Space of modular forms of level 3703, weight 2, and trivial character

• Label: 3703.2.a
• Level: $3703$
• Weight: $2$
• Character orbit: 3703.a
• Rep. character: $\chi_{3703}(1,\cdot)$
• Character field: $\Q$
• Dimension: $253$
• Newform subspaces: $20$
• Sturm bound: $736$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 3703 = 7 \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3703.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(736\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3703))\).

	Total	New	Old
Modular forms	392	253	139
Cusp forms	345	253	92
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.

\(7\)	\(23\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(92\)	\(61\)	\(31\)		\(81\)	\(61\)	\(20\)		\(11\)	\(0\)	\(11\)
\(+\)	\(-\)	\(-\)		\(103\)	\(66\)	\(37\)		\(91\)	\(66\)	\(25\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)		\(104\)	\(71\)	\(33\)		\(92\)	\(71\)	\(21\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)		\(93\)	\(55\)	\(38\)		\(81\)	\(55\)	\(26\)		\(12\)	\(0\)	\(12\)
Plus space		\(+\)		\(185\)	\(116\)	\(69\)		\(162\)	\(116\)	\(46\)		\(23\)	\(0\)	\(23\)
Minus space		\(-\)		\(207\)	\(137\)	\(70\)		\(183\)	\(137\)	\(46\)		\(24\)	\(0\)	\(24\)

