Space of modular forms of level 171, weight 6, and trivial character

• Label: 171.6.a
• Level: $171$
• Weight: $6$
• Character orbit: 171.a
• Rep. character: $\chi_{171}(1,\cdot)$
• Character field: $\Q$
• Dimension: $38$
• Newform subspaces: $12$
• Sturm bound: $120$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	171.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(120\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(171))\).

	Total	New	Old
Modular forms	104	38	66
Cusp forms	96	38	58
Eisenstein series	8	0	8

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

\(3\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(6\)
\(+\)	\(-\)	\(-\)	\(10\)
\(-\)	\(+\)	\(-\)	\(12\)
\(-\)	\(-\)	\(+\)	\(10\)
Plus space		\(+\)	\(16\)
Minus space		\(-\)	\(22\)

Trace form

38 * q - 6 * q^2 + 602 * q^4 + 3 * q^5 + 115 * q^7 + 348 * q^8 + 240 * q^10 - 609 * q^11 - 218 * q^13 + 3408 * q^14 + 12002 * q^16 + 141 * q^17 + 722 * q^19 - 1944 * q^20 - 1344 * q^22 - 7452 * q^23 + 19793 * q^25 - 4506 * q^26 - 1778 * q^28 + 22560 * q^29 - 18596 * q^31 + 15300 * q^32 + 21852 * q^34 + 2889 * q^35 - 5888 * q^37 - 4332 * q^38 + 7284 * q^40 - 29700 * q^41 + 23527 * q^43 + 56016 * q^44 + 66276 * q^46 - 32079 * q^47 + 106257 * q^49 + 79602 * q^50 + 85036 * q^52 + 74094 * q^53 + 54345 * q^55 + 43404 * q^56 + 102330 * q^58 - 28806 * q^59 - 1229 * q^61 - 135348 * q^62 + 123770 * q^64 - 26484 * q^65 - 37832 * q^67 + 57462 * q^68 - 36396 * q^70 + 55074 * q^71 + 9529 * q^73 + 55272 * q^74 + 28880 * q^76 - 72609 * q^77 - 8648 * q^79 - 237156 * q^80 - 136668 * q^82 + 228804 * q^83 + 15009 * q^85 + 191880 * q^86 - 27168 * q^88 - 147222 * q^89 - 109684 * q^91 - 575970 * q^92 - 319440 * q^94 - 35739 * q^95 - 203954 * q^97 + 147150 * q^98

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(171))\) 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	19
171.6.a.a	$171$	$6$	171.a	1.a	$1$	$1$	$1$	$27.426$	\(\Q\)	None	✓				57.6.a.b	$1$	$0$	\(-11\)	\(0\)	\(-6\)	\(-176\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-11q^{2}+89q^{4}-6q^{5}-176q^{7}+\cdots\)
171.6.a.b	$171$	$6$	171.a	1.a	$1$	$1$	$1$	$27.426$	\(\Q\)	None	✓				19.6.a.b	$1$	$0$	\(2\)	\(0\)	\(24\)	\(-167\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-28q^{4}+24q^{5}-167q^{7}+\cdots\)
171.6.a.c	$171$	$6$	171.a	1.a	$1$	$1$	$1$	$27.426$	\(\Q\)	None	✓				57.6.a.a	$1$	$0$	\(2\)	\(0\)	\(98\)	\(240\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-28q^{4}+98q^{5}+240q^{7}+\cdots\)
171.6.a.d	$171$	$6$	171.a	1.a	$1$	$1$	$1$	$27.426$	\(\Q\)	None	✓				19.6.a.a	$1$	$1$	\(6\)	\(0\)	\(-54\)	\(248\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{2}+4q^{4}-54q^{5}+248q^{7}-168q^{8}+\cdots\)
171.6.a.e	$171$	$6$	171.a	1.a	$1$	$2$	$2$	$27.426$	\(\Q(\sqrt{17}) \)	None	✓				57.6.a.c	$1$	$1$	\(3\)	\(0\)	\(87\)	\(-251\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(7+3\beta )q^{4}+(46-5\beta )q^{5}+\cdots\)
171.6.a.f	$171$	$6$	171.a	1.a	$1$	$2$	$2$	$27.426$	\(\Q(\sqrt{177}) \)	None	✓				19.6.a.c	$1$	$0$	\(7\)	\(0\)	\(133\)	\(72\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(4-\beta )q^{2}+(28-7\beta )q^{4}+(69-5\beta )q^{5}+\cdots\)
171.6.a.g	$171$	$6$	171.a	1.a	$1$	$3$	$3$	$27.426$	3.3.9153.1	None	✓				57.6.a.e	$1$	$1$	\(-9\)	\(0\)	\(9\)	\(141\)		$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+(7-7\beta _{1}+2\beta _{2})q^{4}+\cdots\)
171.6.a.h	$171$	$6$	171.a	1.a	$1$	$3$	$3$	$27.426$	3.3.616092.1	None	✓				57.6.a.d	$1$	$0$	\(4\)	\(0\)	\(-206\)	\(186\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(26+4\beta _{1}+2\beta _{2})q^{4}+\cdots\)
171.6.a.i	$171$	$6$	171.a	1.a	$1$	$4$	$4$	$27.426$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		19.6.a.d	$1$	$1$	\(-9\)	\(0\)	\(-90\)	\(-190\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1}-\beta _{2})q^{2}+(23-2\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
171.6.a.j	$171$	$6$	171.a	1.a	$1$	$4$	$4$	$27.426$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		57.6.a.f	$1$	$0$	\(-1\)	\(0\)	\(8\)	\(-142\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(14-\beta _{1}-\beta _{2}+\beta _{3})q^{4}+\cdots\)
171.6.a.k	$171$	$6$	171.a	1.a	$1$	$6$	$6$	$27.426$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓	✓	171.6.a.k	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-10\)		$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(7+\beta _{2})q^{4}+(-3\beta _{1}-\beta _{3}+\cdots)q^{5}+\cdots\)
171.6.a.l	$171$	$6$	171.a	1.a	$1$	$10$	$10$	$27.426$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓	✓	171.6.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(164\)		$+$	$-$	$-$	$2^{8}\cdot 3^{8}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(20+\beta _{2})q^{4}+(3\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(171))\) into lower level spaces

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