Space of modular forms of level 3174, weight 2, and trivial character

• Label: 3174.2.a
• Level: $3174$
• Weight: $2$
• Character orbit: 3174.a
• Rep. character: $\chi_{3174}(1,\cdot)$
• Character field: $\Q$
• Dimension: $83$
• Newform subspaces: $30$
• Sturm bound: $1104$
• Trace bound: $25$

Defining parameters

Level:	\( N \)	\(=\)	\( 3174 = 2 \cdot 3 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3174.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 30 \)
Sturm bound:			\(1104\)
Trace bound:			\(25\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3174))\).

	Total	New	Old
Modular forms	600	83	517
Cusp forms	505	83	422
Eisenstein series	95	0	95

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

\(2\)	\(3\)	\(23\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(66\)	\(7\)	\(59\)		\(55\)	\(7\)	\(48\)		\(11\)	\(0\)	\(11\)
\(+\)	\(+\)	\(-\)	\(-\)		\(84\)	\(14\)	\(70\)		\(72\)	\(14\)	\(58\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)		\(78\)	\(11\)	\(67\)		\(66\)	\(11\)	\(55\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)		\(72\)	\(10\)	\(62\)		\(60\)	\(10\)	\(50\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)		\(78\)	\(13\)	\(65\)		\(66\)	\(13\)	\(53\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)		\(72\)	\(8\)	\(64\)		\(60\)	\(8\)	\(52\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)		\(78\)	\(5\)	\(73\)		\(66\)	\(5\)	\(61\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)		\(72\)	\(15\)	\(57\)		\(60\)	\(15\)	\(45\)		\(12\)	\(0\)	\(12\)
Plus space			\(+\)		\(288\)	\(30\)	\(258\)		\(241\)	\(30\)	\(211\)		\(47\)	\(0\)	\(47\)
Minus space			\(-\)		\(312\)	\(53\)	\(259\)		\(264\)	\(53\)	\(211\)		\(48\)	\(0\)	\(48\)