Trace form

253 * q + q^2 + 249 * q^4 + 2 * q^5 - 8 * q^6 - q^7 - 3 * q^8 + 261 * q^9 + 10 * q^10 - 4 * q^11 + 2 * q^13 - 3 * q^14 - 20 * q^15 + 245 * q^16 + 10 * q^17 + 9 * q^18 - 8 * q^19 + 6 * q^20 - 8 * q^22 + 12 * q^24 + 251 * q^25 + 2 * q^26 - 24 * q^27 - 7 * q^28 + 2 * q^29 + 48 * q^30 - 24 * q^31 + 5 * q^32 + 24 * q^33 - 2 * q^34 + 2 * q^35 + 273 * q^36 + 2 * q^37 + 44 * q^38 - 20 * q^39 + 22 * q^40 + 14 * q^41 + 8 * q^42 + 12 * q^43 + 24 * q^44 + 6 * q^45 - 24 * q^47 - 36 * q^48 + 253 * q^49 - 17 * q^50 + 8 * q^51 - 18 * q^52 - 30 * q^53 - 24 * q^54 - 12 * q^55 - 15 * q^56 - 4 * q^57 - 22 * q^58 - 12 * q^59 - 20 * q^60 + 6 * q^61 - 8 * q^62 - 13 * q^63 + 197 * q^64 + 32 * q^65 - 4 * q^66 - 24 * q^67 - 34 * q^68 + 10 * q^70 - 8 * q^71 + q^72 + 22 * q^73 - 38 * q^74 + 28 * q^75 - 64 * q^76 + 4 * q^77 - 56 * q^78 - 44 * q^79 + 34 * q^80 + 261 * q^81 - 6 * q^82 - 28 * q^83 - 12 * q^84 + 20 * q^85 - 44 * q^86 + 16 * q^87 + 36 * q^88 + 10 * q^89 - 38 * q^90 - 2 * q^91 - 28 * q^93 + 4 * q^94 - 32 * q^95 - 32 * q^96 + 22 * q^97 + q^98 + 20 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3703))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7	23
3703.2.a.a	$3703$	$2$	3703.a	1.a	$1$	$1$	$1$	$29.569$	\(\Q\)	None	✓				161.2.a.a	$1$	$0$	\(-1\)	\(0\)	\(-2\)	\(-1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{4}-2q^{5}-q^{7}+3q^{8}-3q^{9}+\cdots\)
3703.2.a.b	$3703$	$2$	3703.a	1.a	$1$	$2$	$2$	$29.569$	\(\Q(\sqrt{5}) \)	None	✓				161.2.a.b	$1$	$1$	\(-1\)	\(-2\)	\(2\)	\(2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+(2-2\beta )q^{5}+\cdots\)
3703.2.a.c	$3703$	$2$	3703.a	1.a	$1$	$3$	$3$	$29.569$	3.3.148.1	None	✓				161.2.a.c	$1$	$1$	\(-1\)	\(2\)	\(-2\)	\(3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{2}+(1-\beta _{1})q^{3}+(1+2\beta _{1}+\cdots)q^{4}+\cdots\)
3703.2.a.d	$3703$	$2$	3703.a	1.a	$1$	$5$	$5$	$29.569$	5.5.70601.1	None	✓	✓			3703.2.a.d	$1$	$1$	\(-2\)	\(0\)	\(-6\)	\(5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}+(\beta _{2}-\beta _{3})q^{3}+(1-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
3703.2.a.e	$3703$	$2$	3703.a	1.a	$1$	$5$	$5$	$29.569$	5.5.70601.1	None	✓	✓			3703.2.a.d	$1$	$0$	\(-2\)	\(0\)	\(6\)	\(-5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}+(\beta _{2}-\beta _{3})q^{3}+(1-\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
3703.2.a.f	$3703$	$2$	3703.a	1.a	$1$	$5$	$5$	$29.569$	5.5.1090433.1	None	✓	✓			3703.2.a.f	$1$	$1$	\(-2\)	\(2\)	\(-2\)	\(5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{4}q^{3}+(\beta _{1}+\beta _{2})q^{4}-\beta _{2}q^{5}+\cdots\)
3703.2.a.g	$3703$	$2$	3703.a	1.a	$1$	$5$	$5$	$29.569$	5.5.1090433.1	None	✓	✓			3703.2.a.f	$1$	$0$	\(-2\)	\(2\)	\(2\)	\(-5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{4}q^{3}+(\beta _{1}+\beta _{2})q^{4}+\beta _{2}q^{5}+\cdots\)
3703.2.a.h	$3703$	$2$	3703.a	1.a	$1$	$5$	$5$	$29.569$	5.5.220036.1	None	✓	✓			3703.2.a.h	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(5\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-\beta _{3}q^{3}+(\beta _{1}-\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots\)
3703.2.a.i	$3703$	$2$	3703.a	1.a	$1$	$5$	$5$	$29.569$	5.5.220036.1	None	✓	✓			3703.2.a.h	$1$	$0$	\(0\)	\(2\)	\(4\)	\(-5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-\beta _{3}q^{3}+(\beta _{1}-\beta _{3})q^{4}+(1+\cdots)q^{5}+\cdots\)
3703.2.a.j	$3703$	$2$	3703.a	1.a	$1$	$5$	$5$	$29.569$	5.5.2147108.1	None	✓				161.2.a.d	$1$	$0$	\(2\)	\(0\)	\(4\)	\(-5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(2+\beta _{2})q^{4}+(1+\beta _{4})q^{5}+\cdots\)
3703.2.a.k	$3703$	$2$	3703.a	1.a	$1$	$10$	$10$	$29.569$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			3703.2.a.k	$1$	$1$	\(0\)	\(-2\)	\(-8\)	\(10\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{1}+\beta _{7}+\beta _{8}+\cdots)q^{4}+\cdots\)
3703.2.a.l	$3703$	$2$	3703.a	1.a	$1$	$10$	$10$	$29.569$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	✓			3703.2.a.k	$1$	$0$	\(0\)	\(-2\)	\(8\)	\(-10\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{1}+\beta _{7}+\beta _{8}+\cdots)q^{4}+\cdots\)
3703.2.a.m	$3703$	$2$	3703.a	1.a	$1$	$12$	$12$	$29.569$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			3703.2.a.m	$1$	$1$	\(2\)	\(0\)	\(-8\)	\(-12\)		$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(2+\beta _{2}+\beta _{5}+\beta _{10}+\cdots)q^{4}+\cdots\)
3703.2.a.n	$3703$	$2$	3703.a	1.a	$1$	$12$	$12$	$29.569$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓			3703.2.a.m	$1$	$0$	\(2\)	\(0\)	\(8\)	\(12\)		$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(2+\beta _{2}+\beta _{5}+\beta _{10}+\cdots)q^{4}+\cdots\)
3703.2.a.o	$3703$	$2$	3703.a	1.a	$1$	$24$	$24$	$29.569$		None	✓	✓			3703.2.a.o	$1$	$1$	\(4\)	\(0\)	\(-16\)	\(-24\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
3703.2.a.p	$3703$	$2$	3703.a	1.a	$1$	$24$	$24$	$29.569$		None	✓	✓			3703.2.a.o	$1$	$0$	\(4\)	\(0\)	\(16\)	\(24\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
3703.2.a.q	$3703$	$2$	3703.a	1.a	$1$	$25$	$25$	$29.569$		None	✓		✓		161.2.i.a	$1$	$1$	\(-10\)	\(-11\)	\(0\)	\(-25\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
3703.2.a.r	$3703$	$2$	3703.a	1.a	$1$	$25$	$25$	$29.569$		None	✓		✓		161.2.i.a	$1$	$1$	\(-10\)	\(-11\)	\(0\)	\(25\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
3703.2.a.s	$3703$	$2$	3703.a	1.a	$1$	$35$	$35$	$29.569$		None	✓		✓		161.2.i.b	$1$	$0$	\(9\)	\(9\)	\(-2\)	\(35\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
3703.2.a.t	$3703$	$2$	3703.a	1.a	$1$	$35$	$35$	$29.569$		None	✓		✓		161.2.i.b	$1$	$0$	\(9\)	\(9\)	\(2\)	\(-35\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3703))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3703)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(529))\)\(^{\oplus 2}\)