Space of modular forms of level 2738, weight 2, and trivial character

• Label: 2738.2.a
• Level: $2738$
• Weight: $2$
• Character orbit: 2738.a
• Rep. character: $\chi_{2738}(1,\cdot)$
• Character field: $\Q$
• Dimension: $110$
• Newform subspaces: $24$
• Sturm bound: $703$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 2738 = 2 \cdot 37^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2738.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(703\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2738))\).

	Total	New	Old
Modular forms	389	110	279
Cusp forms	314	110	204
Eisenstein series	75	0	75

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

\(2\)	\(37\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(90\)	\(25\)	\(65\)		\(72\)	\(25\)	\(47\)		\(18\)	\(0\)	\(18\)
\(+\)	\(-\)	\(-\)		\(104\)	\(30\)	\(74\)		\(85\)	\(30\)	\(55\)		\(19\)	\(0\)	\(19\)
\(-\)	\(+\)	\(-\)		\(100\)	\(34\)	\(66\)		\(81\)	\(34\)	\(47\)		\(19\)	\(0\)	\(19\)
\(-\)	\(-\)	\(+\)		\(95\)	\(21\)	\(74\)		\(76\)	\(21\)	\(55\)		\(19\)	\(0\)	\(19\)
Plus space		\(+\)		\(185\)	\(46\)	\(139\)		\(148\)	\(46\)	\(102\)		\(37\)	\(0\)	\(37\)
Minus space		\(-\)		\(204\)	\(64\)	\(140\)		\(166\)	\(64\)	\(102\)		\(38\)	\(0\)	\(38\)

