Space of modular forms of level 1449, weight 2, and trivial character

• Label: 1449.2.a
• Level: $1449$
• Weight: $2$
• Character orbit: 1449.a
• Rep. character: $\chi_{1449}(1,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $20$
• Sturm bound: $384$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 20 \)
Sturm bound:			\(384\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1449))\).

	Total	New	Old
Modular forms	200	54	146
Cusp forms	185	54	131
Eisenstein series	15	0	15

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

\(3\)	\(7\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	\(5\)
\(+\)	\(-\)	\(+\)	\(-\)	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(9\)
\(-\)	\(+\)	\(-\)	\(+\)	\(8\)
\(-\)	\(-\)	\(+\)	\(+\)	\(5\)
\(-\)	\(-\)	\(-\)	\(-\)	\(12\)
Plus space			\(+\)	\(23\)
Minus space			\(-\)	\(31\)

Trace form

54 * q + 52 * q^4 - 8 * q^5 + 6 * q^8 + 12 * q^10 + 8 * q^11 - 8 * q^13 + 64 * q^16 - 4 * q^17 - 12 * q^19 + 8 * q^20 + 4 * q^22 + 6 * q^23 + 34 * q^25 + 26 * q^26 + 8 * q^31 + 12 * q^32 - 44 * q^34 + 8 * q^35 + 8 * q^38 + 8 * q^40 + 24 * q^43 + 12 * q^44 - 4 * q^46 + 8 * q^47 + 54 * q^49 - 16 * q^50 - 42 * q^52 - 24 * q^53 + 28 * q^55 - 30 * q^58 + 20 * q^59 + 4 * q^61 + 30 * q^62 + 78 * q^64 - 20 * q^65 - 12 * q^67 - 36 * q^68 + 20 * q^70 + 16 * q^71 - 72 * q^73 - 20 * q^74 + 4 * q^76 - 4 * q^79 + 76 * q^80 + 6 * q^82 - 16 * q^83 - 64 * q^85 + 40 * q^86 + 8 * q^88 - 36 * q^89 - 12 * q^91 + 12 * q^92 - 70 * q^94 - 48 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1449))\) 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	7	23
1449.2.a.a	$1449$	$2$	1449.a	1.a	$1$	$1$	$1$	$11.570$	\(\Q\)	None	✓				483.2.a.b	$1$	$1$	\(-2\)	\(0\)	\(-4\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-4q^{5}-q^{7}+8q^{10}+\cdots\)
1449.2.a.b	$1449$	$2$	1449.a	1.a	$1$	$1$	$1$	$11.570$	\(\Q\)	None	✓	✓			1449.2.a.b	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}-2q^{5}+q^{7}+4q^{10}+\cdots\)
1449.2.a.c	$1449$	$2$	1449.a	1.a	$1$	$1$	$1$	$11.570$	\(\Q\)	None	✓				483.2.a.a	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+2q^{4}+q^{7}-q^{11}+2q^{13}+\cdots\)
1449.2.a.d	$1449$	$2$	1449.a	1.a	$1$	$1$	$1$	$11.570$	\(\Q\)	None	✓				161.2.a.a	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{4}-2q^{5}+q^{7}-3q^{8}-2q^{10}+\cdots\)
1449.2.a.e	$1449$	$2$	1449.a	1.a	$1$	$1$	$1$	$11.570$	\(\Q\)	None	✓	✓			1449.2.a.b	$1$	$0$	\(2\)	\(0\)	\(2\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{4}+2q^{5}+q^{7}+4q^{10}+\cdots\)
1449.2.a.f	$1449$	$2$	1449.a	1.a	$1$	$2$	$2$	$11.570$	\(\Q(\sqrt{5}) \)	None	✓				483.2.a.g	$1$	$1$	\(-1\)	\(0\)	\(3\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+(-1+\beta )q^{4}+(1+\beta )q^{5}+q^{7}+\cdots\)
1449.2.a.g	$1449$	$2$	1449.a	1.a	$1$	$2$	$2$	$11.570$	\(\Q(\sqrt{5}) \)	None	✓				483.2.a.e	$1$	$1$	\(1\)	\(0\)	\(-5\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(-3+\beta )q^{5}+\cdots\)
