Space of modular forms of level 27, weight 6, and trivial character

• Label: 27.6.a
• Level: $27$
• Weight: $6$
• Character orbit: 27.a
• Rep. character: $\chi_{27}(1,\cdot)$
• Character field: $\Q$
• Dimension: $7$
• Newform subspaces: $4$
• Sturm bound: $18$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(18\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(27))\).

	Total	New	Old
Modular forms	18	7	11
Cusp forms	12	7	5
Eisenstein series	6	0	6

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

\(3\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(8\)	\(3\)	\(5\)		\(5\)	\(3\)	\(2\)		\(3\)	\(0\)	\(3\)
\(-\)		\(10\)	\(4\)	\(6\)		\(7\)	\(4\)	\(3\)		\(3\)	\(0\)	\(3\)

Trace form

7 * q + 118 * q^4 + 107 * q^7 - 18 * q^10 - 2653 * q^13 + 6250 * q^16 + 185 * q^19 - 11430 * q^22 + 5521 * q^25 - 11110 * q^28 - 5452 * q^31 + 30240 * q^34 + 34349 * q^37 - 16182 * q^40 - 1576 * q^43 - 41868 * q^46 + 32286 * q^49 - 121996 * q^52 - 3312 * q^55 + 143388 * q^58 - 132895 * q^61 + 56338 * q^64 + 66845 * q^67 + 172602 * q^70 + 27239 * q^73 - 42496 * q^76 - 255331 * q^79 - 126972 * q^82 - 220752 * q^85 + 274518 * q^88 + 288955 * q^91 + 195660 * q^94 + 197387 * q^97

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(27))\) 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
27.6.a.a	$27$	$6$	27.a	1.a	$1$	$1$	$1$	$4.330$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			27.6.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-211\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2^{5}q^{4}-211q^{7}-775q^{13}+2^{10}q^{16}+\cdots\)
27.6.a.b	$27$	$6$	27.a	1.a	$1$	$2$	$2$	$4.330$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓		27.6.a.b	$1$	$1$	\(-9\)	\(0\)	\(-72\)	\(-8\)		$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q+(-4-\beta )q^{2}+(22+9\beta )q^{4}+(-41+\cdots)q^{5}+\cdots\)
27.6.a.c	$27$	$6$	27.a	1.a	$1$	$2$	$2$	$4.330$	\(\Q(\sqrt{6}) \)	None	✓	✓			27.6.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(334\)		$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+22q^{4}+8\beta q^{5}+167q^{7}+\cdots\)
27.6.a.d	$27$	$6$	27.a	1.a	$1$	$2$	$2$	$4.330$	\(\Q(\sqrt{17}) \)	None	✓	✓			27.6.a.b	$1$	$0$	\(9\)	\(0\)	\(72\)	\(-8\)		$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+(4+\beta )q^{2}+(22+9\beta )q^{4}+(41-10\beta )q^{5}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(27))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(27)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)