Space of modular forms of level 175, weight 5, and Character 175.d

• Label: 175.5.d
• Level: $175$
• Weight: $5$
• Character orbit: 175.d
• Rep. character: $\chi_{175}(76,\cdot)$
• Character field: $\Q$
• Dimension: $47$
• Newform subspaces: $9$
• Sturm bound: $100$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	175.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(100\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(175, [\chi])\).

	Total	New	Old
Modular forms	86	53	33
Cusp forms	74	47	27
Eisenstein series	12	6	6

Trace form

47 * q + 5 * q^2 + 341 * q^4 + q^7 + 217 * q^8 - 1053 * q^9 + 294 * q^11 - 491 * q^14 + 1725 * q^16 - 815 * q^18 - 60 * q^21 - 2058 * q^22 + 1490 * q^23 - 679 * q^28 + 14 * q^29 + 7977 * q^32 - 6219 * q^36 - 4270 * q^37 - 8700 * q^39 - 5580 * q^42 - 3610 * q^43 + 12592 * q^44 - 6992 * q^46 + 4447 * q^49 - 13620 * q^51 - 6178 * q^53 + 2123 * q^56 + 12900 * q^57 + 17262 * q^58 + 341 * q^63 + 579 * q^64 + 19150 * q^67 + 12834 * q^71 - 36703 * q^72 - 3232 * q^74 - 16810 * q^77 - 20100 * q^78 - 16786 * q^79 + 30447 * q^81 + 14280 * q^84 - 34072 * q^86 + 882 * q^88 + 48700 * q^91 + 49242 * q^92 + 70980 * q^93 - 60115 * q^98 - 27126 * q^99

Decomposition of \(S_{5}^{\mathrm{new}}(175, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
175.5.d.a	$175$	$5$	175.d	7.b	$2$	$1$	$1$	$18.090$	\(\Q\)	\(\Q(\sqrt{-7}) \)	✓				7.5.b.a	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(-49\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-q^{2}-15q^{4}-7^{2}q^{7}+31q^{8}+3^{4}q^{9}+\cdots\)
175.5.d.b	$175$	$5$	175.d	7.b	$2$	$2$	$2$	$18.090$	\(\Q(\sqrt{21}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			175.5.d.b	$2$	$0$	\(-1\)	\(0\)	\(0\)	\(98\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q-\beta q^{2}+(31+\beta )q^{4}+7^{2}q^{7}+(-47+\cdots)q^{8}+\cdots\)
175.5.d.c	$175$	$5$	175.d	7.b	$2$	$2$	$2$	$18.090$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-35}) \)					35.5.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+17 i q^{3}-16 q^{4}+49 i q^{7}-208 q^{9}+\cdots\)
175.5.d.d	$175$	$5$	175.d	7.b	$2$	$2$	$2$	$18.090$	\(\Q(\sqrt{21}) \)	\(\Q(\sqrt{-7}) \)	✓	✓			175.5.d.b	$2$	$0$	\(1\)	\(0\)	\(0\)	\(-98\)	$3$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{2}+(31+\beta )q^{4}-7^{2}q^{7}+(47+2^{4}\beta )q^{8}+\cdots\)
175.5.d.e	$175$	$5$	175.d	7.b	$2$	$4$	$4$	$18.090$	\(\Q(i, \sqrt{6})\)	None					35.5.c.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-\beta _{1}q^{3}-10q^{4}+\beta _{3}q^{6}+\cdots\)
175.5.d.f	$175$	$5$	175.d	7.b	$2$	$8$	$8$	$18.090$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			175.5.d.f	$2$	$0$	\(-4\)	\(0\)	\(0\)	\(128\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{2})q^{2}-\beta _{1}q^{3}+(1+\beta _{2}+\beta _{6}+\cdots)q^{4}+\cdots\)
175.5.d.g	$175$	$5$	175.d	7.b	$2$	$8$	$8$	$18.090$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					35.5.c.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+3\beta _{3}q^{3}+(22+\beta _{1})q^{4}-3\beta _{2}q^{6}+\cdots\)
175.5.d.h	$175$	$5$	175.d	7.b	$2$	$8$	$8$	$18.090$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			175.5.d.f	$2$	$0$	\(4\)	\(0\)	\(0\)	\(-128\)	$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+(1+\beta _{2}+\beta _{6}+\cdots)q^{4}+\cdots\)
175.5.d.i	$175$	$5$	175.d	7.b	$2$	$12$	$12$	$18.090$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None			✓		35.5.d.a	$2$	$0$	\(6\)	\(0\)	\(0\)	\(50\)	$2^{4}\cdot 3^{2}\cdot 5^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}-\beta _{3}q^{3}+(10-\beta _{1}-\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(175, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(175, [\chi]) \simeq \) \(S_{5}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 2}\)