LMFDB - Space of modular forms of level 441, weight 8, and trivial character

Defining parameters

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	441.a (trivial)
Character field:			$\Q$
Newform subspaces:			$ 30 $
Sturm bound:			$448$
Trace bound:			$13$
Distinguishing $T_p$:			$2$, $5$, $13$

Dimensions

The following table gives the dimensions of various subspaces of $M_{8}(\Gamma_0(441))$.

	Total	New	Old
Modular forms	408	122	286
Cusp forms	376	117	259
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
$+$	$+$	$+$	$25$
$+$	$-$	$-$	$23$
$-$	$+$	$-$	$33$
$-$	$-$	$+$	$36$
Plus space		$+$	$61$
Minus space		$-$	$56$

Trace form

$ 117 q + 6 q^{2} + 7232 q^{4} - 140 q^{5} + 1284 q^{8} + O(q^{10}) $

$ 117 q + 6 q^{2} + 7232 q^{4} - 140 q^{5} + 1284 q^{8} - 4032 q^{10} - 4592 q^{11} - 10108 q^{13} + 440636 q^{16} - 28056 q^{17} + 3584 q^{19} + 15260 q^{20} - 4708 q^{22} - 31248 q^{23} + 1628559 q^{25} - 273448 q^{26} - 241750 q^{29} - 360472 q^{31} + 775468 q^{32} - 281694 q^{34} - 19978 q^{37} + 382298 q^{38} - 608664 q^{40} + 1345736 q^{41} + 1999040 q^{43} - 3060744 q^{44} - 1723916 q^{46} + 331296 q^{47} + 415966 q^{50} - 824908 q^{52} - 673074 q^{53} - 2723784 q^{55} + 1221772 q^{58} + 2294712 q^{59} + 2080988 q^{61} + 9890412 q^{62} + 29392788 q^{64} - 12490548 q^{65} + 1719184 q^{67} - 7540134 q^{68} - 992900 q^{71} + 759752 q^{73} + 2503388 q^{74} + 3362702 q^{76} - 7278012 q^{79} + 24639020 q^{80} + 17352090 q^{82} - 1067304 q^{83} + 22120104 q^{85} + 33305068 q^{86} + 26568912 q^{88} + 6448624 q^{89} - 31914816 q^{92} - 30970044 q^{94} - 24730052 q^{95} + 7436912 q^{97} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{8}^{\mathrm{new}}(\Gamma_0(441))$ into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	3	7	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
441.8.a.a	$441$	$8$	441.a	1.a	$1$	$1$	$1$	$137.762$	$\Q$	None	✓	$1$	$0$	$-6$	$0$	$390$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-6q^{2}-92q^{4}+390q^{5}+1320q^{8}+\cdots$
441.8.a.b	$441$	$8$	441.a	1.a	$1$	$1$	$1$	$137.762$	$\Q$	None	✓	$1$	$0$	$-2$	$0$	$-278$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-124q^{4}-278q^{5}+504q^{8}+\cdots$
441.8.a.c	$441$	$8$	441.a	1.a	$1$	$1$	$1$	$137.762$	$\Q$	$\Q(\sqrt{-3}) $	✓	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$-$	$-$	$1$	$N(\mathrm{U}(1))$	$q-2^{7}q^{4}-2009q^{13}+2^{14}q^{16}-14357q^{19}+\cdots$
441.8.a.d	$441$	$8$	441.a	1.a	$1$	$1$	$1$	$137.762$	$\Q$	$\Q(\sqrt{-3}) $	✓	$2$	$0$	$0$	$0$	$0$	$0$	$+$	$+$	$+$	$1$	$N(\mathrm{U}(1))$	$q-2^{7}q^{4}+2009q^{13}+2^{14}q^{16}+14357q^{19}+\cdots$
441.8.a.e	$441$	$8$	441.a	1.a	$1$	$1$	$1$	$137.762$	$\Q$	None	✓	$1$	$0$	$6$	$0$	$-84$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+6q^{2}-92q^{4}-84q^{5}-1320q^{8}+\cdots$
441.8.a.f	$441$	$8$	441.a	1.a	$1$	$1$	$1$	$137.762$	$\Q$	$\Q(\sqrt{-7}) $	✓	$2$	$0$	$13$	$0$	$0$	$0$	$-$	$-$	$+$	$1$	$N(\mathrm{U}(1))$	$q+13q^{2}+41q^{4}-1131q^{8}-8684q^{11}+\cdots$
441.8.a.g	$441$	$8$	441.a	1.a	$1$	$2$	$2$	$137.762$	$\Q(\sqrt{690}) $	None	✓	$2$	$0$	$-20$	$0$	$0$	$0$	$-$	$-$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	$q-10q^{2}-28q^{4}-\beta q^{5}+1560q^{8}+\cdots$
441.8.a.h	$441$	$8$	441.a	1.a	$1$	$2$	$2$	$137.762$	$\Q(\sqrt{67}) $	None	✓	$1$	$0$	$-12$	$0$	$-24$	$0$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	$q+(-6+\beta )q^{2}+(176-12\beta )q^{4}+(-12+\cdots)q^{5}+\cdots$
