Space of modular forms of level 342, weight 10, and trivial character

• Label: 342.10.a
• Level: $342$
• Weight: $10$
• Character orbit: 342.a
• Rep. character: $\chi_{342}(1,\cdot)$
• Character field: $\Q$
• Dimension: $67$
• Newform subspaces: $17$
• Sturm bound: $600$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(600\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(342))\).

	Total	New	Old
Modular forms	548	67	481
Cusp forms	532	67	465
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
\(+\)	\(+\)	\(+\)	$+$	\(7\)
\(+\)	\(+\)	\(-\)	$-$	\(6\)
\(+\)	\(-\)	\(+\)	$-$	\(11\)
\(+\)	\(-\)	\(-\)	$+$	\(10\)
\(-\)	\(+\)	\(+\)	$-$	\(7\)
\(-\)	\(+\)	\(-\)	$+$	\(6\)
\(-\)	\(-\)	\(+\)	$+$	\(9\)
\(-\)	\(-\)	\(-\)	$-$	\(11\)
Plus space			\(+\)	\(32\)
Minus space			\(-\)	\(35\)

Trace form

\( 67 q - 16 q^{2} + 17152 q^{4} + 6560 q^{5} - 13568 q^{7} - 4096 q^{8} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(342))\) 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
342.10.a.a	$342$	$10$	342.a	1.a	$1$	$1$	$1$	$176.142$	\(\Q\)	None	✓	$1$	$1$	\(-16\)	\(0\)	\(684\)	\(9149\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+684q^{5}+9149q^{7}+\cdots\)
342.10.a.b	$342$	$10$	342.a	1.a	$1$	$1$	$1$	$176.142$	\(\Q\)	None	✓	$1$	$1$	\(-16\)	\(0\)	\(1581\)	\(-4865\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+1581q^{5}-4865q^{7}+\cdots\)
342.10.a.c	$342$	$10$	342.a	1.a	$1$	$3$	$3$	$176.142$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(-48\)	\(0\)	\(108\)	\(-3190\)		$+$	$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(6^{2}-\beta _{2})q^{5}+(-1062+\cdots)q^{7}+\cdots\)
342.10.a.d	$342$	$10$	342.a	1.a	$1$	$3$	$3$	$176.142$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(-48\)	\(0\)	\(2733\)	\(-2145\)		$+$	$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(911-5\beta _{1}-2\beta _{2})q^{5}+\cdots\)
342.10.a.e	$342$	$10$	342.a	1.a	$1$	$3$	$3$	$176.142$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(48\)	\(0\)	\(-486\)	\(-13317\)		$-$	$-$	$+$	$+$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(-162-2\beta _{1}-\beta _{2})q^{5}+\cdots\)
342.10.a.f	$342$	$10$	342.a	1.a	$1$	$3$	$3$	$176.142$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$0$	\(48\)	\(0\)	\(324\)	\(-3190\)		$-$	$-$	$-$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(108+\beta _{1}-\beta _{2})q^{5}+\cdots\)
342.10.a.g	$342$	$10$	342.a	1.a	$1$	$3$	$3$	$176.142$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(48\)	\(0\)	\(1461\)	\(-2145\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(487+2\beta _{1}-\beta _{2})q^{5}+\cdots\)
342.10.a.h	$342$	$10$	342.a	1.a	$1$	$3$	$3$	$176.142$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓	$1$	$1$	\(48\)	\(0\)	\(2205\)	\(2415\)		$-$	$-$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(735-2\beta _{1}-3\beta _{2})q^{5}+\cdots\)
342.10.a.i	$342$	$10$	342.a	1.a	$1$	$4$	$4$	$176.142$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-64\)	\(0\)	\(-866\)	\(2670\)		$+$	$-$	$+$	$-$	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(-6^{3}+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
342.10.a.j	$342$	$10$	342.a	1.a	$1$	$4$	$4$	$176.142$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(-64\)	\(0\)	\(739\)	\(2415\)		$+$	$-$	$+$	$-$	$2^{2}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(185-\beta _{2})q^{5}+(602+\cdots)q^{7}+\cdots\)
342.10.a.k	$342$	$10$	342.a	1.a	$1$	$4$	$4$	$176.142$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(64\)	\(0\)	\(-420\)	\(1854\)		$-$	$-$	$-$	$-$	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(-105-\beta _{1})q^{5}+\cdots\)
342.10.a.l	$342$	$10$	342.a	1.a	$1$	$4$	$4$	$176.142$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$0$	\(64\)	\(0\)	\(1395\)	\(12307\)		$-$	$-$	$-$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(350+3\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
342.10.a.m	$342$	$10$	342.a	1.a	$1$	$5$	$5$	$176.142$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-80\)	\(0\)	\(-2898\)	\(1854\)		$+$	$-$	$-$	$+$	$2^{6}\cdot 3^{4}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(-580-\beta _{1})q^{5}+\cdots\)
342.10.a.n	$342$	$10$	342.a	1.a	$1$	$6$	$6$	$176.142$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-96\)	\(0\)	\(384\)	\(-9950\)		$+$	$+$	$-$	$-$	$2^{5}\cdot 3^{5}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(2^{6}+\beta _{1})q^{5}+(-1658+\cdots)q^{7}+\cdots\)
342.10.a.o	$342$	$10$	342.a	1.a	$1$	$6$	$6$	$176.142$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(96\)	\(0\)	\(-384\)	\(-9950\)		$-$	$+$	$-$	$+$	$2^{5}\cdot 3^{5}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(-2^{6}-\beta _{1})q^{5}+\cdots\)
342.10.a.p	$342$	$10$	342.a	1.a	$1$	$7$	$7$	$176.142$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-112\)	\(0\)	\(-866\)	\(1260\)		$+$	$+$	$+$	$+$	$2^{9}\cdot 3^{7}$	$\mathrm{SU}(2)$	\(q-2^{4}q^{2}+2^{8}q^{4}+(-124+\beta _{1})q^{5}+\cdots\)
342.10.a.q	$342$	$10$	342.a	1.a	$1$	$7$	$7$	$176.142$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(112\)	\(0\)	\(866\)	\(1260\)		$-$	$+$	$+$	$-$	$2^{9}\cdot 3^{7}$	$\mathrm{SU}(2)$	\(q+2^{4}q^{2}+2^{8}q^{4}+(124-\beta _{1})q^{5}+(180+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(342))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(342)) \cong \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 2}\)