Space of modular forms of level 270, weight 6, and trivial character

• Label: 270.6.a
• Level: $270$
• Weight: $6$
• Character orbit: 270.a
• Rep. character: $\chi_{270}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $16$
• Sturm bound: $324$
• Trace bound: $17$

Defining parameters

Level:	\( N \)	\(=\)	\( 270 = 2 \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	270.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(324\)
Trace bound:			\(17\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(270))\).

	Total	New	Old
Modular forms	282	28	254
Cusp forms	258	28	230
Eisenstein series	24	0	24

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

\(2\)	\(3\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(3\)
\(+\)	\(+\)	\(-\)	$-$	\(4\)
\(+\)	\(-\)	\(+\)	$-$	\(4\)
\(+\)	\(-\)	\(-\)	$+$	\(3\)
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(-\)	$+$	\(3\)
\(-\)	\(-\)	\(+\)	$+$	\(3\)
\(-\)	\(-\)	\(-\)	$-$	\(4\)
Plus space			\(+\)	\(12\)
Minus space			\(-\)	\(16\)

Trace form

\( 28 q + 448 q^{4} + 152 q^{7} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(270))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	5
270.6.a.a	$270$	$6$	270.a	1.a	$1$	$1$	$1$	$43.304$	\(\Q\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(-25\)	\(-91\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}-91q^{7}-2^{6}q^{8}+\cdots\)
270.6.a.b	$270$	$6$	270.a	1.a	$1$	$1$	$1$	$43.304$	\(\Q\)	None	✓	$1$	$1$	\(-4\)	\(0\)	\(25\)	\(71\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}+71q^{7}-2^{6}q^{8}+\cdots\)
270.6.a.c	$270$	$6$	270.a	1.a	$1$	$1$	$1$	$43.304$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(0\)	\(-25\)	\(71\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}+71q^{7}+2^{6}q^{8}+\cdots\)
270.6.a.d	$270$	$6$	270.a	1.a	$1$	$1$	$1$	$43.304$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(0\)	\(25\)	\(-91\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}-91q^{7}+2^{6}q^{8}+\cdots\)
270.6.a.e	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{14}) \)	None	✓	$1$	$0$	\(-8\)	\(0\)	\(-50\)	\(-200\)		$+$	$-$	$+$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}+(-10^{2}+7\beta )q^{7}+\cdots\)
270.6.a.f	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{659}) \)	None	✓	$1$	$1$	\(-8\)	\(0\)	\(-50\)	\(160\)		$+$	$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}+(80+\beta )q^{7}+\cdots\)
270.6.a.g	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{9681}) \)	None	✓	$1$	$0$	\(-8\)	\(0\)	\(-50\)	\(169\)		$+$	$-$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}-5^{2}q^{5}+(85-\beta )q^{7}+\cdots\)
270.6.a.h	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{74}) \)	None	✓	$1$	$1$	\(-8\)	\(0\)	\(50\)	\(-20\)		$+$	$-$	$-$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}+(-10+\beta )q^{7}+\cdots\)
270.6.a.i	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{11}) \)	None	✓	$1$	$0$	\(-8\)	\(0\)	\(50\)	\(-20\)		$+$	$+$	$-$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}+(-10+\beta )q^{7}+\cdots\)
270.6.a.j	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{1401}) \)	None	✓	$1$	$0$	\(-8\)	\(0\)	\(50\)	\(7\)		$+$	$+$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q-4q^{2}+2^{4}q^{4}+5^{2}q^{5}+(4+\beta )q^{7}+\cdots\)
270.6.a.k	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{11}) \)	None	✓	$1$	$1$	\(8\)	\(0\)	\(-50\)	\(-20\)		$-$	$-$	$+$	$+$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}+(-10+\beta )q^{7}+\cdots\)
270.6.a.l	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{74}) \)	None	✓	$1$	$0$	\(8\)	\(0\)	\(-50\)	\(-20\)		$-$	$+$	$+$	$-$	$2\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}+(-10+\beta )q^{7}+\cdots\)
270.6.a.m	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{1401}) \)	None	✓	$1$	$0$	\(8\)	\(0\)	\(-50\)	\(7\)		$-$	$+$	$+$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}-5^{2}q^{5}+(4+\beta )q^{7}+\cdots\)
270.6.a.n	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{14}) \)	None	✓	$1$	$1$	\(8\)	\(0\)	\(50\)	\(-200\)		$-$	$+$	$-$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}+(-10^{2}+7\beta )q^{7}+\cdots\)
270.6.a.o	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{659}) \)	None	✓	$1$	$0$	\(8\)	\(0\)	\(50\)	\(160\)		$-$	$-$	$-$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}+(80+\beta )q^{7}+\cdots\)
270.6.a.p	$270$	$6$	270.a	1.a	$1$	$2$	$2$	$43.304$	\(\Q(\sqrt{9681}) \)	None	✓	$1$	$0$	\(8\)	\(0\)	\(50\)	\(169\)		$-$	$-$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+4q^{2}+2^{4}q^{4}+5^{2}q^{5}+(85-\beta )q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(270))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(270)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 12}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(45))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(90))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(135))\)\(^{\oplus 2}\)