Space of modular forms of level 1421, weight 2, and trivial character

• Label: 1421.2.a
• Level: $1421$
• Weight: $2$
• Character orbit: 1421.a
• Rep. character: $\chi_{1421}(1,\cdot)$
• Character field: $\Q$
• Dimension: $95$
• Newform subspaces: $23$
• Sturm bound: $280$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1421 = 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1421.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(280\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1421))\).

	Total	New	Old
Modular forms	148	95	53
Cusp forms	133	95	38
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.

\(7\)	\(29\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(17\)
\(+\)	\(-\)	$-$	\(29\)
\(-\)	\(+\)	$-$	\(29\)
\(-\)	\(-\)	$+$	\(20\)
Plus space		\(+\)	\(37\)
Minus space		\(-\)	\(58\)

Trace form

\( 95 q + q^{2} + 2 q^{3} + 97 q^{4} + 6 q^{6} - 3 q^{8} + 101 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1421))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	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	29
1421.2.a.a	$1421$	$2$	1421.a	1.a	$1$	$1$	$1$	$11.347$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(-1\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+2q^{4}-2q^{5}+2q^{6}+\cdots\)
1421.2.a.b	$1421$	$2$	1421.a	1.a	$1$	$1$	$1$	$11.347$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}+2q^{5}-2q^{6}+\cdots\)
1421.2.a.c	$1421$	$2$	1421.a	1.a	$1$	$1$	$1$	$11.347$	\(\Q\)	None	✓	$1$	$0$	\(-2\)	\(1\)	\(4\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}+4q^{5}-2q^{6}+\cdots\)
1421.2.a.d	$1421$	$2$	1421.a	1.a	$1$	$1$	$1$	$11.347$	\(\Q\)	None	✓	$1$	$1$	\(-1\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}+3q^{8}+\cdots\)
1421.2.a.e	$1421$	$2$	1421.a	1.a	$1$	$1$	$1$	$11.347$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-3\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{4}-3q^{5}+q^{9}+6q^{11}+\cdots\)
1421.2.a.f	$1421$	$2$	1421.a	1.a	$1$	$1$	$1$	$11.347$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{4}+3q^{5}+q^{9}+6q^{11}+\cdots\)
1421.2.a.g	$1421$	$2$	1421.a	1.a	$1$	$1$	$1$	$11.347$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(-2\)	\(-2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}-q^{4}-2q^{5}-2q^{6}-3q^{8}+\cdots\)
1421.2.a.h	$1421$	$2$	1421.a	1.a	$1$	$1$	$1$	$11.347$	\(\Q\)	None	✓	$1$	$1$	\(2\)	\(-2\)	\(-1\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-2q^{3}+2q^{4}-q^{5}-4q^{6}+\cdots\)
1421.2.a.i	$1421$	$2$	1421.a	1.a	$1$	$1$	$1$	$11.347$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(2\)	\(1\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+2q^{3}+2q^{4}+q^{5}+4q^{6}+\cdots\)
1421.2.a.j	$1421$	$2$	1421.a	1.a	$1$	$2$	$2$	$11.347$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(-2\)	\(2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(-1+\beta )q^{3}+(1-2\beta )q^{4}+\cdots\)
1421.2.a.k	$1421$	$2$	1421.a	1.a	$1$	$2$	$2$	$11.347$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$1$	\(-2\)	\(1\)	\(-3\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1-\beta )q^{3}-q^{4}+(-1-\beta )q^{5}+\cdots\)
1421.2.a.l	$1421$	$2$	1421.a	1.a	$1$	$2$	$2$	$11.347$	\(\Q(\sqrt{7}) \)	None	✓	$2$	$1$	\(2\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta q^{3}-q^{4}+\beta q^{5}-\beta q^{6}-3q^{8}+\cdots\)
1421.2.a.m	$1421$	$2$	1421.a	1.a	$1$	$2$	$2$	$11.347$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(4\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+(-1+\beta )q^{3}+2q^{4}-2\beta q^{5}+\cdots\)
1421.2.a.n	$1421$	$2$	1421.a	1.a	$1$	$3$	$3$	$11.347$	3.3.148.1	None	✓	$1$	$0$	\(-1\)	\(3\)	\(5\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{2})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
1421.2.a.o	$1421$	$2$	1421.a	1.a	$1$	$4$	$4$	$11.347$	\(\Q(\sqrt{2}, \sqrt{7})\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-2q^{4}+\beta _{2}q^{5}+(1+\beta _{3})q^{9}+\cdots\)
1421.2.a.p	$1421$	$2$	1421.a	1.a	$1$	$4$	$4$	$11.347$	\(\Q(\zeta_{24})^+\)	None	✓	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}+\beta _{3}q^{3}+q^{4}-\beta _{3}q^{5}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
1421.2.a.q	$1421$	$2$	1421.a	1.a	$1$	$5$	$5$	$11.347$	5.5.2626356.1	None	✓	$1$	$0$	\(2\)	\(2\)	\(-5\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(2+\beta _{2})q^{4}+(-1+\cdots)q^{5}+\cdots\)
1421.2.a.r	$1421$	$2$	1421.a	1.a	$1$	$8$	$8$	$11.347$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(-2\)	\(-5\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1421.2.a.s	$1421$	$2$	1421.a	1.a	$1$	$8$	$8$	$11.347$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(-2\)	\(5\)	\(2\)	\(0\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1-\beta _{4})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
1421.2.a.t	$1421$	$2$	1421.a	1.a	$1$	$8$	$8$	$11.347$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-2\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots\)
1421.2.a.u	$1421$	$2$	1421.a	1.a	$1$	$8$	$8$	$11.347$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(2\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-\beta _{5}q^{3}+(1+\beta _{2})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
1421.2.a.v	$1421$	$2$	1421.a	1.a	$1$	$10$	$10$	$11.347$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None	✓	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{2}-\beta _{1}q^{3}+(2-\beta _{7})q^{4}+\beta _{6}q^{5}+\cdots\)
1421.2.a.w	$1421$	$2$	1421.a	1.a	$1$	$20$	$20$	$11.347$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓	$2$	$0$	\(4\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{6}$	$\mathrm{SU}(2)$	\(q+\beta _{6}q^{2}-\beta _{1}q^{3}+(2+\beta _{9})q^{4}+\beta _{12}q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1421))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1421)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(29))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(203))\)\(^{\oplus 2}\)