Space of modular forms of level 4698, weight 2, and trivial character

• Label: 4698.2.a
• Level: $4698$
• Weight: $2$
• Character orbit: 4698.a
• Rep. character: $\chi_{4698}(1,\cdot)$
• Character field: $\Q$
• Dimension: $112$
• Newform subspaces: $38$
• Sturm bound: $1620$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 4698 = 2 \cdot 3^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4698.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 38 \)
Sturm bound:			\(1620\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4698))\).

	Total	New	Old
Modular forms	834	112	722
Cusp forms	787	112	675
Eisenstein series	47	0	47

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

\(2\)	\(3\)	\(29\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(99\)	\(15\)	\(84\)		\(94\)	\(15\)	\(79\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(108\)	\(14\)	\(94\)		\(102\)	\(14\)	\(88\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(108\)	\(13\)	\(95\)		\(102\)	\(13\)	\(89\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(102\)	\(14\)	\(88\)		\(96\)	\(14\)	\(82\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(102\)	\(16\)	\(86\)		\(96\)	\(16\)	\(80\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(105\)	\(11\)	\(94\)		\(99\)	\(11\)	\(88\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(105\)	\(12\)	\(93\)		\(99\)	\(12\)	\(87\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(105\)	\(17\)	\(88\)		\(99\)	\(17\)	\(82\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(411\)	\(52\)	\(359\)		\(388\)	\(52\)	\(336\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(423\)	\(60\)	\(363\)		\(399\)	\(60\)	\(339\)		\(24\)	\(0\)	\(24\)

