Space of modular forms of level 169, weight 6, and trivial character

• Label: 169.6.a
• Level: $169$
• Weight: $6$
• Character orbit: 169.a
• Rep. character: $\chi_{169}(1,\cdot)$
• Character field: $\Q$
• Dimension: $59$
• Newform subspaces: $8$
• Sturm bound: $91$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 169 = 13^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	169.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(91\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(169))\).

	Total	New	Old
Modular forms	83	70	13
Cusp forms	69	59	10
Eisenstein series	14	11	3

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

\(13\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(40\)	\(33\)	\(7\)		\(33\)	\(28\)	\(5\)		\(7\)	\(5\)	\(2\)
\(-\)		\(43\)	\(37\)	\(6\)		\(36\)	\(31\)	\(5\)		\(7\)	\(6\)	\(1\)

Trace form

59 * q - 2 * q^2 + 20 * q^3 + 850 * q^4 - 14 * q^5 + 180 * q^6 + 96 * q^7 - 552 * q^8 + 4191 * q^9 + 1130 * q^10 - 180 * q^11 - 162 * q^12 + 1374 * q^14 - 520 * q^15 + 7250 * q^16 + 1180 * q^17 - 1874 * q^18 + 164 * q^19 + 1564 * q^20 - 5464 * q^21 + 596 * q^22 - 78 * q^23 + 14436 * q^24 + 14807 * q^25 + 11522 * q^27 + 2912 * q^28 + 6890 * q^29 - 3062 * q^30 - 5112 * q^31 - 29696 * q^32 - 7720 * q^33 - 21476 * q^34 + 8104 * q^35 + 72330 * q^36 + 37490 * q^37 - 29726 * q^38 + 78134 * q^40 + 3158 * q^41 - 33500 * q^42 - 8202 * q^43 - 61444 * q^44 - 38750 * q^45 - 15188 * q^46 - 44232 * q^47 + 62760 * q^48 + 116073 * q^49 + 99898 * q^50 - 67790 * q^51 + 65834 * q^53 - 54420 * q^54 - 44060 * q^55 - 6328 * q^56 - 4656 * q^57 + 16688 * q^58 - 25012 * q^59 - 160128 * q^60 - 10036 * q^61 + 83786 * q^62 + 105424 * q^63 - 56636 * q^64 + 137382 * q^66 + 13796 * q^67 - 50220 * q^68 + 47452 * q^69 - 105284 * q^70 - 144976 * q^71 + 5904 * q^72 - 31386 * q^73 + 77248 * q^74 + 562 * q^75 + 94120 * q^76 - 22616 * q^77 + 114294 * q^79 + 146948 * q^80 + 90163 * q^81 - 57518 * q^82 - 22380 * q^83 + 29108 * q^84 + 43516 * q^85 + 35844 * q^86 - 175584 * q^87 - 127176 * q^88 - 150202 * q^89 + 34772 * q^90 - 329960 * q^92 - 20160 * q^93 + 97282 * q^94 + 178626 * q^95 + 202148 * q^96 - 41578 * q^97 + 54926 * q^98 + 220140 * q^99

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(169))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		13
169.6.a.a	$169$	$6$	169.a	1.a	$1$	$2$	$2$	$27.105$	\(\Q(\sqrt{17}) \)	None	✓				13.6.a.a	$1$	$1$	\(5\)	\(-28\)	\(42\)	\(36\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{2}+(-11-6\beta )q^{3}+(-19+\cdots)q^{4}+\cdots\)
169.6.a.b	$169$	$6$	169.a	1.a	$1$	$3$	$3$	$27.105$	3.3.168897.1	None	✓				13.6.a.b	$1$	$1$	\(-7\)	\(8\)	\(-56\)	\(60\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(3-\beta _{1}-\beta _{2})q^{3}+\cdots\)
169.6.a.c	$169$	$6$	169.a	1.a	$1$	$4$	$4$	$27.105$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				13.6.c.a	$1$	$1$	\(-5\)	\(-8\)	\(10\)	\(68\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{2}+(-2+\beta _{2})q^{3}+(4+\cdots)q^{4}+\cdots\)
169.6.a.d	$169$	$6$	169.a	1.a	$1$	$4$	$4$	$27.105$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				13.6.c.a	$1$	$1$	\(5\)	\(-8\)	\(-10\)	\(-68\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(-2+\beta _{2})q^{3}+(4+\beta _{1}+\cdots)q^{4}+\cdots\)
169.6.a.e	$169$	$6$	169.a	1.a	$1$	$6$	$6$	$27.105$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				13.6.b.a	$2$	$0$	\(0\)	\(16\)	\(0\)	\(0\)		$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{2})q^{3}+(22+\beta _{2}+\beta _{5})q^{4}+\cdots\)
169.6.a.f	$169$	$6$	169.a	1.a	$1$	$10$	$10$	$27.105$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓				13.6.e.a	$2$	$0$	\(0\)	\(20\)	\(0\)	\(0\)		$-$	$-$	$2^{6}\cdot 3\cdot 13^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2-\beta _{4})q^{3}+(13+\beta _{3}+\beta _{5}+\cdots)q^{4}+\cdots\)
169.6.a.g	$169$	$6$	169.a	1.a	$1$	$15$	$15$	$27.105$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓	✓		169.6.a.g	$1$	$1$	\(-13\)	\(10\)	\(-168\)	\(-391\)		$+$	$+$	$2^{3}\cdot 13^{7}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{4})q^{3}+(2^{4}-\beta _{1}+\cdots)q^{4}+\cdots\)
169.6.a.h	$169$	$6$	169.a	1.a	$1$	$15$	$15$	$27.105$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	✓	✓		169.6.a.g	$1$	$0$	\(13\)	\(10\)	\(168\)	\(391\)		$-$	$-$	$2^{3}\cdot 13^{7}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1-\beta _{4})q^{3}+(2^{4}-\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(169))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(169)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 2}\)