441.8.a.i	$441$	$8$	441.a	1.a	$1$	$2$	$2$	$137.762$	$\Q(\sqrt{21}) $	None	✓	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q-\beta q^{2}-44q^{4}+7\beta q^{5}+172\beta q^{8}+\cdots$
441.8.a.j	$441$	$8$	441.a	1.a	$1$	$2$	$2$	$137.762$	$\Q(\sqrt{7}) $	$\Q(\sqrt{-7}) $	✓	$4$	$1$	$0$	$0$	$0$	$0$	$+$	$-$	$-$	$7$	$N(\mathrm{U}(1))$	$q+\beta q^{2}+215q^{4}+87\beta q^{8}-86\beta q^{11}+\cdots$
441.8.a.k	$441$	$8$	441.a	1.a	$1$	$2$	$2$	$137.762$	$\Q(\sqrt{10}) $	None	✓	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	$q+\beta q^{2}+232q^{4}+2^{4}\beta q^{5}+104\beta q^{8}+\cdots$
441.8.a.l	$441$	$8$	441.a	1.a	$1$	$2$	$2$	$137.762$	$\Q(\sqrt{865}) $	None	✓	$1$	$0$	$3$	$0$	$330$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{2}+(89+3\beta )q^{4}+(160+10\beta )q^{5}+\cdots$
441.8.a.m	$441$	$8$	441.a	1.a	$1$	$2$	$2$	$137.762$	$\Q(\sqrt{1065}) $	None	✓	$1$	$0$	$9$	$0$	$-360$	$0$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	$q+(5-\beta )q^{2}+(163-9\beta )q^{4}+(-170+\cdots)q^{5}+\cdots$
441.8.a.n	$441$	$8$	441.a	1.a	$1$	$3$	$3$	$137.762$	3.3.2910828.1	None	✓	$1$	$0$	$3$	$0$	$-114$	$0$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{2}+(75+\beta _{2})q^{4}+(-38+\cdots)q^{5}+\cdots$
441.8.a.o	$441$	$8$	441.a	1.a	$1$	$4$	$4$	$137.762$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$4$	$1$	$0$	$0$	$0$	$0$	$+$	$-$	$-$	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-76q^{4}+\beta _{2}q^{5}-204\beta _{1}q^{8}+\cdots$
441.8.a.p	$441$	$8$	441.a	1.a	$1$	$4$	$4$	$137.762$	4.4.2391090268.1	None	✓	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$-$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}+(3^{3}-\beta _{3})q^{4}+(-5\beta _{1}+20\beta _{2}+\cdots)q^{5}+\cdots$
441.8.a.q	$441$	$8$	441.a	1.a	$1$	$4$	$4$	$137.762$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$1$	$1$	$0$	$-196$	$0$	$-$	$+$	$-$	$2\cdot 3\cdot 7$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(5^{2}+2\beta _{1}+\beta _{3})q^{4}+(-46+\cdots)q^{5}+\cdots$
441.8.a.r	$441$	$8$	441.a	1.a	$1$	$4$	$4$	$137.762$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$0$	$1$	$0$	$196$	$0$	$-$	$-$	$+$	$2\cdot 3\cdot 7$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(5^{2}+2\beta _{1}+\beta _{3})q^{4}+(46+\cdots)q^{5}+\cdots$
441.8.a.s	$441$	$8$	441.a	1.a	$1$	$4$	$4$	$137.762$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$1$	$6$	$0$	$-252$	$0$	$-$	$+$	$-$	$2^{4}\cdot 7$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{2}+(86-\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots$
441.8.a.t	$441$	$8$	441.a	1.a	$1$	$4$	$4$	$137.762$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$0$	$6$	$0$	$252$	$0$	$-$	$-$	$+$	$2^{4}\cdot 7$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{2}+(86-\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots$
441.8.a.u	$441$	$8$	441.a	1.a	$1$	$4$	$4$	$137.762$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$0$	$15$	$0$	$-504$	$0$	$-$	$-$	$+$	$2^{2}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	$q+(4+\beta _{1})q^{2}+(110+3\beta _{1}+\beta _{3})q^{4}+\cdots$
441.8.a.v	$441$	$8$	441.a	1.a	$1$	$4$	$4$	$137.762$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$0$	$15$	$0$	$504$	$0$	$-$	$-$	$+$	$2^{2}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	$q+(4+\beta _{1})q^{2}+(110+3\beta _{1}+\beta _{3})q^{4}+\cdots$