Trace form

112 * q + 112 * q^4 + 8 * q^7 + 8 * q^13 + 112 * q^16 + 20 * q^19 + 12 * q^22 + 136 * q^25 + 8 * q^28 + 32 * q^31 + 12 * q^34 + 32 * q^37 - 28 * q^43 + 72 * q^49 + 8 * q^52 - 24 * q^55 - 40 * q^61 + 112 * q^64 - 4 * q^67 - 76 * q^73 + 20 * q^76 + 8 * q^79 + 12 * q^82 - 48 * q^85 + 12 * q^88 - 80 * q^91 - 48 * q^94 - 4 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4698))\) 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	29
4698.2.a.a	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓				522.2.e.b	$1$	$1$	\(-1\)	\(0\)	\(-4\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{5}+4q^{7}-q^{8}+4q^{10}+\cdots\)
4698.2.a.b	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓	✓			4698.2.a.b	$1$	$0$	\(-1\)	\(0\)	\(-3\)	\(3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-3q^{5}+3q^{7}-q^{8}+3q^{10}+\cdots\)
4698.2.a.c	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓	✓			4698.2.a.c	$1$	$0$	\(-1\)	\(0\)	\(-1\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{5}+q^{7}-q^{8}+q^{10}+\cdots\)
4698.2.a.d	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓				522.2.e.d	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{11}-4q^{13}+\cdots\)
4698.2.a.e	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓				522.2.e.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+3q^{7}-q^{8}-3q^{11}-3q^{14}+\cdots\)
4698.2.a.f	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓	✓			4698.2.a.f	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-q^{10}+\cdots\)
4698.2.a.g	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓	✓			4698.2.a.g	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}-2q^{7}-q^{8}-2q^{10}+\cdots\)
4698.2.a.h	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓				522.2.e.c	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}+3q^{7}-q^{8}-2q^{10}+\cdots\)
4698.2.a.i	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓	✓			4698.2.a.i	$1$	$0$	\(-1\)	\(0\)	\(3\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+3q^{5}+5q^{7}-q^{8}-3q^{10}+\cdots\)
4698.2.a.j	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓	✓			4698.2.a.i	$1$	$0$	\(1\)	\(0\)	\(-3\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-3q^{5}+5q^{7}+q^{8}-3q^{10}+\cdots\)
4698.2.a.k	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓	✓			4698.2.a.g	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}-2q^{7}+q^{8}-2q^{10}+\cdots\)
4698.2.a.l	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓				522.2.e.c	$1$	$1$	\(1\)	\(0\)	\(-2\)	\(3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{5}+3q^{7}+q^{8}-2q^{10}+\cdots\)
4698.2.a.m	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓	✓			4698.2.a.f	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{5}+q^{7}+q^{8}-q^{10}+\cdots\)
4698.2.a.n	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓				522.2.e.d	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}+3q^{11}-4q^{13}+\cdots\)
4698.2.a.o	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓				522.2.e.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+3q^{7}+q^{8}+3q^{11}+3q^{14}+\cdots\)
4698.2.a.p	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓	✓			4698.2.a.c	$1$	$0$	\(1\)	\(0\)	\(1\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{5}+q^{7}+q^{8}+q^{10}+\cdots\)
4698.2.a.q	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓	✓			4698.2.a.b	$1$	$0$	\(1\)	\(0\)	\(3\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+3q^{5}+3q^{7}+q^{8}+3q^{10}+\cdots\)
4698.2.a.r	$4698$	$2$	4698.a	1.a	$1$	$1$	$1$	$37.514$	\(\Q\)	None	✓				522.2.e.b	$1$	$0$	\(1\)	\(0\)	\(4\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{5}+4q^{7}+q^{8}+4q^{10}+\cdots\)
4698.2.a.s	$4698$	$2$	4698.a	1.a	$1$	$2$	$2$	$37.514$	\(\Q(\sqrt{3}) \)	None	✓	✓			4698.2.a.s	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta q^{5}+(-2+\beta )q^{7}-q^{8}+\cdots\)
4698.2.a.t	$4698$	$2$	4698.a	1.a	$1$	$2$	$2$	$37.514$	\(\Q(\sqrt{6}) \)	None	✓				522.2.e.e	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-\beta q^{7}-q^{8}+(3-\beta )q^{11}+\cdots\)
4698.2.a.u	$4698$	$2$	4698.a	1.a	$1$	$2$	$2$	$37.514$	\(\Q(\sqrt{3}) \)	None	✓	✓			4698.2.a.s	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{5}+(-2-\beta )q^{7}+q^{8}+\cdots\)
4698.2.a.v	$4698$	$2$	4698.a	1.a	$1$	$2$	$2$	$37.514$	\(\Q(\sqrt{6}) \)	None	✓				522.2.e.e	$1$	$1$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{7}+q^{8}+(-3-\beta )q^{11}+\cdots\)
