Space of modular forms of level 2299, weight 2, and trivial character

• Label: 2299.2.a
• Level: $2299$
• Weight: $2$
• Character orbit: 2299.a
• Rep. character: $\chi_{2299}(1,\cdot)$
• Character field: $\Q$
• Dimension: $164$
• Newform subspaces: $25$
• Sturm bound: $440$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 2299 = 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2299.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 25 \)
Sturm bound:			\(440\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\), \(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2299))\).

	Total	New	Old
Modular forms	232	164	68
Cusp forms	209	164	45
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.

\(11\)	\(19\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(52\)	\(38\)	\(14\)		\(47\)	\(38\)	\(9\)		\(5\)	\(0\)	\(5\)
\(+\)	\(-\)	\(-\)		\(64\)	\(46\)	\(18\)		\(58\)	\(46\)	\(12\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)		\(62\)	\(45\)	\(17\)		\(56\)	\(45\)	\(11\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)		\(54\)	\(35\)	\(19\)		\(48\)	\(35\)	\(13\)		\(6\)	\(0\)	\(6\)
Plus space		\(+\)		\(106\)	\(73\)	\(33\)		\(95\)	\(73\)	\(22\)		\(11\)	\(0\)	\(11\)
Minus space		\(-\)		\(126\)	\(91\)	\(35\)		\(114\)	\(91\)	\(23\)		\(12\)	\(0\)	\(12\)

