Space of modular forms of level 345, weight 2, and trivial character

• Label: 345.2.a
• Level: $345$
• Weight: $2$
• Character orbit: 345.a
• Rep. character: $\chi_{345}(1,\cdot)$
• Character field: $\Q$
• Dimension: $15$
• Newform subspaces: $10$
• Sturm bound: $96$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 345 = 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	345.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 10 \)
Sturm bound:			\(96\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(2\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(345))\).

	Total	New	Old
Modular forms	52	15	37
Cusp forms	45	15	30
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.

\(3\)	\(5\)	\(23\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(+\)	\(-\)	$-$	\(1\)
\(+\)	\(-\)	\(+\)	$-$	\(1\)
\(+\)	\(-\)	\(-\)	$+$	\(2\)
\(-\)	\(+\)	\(+\)	$-$	\(4\)
\(-\)	\(+\)	\(-\)	$+$	\(1\)
\(-\)	\(-\)	\(+\)	$+$	\(1\)
\(-\)	\(-\)	\(-\)	$-$	\(3\)
Plus space			\(+\)	\(6\)
Minus space			\(-\)	\(9\)

Trace form

\( 15 q - 3 q^{2} + 3 q^{3} + 13 q^{4} - q^{5} - 3 q^{6} - 15 q^{8} + 15 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(345))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	23
345.2.a.a	$345$	$2$	345.a	1.a	$1$	$1$	$1$	$2.755$	\(\Q\)	None	✓	$1$	$1$	\(-2\)	\(1\)	\(1\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}+q^{5}-2q^{6}-5q^{7}+\cdots\)
345.2.a.b	$345$	$2$	345.a	1.a	$1$	$1$	$1$	$2.755$	\(\Q\)	None	✓	$1$	$0$	\(-1\)	\(1\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{5}-q^{6}+4q^{7}+\cdots\)
345.2.a.c	$345$	$2$	345.a	1.a	$1$	$1$	$1$	$2.755$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}-q^{5}+q^{7}+q^{9}+4q^{11}+\cdots\)
345.2.a.d	$345$	$2$	345.a	1.a	$1$	$1$	$1$	$2.755$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}-q^{5}-3q^{7}+q^{9}-4q^{11}+\cdots\)
345.2.a.e	$345$	$2$	345.a	1.a	$1$	$1$	$1$	$2.755$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(1\)	\(-1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}-q^{5}+q^{6}+4q^{7}+\cdots\)
345.2.a.f	$345$	$2$	345.a	1.a	$1$	$1$	$1$	$2.755$	\(\Q\)	None	✓	$1$	$0$	\(2\)	\(-1\)	\(1\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}+q^{5}-2q^{6}+3q^{7}+\cdots\)
345.2.a.g	$345$	$2$	345.a	1.a	$1$	$2$	$2$	$2.755$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$1$	\(-2\)	\(-2\)	\(2\)	\(-6\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-q^{3}+(2-2\beta )q^{4}+q^{5}+\cdots\)
345.2.a.h	$345$	$2$	345.a	1.a	$1$	$2$	$2$	$2.755$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}-q^{5}-\beta q^{6}+(-1-2\beta )q^{7}+\cdots\)
345.2.a.i	$345$	$2$	345.a	1.a	$1$	$2$	$2$	$2.755$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+4q^{4}-q^{5}+\beta q^{6}-q^{7}+\cdots\)
345.2.a.j	$345$	$2$	345.a	1.a	$1$	$3$	$3$	$2.755$	3.3.316.1	None	✓	$1$	$0$	\(-1\)	\(3\)	\(3\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(1+\beta _{2})q^{4}+q^{5}-\beta _{1}q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(345))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(345)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(115))\)\(^{\oplus 2}\)