4698.2.a.w	$4698$	$2$	4698.a	1.a	$1$	$3$	$3$	$37.514$	3.3.564.1	None	✓	✓			4698.2.a.w	$1$	$1$	\(-3\)	\(0\)	\(5\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(2-\beta _{1})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
4698.2.a.x	$4698$	$2$	4698.a	1.a	$1$	$3$	$3$	$37.514$	\(\Q(\zeta_{18})^+\)	None	✓				522.2.e.f	$1$	$0$	\(-3\)	\(0\)	\(6\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(2+2\beta _{1}-\beta _{2})q^{5}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
4698.2.a.y	$4698$	$2$	4698.a	1.a	$1$	$3$	$3$	$37.514$	\(\Q(\zeta_{18})^+\)	None	✓				522.2.e.f	$1$	$1$	\(3\)	\(0\)	\(-6\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-2-2\beta _{1}+\beta _{2})q^{5}+\cdots\)
4698.2.a.z	$4698$	$2$	4698.a	1.a	$1$	$3$	$3$	$37.514$	3.3.564.1	None	✓	✓			4698.2.a.w	$1$	$1$	\(3\)	\(0\)	\(-5\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-2+\beta _{1})q^{5}+(-1+\beta _{1}+\cdots)q^{7}+\cdots\)
4698.2.a.ba	$4698$	$2$	4698.a	1.a	$1$	$4$	$4$	$37.514$	4.4.116532.1	None	✓	✓	✓		4698.2.a.ba	$1$	$0$	\(-4\)	\(0\)	\(-3\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1-\beta _{2})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
4698.2.a.bb	$4698$	$2$	4698.a	1.a	$1$	$4$	$4$	$37.514$	4.4.45729.1	None	✓				522.2.e.g	$1$	$0$	\(-4\)	\(0\)	\(2\)	\(-8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(\beta _{1}+\beta _{3})q^{5}-2q^{7}-q^{8}+\cdots\)
4698.2.a.bc	$4698$	$2$	4698.a	1.a	$1$	$4$	$4$	$37.514$	4.4.45729.1	None	✓		✓		522.2.e.g	$1$	$1$	\(4\)	\(0\)	\(-2\)	\(-8\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(-\beta _{1}-\beta _{3})q^{5}-2q^{7}+\cdots\)
4698.2.a.bd	$4698$	$2$	4698.a	1.a	$1$	$4$	$4$	$37.514$	4.4.116532.1	None	✓	✓			4698.2.a.ba	$1$	$0$	\(4\)	\(0\)	\(3\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1+\beta _{2})q^{5}+(1+\beta _{1})q^{7}+\cdots\)
4698.2.a.be	$4698$	$2$	4698.a	1.a	$1$	$6$	$6$	$37.514$	6.6.205267392.1	None	✓				522.2.e.h	$1$	$1$	\(-6\)	\(0\)	\(-2\)	\(-8\)		$+$	$-$	$-$	$+$	$3$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-\beta _{2}q^{5}+(-1+\beta _{4})q^{7}+\cdots\)
4698.2.a.bf	$4698$	$2$	4698.a	1.a	$1$	$6$	$6$	$37.514$	6.6.31083264.1	None	✓	✓			4698.2.a.bf	$1$	$1$	\(-6\)	\(0\)	\(2\)	\(-8\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta _{2}q^{5}+(-1-\beta _{1})q^{7}+\cdots\)
4698.2.a.bg	$4698$	$2$	4698.a	1.a	$1$	$6$	$6$	$37.514$	6.6.31083264.1	None	✓	✓	✓		4698.2.a.bf	$1$	$1$	\(6\)	\(0\)	\(-2\)	\(-8\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-\beta _{2}q^{5}+(-1-\beta _{1})q^{7}+\cdots\)
4698.2.a.bh	$4698$	$2$	4698.a	1.a	$1$	$6$	$6$	$37.514$	6.6.205267392.1	None	✓		✓		522.2.e.h	$1$	$0$	\(6\)	\(0\)	\(2\)	\(-8\)		$-$	$+$	$+$	$-$	$3$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{2}q^{5}+(-1+\beta _{4})q^{7}+\cdots\)
4698.2.a.bi	$4698$	$2$	4698.a	1.a	$1$	$8$	$8$	$37.514$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		4698.2.a.bi	$1$	$0$	\(-8\)	\(0\)	\(-6\)	\(8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+(-1+\beta _{1})q^{5}+(1-\beta _{3}+\cdots)q^{7}+\cdots\)
4698.2.a.bj	$4698$	$2$	4698.a	1.a	$1$	$8$	$8$	$37.514$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			4698.2.a.bi	$1$	$0$	\(8\)	\(0\)	\(6\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+(1-\beta _{1})q^{5}+(1-\beta _{3})q^{7}+\cdots\)
4698.2.a.bk	$4698$	$2$	4698.a	1.a	$1$	$9$	$9$	$37.514$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		522.2.e.i	$1$	$1$	\(-9\)	\(0\)	\(-4\)	\(3\)		$+$	$+$	$+$	$+$	$3$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-\beta _{2}q^{5}-\beta _{6}q^{7}-q^{8}+\cdots\)
4698.2.a.bl	$4698$	$2$	4698.a	1.a	$1$	$9$	$9$	$37.514$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		522.2.e.i	$1$	$0$	\(9\)	\(0\)	\(4\)	\(3\)		$-$	$-$	$-$	$-$	$3$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta _{2}q^{5}-\beta _{6}q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4698))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4698)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(58))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(174))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(522))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(783))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1566))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2349))\)\(^{\oplus 2}\)