Space of modular forms of level 27, weight 12, and trivial character

• Label: 27.12.a
• Level: $27$
• Weight: $12$
• Character orbit: 27.a
• Rep. character: $\chi_{27}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $5$
• Sturm bound: $36$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 27 = 3^{3} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	27.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(36\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(27))\).

	Total	New	Old
Modular forms	36	15	21
Cusp forms	30	15	15
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
\(+\)		\(19\)	\(8\)	\(11\)		\(16\)	\(8\)	\(8\)		\(3\)	\(0\)	\(3\)
\(-\)		\(17\)	\(7\)	\(10\)		\(14\)	\(7\)	\(7\)		\(3\)	\(0\)	\(3\)

Trace form

15 * q + 14406 * q^4 - 18681 * q^7 + 519606 * q^10 - 4254369 * q^13 + 16691370 * q^16 + 28469829 * q^19 - 125905230 * q^22 + 198116421 * q^25 + 172835130 * q^28 + 148121940 * q^31 - 415028880 * q^34 - 343006647 * q^37 + 4474241802 * q^40 - 4208819208 * q^43 - 1691618940 * q^46 + 5381935974 * q^49 - 10596586812 * q^52 - 3811589496 * q^55 - 1573243956 * q^58 - 7558827171 * q^61 + 21525481122 * q^64 + 56200354545 * q^67 - 134560554174 * q^70 - 53248502193 * q^73 + 288888124512 * q^76 - 131380106487 * q^79 + 32608482516 * q^82 + 400274468208 * q^85 - 667610779626 * q^88 - 137917872489 * q^91 + 573300550476 * q^94 - 214230528861 * q^97

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(27))\) 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
27.12.a.a	$27$	$12$	27.a	1.a	$1$	$1$	$1$	$20.745$	\(\Q\)	\(\Q(\sqrt{-3}) \)	✓	✓			27.12.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(77153\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-2^{11}q^{4}+77153q^{7}-2248615q^{13}+\cdots\)
27.12.a.b	$27$	$12$	27.a	1.a	$1$	$2$	$2$	$20.745$	\(\Q(\sqrt{93}) \)	None	✓	✓			27.12.a.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-100142\)		$-$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+1300q^{4}+10\beta q^{5}-50071q^{7}+\cdots\)
27.12.a.c	$27$	$12$	27.a	1.a	$1$	$4$	$4$	$20.745$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		27.12.a.c	$1$	$1$	\(-9\)	\(0\)	\(-12060\)	\(-12076\)		$-$	$-$	$2^{2}\cdot 3^{8}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(1522-22\beta _{1}+\beta _{3})q^{4}+\cdots\)
27.12.a.d	$27$	$12$	27.a	1.a	$1$	$4$	$4$	$20.745$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			27.12.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(28460\)		$+$	$+$	$2^{3}\cdot 3^{9}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(409-\beta _{3})q^{4}+(-2^{4}\beta _{1}+\cdots)q^{5}+\cdots\)
27.12.a.e	$27$	$12$	27.a	1.a	$1$	$4$	$4$	$20.745$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			27.12.a.c	$1$	$0$	\(9\)	\(0\)	\(12060\)	\(-12076\)		$+$	$+$	$2^{2}\cdot 3^{8}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{2}+(1522-22\beta _{1}+\beta _{3})q^{4}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(27))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(27)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 2}\)