Space of modular forms of level 275, weight 6, and trivial character

• Label: 275.6.a
• Level: $275$
• Weight: $6$
• Character orbit: 275.a
• Rep. character: $\chi_{275}(1,\cdot)$
• Character field: $\Q$
• Dimension: $78$
• Newform subspaces: $12$
• Sturm bound: $180$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 275 = 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	275.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(180\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(275))\).

	Total	New	Old
Modular forms	156	78	78
Cusp forms	144	78	66
Eisenstein series	12	0	12

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

\(5\)	\(11\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(18\)
\(+\)	\(-\)	$-$	\(20\)
\(-\)	\(+\)	$-$	\(22\)
\(-\)	\(-\)	$+$	\(18\)
Plus space		\(+\)	\(36\)
Minus space		\(-\)	\(42\)

Trace form

\( 78 q + 4 q^{2} + 17 q^{3} + 1180 q^{4} + 374 q^{6} + 94 q^{7} - 480 q^{8} + 6343 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(275))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	11
275.6.a.a	$275$	$6$	275.a	1.a	$1$	$1$	$1$	$44.106$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(15\)	\(0\)	\(-10\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+15q^{3}-2^{4}q^{4}+60q^{6}-10q^{7}+\cdots\)
275.6.a.b	$275$	$6$	275.a	1.a	$1$	$3$	$3$	$44.106$	3.3.54492.1	None	✓	$1$	$0$	\(0\)	\(-34\)	\(0\)	\(-84\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+(-11-\beta _{1}+\beta _{2})q^{3}+(30+\cdots)q^{4}+\cdots\)
275.6.a.c	$275$	$6$	275.a	1.a	$1$	$3$	$3$	$44.106$	3.3.21865.1	None	✓	$1$	$0$	\(7\)	\(36\)	\(0\)	\(102\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta _{2})q^{2}+(11+3\beta _{1})q^{3}+(12+4\beta _{1}+\cdots)q^{4}+\cdots\)
275.6.a.d	$275$	$6$	275.a	1.a	$1$	$4$	$4$	$44.106$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	$1$	$1$	\(5\)	\(0\)	\(0\)	\(90\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{2}+(-\beta _{1}+\beta _{3})q^{3}+(15+\cdots)q^{4}+\cdots\)
275.6.a.e	$275$	$6$	275.a	1.a	$1$	$5$	$5$	$44.106$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(-9\)	\(0\)	\(0\)	\(-70\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{2}+(-\beta _{1}+\beta _{3})q^{3}+(24+\cdots)q^{4}+\cdots\)
275.6.a.f	$275$	$6$	275.a	1.a	$1$	$6$	$6$	$44.106$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(66\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(\beta _{1}+\beta _{3})q^{3}+(20+\cdots)q^{4}+\cdots\)
275.6.a.g	$275$	$6$	275.a	1.a	$1$	$8$	$8$	$44.106$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(-27\)	\(0\)	\(-359\)		$-$	$-$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-3-\beta _{3})q^{3}+(11+\beta _{1}+\cdots)q^{4}+\cdots\)
275.6.a.h	$275$	$6$	275.a	1.a	$1$	$8$	$8$	$44.106$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(-9\)	\(0\)	\(-155\)		$+$	$+$	$+$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-2+\beta _{1}+\beta _{2})q^{3}+\cdots\)
275.6.a.i	$275$	$6$	275.a	1.a	$1$	$8$	$8$	$44.106$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(9\)	\(0\)	\(155\)		$-$	$+$	$-$	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{1}-\beta _{2})q^{3}+(14+\cdots)q^{4}+\cdots\)
275.6.a.j	$275$	$6$	275.a	1.a	$1$	$8$	$8$	$44.106$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(27\)	\(0\)	\(359\)		$+$	$-$	$-$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(3+\beta _{3})q^{3}+(11+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
275.6.a.k	$275$	$6$	275.a	1.a	$1$	$10$	$10$	$44.106$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$2^{4}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{7})q^{3}+(12+\beta _{2}+\cdots)q^{4}+\cdots\)
275.6.a.l	$275$	$6$	275.a	1.a	$1$	$14$	$14$	$44.106$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{9}\cdot 3^{2}\cdot 5^{8}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{9}q^{3}+(18+\beta _{2})q^{4}+(1+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(275))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(275)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 2}\)