Space of modular forms of level 1075, weight 4, and trivial character

• Label: 1075.4.a
• Level: $1075$
• Weight: $4$
• Character orbit: 1075.a
• Rep. character: $\chi_{1075}(1,\cdot)$
• Character field: $\Q$
• Dimension: $200$
• Newform subspaces: $12$
• Sturm bound: $440$
• Trace bound: $8$

Defining parameters

Level:	\( N \)	\(=\)	\( 1075 = 5^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1075.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(440\)
Trace bound:			\(8\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1075))\).

	Total	New	Old
Modular forms	336	200	136
Cusp forms	324	200	124
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.

\(5\)	\(43\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(50\)
\(+\)	\(-\)	$-$	\(44\)
\(-\)	\(+\)	$-$	\(51\)
\(-\)	\(-\)	$+$	\(55\)
Plus space		\(+\)	\(105\)
Minus space		\(-\)	\(95\)

Trace form

\( 200 q - 2 q^{2} + 4 q^{3} + 816 q^{4} - 30 q^{6} + 12 q^{7} + 24 q^{8} + 1818 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1075))\) 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	43
1075.4.a.a	$1075$	$4$	1075.a	1.a	$1$	$4$	$4$	$63.427$	4.4.45868.1	None	✓	$1$	$0$	\(4\)	\(11\)	\(0\)	\(20\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{3})q^{2}+(3-\beta _{2}+\beta _{3})q^{3}+(1+\cdots)q^{4}+\cdots\)
1075.4.a.b	$1075$	$4$	1075.a	1.a	$1$	$6$	$6$	$63.427$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(-6\)	\(-7\)	\(0\)	\(-8\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-1+\beta _{3})q^{3}+(4+\cdots)q^{4}+\cdots\)
1075.4.a.c	$1075$	$4$	1075.a	1.a	$1$	$6$	$6$	$63.427$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$1$	\(5\)	\(5\)	\(0\)	\(34\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(1+\beta _{5})q^{3}+(1-\beta _{1}+\cdots)q^{4}+\cdots\)
1075.4.a.d	$1075$	$4$	1075.a	1.a	$1$	$8$	$8$	$63.427$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(1\)	\(7\)	\(0\)	\(36\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(3+\beta _{2})q^{4}+\cdots\)
1075.4.a.e	$1075$	$4$	1075.a	1.a	$1$	$13$	$13$	$63.427$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓	$1$	$1$	\(-5\)	\(-5\)	\(0\)	\(-34\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(5+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{6}+\cdots\)
1075.4.a.f	$1075$	$4$	1075.a	1.a	$1$	$15$	$15$	$63.427$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$0$	\(-1\)	\(-7\)	\(0\)	\(-36\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{5}q^{3}+(6+\beta _{2})q^{4}+(3+\beta _{5}+\cdots)q^{6}+\cdots\)
1075.4.a.g	$1075$	$4$	1075.a	1.a	$1$	$19$	$19$	$63.427$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(4+\beta _{2})q^{4}+(-2+\cdots)q^{6}+\cdots\)
1075.4.a.h	$1075$	$4$	1075.a	1.a	$1$	$19$	$19$	$63.427$	\(\mathbb{Q}[x]/(x^{19} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{5}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{4}q^{3}+(4+\beta _{2})q^{4}+(-2+\cdots)q^{6}+\cdots\)
1075.4.a.i	$1075$	$4$	1075.a	1.a	$1$	$23$	$23$	$63.427$		None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
1075.4.a.j	$1075$	$4$	1075.a	1.a	$1$	$23$	$23$	$63.427$		None	✓	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
1075.4.a.k	$1075$	$4$	1075.a	1.a	$1$	$32$	$32$	$63.427$		None	✓	$1$	$1$	\(-10\)	\(-18\)	\(0\)	\(-28\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
1075.4.a.l	$1075$	$4$	1075.a	1.a	$1$	$32$	$32$	$63.427$		None	✓	$1$	$0$	\(10\)	\(18\)	\(0\)	\(28\)		$-$	$-$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1075))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1075)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(215))\)\(^{\oplus 2}\)