441.8.a.w	$441$	$8$	441.a	1.a	$1$	$5$	$5$	$137.762$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓	$1$	$0$	$-15$	$0$	$-198$	$0$	$-$	$-$	$+$	$2\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)$	$q+(-3+\beta _{1})q^{2}+(46-6\beta _{1}+\beta _{2})q^{4}+\cdots$
441.8.a.x	$441$	$8$	441.a	1.a	$1$	$5$	$5$	$137.762$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓	$1$	$1$	$-15$	$0$	$198$	$0$	$-$	$+$	$-$	$2\cdot 3\cdot 7^{2}$	$\mathrm{SU}(2)$	$q+(-3+\beta _{1})q^{2}+(46-6\beta _{1}+\beta _{2})q^{4}+\cdots$
441.8.a.y	$441$	$8$	441.a	1.a	$1$	$6$	$6$	$137.762$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	$1$	$1$	$14$	$0$	$-500$	$0$	$-$	$+$	$-$	$2^{4}\cdot 7^{3}$	$\mathrm{SU}(2)$	$q+(2-\beta _{1})q^{2}+(72-2\beta _{1}+7\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots$
441.8.a.z	$441$	$8$	441.a	1.a	$1$	$6$	$6$	$137.762$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	$1$	$1$	$14$	$0$	$500$	$0$	$-$	$+$	$-$	$2^{4}\cdot 7^{3}$	$\mathrm{SU}(2)$	$q+(2-\beta _{1})q^{2}+(72-2\beta _{1}+7\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots$
441.8.a.ba	$441$	$8$	441.a	1.a	$1$	$8$	$8$	$137.762$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$2$	$1$	$-30$	$0$	$0$	$0$	$-$	$+$	$-$	$2^{6}\cdot 7^{6}$	$\mathrm{SU}(2)$	$q+(-4-\beta _{2})q^{2}+(59+5\beta _{2}+\beta _{5})q^{4}+\cdots$
441.8.a.bb	$441$	$8$	441.a	1.a	$1$	$8$	$8$	$137.762$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$-$	$-$	$2^{5}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(103+\beta _{3})q^{4}+(-6\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$
441.8.a.bc	$441$	$8$	441.a	1.a	$1$	$8$	$8$	$137.762$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$2$	$0$	$0$	$0$	$0$	$0$	$+$	$+$	$+$	$2^{5}\cdot 3^{4}\cdot 7^{2}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(103+\beta _{3})q^{4}+(6\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$
441.8.a.bd	$441$	$8$	441.a	1.a	$1$	$16$	$16$	$137.762$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None	✓	$4$	$0$	$0$	$0$	$0$	$0$	$+$	$+$	$+$	$2^{21}\cdot 3^{8}\cdot 7^{12}$	$\mathrm{SU}(2)$	$q+\beta _{8}q^{2}+(57+\beta _{2})q^{4}+\beta _{1}q^{5}+(57\beta _{8}+\cdots)q^{8}+\cdots$

Decomposition of $S_{8}^{\mathrm{old}}(\Gamma_0(441))$ into lower level spaces

$ S_{8}^{\mathrm{old}}(\Gamma_0(441)) \cong $ $S_{8}^{\mathrm{new}}(\Gamma_0(3))$$^{\oplus 6}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(7))$$^{\oplus 6}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(9))$$^{\oplus 3}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(21))$$^{\oplus 4}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(49))$$^{\oplus 3}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(63))$$^{\oplus 2}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(147))$$^{\oplus 2}$

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Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(448\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(2\), \(5\), \(13\)

\(3\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(25\)
\(+\)	\(-\)	$-$	\(23\)
\(-\)	\(+\)	$-$	\(33\)
\(-\)	\(-\)	$+$	\(36\)
Plus space		\(+\)	\(61\)
Minus space		\(-\)	\(56\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(441))\) into newform subspaces

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(441))\) into lower level spaces

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	441.8.a

Level	$441$
Weight	$8$
Character orbit	441.a
Rep. character	$\chi_{441}(1,\cdot)$
Character field	$\Q$
Dimension	$117$
Newform subspaces	$30$
Sturm bound	$448$
Trace bound	$13$