Trace form

110 * q - 2 * q^3 + 110 * q^4 + 4 * q^6 + 112 * q^9 - 2 * q^10 + 6 * q^11 - 2 * q^12 + 4 * q^14 + 16 * q^15 + 110 * q^16 + 12 * q^17 + 8 * q^18 - 4 * q^19 + 4 * q^21 + 4 * q^22 + 4 * q^23 + 4 * q^24 + 104 * q^25 - 2 * q^26 - 20 * q^27 - 20 * q^31 + 8 * q^33 - 12 * q^34 + 4 * q^35 + 112 * q^36 + 2 * q^38 - 12 * q^39 - 2 * q^40 - 28 * q^41 - 16 * q^42 + 12 * q^43 + 6 * q^44 + 16 * q^45 - 4 * q^46 - 4 * q^47 - 2 * q^48 + 102 * q^49 - 16 * q^50 + 8 * q^51 + 18 * q^53 + 16 * q^54 - 12 * q^55 + 4 * q^56 + 4 * q^57 + 6 * q^58 + 16 * q^60 - 16 * q^61 - 16 * q^62 + 36 * q^63 + 110 * q^64 + 20 * q^65 - 8 * q^66 - 10 * q^67 + 12 * q^68 - 8 * q^69 + 28 * q^70 - 8 * q^71 + 8 * q^72 + 16 * q^73 + 2 * q^75 - 4 * q^76 - 20 * q^77 - 12 * q^78 + 4 * q^79 + 78 * q^81 - 8 * q^82 - 10 * q^83 + 4 * q^84 + 16 * q^85 - 2 * q^86 - 4 * q^87 + 4 * q^88 + 16 * q^89 - 38 * q^90 + 4 * q^92 + 8 * q^93 - 36 * q^95 + 4 * q^96 - 4 * q^97 + 16 * q^98 + 2 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2738))\) 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	37
2738.2.a.a	$2738$	$2$	2738.a	1.a	$1$	$1$	$1$	$21.863$	\(\Q\)	None	✓				74.2.c.b	$1$	$1$	\(-1\)	\(-2\)	\(3\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}+3q^{5}+2q^{6}-4q^{7}+\cdots\)
2738.2.a.b	$2738$	$2$	2738.a	1.a	$1$	$1$	$1$	$21.863$	\(\Q\)	None	✓				74.2.c.a	$1$	$1$	\(-1\)	\(2\)	\(-1\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}+q^{4}-q^{5}-2q^{6}-q^{8}+\cdots\)
2738.2.a.c	$2738$	$2$	2738.a	1.a	$1$	$1$	$1$	$21.863$	\(\Q\)	None	✓				74.2.c.b	$1$	$2$	\(1\)	\(-2\)	\(-3\)	\(-4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-3q^{5}-2q^{6}-4q^{7}+\cdots\)
2738.2.a.d	$2738$	$2$	2738.a	1.a	$1$	$1$	$1$	$21.863$	\(\Q\)	None	✓				74.2.c.a	$1$	$0$	\(1\)	\(2\)	\(1\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+q^{5}+2q^{6}+q^{8}+\cdots\)
2738.2.a.e	$2738$	$2$	2738.a	1.a	$1$	$2$	$2$	$21.863$	\(\Q(\sqrt{3}) \)	None	✓				74.2.e.b	$1$	$0$	\(-2\)	\(-2\)	\(0\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta )q^{3}+q^{4}+\beta q^{5}+(1+\cdots)q^{6}+\cdots\)
2738.2.a.f	$2738$	$2$	2738.a	1.a	$1$	$2$	$2$	$21.863$	\(\Q(\sqrt{3}) \)	None	✓				74.2.e.a	$1$	$0$	\(-2\)	\(-2\)	\(0\)	\(8\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta )q^{3}+q^{4}+\beta q^{5}+(1+\cdots)q^{6}+\cdots\)
2738.2.a.g	$2738$	$2$	2738.a	1.a	$1$	$2$	$2$	$21.863$	\(\Q(\sqrt{5}) \)	None	✓				74.2.a.b	$1$	$1$	\(-2\)	\(-1\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta q^{3}+q^{4}+(1-3\beta )q^{5}+\beta q^{6}+\cdots\)
2738.2.a.h	$2738$	$2$	2738.a	1.a	$1$	$2$	$2$	$21.863$	\(\Q(\sqrt{21}) \)	None	✓				74.2.b.a	$1$	$0$	\(-2\)	\(1\)	\(3\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}+(1+\beta )q^{5}-\beta q^{6}+\cdots\)
2738.2.a.i	$2738$	$2$	2738.a	1.a	$1$	$2$	$2$	$21.863$	\(\Q(\sqrt{3}) \)	None	✓				74.2.e.b	$1$	$1$	\(2\)	\(-2\)	\(0\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta )q^{3}+q^{4}-\beta q^{5}+(-1+\cdots)q^{6}+\cdots\)
2738.2.a.j	$2738$	$2$	2738.a	1.a	$1$	$2$	$2$	$21.863$	\(\Q(\sqrt{3}) \)	None	✓				74.2.e.a	$1$	$1$	\(2\)	\(-2\)	\(0\)	\(8\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta )q^{3}+q^{4}-\beta q^{5}+(-1+\cdots)q^{6}+\cdots\)
2738.2.a.k	$2738$	$2$	2738.a	1.a	$1$	$2$	$2$	$21.863$	\(\Q(\sqrt{21}) \)	None	✓				74.2.b.a	$1$	$1$	\(2\)	\(1\)	\(-3\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+(-1-\beta )q^{5}+\beta q^{6}+\cdots\)
