Space of modular forms of level 3440, weight 2, and trivial character

• Label: 3440.2.a
• Level: $3440$
• Weight: $2$
• Character orbit: 3440.a
• Rep. character: $\chi_{3440}(1,\cdot)$
• Character field: $\Q$
• Dimension: $84$
• Newform subspaces: $26$
• Sturm bound: $1056$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 3440 = 2^{4} \cdot 5 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3440.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 26 \)
Sturm bound:			\(1056\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3440))\).

	Total	New	Old
Modular forms	540	84	456
Cusp forms	517	84	433
Eisenstein series	23	0	23

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

\(2\)	\(5\)	\(43\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	$+$	\(12\)
\(+\)	\(+\)	\(-\)	$-$	\(9\)
\(+\)	\(-\)	\(+\)	$-$	\(12\)
\(+\)	\(-\)	\(-\)	$+$	\(9\)
\(-\)	\(+\)	\(+\)	$-$	\(14\)
\(-\)	\(+\)	\(-\)	$+$	\(7\)
\(-\)	\(-\)	\(+\)	$+$	\(7\)
\(-\)	\(-\)	\(-\)	$-$	\(14\)
Plus space			\(+\)	\(35\)
Minus space			\(-\)	\(49\)

Trace form

\( 84 q + 84 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3440))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	5	43
3440.2.a.a	$3440$	$2$	3440.a	1.a	$1$	$1$	$1$	$27.469$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-q^{7}-3q^{9}+4q^{11}-q^{13}+\cdots\)
3440.2.a.b	$3440$	$2$	3440.a	1.a	$1$	$1$	$1$	$27.469$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}+2q^{7}-3q^{9}+q^{11}-q^{13}+\cdots\)
3440.2.a.c	$3440$	$2$	3440.a	1.a	$1$	$1$	$1$	$27.469$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+3q^{7}-3q^{9}-3q^{13}-4q^{17}+\cdots\)
3440.2.a.d	$3440$	$2$	3440.a	1.a	$1$	$1$	$1$	$27.469$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-q^{5}+q^{7}+q^{9}+6q^{11}+5q^{13}+\cdots\)
3440.2.a.e	$3440$	$2$	3440.a	1.a	$1$	$1$	$1$	$27.469$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(2\)	\(1\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-3q^{7}+q^{9}+2q^{11}+\cdots\)
3440.2.a.f	$3440$	$2$	3440.a	1.a	$1$	$1$	$1$	$27.469$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(2\)	\(1\)	\(5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}+5q^{7}+q^{9}+2q^{11}+\cdots\)
3440.2.a.g	$3440$	$2$	3440.a	1.a	$1$	$2$	$2$	$27.469$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}+q^{5}+(-1-2\beta )q^{7}+\cdots\)
3440.2.a.h	$3440$	$2$	3440.a	1.a	$1$	$2$	$2$	$27.469$	\(\Q(\sqrt{6}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-q^{5}-q^{7}+3q^{9}+(-2-\beta )q^{11}+\cdots\)
3440.2.a.i	$3440$	$2$	3440.a	1.a	$1$	$2$	$2$	$27.469$	\(\Q(\sqrt{17}) \)	None	✓	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}+(-1-\beta )q^{7}-3q^{9}+(4-\beta )q^{11}+\cdots\)
3440.2.a.j	$3440$	$2$	3440.a	1.a	$1$	$2$	$2$	$27.469$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+q^{5}-q^{7}-q^{9}+(-2-\beta )q^{11}+\cdots\)
3440.2.a.k	$3440$	$2$	3440.a	1.a	$1$	$3$	$3$	$27.469$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$1$	\(0\)	\(-3\)	\(-3\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}-q^{5}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
3440.2.a.l	$3440$	$2$	3440.a	1.a	$1$	$3$	$3$	$27.469$	3.3.321.1	None	✓	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+(1+\beta _{1}+\beta _{2})q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
3440.2.a.m	$3440$	$2$	3440.a	1.a	$1$	$3$	$3$	$27.469$	\(\Q(\zeta_{14})^+\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(5\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+(2+\beta _{2})q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