Trace form

164 * q - q^2 - 2 * q^3 + 163 * q^4 + 3 * q^5 + 8 * q^6 - 7 * q^7 + 3 * q^8 + 164 * q^9 - 6 * q^10 + 6 * q^13 + 4 * q^14 - 18 * q^15 + 137 * q^16 + q^17 + 3 * q^18 - 2 * q^19 - 4 * q^20 - 2 * q^21 - 12 * q^23 + 8 * q^24 + 159 * q^25 - 6 * q^26 + 4 * q^27 - 26 * q^28 + 12 * q^29 + 32 * q^30 - 20 * q^31 + 35 * q^32 - 2 * q^34 - 13 * q^35 + 167 * q^36 - 8 * q^37 + 3 * q^38 - 28 * q^39 + 18 * q^40 + 8 * q^41 + 48 * q^42 - 23 * q^43 - 5 * q^45 + 8 * q^46 - 9 * q^47 + 44 * q^48 + 155 * q^49 + 65 * q^50 + 2 * q^51 + 50 * q^52 - 38 * q^53 + 32 * q^54 - 20 * q^56 - 2 * q^57 + 26 * q^58 - 6 * q^59 - 48 * q^60 + 15 * q^61 - 4 * q^62 - 23 * q^63 + 109 * q^64 + 28 * q^65 - 32 * q^67 - 12 * q^68 - 28 * q^69 - 12 * q^70 - 58 * q^71 - 25 * q^72 - 3 * q^73 - 54 * q^74 - 48 * q^75 - 5 * q^76 - 12 * q^78 - 56 * q^79 - 14 * q^80 + 160 * q^81 + 6 * q^82 + 8 * q^83 - 12 * q^84 + 37 * q^85 - 68 * q^86 - 32 * q^87 + 2 * q^89 - 38 * q^90 - 60 * q^91 - 60 * q^92 - 28 * q^93 - 24 * q^94 - 9 * q^95 - 56 * q^96 - 6 * q^97 - 25 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2299))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		11	19
2299.2.a.a	$2299$	$2$	2299.a	1.a	$1$	$1$	$1$	$18.358$	\(\Q\)	None	✓	✓			2299.2.a.a	$1$	$2$	\(-2\)	\(-3\)	\(-3\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-3q^{3}+2q^{4}-3q^{5}+6q^{6}+\cdots\)
2299.2.a.b	$2299$	$2$	2299.a	1.a	$1$	$1$	$1$	$18.358$	\(\Q\)	None	✓				19.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(3\)	\(1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{4}+3q^{5}+q^{7}+q^{9}+4q^{12}+\cdots\)
2299.2.a.c	$2299$	$2$	2299.a	1.a	$1$	$1$	$1$	$18.358$	\(\Q\)	None	✓				209.2.a.a	$1$	$0$	\(0\)	\(1\)	\(-3\)	\(4\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-3q^{5}+4q^{7}-2q^{9}+\cdots\)
2299.2.a.d	$2299$	$2$	2299.a	1.a	$1$	$1$	$1$	$18.358$	\(\Q\)	None	✓	✓			2299.2.a.a	$1$	$1$	\(2\)	\(-3\)	\(-3\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-3q^{3}+2q^{4}-3q^{5}-6q^{6}+\cdots\)
2299.2.a.e	$2299$	$2$	2299.a	1.a	$1$	$2$	$2$	$18.358$	\(\Q(\sqrt{3}) \)	None	✓	✓			2299.2.a.e	$1$	$1$	\(-2\)	\(0\)	\(6\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+\beta q^{3}+(2-2\beta )q^{4}+\cdots\)
2299.2.a.f	$2299$	$2$	2299.a	1.a	$1$	$2$	$2$	$18.358$	\(\Q(\sqrt{2}) \)	None	✓				209.2.a.b	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{3}-q^{5}+(2-\beta )q^{6}+\cdots\)
2299.2.a.g	$2299$	$2$	2299.a	1.a	$1$	$2$	$2$	$18.358$	\(\Q(\sqrt{5}) \)	None	✓				209.2.f.a	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{2}+2\beta q^{3}+3q^{4}-\beta q^{5}+\cdots\)
2299.2.a.h	$2299$	$2$	2299.a	1.a	$1$	$2$	$2$	$18.358$	\(\Q(\sqrt{5}) \)	None	✓				209.2.f.a	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{2}+(2-2\beta )q^{3}+3q^{4}+(-1+\cdots)q^{5}+\cdots\)
2299.2.a.i	$2299$	$2$	2299.a	1.a	$1$	$2$	$2$	$18.358$	\(\Q(\sqrt{5}) \)	None	✓	✓			2299.2.a.i	$1$	$1$	\(0\)	\(3\)	\(-1\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{2}+(1+\beta )q^{3}+3q^{4}+(-1+\cdots)q^{5}+\cdots\)
2299.2.a.j	$2299$	$2$	2299.a	1.a	$1$	$2$	$2$	$18.358$	\(\Q(\sqrt{5}) \)	None	✓	✓			2299.2.a.i	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-2\beta )q^{2}+(2-\beta )q^{3}+3q^{4}-\beta q^{5}+\cdots\)
2299.2.a.k	$2299$	$2$	2299.a	1.a	$1$	$2$	$2$	$18.358$	\(\Q(\sqrt{3}) \)	None	✓	✓			2299.2.a.e	$1$	$0$	\(2\)	\(0\)	\(6\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-\beta q^{3}+(2+2\beta )q^{4}+3q^{5}+\cdots\)
2299.2.a.l	$2299$	$2$	2299.a	1.a	$1$	$3$	$3$	$18.358$	3.3.1229.1	None	✓	✓			2299.2.a.l	$1$	$1$	\(-3\)	\(-1\)	\(1\)	\(3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}-q^{4}+\beta _{1}q^{5}+\beta _{1}q^{6}+\cdots\)
2299.2.a.m	$2299$	$2$	2299.a	1.a	$1$	$3$	$3$	$18.358$	3.3.1229.1	None	✓	✓			2299.2.a.l	$1$	$0$	\(3\)	\(-1\)	\(1\)	\(-3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{1}q^{3}-q^{4}+\beta _{1}q^{5}-\beta _{1}q^{6}+\cdots\)
2299.2.a.n	$2299$	$2$	2299.a	1.a	$1$	$5$	$5$	$18.358$	5.5.246832.1	None	✓				209.2.a.c	$1$	$1$	\(-2\)	\(1\)	\(-5\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4})q^{4}+\cdots\)
2299.2.a.o	$2299$	$2$	2299.a	1.a	$1$	$7$	$7$	$18.358$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			2299.2.a.o	$1$	$1$	\(-4\)	\(-1\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-\beta _{5}q^{3}+(1-\beta _{5}+\beta _{6})q^{4}+\cdots\)
2299.2.a.p	$2299$	$2$	2299.a	1.a	$1$	$7$	$7$	$18.358$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			2299.2.a.p	$1$	$1$	\(-2\)	\(-1\)	\(-1\)	\(-2\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{3}q^{3}+(1+\beta _{1}+\beta _{4}+\beta _{6})q^{4}+\cdots\)
2299.2.a.q	$2299$	$2$	2299.a	1.a	$1$	$7$	$7$	$18.358$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓				209.2.a.d	$1$	$0$	\(1\)	\(2\)	\(2\)	\(-10\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+\cdots\)
2299.2.a.r	$2299$	$2$	2299.a	1.a	$1$	$7$	$7$	$18.358$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			2299.2.a.p	$1$	$0$	\(2\)	\(-1\)	\(-1\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(1+\beta _{1}+\beta _{4}+\beta _{6})q^{4}+\cdots\)
2299.2.a.s	$2299$	$2$	2299.a	1.a	$1$	$7$	$7$	$18.358$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			2299.2.a.o	$1$	$0$	\(4\)	\(-1\)	\(-1\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-\beta _{5}q^{3}+(1-\beta _{5}+\beta _{6})q^{4}+\cdots\)
2299.2.a.t	$2299$	$2$	2299.a	1.a	$1$	$14$	$14$	$18.358$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓				209.2.f.b	$1$	$1$	\(-1\)	\(-2\)	\(-14\)	\(4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{2}q^{3}+(\beta _{4}-\beta _{5})q^{4}+(-1+\cdots)q^{5}+\cdots\)
2299.2.a.u	$2299$	$2$	2299.a	1.a	$1$	$14$	$14$	$18.358$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓		✓		209.2.f.b	$1$	$1$	\(1\)	\(-2\)	\(-14\)	\(-4\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{2}q^{3}+(\beta _{4}-\beta _{5})q^{4}+(-1+\cdots)q^{5}+\cdots\)
2299.2.a.v	$2299$	$2$	2299.a	1.a	$1$	$16$	$16$	$18.358$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓	✓		2299.2.a.v	$1$	$1$	\(-6\)	\(-2\)	\(0\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
2299.2.a.w	$2299$	$2$	2299.a	1.a	$1$	$16$	$16$	$18.358$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None	✓	✓			2299.2.a.v	$1$	$0$	\(6\)	\(-2\)	\(0\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\cdots\)
2299.2.a.x	$2299$	$2$	2299.a	1.a	$1$	$20$	$20$	$18.358$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		209.2.f.c	$1$	$0$	\(-1\)	\(4\)	\(18\)	\(2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{3}+\cdots)q^{5}+\cdots\)
2299.2.a.y	$2299$	$2$	2299.a	1.a	$1$	$20$	$20$	$18.358$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓		✓		209.2.f.c	$1$	$0$	\(1\)	\(4\)	\(18\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(1+\beta _{2})q^{4}+(1-\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2299))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2299)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(121))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 2}\)