Space of modular forms of level 342, weight 6, and trivial character

• Label: 342.6.a
• Level: $342$
• Weight: $6$
• Character orbit: 342.a
• Rep. character: $\chi_{342}(1,\cdot)$
• Character field: $\Q$
• Dimension: $37$
• Newform subspaces: $17$
• Sturm bound: $360$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	342.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(360\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(342))\).

	Total	New	Old
Modular forms	308	37	271
Cusp forms	292	37	255
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
\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(5\)
\(-\)	\(-\)	\(-\)	\(-\)	\(7\)
Plus space			\(+\)	\(17\)
Minus space			\(-\)	\(20\)

Trace form

37 * q + 4 * q^2 + 592 * q^4 - 110 * q^5 + 42 * q^7 + 64 * q^8 - 808 * q^10 - 716 * q^11 + 542 * q^13 - 80 * q^14 + 9472 * q^16 + 996 * q^17 - 361 * q^19 - 1760 * q^20 + 1424 * q^22 + 8314 * q^23 + 15263 * q^25 - 7408 * q^26 + 672 * q^28 - 7846 * q^29 + 25708 * q^31 + 1024 * q^32 + 3384 * q^34 - 13764 * q^35 + 1306 * q^37 + 4332 * q^38 - 12928 * q^40 + 22058 * q^41 - 15992 * q^43 - 11456 * q^44 - 38608 * q^46 + 220 * q^47 + 118043 * q^49 - 16260 * q^50 + 8672 * q^52 - 89558 * q^53 + 60208 * q^55 - 1280 * q^56 - 25328 * q^58 + 83220 * q^59 - 58522 * q^61 - 37584 * q^62 + 151552 * q^64 + 209552 * q^65 + 67764 * q^67 + 15936 * q^68 + 220560 * q^70 + 13180 * q^71 - 17508 * q^73 + 60552 * q^74 - 5776 * q^76 + 179712 * q^77 - 152476 * q^79 - 28160 * q^80 - 237320 * q^82 - 169476 * q^83 + 78612 * q^85 + 127200 * q^86 + 22784 * q^88 + 76590 * q^89 - 113400 * q^91 + 133024 * q^92 + 219776 * q^94 - 72922 * q^95 - 8078 * q^97 + 168388 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(342))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	19
342.6.a.a	$342$	$6$	342.a	1.a	$1$	$1$	$1$	$54.851$	\(\Q\)	None	✓				114.6.a.c	$1$	$1$	\(-4\)	\(0\)	\(-21\)	\(-143\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}-21q^{5}-143q^{7}+\cdots\)
342.6.a.b	$342$	$6$	342.a	1.a	$1$	$1$	$1$	$54.851$	\(\Q\)	None	✓				38.6.a.b	$1$	$1$	\(-4\)	\(0\)	\(45\)	\(-121\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+45q^{5}-11^{2}q^{7}+\cdots\)
342.6.a.c	$342$	$6$	342.a	1.a	$1$	$1$	$1$	$54.851$	\(\Q\)	None	✓				114.6.a.d	$1$	$0$	\(-4\)	\(0\)	\(91\)	\(-33\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+91q^{5}-33q^{7}-2^{6}q^{8}+\cdots\)
342.6.a.d	$342$	$6$	342.a	1.a	$1$	$1$	$1$	$54.851$	\(\Q\)	None	✓				114.6.a.b	$1$	$0$	\(4\)	\(0\)	\(-81\)	\(-247\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-3^{4}q^{5}-247q^{7}+\cdots\)
342.6.a.e	$342$	$6$	342.a	1.a	$1$	$1$	$1$	$54.851$	\(\Q\)	None	✓				38.6.a.a	$1$	$1$	\(4\)	\(0\)	\(-31\)	\(-27\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-31q^{5}-3^{3}q^{7}+2^{6}q^{8}+\cdots\)
