Space of modular forms of level 1414, weight 2, and trivial character

• Label: 1414.2.a
• Level: $1414$
• Weight: $2$
• Character orbit: 1414.a
• Rep. character: $\chi_{1414}(1,\cdot)$
• Character field: $\Q$
• Dimension: $49$
• Newform subspaces: $9$
• Sturm bound: $408$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 1414 = 2 \cdot 7 \cdot 101 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1414.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(408\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1414))\).

	Total	New	Old
Modular forms	208	49	159
Cusp forms	201	49	152
Eisenstein series	7	0	7

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

\(2\)	\(7\)	\(101\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	\(7\)
\(+\)	\(-\)	\(+\)	\(-\)	\(7\)
\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(9\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(4\)
\(-\)	\(-\)	\(-\)	\(-\)	\(8\)
Plus space			\(+\)	\(18\)
Minus space			\(-\)	\(31\)

Trace form

\( 49 q + q^{2} + 8 q^{3} + 49 q^{4} + 2 q^{5} - q^{7} + q^{8} + 57 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1414))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	7	101
1414.2.a.a	$1414$	$2$	1414.a	1.a	$1$	$1$	$1$	$11.291$	\(\Q\)	None	✓	✓			1414.2.a.a	$1$	$1$	\(-1\)	\(1\)	\(2\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}+2q^{5}-q^{6}-q^{7}+\cdots\)
1414.2.a.b	$1414$	$2$	1414.a	1.a	$1$	$4$	$4$	$11.291$	4.4.4525.1	None	✓	✓	✓		1414.2.a.b	$1$	$1$	\(-4\)	\(1\)	\(-6\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta _{1}q^{3}+q^{4}+(-2-\beta _{3})q^{5}+\cdots\)
1414.2.a.c	$1414$	$2$	1414.a	1.a	$1$	$4$	$4$	$11.291$	4.4.1957.1	None	✓	✓	✓	✓	1414.2.a.c	$1$	$1$	\(4\)	\(-3\)	\(-8\)	\(4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots\)
1414.2.a.d	$1414$	$2$	1414.a	1.a	$1$	$4$	$4$	$11.291$	\(\Q(\zeta_{15})^+\)	None	✓	✓	✓	✓	1414.2.a.d	$1$	$1$	\(4\)	\(-3\)	\(-2\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1-\beta _{2}-\beta _{3})q^{3}+q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
1414.2.a.e	$1414$	$2$	1414.a	1.a	$1$	$5$	$5$	$11.291$	5.5.301909.1	None	✓	✓	✓	✓	1414.2.a.e	$1$	$1$	\(-5\)	\(0\)	\(-6\)	\(5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{3}q^{3}+q^{4}+(-1+\beta _{4})q^{5}+\cdots\)
1414.2.a.f	$1414$	$2$	1414.a	1.a	$1$	$7$	$7$	$11.291$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	1414.2.a.f	$1$	$0$	\(-7\)	\(-1\)	\(3\)	\(-7\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{4}q^{3}+q^{4}+\beta _{5}q^{5}+\beta _{4}q^{6}+\cdots\)
1414.2.a.g	$1414$	$2$	1414.a	1.a	$1$	$7$	$7$	$11.291$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓	✓	1414.2.a.g	$1$	$0$	\(-7\)	\(3\)	\(11\)	\(7\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta _{3}q^{3}+q^{4}+(2-\beta _{6})q^{5}-\beta _{3}q^{6}+\cdots\)
1414.2.a.h	$1414$	$2$	1414.a	1.a	$1$	$8$	$8$	$11.291$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	1414.2.a.h	$1$	$0$	\(8\)	\(6\)	\(7\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+(1-\beta _{3})q^{5}+\cdots\)
1414.2.a.i	$1414$	$2$	1414.a	1.a	$1$	$9$	$9$	$11.291$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓	✓	1414.2.a.i	$1$	$0$	\(9\)	\(4\)	\(1\)	\(-9\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{3}q^{5}+\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1414))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1414)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(101))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(202))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(707))\)\(^{\oplus 2}\)