1449.2.a.h	$1449$	$2$	1449.a	1.a	$1$	$2$	$2$	$11.570$	\(\Q(\sqrt{5}) \)	None	✓				483.2.a.d	$1$	$1$	\(1\)	\(0\)	\(-1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(-1+\beta )q^{5}+\cdots\)
1449.2.a.i	$1449$	$2$	1449.a	1.a	$1$	$2$	$2$	$11.570$	\(\Q(\sqrt{5}) \)	None	✓				161.2.a.b	$1$	$1$	\(1\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(-1+\beta )q^{4}+(2-2\beta )q^{5}+\cdots\)
1449.2.a.j	$1449$	$2$	1449.a	1.a	$1$	$2$	$2$	$11.570$	\(\Q(\sqrt{13}) \)	None	✓				483.2.a.f	$1$	$0$	\(1\)	\(0\)	\(5\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1+\beta )q^{4}+(3-\beta )q^{5}-q^{7}+\cdots\)
1449.2.a.k	$1449$	$2$	1449.a	1.a	$1$	$2$	$2$	$11.570$	\(\Q(\sqrt{5}) \)	None	✓				483.2.a.c	$1$	$0$	\(3\)	\(0\)	\(5\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+3\beta q^{4}+(2+\beta )q^{5}+q^{7}+\cdots\)
1449.2.a.l	$1449$	$2$	1449.a	1.a	$1$	$3$	$3$	$11.570$	3.3.837.1	None	✓		✓		483.2.a.h	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(-1+\beta _{1})q^{5}+\cdots\)
1449.2.a.m	$1449$	$2$	1449.a	1.a	$1$	$3$	$3$	$11.570$	3.3.148.1	None	✓				161.2.a.c	$1$	$0$	\(1\)	\(0\)	\(-2\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{2})q^{2}+(1+2\beta _{1})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1449.2.a.n	$1449$	$2$	1449.a	1.a	$1$	$4$	$4$	$11.570$	4.4.2624.1	None	✓	✓	✓		1449.2.a.n	$1$	$1$	\(-2\)	\(0\)	\(-2\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-\beta _{3})q^{2}+(2-2\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
1449.2.a.o	$1449$	$2$	1449.a	1.a	$1$	$4$	$4$	$11.570$	4.4.15317.1	None	✓		✓		483.2.a.j	$1$	$0$	\(-2\)	\(0\)	\(-5\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-2-\beta _{2}+\cdots)q^{5}+\cdots\)
1449.2.a.p	$1449$	$2$	1449.a	1.a	$1$	$4$	$4$	$11.570$	4.4.24197.1	None	✓				483.2.a.i	$1$	$0$	\(0\)	\(0\)	\(-5\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+\beta _{3})q^{5}+\cdots\)
1449.2.a.q	$1449$	$2$	1449.a	1.a	$1$	$4$	$4$	$11.570$	4.4.2624.1	None	✓	✓	✓		1449.2.a.n	$1$	$0$	\(2\)	\(0\)	\(2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}+\beta _{3})q^{2}+(2-2\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
1449.2.a.r	$1449$	$2$	1449.a	1.a	$1$	$5$	$5$	$11.570$	5.5.2147108.1	None	✓		✓		161.2.a.d	$1$	$0$	\(-2\)	\(0\)	\(4\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(2+\beta _{2})q^{4}+(1+\beta _{4})q^{5}+\cdots\)
1449.2.a.s	$1449$	$2$	1449.a	1.a	$1$	$5$	$5$	$11.570$	5.5.1337792.1	None	✓	✓	✓	✓	1449.2.a.s	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
1449.2.a.t	$1449$	$2$	1449.a	1.a	$1$	$5$	$5$	$11.570$	5.5.1337792.1	None	✓	✓	✓	✓	1449.2.a.s	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(1+\beta _{2})q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1449))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1449)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(63))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(161))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(483))\)\(^{\oplus 2}\)