Space of modular forms of level 7399, weight 2, and trivial character

• Label: 7399.2.a
• Level: $7399$
• Weight: $2$
• Character orbit: 7399.a
• Rep. character: $\chi_{7399}(1,\cdot)$
• Character field: $\Q$
• Dimension: $513$
• Newform subspaces: $17$
• Sturm bound: $1418$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 7399 = 7^{2} \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7399.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(1418\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(7399))\).

	Total	New	Old
Modular forms	716	513	203
Cusp forms	701	513	188
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.

\(7\)	\(151\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(112\)
\(+\)	\(-\)	\(-\)	\(140\)
\(-\)	\(+\)	\(-\)	\(141\)
\(-\)	\(-\)	\(+\)	\(120\)
Plus space		\(+\)	\(232\)
Minus space		\(-\)	\(281\)

Trace form

\( 513 q + 2 q^{2} + 4 q^{3} + 514 q^{4} + 2 q^{5} - 2 q^{6} + 519 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(7399))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		7	151
7399.2.a.a	$7399$	$2$	7399.a	1.a	$1$	$2$	$2$	$59.081$	\(\Q(\sqrt{5}) \)	None	✓				1057.2.a.a	$1$	$1$	\(-4\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+\beta q^{3}+2q^{4}+(-1+2\beta )q^{5}+\cdots\)
7399.2.a.b	$7399$	$2$	7399.a	1.a	$1$	$3$	$3$	$59.081$	\(\Q(\zeta_{14})^+\)	None	✓				151.2.a.a	$1$	$0$	\(-2\)	\(1\)	\(7\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{2}-\beta _{2}q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
7399.2.a.c	$7399$	$2$	7399.a	1.a	$1$	$3$	$3$	$59.081$	3.3.257.1	None	✓				151.2.a.b	$1$	$1$	\(0\)	\(-6\)	\(-5\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{2}-2q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
7399.2.a.d	$7399$	$2$	7399.a	1.a	$1$	$3$	$3$	$59.081$	3.3.257.1	None	✓				1057.2.a.b	$1$	$1$	\(0\)	\(3\)	\(-5\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+(-2+\cdots)q^{5}+\cdots\)
7399.2.a.e	$7399$	$2$	7399.a	1.a	$1$	$6$	$6$	$59.081$	6.6.4838537.1	None	✓				151.2.a.c	$1$	$1$	\(1\)	\(5\)	\(-6\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(1+\beta _{2}-\beta _{3}+\beta _{4})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots\)
7399.2.a.f	$7399$	$2$	7399.a	1.a	$1$	$12$	$12$	$59.081$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓				1057.2.a.c	$1$	$1$	\(-4\)	\(6\)	\(7\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+(1+\beta _{9})q^{3}+(\beta _{1}+\beta _{2})q^{4}+\cdots\)
7399.2.a.g	$7399$	$2$	7399.a	1.a	$1$	$14$	$14$	$59.081$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None	✓				1057.2.a.d	$1$	$1$	\(3\)	\(-10\)	\(-2\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+(-1-\beta _{7})q^{3}+(1+\beta _{2})q^{4}+\cdots\)
7399.2.a.h	$7399$	$2$	7399.a	1.a	$1$	$20$	$20$	$59.081$	\(\mathbb{Q}[x]/(x^{20} - \cdots)\)	None	✓				1057.2.a.e	$1$	$0$	\(-5\)	\(9\)	\(11\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+\beta _{15}q^{3}+(1+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
7399.2.a.i	$7399$	$2$	7399.a	1.a	$1$	$24$	$24$	$59.081$		None	✓				1057.2.a.f	$1$	$0$	\(9\)	\(-5\)	\(-5\)	\(0\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
7399.2.a.j	$7399$	$2$	7399.a	1.a	$1$	$30$	$30$	$59.081$		None	✓	✓			7399.2.a.j	$2$	$1$	\(-6\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
7399.2.a.k	$7399$	$2$	7399.a	1.a	$1$	$44$	$44$	$59.081$		None	✓	✓			7399.2.a.k	$2$	$0$	\(6\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
7399.2.a.l	$7399$	$2$	7399.a	1.a	$1$	$50$	$50$	$59.081$		None	✓		✓		1057.2.f.a	$1$	$1$	\(0\)	\(-6\)	\(-25\)	\(0\)		$-$	$-$	$+$		$\mathrm{SU}(2)$
7399.2.a.m	$7399$	$2$	7399.a	1.a	$1$	$50$	$50$	$59.081$		None	✓				1057.2.f.b	$1$	$1$	\(0\)	\(-6\)	\(-23\)	\(0\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
7399.2.a.n	$7399$	$2$	7399.a	1.a	$1$	$50$	$50$	$59.081$		None	✓		✓		1057.2.f.b	$1$	$0$	\(0\)	\(6\)	\(23\)	\(0\)		$-$	$+$	$-$		$\mathrm{SU}(2)$
7399.2.a.o	$7399$	$2$	7399.a	1.a	$1$	$50$	$50$	$59.081$		None	✓				1057.2.f.a	$1$	$0$	\(0\)	\(6\)	\(25\)	\(0\)		$+$	$-$	$-$		$\mathrm{SU}(2)$
7399.2.a.p	$7399$	$2$	7399.a	1.a	$1$	$62$	$62$	$59.081$		None	✓	✓	✓		7399.2.a.p	$2$	$1$	\(-10\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$		$\mathrm{SU}(2)$
7399.2.a.q	$7399$	$2$	7399.a	1.a	$1$	$90$	$90$	$59.081$		None	✓	✓	✓		7399.2.a.q	$2$	$0$	\(14\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$		$\mathrm{SU}(2)$

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(7399))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(7399)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(151))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1057))\)\(^{\oplus 2}\)