
441.6.a.bb	$441$	$6$	441.a	1.a	$1$	$6$	$6$	$70.729$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				147.6.a.n	$1$	$0$	\(-2\)	\(0\)	\(100\)	\(0\)		$-$	$+$	$-$	$2^{4}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+(5^{2}-\beta _{1}+\beta _{4})q^{4}+(2^{4}+2\beta _{1}+\cdots)q^{5}+\cdots\)
441.6.a.bc	$441$	$6$	441.a	1.a	$1$	$6$	$6$	$70.729$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				63.6.e.f	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(30+\beta _{3})q^{4}+(-2\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
441.6.a.bd	$441$	$6$	441.a	1.a	$1$	$6$	$6$	$70.729$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		63.6.e.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(30+\beta _{3})q^{4}+(2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
441.6.a.be	$441$	$6$	441.a	1.a	$1$	$8$	$8$	$70.729$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		441.6.a.be	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$2^{5}\cdot 7^{4}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}+(3-\beta _{2})q^{4}+\beta _{1}q^{5}+(-\beta _{4}+\cdots)q^{8}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(441))\) into lower level spaces

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