Space of modular forms of level 2349, weight 2, and trivial character

• Label: 2349.2.a
• Level: $2349$
• Weight: $2$
• Character orbit: 2349.a
• Rep. character: $\chi_{2349}(1,\cdot)$
• Character field: $\Q$
• Dimension: $112$
• Newform subspaces: $10$
• Sturm bound: $540$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 2349 = 3^{4} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2349.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(540\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2349))\).

	Total	New	Old
Modular forms	282	112	170
Cusp forms	259	112	147
Eisenstein series	23	0	23

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

\(3\)	\(29\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(63\)	\(25\)	\(38\)		\(58\)	\(25\)	\(33\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(75\)	\(31\)	\(44\)		\(69\)	\(31\)	\(38\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(75\)	\(31\)	\(44\)		\(69\)	\(31\)	\(38\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(69\)	\(25\)	\(44\)		\(63\)	\(25\)	\(38\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(132\)	\(50\)	\(82\)		\(121\)	\(50\)	\(71\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(150\)	\(62\)	\(88\)		\(138\)	\(62\)	\(76\)		\(12\)	\(0\)	\(12\)

Trace form

112 * q + 112 * q^4 - 4 * q^7 - 4 * q^13 + 112 * q^16 - 16 * q^19 - 12 * q^22 + 100 * q^25 - 16 * q^28 - 28 * q^31 - 36 * q^34 - 16 * q^37 - 24 * q^40 - 4 * q^43 - 24 * q^46 + 132 * q^49 - 16 * q^52 + 12 * q^55 + 20 * q^61 + 88 * q^64 - 4 * q^67 + 72 * q^70 + 8 * q^73 - 4 * q^76 - 4 * q^79 + 60 * q^82 + 24 * q^85 - 84 * q^88 + 52 * q^91 - 28 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2349))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	29
2349.2.a.a	$2349$	$2$	2349.a	1.a	$1$	$7$	$7$	$18.757$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			2349.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(-6\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{6})q^{5}+\cdots\)
2349.2.a.b	$2349$	$2$	2349.a	1.a	$1$	$7$	$7$	$18.757$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			2349.2.a.b	$1$	$1$	\(-1\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{2}+(1+\beta _{2}-\beta _{4})q^{4}+(-\beta _{3}+\cdots)q^{5}+\cdots\)
2349.2.a.c	$2349$	$2$	2349.a	1.a	$1$	$7$	$7$	$18.757$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			2349.2.a.b	$1$	$0$	\(1\)	\(0\)	\(2\)	\(-2\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{2}+(1+\beta _{2}-\beta _{4})q^{4}+(\beta _{3}+\beta _{6})q^{5}+\cdots\)
2349.2.a.d	$2349$	$2$	2349.a	1.a	$1$	$7$	$7$	$18.757$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			2349.2.a.a	$1$	$0$	\(1\)	\(0\)	\(6\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{6})q^{5}+\cdots\)
2349.2.a.e	$2349$	$2$	2349.a	1.a	$1$	$11$	$11$	$18.757$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓				261.2.e.a	$1$	$1$	\(-1\)	\(0\)	\(1\)	\(-7\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{2}q^{4}+\beta _{3}q^{5}+(-1-\beta _{8}+\cdots)q^{7}+\cdots\)
2349.2.a.f	$2349$	$2$	2349.a	1.a	$1$	$11$	$11$	$18.757$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		261.2.e.a	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{3}q^{5}+(-1-\beta _{8}+\cdots)q^{7}+\cdots\)
2349.2.a.g	$2349$	$2$	2349.a	1.a	$1$	$14$	$14$	$18.757$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	✓	✓		2349.2.a.g	$1$	$1$	\(-2\)	\(0\)	\(-8\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{11}+\cdots)q^{5}+\cdots\)
2349.2.a.h	$2349$	$2$	2349.a	1.a	$1$	$14$	$14$	$18.757$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓	✓			2349.2.a.g	$1$	$0$	\(2\)	\(0\)	\(8\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1-\beta _{11})q^{5}+\cdots\)
2349.2.a.i	$2349$	$2$	2349.a	1.a	$1$	$17$	$17$	$18.757$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓		✓		261.2.e.b	$1$	$0$	\(-1\)	\(0\)	\(1\)	\(9\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{5}q^{5}+(1+\beta _{14}+\cdots)q^{7}+\cdots\)
2349.2.a.j	$2349$	$2$	2349.a	1.a	$1$	$17$	$17$	$18.757$	\(\mathbb{Q}[x]/(x^{17} - \cdots)\)	None	✓		✓		261.2.e.b	$1$	$0$	\(1\)	\(0\)	\(-1\)	\(9\)		$+$	$-$	$-$	$2^{3}\cdot 3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{5}q^{5}+(1+\beta _{14}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2349))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2349)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(87))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(261))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(783))\)\(^{\oplus 2}\)