Space of modular forms of level 342, weight 8, and trivial character

• Label: 342.8.a
• Level: $342$
• Weight: $8$
• Character orbit: 342.a
• Rep. character: $\chi_{342}(1,\cdot)$
• Character field: $\Q$
• Dimension: $53$
• Newform subspaces: $19$
• Sturm bound: $480$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(480\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(342))\).

	Total	New	Old
Modular forms	428	53	375
Cusp forms	412	53	359
Eisenstein series	16	0	16

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

\(2\)	\(3\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(5\)
\(+\)	\(+\)	\(-\)	$-$	\(6\)
\(+\)	\(-\)	\(+\)	$-$	\(7\)
\(+\)	\(-\)	\(-\)	$+$	\(8\)
\(-\)	\(+\)	\(+\)	$-$	\(5\)
\(-\)	\(+\)	\(-\)	$+$	\(6\)
\(-\)	\(-\)	\(+\)	$+$	\(9\)
\(-\)	\(-\)	\(-\)	$-$	\(7\)
Plus space			\(+\)	\(28\)
Minus space			\(-\)	\(25\)

Trace form

\( 53 q + 8 q^{2} + 3392 q^{4} - 646 q^{5} + 954 q^{7} + 512 q^{8} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(342))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	19
342.8.a.a	$342$	$8$	342.a	1.a	$1$	$1$	$1$	$106.836$	\(\Q\)	None	✓	$1$	$1$	\(-8\)	\(0\)	\(47\)	\(405\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+47q^{5}+405q^{7}+\cdots\)
342.8.a.b	$342$	$8$	342.a	1.a	$1$	$1$	$1$	$106.836$	\(\Q\)	None	✓	$1$	$0$	\(-8\)	\(0\)	\(75\)	\(-497\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+75q^{5}-497q^{7}+\cdots\)
342.8.a.c	$342$	$8$	342.a	1.a	$1$	$1$	$1$	$106.836$	\(\Q\)	None	✓	$1$	$1$	\(-8\)	\(0\)	\(140\)	\(-60\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+140q^{5}-60q^{7}+\cdots\)
342.8.a.d	$342$	$8$	342.a	1.a	$1$	$1$	$1$	$106.836$	\(\Q\)	None	✓	$1$	$1$	\(8\)	\(0\)	\(-450\)	\(-568\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}-450q^{5}-568q^{7}+\cdots\)
342.8.a.e	$342$	$8$	342.a	1.a	$1$	$1$	$1$	$106.836$	\(\Q\)	None	✓	$1$	$0$	\(8\)	\(0\)	\(-440\)	\(951\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}-440q^{5}+951q^{7}+\cdots\)
342.8.a.f	$342$	$8$	342.a	1.a	$1$	$1$	$1$	$106.836$	\(\Q\)	None	✓	$1$	$1$	\(8\)	\(0\)	\(135\)	\(71\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+135q^{5}+71q^{7}+\cdots\)
342.8.a.g	$342$	$8$	342.a	1.a	$1$	$2$	$2$	$106.836$	\(\Q(\sqrt{633}) \)	None	✓	$1$	$1$	\(-16\)	\(0\)	\(-155\)	\(-2238\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(-62-31\beta )q^{5}+\cdots\)
342.8.a.h	$342$	$8$	342.a	1.a	$1$	$2$	$2$	$106.836$	\(\Q(\sqrt{2737}) \)	None	✓	$1$	$0$	\(16\)	\(0\)	\(-175\)	\(-2592\)		$-$	$-$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(-90-5\beta )q^{5}+(-1295+\cdots)q^{7}+\cdots\)
342.8.a.i	$342$	$8$	342.a	1.a	$1$	$2$	$2$	$106.836$	\(\Q(\sqrt{17953}) \)	None	✓	$1$	$1$	\(16\)	\(0\)	\(69\)	\(-348\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(6^{2}-3\beta )q^{5}+(-181+\cdots)q^{7}+\cdots\)
342.8.a.j	$342$	$8$	342.a	1.a	$1$	$3$	$3$	$106.836$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(-24\)	\(0\)	\(-747\)	\(155\)		$+$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(-249-\beta _{1})q^{5}+\cdots\)
342.8.a.k	$342$	$8$	342.a	1.a	$1$	$3$	$3$	$106.836$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(-24\)	\(0\)	\(9\)	\(1065\)		$+$	$-$	$+$	$-$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(3-\beta _{1}+2\beta _{2})q^{5}+\cdots\)
342.8.a.l	$342$	$8$	342.a	1.a	$1$	$3$	$3$	$106.836$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(24\)	\(0\)	\(-243\)	\(155\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(-3^{4}-\beta _{1}+\beta _{2})q^{5}+\cdots\)
342.8.a.m	$342$	$8$	342.a	1.a	$1$	$3$	$3$	$106.836$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(24\)	\(0\)	\(369\)	\(1065\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(123-\beta _{2})q^{5}+(355+\cdots)q^{7}+\cdots\)
342.8.a.n	$342$	$8$	342.a	1.a	$1$	$3$	$3$	$106.836$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(24\)	\(0\)	\(441\)	\(345\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(147+\beta _{1}+\beta _{2})q^{5}+\cdots\)
342.8.a.o	$342$	$8$	342.a	1.a	$1$	$4$	$4$	$106.836$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-32\)	\(0\)	\(279\)	\(2485\)		$+$	$-$	$-$	$+$	$2\cdot 3^{3}$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(70+\beta _{1}+\beta _{2})q^{5}+\cdots\)
342.8.a.p	$342$	$8$	342.a	1.a	$1$	$5$	$5$	$106.836$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(-40\)	\(0\)	\(250\)	\(330\)		$+$	$+$	$+$	$+$	$2^{3}\cdot 3^{5}$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}+(50-\beta _{3})q^{5}+(66+\cdots)q^{7}+\cdots\)
342.8.a.q	$342$	$8$	342.a	1.a	$1$	$5$	$5$	$106.836$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(40\)	\(0\)	\(-250\)	\(330\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3^{5}$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+(-50+\beta _{3})q^{5}+(66+\cdots)q^{7}+\cdots\)
342.8.a.r	$342$	$8$	342.a	1.a	$1$	$6$	$6$	$106.836$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(-48\)	\(0\)	\(0\)	\(-50\)		$+$	$+$	$-$	$-$	$2^{5}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q-8q^{2}+2^{6}q^{4}-\beta _{1}q^{5}+(-8+\beta _{1}+\cdots)q^{7}+\cdots\)
342.8.a.s	$342$	$8$	342.a	1.a	$1$	$6$	$6$	$106.836$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(48\)	\(0\)	\(0\)	\(-50\)		$-$	$+$	$-$	$+$	$2^{5}\cdot 3^{6}$	$\mathrm{SU}(2)$	\(q+8q^{2}+2^{6}q^{4}+\beta _{1}q^{5}+(-8+\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(342))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(342)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 2}\)