Space of modular forms of level 175, weight 10, and trivial character

• Label: 175.10.a
• Level: $175$
• Weight: $10$
• Character orbit: 175.a
• Rep. character: $\chi_{175}(1,\cdot)$
• Character field: $\Q$
• Dimension: $85$
• Newform subspaces: $13$
• Sturm bound: $200$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(200\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(175))\).

	Total	New	Old
Modular forms	186	85	101
Cusp forms	174	85	89
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.

\(5\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(19\)
\(+\)	\(-\)	\(-\)	\(22\)
\(-\)	\(+\)	\(-\)	\(23\)
\(-\)	\(-\)	\(+\)	\(21\)
Plus space		\(+\)	\(40\)
Minus space		\(-\)	\(45\)

Trace form

85 * q - 17 * q^2 + 294 * q^3 + 22273 * q^4 - 11386 * q^6 + 2401 * q^7 - 7905 * q^8 + 522233 * q^9 - 54432 * q^11 + 480838 * q^12 - 48136 * q^13 - 88837 * q^14 + 5786121 * q^16 + 711878 * q^17 - 2153581 * q^18 - 640366 * q^19 - 408170 * q^21 + 2328236 * q^22 - 695176 * q^23 - 8763666 * q^24 - 3625100 * q^26 + 4282140 * q^27 + 5226977 * q^28 - 12919774 * q^29 - 356700 * q^31 + 2428023 * q^32 - 9834072 * q^33 + 10780618 * q^34 + 121310165 * q^36 - 34821542 * q^37 - 28892630 * q^38 + 53917784 * q^39 + 39695746 * q^41 - 8926918 * q^42 + 78756864 * q^43 + 10878162 * q^44 - 130524998 * q^46 + 87054668 * q^47 + 175660814 * q^48 + 490008085 * q^49 + 251405632 * q^51 + 98290868 * q^52 + 107791454 * q^53 - 350769592 * q^54 - 112316379 * q^56 + 128646980 * q^57 - 484167470 * q^58 - 517888282 * q^59 - 243558728 * q^61 + 529325916 * q^62 + 166900713 * q^63 + 1343015567 * q^64 + 1377678316 * q^66 - 526332 * q^67 - 511592234 * q^68 + 484724056 * q^69 - 774052540 * q^71 - 848550385 * q^72 + 317883234 * q^73 + 578853308 * q^74 + 340259166 * q^76 + 168242872 * q^77 - 370287692 * q^78 + 881409660 * q^79 + 1567163561 * q^81 - 1592705654 * q^82 + 989257114 * q^83 - 1017394938 * q^84 + 3261763934 * q^86 - 3212160700 * q^87 - 2388172120 * q^88 - 548591634 * q^89 + 326948972 * q^91 + 2656347768 * q^92 + 722379888 * q^93 + 8394540344 * q^94 + 3715282870 * q^96 + 2220934678 * q^97 - 98001617 * q^98 + 3534938932 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(175))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	7
175.10.a.a	$175$	$10$	175.a	1.a	$1$	$1$	$1$	$90.131$	\(\Q\)	None	✓				35.10.a.a	$1$	$1$	\(-28\)	\(116\)	\(0\)	\(-2401\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-28q^{2}+116q^{3}+272q^{4}-3248q^{6}+\cdots\)
175.10.a.b	$175$	$10$	175.a	1.a	$1$	$2$	$2$	$90.131$	\(\Q(\sqrt{193}) \)	None	✓				7.10.a.a	$1$	$0$	\(6\)	\(86\)	\(0\)	\(4802\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(3+\beta )q^{2}+(43-11\beta )q^{3}+(-310+\cdots)q^{4}+\cdots\)
175.10.a.c	$175$	$10$	175.a	1.a	$1$	$2$	$2$	$90.131$	\(\Q(\sqrt{2}) \)	None	✓				35.10.a.b	$1$	$1$	\(24\)	\(174\)	\(0\)	\(-4802\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(12+\beta )q^{2}+(87+54\beta )q^{3}+(-360+\cdots)q^{4}+\cdots\)
