Space of modular forms of level 175, weight 4, and trivial character

• Label: 175.4.a
• Level: $175$
• Weight: $4$
• Character orbit: 175.a
• Rep. character: $\chi_{175}(1,\cdot)$
• Character field: $\Q$
• Dimension: $29$
• Newform subspaces: $10$
• Sturm bound: $80$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 175 = 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	175.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(80\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(175))\).

	Total	New	Old
Modular forms	66	29	37
Cusp forms	54	29	25
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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(18\)	\(8\)	\(10\)		\(15\)	\(8\)	\(7\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)		\(15\)	\(5\)	\(10\)		\(12\)	\(5\)	\(7\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)		\(15\)	\(7\)	\(8\)		\(12\)	\(7\)	\(5\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)		\(18\)	\(9\)	\(9\)		\(15\)	\(9\)	\(6\)		\(3\)	\(0\)	\(3\)
Plus space		\(+\)		\(36\)	\(17\)	\(19\)		\(30\)	\(17\)	\(13\)		\(6\)	\(0\)	\(6\)
Minus space		\(-\)		\(30\)	\(12\)	\(18\)		\(24\)	\(12\)	\(12\)		\(6\)	\(0\)	\(6\)

Trace form

29 * q - 5 * q^2 + 6 * q^3 + 137 * q^4 - 22 * q^6 - 7 * q^7 - 33 * q^8 + 233 * q^9 + 88 * q^11 + 190 * q^12 - 44 * q^13 + 63 * q^14 + 369 * q^16 + 142 * q^17 + 335 * q^18 - 110 * q^19 - 14 * q^21 - 20 * q^22 - 200 * q^23 + 30 * q^24 + 168 * q^26 - 276 * q^27 + 49 * q^28 - 510 * q^29 - 12 * q^31 - 489 * q^32 - 312 * q^33 - 46 * q^34 + 1829 * q^36 + 170 * q^37 - 10 * q^38 + 776 * q^39 + 98 * q^41 - 154 * q^42 + 216 * q^43 + 154 * q^44 - 1702 * q^46 + 220 * q^47 + 1982 * q^48 + 1421 * q^49 - 1472 * q^51 - 2180 * q^52 + 86 * q^53 - 3160 * q^54 + 525 * q^56 + 1724 * q^57 - 502 * q^58 + 390 * q^59 + 1988 * q^61 + 996 * q^62 - 903 * q^63 + 7 * q^64 - 2684 * q^66 - 132 * q^67 - 2106 * q^68 - 2024 * q^69 - 572 * q^71 + 935 * q^72 - 366 * q^73 + 2964 * q^74 - 5130 * q^76 + 280 * q^77 - 4028 * q^78 - 1140 * q^79 + 329 * q^81 + 602 * q^82 + 3786 * q^83 - 882 * q^84 + 5118 * q^86 + 1364 * q^87 + 5008 * q^88 + 3590 * q^89 + 1036 * q^91 + 200 * q^92 - 6096 * q^93 - 5936 * q^94 - 2762 * q^96 - 4162 * q^97 - 245 * q^98 + 7276 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(175))\) 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
175.4.a.a	$175$	$4$	175.a	1.a	$1$	$1$	$1$	$10.325$	\(\Q\)	None	✓				35.4.a.a	$1$	$0$	\(-1\)	\(8\)	\(0\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+8q^{3}-7q^{4}-8q^{6}-7q^{7}+\cdots\)
175.4.a.b	$175$	$4$	175.a	1.a	$1$	$1$	$1$	$10.325$	\(\Q\)	None	✓				7.4.a.a	$1$	$1$	\(1\)	\(2\)	\(0\)	\(7\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}-7q^{4}+2q^{6}+7q^{7}+\cdots\)
175.4.a.c	$175$	$4$	175.a	1.a	$1$	$2$	$2$	$10.325$	\(\Q(\sqrt{2}) \)	None	✓				35.4.a.b	$1$	$1$	\(-8\)	\(-2\)	\(0\)	\(14\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-4+\beta )q^{2}+(-1-4\beta )q^{3}+(10+\cdots)q^{4}+\cdots\)
175.4.a.d	$175$	$4$	175.a	1.a	$1$	$2$	$2$	$10.325$	\(\Q(\sqrt{41}) \)	None	✓	✓			175.4.a.d	$1$	$1$	\(-1\)	\(5\)	\(0\)	\(-14\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(2+\beta )q^{3}+(2+\beta )q^{4}+(-10+\cdots)q^{6}+\cdots\)
175.4.a.e	$175$	$4$	175.a	1.a	$1$	$2$	$2$	$10.325$	\(\Q(\sqrt{41}) \)	None	✓	✓			175.4.a.d	$1$	$1$	\(1\)	\(-5\)	\(0\)	\(14\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-2-\beta )q^{3}+(2+\beta )q^{4}+(-10+\cdots)q^{6}+\cdots\)
175.4.a.f	$175$	$4$	175.a	1.a	$1$	$3$	$3$	$10.325$	3.3.14360.1	None	✓				35.4.a.c	$1$	$0$	\(3\)	\(-2\)	\(0\)	\(-21\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(-1-\beta _{1}+\beta _{2})q^{3}+\cdots\)
175.4.a.g	$175$	$4$	175.a	1.a	$1$	$4$	$4$	$10.325$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓	✓		175.4.a.g	$1$	$0$	\(-4\)	\(3\)	\(0\)	\(-28\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(1-\beta _{3})q^{3}+(9+\beta _{2}+\cdots)q^{4}+\cdots\)
175.4.a.h	$175$	$4$	175.a	1.a	$1$	$4$	$4$	$10.325$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓	✓			175.4.a.g	$1$	$0$	\(4\)	\(-3\)	\(0\)	\(28\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(-1+\beta _{3})q^{3}+(9+\beta _{2}+\cdots)q^{4}+\cdots\)
175.4.a.i	$175$	$4$	175.a	1.a	$1$	$5$	$5$	$10.325$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		35.4.b.a	$1$	$1$	\(-4\)	\(-10\)	\(0\)	\(-35\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+(-2+\beta _{3})q^{3}+(4+\cdots)q^{4}+\cdots\)
175.4.a.j	$175$	$4$	175.a	1.a	$1$	$5$	$5$	$10.325$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		35.4.b.a	$1$	$0$	\(4\)	\(10\)	\(0\)	\(35\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+(2-\beta _{3})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(175))\) into lower level spaces

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