Trace form

83 * q - q^2 - q^3 + 83 * q^4 + 2 * q^5 - q^6 - q^8 + 83 * q^9 - 2 * q^10 + 12 * q^11 - q^12 + 2 * q^13 + 2 * q^15 + 83 * q^16 + 6 * q^17 - q^18 + 8 * q^19 + 2 * q^20 - 4 * q^21 - q^24 + 93 * q^25 + 2 * q^26 - q^27 + 10 * q^29 + 6 * q^30 + 8 * q^31 - q^32 + 6 * q^34 + 24 * q^35 + 83 * q^36 - 10 * q^37 + 12 * q^38 - 6 * q^39 - 2 * q^40 - 2 * q^41 + 4 * q^42 + 16 * q^43 + 12 * q^44 + 2 * q^45 - q^48 + 83 * q^49 + q^50 + 10 * q^51 + 2 * q^52 - 22 * q^53 - q^54 - 8 * q^57 + 2 * q^58 + 12 * q^59 + 2 * q^60 - 2 * q^61 + 16 * q^62 + 83 * q^64 - 20 * q^65 + 12 * q^66 - 16 * q^67 + 6 * q^68 + 32 * q^70 + 16 * q^71 - q^72 + 22 * q^73 - 30 * q^74 + q^75 + 8 * q^76 - 32 * q^77 + 10 * q^78 + 8 * q^79 + 2 * q^80 + 83 * q^81 + 6 * q^82 - 20 * q^83 - 4 * q^84 + 4 * q^85 - 4 * q^86 + 10 * q^87 - 18 * q^89 - 2 * q^90 + 32 * q^91 - 16 * q^94 - 8 * q^95 - q^96 + 14 * q^97 - 25 * q^98 + 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3174))\) 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	3	23
3174.2.a.a	$3174$	$2$	3174.a	1.a	$1$	$1$	$1$	$25.345$	\(\Q\)	None	✓	✓			3174.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(-2\)	\(-5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}-2q^{5}+q^{6}-5q^{7}+\cdots\)
3174.2.a.b	$3174$	$2$	3174.a	1.a	$1$	$1$	$1$	$25.345$	\(\Q\)	None	✓				138.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(2\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+2q^{7}+\cdots\)
3174.2.a.c	$3174$	$2$	3174.a	1.a	$1$	$1$	$1$	$25.345$	\(\Q\)	None	✓	✓			3174.2.a.a	$1$	$0$	\(-1\)	\(-1\)	\(2\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+2q^{5}+q^{6}+5q^{7}+\cdots\)
3174.2.a.d	$3174$	$2$	3174.a	1.a	$1$	$1$	$1$	$25.345$	\(\Q\)	None	✓				138.2.a.b	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}-2q^{7}-q^{8}+\cdots\)
3174.2.a.e	$3174$	$2$	3174.a	1.a	$1$	$1$	$1$	$25.345$	\(\Q\)	None	✓				138.2.a.c	$1$	$1$	\(1\)	\(-1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}+q^{8}+\cdots\)
3174.2.a.f	$3174$	$2$	3174.a	1.a	$1$	$1$	$1$	$25.345$	\(\Q\)	None	✓	✓			3174.2.a.f	$1$	$1$	\(1\)	\(-1\)	\(-1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
3174.2.a.g	$3174$	$2$	3174.a	1.a	$1$	$1$	$1$	$25.345$	\(\Q\)	None	✓	✓			3174.2.a.f	$1$	$1$	\(1\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
3174.2.a.h	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{2}) \)	None	✓	✓			3174.2.a.h	$2$	$1$	\(-2\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}+\beta q^{7}-q^{8}+\cdots\)
3174.2.a.i	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{2}) \)	None	✓	✓			3174.2.a.i	$2$	$0$	\(-2\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+\beta q^{5}+q^{6}-\beta q^{7}+\cdots\)
3174.2.a.j	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{3}) \)	None	✓	✓			3174.2.a.j	$2$	$1$	\(-2\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+\beta q^{5}-q^{6}-q^{8}+\cdots\)
3174.2.a.k	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{6}) \)	None	✓	✓			3174.2.a.k	$2$	$1$	\(-2\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}+\beta q^{7}-q^{8}+\cdots\)
3174.2.a.l	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{2}) \)	None	✓	✓			3174.2.a.l	$2$	$0$	\(-2\)	\(2\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+\beta q^{5}-q^{6}+3\beta q^{7}+\cdots\)
3174.2.a.m	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{2}) \)	None	✓	✓			3174.2.a.m	$2$	$0$	\(2\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+3\beta q^{5}-q^{6}+\beta q^{7}+\cdots\)
3174.2.a.n	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{2}) \)	None	✓	✓			3174.2.a.n	$2$	$0$	\(2\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+2\beta q^{5}-q^{6}-\beta q^{7}+\cdots\)