175.10.a.d	$175$	$10$	175.a	1.a	$1$	$3$	$3$	$90.131$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓				7.10.a.b	$1$	$1$	\(-21\)	\(-84\)	\(0\)	\(-7203\)		$+$	$+$	$+$	$2\cdot 3$	$\mathrm{SU}(2)$	\(q+(-7+\beta _{2})q^{2}+(-28+\beta _{1}+\beta _{2})q^{3}+\cdots\)
175.10.a.e	$175$	$10$	175.a	1.a	$1$	$4$	$4$	$90.131$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				35.10.a.c	$1$	$0$	\(19\)	\(18\)	\(0\)	\(9604\)		$+$	$-$	$-$	$2\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(5+\beta _{1})q^{2}+(4-\beta _{2})q^{3}+(435+9\beta _{1}+\cdots)q^{4}+\cdots\)
175.10.a.f	$175$	$10$	175.a	1.a	$1$	$5$	$5$	$90.131$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓				35.10.a.d	$1$	$1$	\(-2\)	\(-140\)	\(0\)	\(-12005\)		$+$	$+$	$+$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-28-\beta _{2})q^{3}+(168-4\beta _{1}+\cdots)q^{4}+\cdots\)
175.10.a.g	$175$	$10$	175.a	1.a	$1$	$6$	$6$	$90.131$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓				35.10.a.e	$1$	$0$	\(-15\)	\(124\)	\(0\)	\(14406\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-3+\beta _{1})q^{2}+(20+\beta _{1}-\beta _{2})q^{3}+\cdots\)
175.10.a.h	$175$	$10$	175.a	1.a	$1$	$8$	$8$	$90.131$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			175.10.a.h	$1$	$1$	\(-27\)	\(69\)	\(0\)	\(19208\)		$-$	$-$	$+$	$2^{7}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{2}+(8+\beta _{1}-\beta _{2})q^{3}+\cdots\)
175.10.a.i	$175$	$10$	175.a	1.a	$1$	$8$	$8$	$90.131$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		175.10.a.h	$1$	$1$	\(27\)	\(-69\)	\(0\)	\(-19208\)		$+$	$+$	$+$	$2^{7}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+(3+\beta _{1})q^{2}+(-8-\beta _{1}+\beta _{2})q^{3}+\cdots\)
175.10.a.j	$175$	$10$	175.a	1.a	$1$	$10$	$10$	$90.131$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			175.10.a.j	$1$	$0$	\(-22\)	\(77\)	\(0\)	\(-24010\)		$-$	$+$	$-$	$2^{7}\cdot 3^{4}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(8+\beta _{3})q^{3}+(237+\cdots)q^{4}+\cdots\)
175.10.a.k	$175$	$10$	175.a	1.a	$1$	$10$	$10$	$90.131$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓	✓		175.10.a.j	$1$	$0$	\(22\)	\(-77\)	\(0\)	\(24010\)		$+$	$-$	$-$	$2^{7}\cdot 3^{4}\cdot 5^{6}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(-8-\beta _{3})q^{3}+(237+\cdots)q^{4}+\cdots\)
175.10.a.l	$175$	$10$	175.a	1.a	$1$	$13$	$13$	$90.131$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		35.10.b.a	$1$	$1$	\(-32\)	\(-158\)	\(0\)	\(31213\)		$-$	$-$	$+$	$2^{9}\cdot 3^{2}\cdot 5^{10}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{1})q^{2}+(-12+\beta _{3})q^{3}+(208+\cdots)q^{4}+\cdots\)
175.10.a.m	$175$	$10$	175.a	1.a	$1$	$13$	$13$	$90.131$	\(\mathbb{Q}[x]/(x^{13} - \cdots)\)	None	✓		✓		35.10.b.a	$1$	$0$	\(32\)	\(158\)	\(0\)	\(-31213\)		$-$	$+$	$-$	$2^{9}\cdot 3^{2}\cdot 5^{10}$	$\mathrm{SU}(2)$	\(q+(2+\beta _{1})q^{2}+(12-\beta _{3})q^{3}+(208+5\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(175))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(175)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 2}\)