3440.2.a.n	$3440$	$2$	3440.a	1.a	$1$	$3$	$3$	$27.469$	3.3.316.1	None	✓	$1$	$0$	\(0\)	\(2\)	\(-3\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1}+\beta _{2})q^{3}-q^{5}+(2+\beta _{2})q^{7}+\cdots\)
3440.2.a.o	$3440$	$2$	3440.a	1.a	$1$	$3$	$3$	$27.469$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(3\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+q^{5}+(1-\beta _{2})q^{7}+(2\beta _{1}+\cdots)q^{9}+\cdots\)
3440.2.a.p	$3440$	$2$	3440.a	1.a	$1$	$3$	$3$	$27.469$	\(\Q(\zeta_{18})^+\)	None	✓	$1$	$0$	\(0\)	\(3\)	\(3\)	\(9\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+2\beta _{1}-\beta _{2})q^{3}+q^{5}+(3-\beta _{1}+\cdots)q^{7}+\cdots\)
3440.2.a.q	$3440$	$2$	3440.a	1.a	$1$	$4$	$4$	$27.469$	4.4.65057.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(4\)	\(-7\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+(-2+\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots\)
3440.2.a.r	$3440$	$2$	3440.a	1.a	$1$	$4$	$4$	$27.469$	4.4.6809.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{3})q^{3}-q^{5}+(\beta _{1}-\beta _{2})q^{7}+\cdots\)
3440.2.a.s	$3440$	$2$	3440.a	1.a	$1$	$4$	$4$	$27.469$	4.4.10273.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(7\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}-q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
3440.2.a.t	$3440$	$2$	3440.a	1.a	$1$	$5$	$5$	$27.469$	5.5.1032140.1	None	✓	$1$	$1$	\(0\)	\(-3\)	\(5\)	\(-9\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+q^{5}+(-1-\beta _{1}-\beta _{4})q^{7}+\cdots\)
3440.2.a.u	$3440$	$2$	3440.a	1.a	$1$	$5$	$5$	$27.469$	5.5.1222393.1	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-5\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+(-1+\beta _{1}-\beta _{3}+\beta _{4})q^{7}+\cdots\)
3440.2.a.v	$3440$	$2$	3440.a	1.a	$1$	$5$	$5$	$27.469$	5.5.2625272.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(-5\)	\(5\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}+(1+\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
3440.2.a.w	$3440$	$2$	3440.a	1.a	$1$	$5$	$5$	$27.469$	5.5.1933097.1	None	✓	$1$	$0$	\(0\)	\(1\)	\(5\)	\(-5\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{3})q^{3}+q^{5}+(-1+\beta _{3}-\beta _{4})q^{7}+\cdots\)
3440.2.a.x	$3440$	$2$	3440.a	1.a	$1$	$6$	$6$	$27.469$	6.6.32503921.1	None	✓	$1$	$0$	\(0\)	\(-4\)	\(-6\)	\(-8\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{3}-q^{5}+(-1-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
3440.2.a.y	$3440$	$2$	3440.a	1.a	$1$	$7$	$7$	$27.469$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-1\)	\(-7\)	\(-1\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}-\beta _{5}q^{7}+(2+\beta _{3}-\beta _{4}+\cdots)q^{9}+\cdots\)
3440.2.a.z	$3440$	$2$	3440.a	1.a	$1$	$7$	$7$	$27.469$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(-1\)	\(7\)	\(-3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+\beta _{6}q^{7}+(2+\beta _{2})q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3440))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3440)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(40))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(80))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(86))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(172))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(215))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(344))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(430))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(688))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(860))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1720))\)\(^{\oplus 2}\)