2738.2.a.l	$2738$	$2$	2738.a	1.a	$1$	$2$	$2$	$21.863$	\(\Q(\sqrt{13}) \)	None	✓				74.2.a.a	$1$	$0$	\(2\)	\(3\)	\(1\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}+q^{4}+\beta q^{5}+(1+\beta )q^{6}+\cdots\)
2738.2.a.m	$2738$	$2$	2738.a	1.a	$1$	$3$	$3$	$21.863$	\(\Q(\zeta_{18})^+\)	None	✓				74.2.f.a	$1$	$1$	\(-3\)	\(0\)	\(-6\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}+q^{4}+(-2-\beta _{2})q^{5}+\cdots\)
2738.2.a.n	$2738$	$2$	2738.a	1.a	$1$	$3$	$3$	$21.863$	3.3.404.1	None	✓				74.2.c.c	$1$	$1$	\(-3\)	\(0\)	\(-1\)	\(0\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{1}q^{5}+\beta _{2}q^{6}+\cdots\)
2738.2.a.o	$2738$	$2$	2738.a	1.a	$1$	$3$	$3$	$21.863$	3.3.404.1	None	✓				74.2.c.c	$1$	$0$	\(3\)	\(0\)	\(1\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{2}q^{3}+q^{4}-\beta _{1}q^{5}-\beta _{2}q^{6}+\cdots\)
2738.2.a.p	$2738$	$2$	2738.a	1.a	$1$	$3$	$3$	$21.863$	\(\Q(\zeta_{18})^+\)	None	✓				74.2.f.a	$1$	$0$	\(3\)	\(0\)	\(6\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{1}q^{3}+q^{4}+(2+\beta _{2})q^{5}-\beta _{1}q^{6}+\cdots\)
2738.2.a.q	$2738$	$2$	2738.a	1.a	$1$	$6$	$6$	$21.863$	6.6.37902897.1	None	✓				74.2.f.b	$1$	$1$	\(-6\)	\(0\)	\(0\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{3}q^{5}+\beta _{1}q^{6}+\cdots\)
2738.2.a.r	$2738$	$2$	2738.a	1.a	$1$	$6$	$6$	$21.863$	\(\Q(\zeta_{36})^+\)	None	✓				74.2.h.a	$1$	$0$	\(-6\)	\(0\)	\(6\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(\beta _{2}+\beta _{4})q^{3}+q^{4}+(1+\beta _{3}+\cdots)q^{5}+\cdots\)
2738.2.a.s	$2738$	$2$	2738.a	1.a	$1$	$6$	$6$	$21.863$	\(\Q(\zeta_{36})^+\)	None	✓				74.2.h.a	$1$	$1$	\(6\)	\(0\)	\(-6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(\beta _{2}+\beta _{4})q^{3}+q^{4}+(-1-\beta _{3}+\cdots)q^{5}+\cdots\)
2738.2.a.t	$2738$	$2$	2738.a	1.a	$1$	$6$	$6$	$21.863$	6.6.37902897.1	None	✓				74.2.f.b	$1$	$0$	\(6\)	\(0\)	\(0\)	\(-3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{3}q^{5}-\beta _{1}q^{6}+\cdots\)
2738.2.a.u	$2738$	$2$	2738.a	1.a	$1$	$9$	$9$	$21.863$	\(\Q(\zeta_{38})^+\)	None	✓	✓	✓		2738.2.a.u	$1$	$1$	\(-9\)	\(-7\)	\(7\)	\(-14\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta _{1}-\beta _{4})q^{3}+q^{4}+(1+\cdots)q^{5}+\cdots\)
2738.2.a.v	$2738$	$2$	2738.a	1.a	$1$	$9$	$9$	$21.863$	\(\Q(\zeta_{38})^+\)	None	✓	✓	✓		2738.2.a.u	$1$	$1$	\(9\)	\(-7\)	\(-7\)	\(-14\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta _{1}-\beta _{4})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
2738.2.a.w	$2738$	$2$	2738.a	1.a	$1$	$18$	$18$	$21.863$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓	✓		2738.2.a.w	$1$	$0$	\(-18\)	\(8\)	\(-9\)	\(18\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta _{1}+\beta _{12})q^{3}+q^{4}+(-1+\cdots)q^{5}+\cdots\)
2738.2.a.x	$2738$	$2$	2738.a	1.a	$1$	$18$	$18$	$21.863$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None	✓	✓	✓		2738.2.a.w	$1$	$0$	\(18\)	\(8\)	\(9\)	\(18\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1}+\beta _{12})q^{3}+q^{4}+(1+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2738))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2738)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1369))\)\(^{\oplus 2}\)