3174.2.a.o	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{41}) \)	None	✓	✓			3174.2.a.o	$1$	$0$	\(2\)	\(2\)	\(-1\)	\(-3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}-\beta q^{5}+q^{6}+(-1+\cdots)q^{7}+\cdots\)
3174.2.a.p	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{3}) \)	None	✓	✓			3174.2.a.p	$2$	$0$	\(2\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+\beta q^{5}+q^{6}+q^{8}+\cdots\)
3174.2.a.q	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{3}) \)	None	✓	✓			3174.2.a.q	$2$	$0$	\(2\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}-\beta q^{7}+q^{8}+\cdots\)
3174.2.a.r	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{41}) \)	None	✓	✓			3174.2.a.o	$1$	$0$	\(2\)	\(2\)	\(1\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+\beta q^{5}+q^{6}+(1+\beta )q^{7}+\cdots\)
3174.2.a.s	$3174$	$2$	3174.a	1.a	$1$	$2$	$2$	$25.345$	\(\Q(\sqrt{5}) \)	None	✓				138.2.a.d	$1$	$0$	\(2\)	\(2\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(1+\beta )q^{5}+q^{6}+\cdots\)
3174.2.a.t	$3174$	$2$	3174.a	1.a	$1$	$4$	$4$	$25.345$	\(\Q(\sqrt{3}, \sqrt{35})\)	None	✓	✓			3174.2.a.t	$2$	$0$	\(-4\)	\(-4\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+\beta _{1}q^{5}+q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
3174.2.a.u	$3174$	$2$	3174.a	1.a	$1$	$4$	$4$	$25.345$	\(\Q(\zeta_{24})^+\)	None	✓	✓			3174.2.a.u	$2$	$0$	\(-4\)	\(4\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
3174.2.a.v	$3174$	$2$	3174.a	1.a	$1$	$4$	$4$	$25.345$	\(\Q(\zeta_{24})^+\)	None	✓	✓			3174.2.a.v	$2$	$0$	\(4\)	\(-4\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
3174.2.a.w	$3174$	$2$	3174.a	1.a	$1$	$5$	$5$	$25.345$	\(\Q(\zeta_{22})^+\)	None	✓		✓		138.2.e.d	$1$	$0$	\(-5\)	\(-5\)	\(-1\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(\beta _{2}-\beta _{3}-\beta _{4})q^{5}+\cdots\)
3174.2.a.x	$3174$	$2$	3174.a	1.a	$1$	$5$	$5$	$25.345$	\(\Q(\zeta_{22})^+\)	None	✓		✓		138.2.e.d	$1$	$1$	\(-5\)	\(-5\)	\(1\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+(\beta _{1}+\beta _{2}-\beta _{4})q^{5}+\cdots\)
3174.2.a.y	$3174$	$2$	3174.a	1.a	$1$	$5$	$5$	$25.345$	\(\Q(\zeta_{22})^+\)	None	✓		✓		138.2.e.c	$1$	$1$	\(-5\)	\(5\)	\(-11\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+(-3-\beta _{2}+2\beta _{3}+\cdots)q^{5}+\cdots\)
3174.2.a.z	$3174$	$2$	3174.a	1.a	$1$	$5$	$5$	$25.345$	\(\Q(\zeta_{22})^+\)	None	✓		✓		138.2.e.c	$1$	$0$	\(-5\)	\(5\)	\(11\)	\(-1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+(2-\beta _{1}-\beta _{2}-\beta _{4})q^{5}+\cdots\)
3174.2.a.ba	$3174$	$2$	3174.a	1.a	$1$	$5$	$5$	$25.345$	\(\Q(\zeta_{22})^+\)	None	✓		✓		138.2.e.b	$1$	$0$	\(5\)	\(-5\)	\(-1\)	\(11\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(\beta _{2}+\beta _{3}+\beta _{4})q^{5}+\cdots\)
3174.2.a.bb	$3174$	$2$	3174.a	1.a	$1$	$5$	$5$	$25.345$	\(\Q(\zeta_{22})^+\)	None	✓		✓		138.2.e.b	$1$	$1$	\(5\)	\(-5\)	\(1\)	\(-11\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}+(1-2\beta _{1}+\beta _{2}+\beta _{4})q^{5}+\cdots\)
3174.2.a.bc	$3174$	$2$	3174.a	1.a	$1$	$5$	$5$	$25.345$	\(\Q(\zeta_{22})^+\)	None	✓		✓	✓	138.2.e.a	$1$	$1$	\(5\)	\(5\)	\(-7\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(-1-\beta _{3}+\beta _{4})q^{5}+\cdots\)
3174.2.a.bd	$3174$	$2$	3174.a	1.a	$1$	$5$	$5$	$25.345$	\(\Q(\zeta_{22})^+\)	None	✓		✓		138.2.e.a	$1$	$0$	\(5\)	\(5\)	\(7\)	\(7\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+(1+\beta _{3}-\beta _{4})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3174))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3174)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(529))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1058))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1587))\)\(^{\oplus 2}\)