Space of modular forms of level 845, weight 4, and trivial character

• Label: 845.4.a
• Level: $845$
• Weight: $4$
• Character orbit: 845.a
• Rep. character: $\chi_{845}(1,\cdot)$
• Character field: $\Q$
• Dimension: $155$
• Newform subspaces: $18$
• Sturm bound: $364$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 18 \)
Sturm bound:			\(364\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(845))\).

	Total	New	Old
Modular forms	288	155	133
Cusp forms	260	155	105
Eisenstein series	28	0	28

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

\(5\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(42\)
\(+\)	\(-\)	$-$	\(36\)
\(-\)	\(+\)	$-$	\(35\)
\(-\)	\(-\)	$+$	\(42\)
Plus space		\(+\)	\(84\)
Minus space		\(-\)	\(71\)

Trace form

\( 155 q - 4 q^{2} + 2 q^{3} + 624 q^{4} - 5 q^{5} + 52 q^{6} - 34 q^{7} - 12 q^{8} + 1395 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(845))\) 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	13
845.4.a.a	$845$	$4$	845.a	1.a	$1$	$1$	$1$	$49.857$	\(\Q\)	None	✓	$1$	$1$	\(-5\)	\(2\)	\(5\)	\(12\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-5q^{2}+2q^{3}+17q^{4}+5q^{5}-10q^{6}+\cdots\)
845.4.a.b	$845$	$4$	845.a	1.a	$1$	$1$	$1$	$49.857$	\(\Q\)	None	✓	$1$	$1$	\(4\)	\(2\)	\(5\)	\(-6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{2}+2q^{3}+8q^{4}+5q^{5}+8q^{6}+\cdots\)
845.4.a.c	$845$	$4$	845.a	1.a	$1$	$2$	$2$	$49.857$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(-4\)	\(-10\)	\(10\)	\(36\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{2}+(-5+3\beta )q^{3}+(-1+\cdots)q^{4}+\cdots\)
845.4.a.d	$845$	$4$	845.a	1.a	$1$	$2$	$2$	$49.857$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(-1\)	\(0\)	\(10\)	\(-58\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-2+4\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)
845.4.a.e	$845$	$4$	845.a	1.a	$1$	$2$	$2$	$49.857$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(2\)	\(-4\)	\(-10\)	\(20\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+(-2+\beta )q^{3}+(-1+2\beta )q^{4}+\cdots\)
845.4.a.f	$845$	$4$	845.a	1.a	$1$	$5$	$5$	$49.857$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(8\)	\(-25\)	\(-38\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(2+\beta _{1}+\beta _{3})q^{3}+(7-\beta _{2}+\cdots)q^{4}+\cdots\)
845.4.a.g	$845$	$4$	845.a	1.a	$1$	$7$	$7$	$49.857$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(-8\)	\(35\)	\(-75\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
845.4.a.h	$845$	$4$	845.a	1.a	$1$	$7$	$7$	$49.857$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(-4\)	\(35\)	\(-7\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
845.4.a.i	$845$	$4$	845.a	1.a	$1$	$7$	$7$	$49.857$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(0\)	\(-35\)	\(-54\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(4+\beta _{2})q^{4}-5q^{5}+\cdots\)
845.4.a.j	$845$	$4$	845.a	1.a	$1$	$7$	$7$	$49.857$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(-8\)	\(-35\)	\(75\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
845.4.a.k	$845$	$4$	845.a	1.a	$1$	$7$	$7$	$49.857$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(-4\)	\(-35\)	\(7\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1+\beta _{3})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
845.4.a.l	$845$	$4$	845.a	1.a	$1$	$7$	$7$	$49.857$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(2\)	\(0\)	\(35\)	\(54\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(4+\beta _{2})q^{4}+5q^{5}+\cdots\)
845.4.a.m	$845$	$4$	845.a	1.a	$1$	$14$	$14$	$49.857$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$1$	\(-4\)	\(12\)	\(-70\)	\(-30\)		$+$	$-$	$-$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
845.4.a.n	$845$	$4$	845.a	1.a	$1$	$14$	$14$	$49.857$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	$1$	$0$	\(4\)	\(12\)	\(70\)	\(30\)		$-$	$-$	$+$	$2^{4}\cdot 3$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(4+\beta _{2})q^{4}+\cdots\)
845.4.a.o	$845$	$4$	845.a	1.a	$1$	$15$	$15$	$49.857$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(-17\)	\(75\)	\(-44\)		$-$	$+$	$-$	$13^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1-\beta _{11})q^{3}+(2+\beta _{1}+\cdots)q^{4}+\cdots\)
845.4.a.p	$845$	$4$	845.a	1.a	$1$	$15$	$15$	$49.857$	\(\mathbb{Q}[x]/(x^{15} - \cdots)\)	None	✓	$1$	$1$	\(2\)	\(-17\)	\(-75\)	\(44\)		$+$	$-$	$-$	$13^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{11})q^{3}+(2+\beta _{1}+\cdots)q^{4}+\cdots\)
845.4.a.q	$845$	$4$	845.a	1.a	$1$	$21$	$21$	$49.857$		None	✓	$1$	$0$	\(-8\)	\(19\)	\(-105\)	\(-40\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
845.4.a.r	$845$	$4$	845.a	1.a	$1$	$21$	$21$	$49.857$		None	✓	$1$	$0$	\(8\)	\(19\)	\(105\)	\(40\)		$-$	$-$	$+$		$\mathrm{SU}(2)$

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(845))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(845)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(13))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 2}\)