Space of modular forms of level 160, weight 10, and trivial character

• Label: 160.10.a
• Level: $160$
• Weight: $10$
• Character orbit: 160.a
• Rep. character: $\chi_{160}(1,\cdot)$
• Character field: $\Q$
• Dimension: $36$
• Newform subspaces: $8$
• Sturm bound: $240$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 160 = 2^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	160.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(240\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(160))\).

	Total	New	Old
Modular forms	224	36	188
Cusp forms	208	36	172
Eisenstein series	16	0	16

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

\(2\)	\(5\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(55\)	\(9\)	\(46\)		\(51\)	\(9\)	\(42\)		\(4\)	\(0\)	\(4\)
\(+\)	\(-\)	\(-\)		\(57\)	\(10\)	\(47\)		\(53\)	\(10\)	\(43\)		\(4\)	\(0\)	\(4\)
\(-\)	\(+\)	\(-\)		\(57\)	\(9\)	\(48\)		\(53\)	\(9\)	\(44\)		\(4\)	\(0\)	\(4\)
\(-\)	\(-\)	\(+\)		\(55\)	\(8\)	\(47\)		\(51\)	\(8\)	\(43\)		\(4\)	\(0\)	\(4\)
Plus space		\(+\)		\(110\)	\(17\)	\(93\)		\(102\)	\(17\)	\(85\)		\(8\)	\(0\)	\(8\)
Minus space		\(-\)		\(114\)	\(19\)	\(95\)		\(106\)	\(19\)	\(87\)		\(8\)	\(0\)	\(8\)

Trace form

36 * q + 180716 * q^9 + 389232 * q^13 + 484424 * q^17 - 4710088 * q^21 + 14062500 * q^25 + 3877832 * q^29 - 19095184 * q^33 + 38861424 * q^37 + 28061552 * q^41 + 13805000 * q^45 + 147824348 * q^49 + 41377168 * q^53 - 240601888 * q^57 + 198353920 * q^61 + 175805000 * q^65 + 30770152 * q^69 - 105682104 * q^73 - 733504848 * q^77 + 169601100 * q^81 + 3025259336 * q^89 + 1085500304 * q^93 - 2553262376 * q^97

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(160))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5
160.10.a.a	$160$	$10$	160.a	1.a	$1$	$4$	$4$	$82.406$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			160.10.a.a	$1$	$1$	\(0\)	\(-176\)	\(2500\)	\(-1392\)		$-$	$-$	$+$	$2^{13}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-44-\beta _{1})q^{3}+5^{4}q^{5}+(-348+\cdots)q^{7}+\cdots\)
160.10.a.b	$160$	$10$	160.a	1.a	$1$	$4$	$4$	$82.406$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			160.10.a.b	$2$	$1$	\(0\)	\(0\)	\(-2500\)	\(0\)		$+$	$+$	$+$	$2^{12}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-5^{4}q^{5}+(-14\beta _{1}-7\beta _{2}+\cdots)q^{7}+\cdots\)
160.10.a.c	$160$	$10$	160.a	1.a	$1$	$4$	$4$	$82.406$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			160.10.a.c	$2$	$0$	\(0\)	\(0\)	\(-2500\)	\(0\)		$-$	$+$	$-$	$2^{12}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-5^{4}q^{5}+(34\beta _{1}-\beta _{2})q^{7}+\cdots\)
160.10.a.d	$160$	$10$	160.a	1.a	$1$	$4$	$4$	$82.406$	\(\Q(\sqrt{7}, \sqrt{418})\)	None	✓	✓			160.10.a.d	$2$	$1$	\(0\)	\(0\)	\(2500\)	\(0\)		$-$	$-$	$+$	$2^{11}\cdot 5\cdot 13$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+5^{4}q^{5}+(-8\beta _{1}+9\beta _{2})q^{7}+\cdots\)
160.10.a.e	$160$	$10$	160.a	1.a	$1$	$4$	$4$	$82.406$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			160.10.a.a	$1$	$0$	\(0\)	\(176\)	\(2500\)	\(1392\)		$+$	$-$	$-$	$2^{13}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(44+\beta _{1})q^{3}+5^{4}q^{5}+(348-14\beta _{1}+\cdots)q^{7}+\cdots\)
160.10.a.f	$160$	$10$	160.a	1.a	$1$	$5$	$5$	$82.406$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		160.10.a.f	$1$	$0$	\(0\)	\(-14\)	\(-3125\)	\(3410\)		$-$	$+$	$-$	$2^{23}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{3}-5^{4}q^{5}+(683-3\beta _{1}+\cdots)q^{7}+\cdots\)
160.10.a.g	$160$	$10$	160.a	1.a	$1$	$5$	$5$	$82.406$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓		160.10.a.f	$1$	$1$	\(0\)	\(14\)	\(-3125\)	\(-3410\)		$+$	$+$	$+$	$2^{23}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+(3-\beta _{1})q^{3}-5^{4}q^{5}+(-683+3\beta _{1}+\cdots)q^{7}+\cdots\)
160.10.a.h	$160$	$10$	160.a	1.a	$1$	$6$	$6$	$82.406$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓	✓		160.10.a.h	$2$	$0$	\(0\)	\(0\)	\(3750\)	\(0\)		$+$	$-$	$-$	$2^{33}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+5^{4}q^{5}+(3\beta _{1}-\beta _{2})q^{7}+(11920+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(160))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(160)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 10}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 5}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(32))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)