342.6.a.f	$342$	$6$	342.a	1.a	$1$	$1$	$1$	$54.851$	\(\Q\)	None	✓				114.6.a.a	$1$	$0$	\(4\)	\(0\)	\(54\)	\(104\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+54q^{5}+104q^{7}+\cdots\)
342.6.a.g	$342$	$6$	342.a	1.a	$1$	$2$	$2$	$54.851$	\(\Q(\sqrt{2441}) \)	None	✓				114.6.a.g	$1$	$0$	\(-8\)	\(0\)	\(5\)	\(-105\)		$+$	$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+(3+\beta )q^{5}+(-53+\cdots)q^{7}+\cdots\)
342.6.a.h	$342$	$6$	342.a	1.a	$1$	$2$	$2$	$54.851$	\(\Q(\sqrt{201}) \)	None	✓				114.6.a.e	$1$	$1$	\(8\)	\(0\)	\(13\)	\(-33\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+(4+5\beta )q^{5}+(-22+\cdots)q^{7}+\cdots\)
342.6.a.i	$342$	$6$	342.a	1.a	$1$	$2$	$2$	$54.851$	\(\Q(\sqrt{1441}) \)	None	✓				38.6.a.c	$1$	$0$	\(8\)	\(0\)	\(45\)	\(114\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+(24-3\beta )q^{5}+(59+\cdots)q^{7}+\cdots\)
342.6.a.j	$342$	$6$	342.a	1.a	$1$	$2$	$2$	$54.851$	\(\Q(\sqrt{4089}) \)	None	✓				114.6.a.f	$1$	$1$	\(8\)	\(0\)	\(49\)	\(-105\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+(5^{2}-\beta )q^{5}+(-55+\cdots)q^{7}+\cdots\)
342.6.a.k	$342$	$6$	342.a	1.a	$1$	$3$	$3$	$54.851$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		114.6.a.i	$1$	$1$	\(-12\)	\(0\)	\(-135\)	\(125\)		$+$	$-$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+(-45+\beta _{1})q^{5}+(42+\cdots)q^{7}+\cdots\)
342.6.a.l	$342$	$6$	342.a	1.a	$1$	$3$	$3$	$54.851$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		38.6.a.d	$1$	$0$	\(-12\)	\(0\)	\(-81\)	\(228\)		$+$	$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+(-3^{3}-\beta _{1})q^{5}+(76+\cdots)q^{7}+\cdots\)
342.6.a.m	$342$	$6$	342.a	1.a	$1$	$3$	$3$	$54.851$	3.3.364092.1	None	✓	✓	✓	✓	342.6.a.m	$1$	$0$	\(-12\)	\(0\)	\(96\)	\(2\)		$+$	$+$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+(33-3\beta _{1}+\beta _{2})q^{5}+\cdots\)
342.6.a.n	$342$	$6$	342.a	1.a	$1$	$3$	$3$	$54.851$	3.3.364092.1	None	✓	✓	✓	✓	342.6.a.m	$1$	$1$	\(12\)	\(0\)	\(-96\)	\(2\)		$-$	$+$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+(-33+3\beta _{1}-\beta _{2})q^{5}+\cdots\)
342.6.a.o	$342$	$6$	342.a	1.a	$1$	$3$	$3$	$54.851$	3.3.2922585.1	None	✓		✓		114.6.a.h	$1$	$0$	\(12\)	\(0\)	\(-63\)	\(125\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+(-21-\beta _{1}+\beta _{2})q^{5}+\cdots\)
342.6.a.p	$342$	$6$	342.a	1.a	$1$	$4$	$4$	$54.851$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	342.6.a.p	$1$	$1$	\(-16\)	\(0\)	\(46\)	\(78\)		$+$	$+$	$+$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+(11+\beta _{2})q^{5}+(20+\cdots)q^{7}+\cdots\)
342.6.a.q	$342$	$6$	342.a	1.a	$1$	$4$	$4$	$54.851$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓	✓	342.6.a.p	$1$	$0$	\(16\)	\(0\)	\(-46\)	\(78\)		$-$	$+$	$+$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+(-11-\beta _{2})q^{5}+(20+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(342))\) into lower level spaces

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