Space of modular forms of level 3380, weight 2, and trivial character

• Label: 3380.2.a
• Level: $3380$
• Weight: $2$
• Character orbit: 3380.a
• Rep. character: $\chi_{3380}(1,\cdot)$
• Character field: $\Q$
• Dimension: $51$
• Newform subspaces: $19$
• Sturm bound: $1092$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3380.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(1092\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3380))\).

	Total	New	Old
Modular forms	588	51	537
Cusp forms	505	51	454
Eisenstein series	83	0	83

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

\(2\)	\(5\)	\(13\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(70\)	\(0\)	\(70\)		\(57\)	\(0\)	\(57\)		\(13\)	\(0\)	\(13\)
\(+\)	\(+\)	\(-\)	\(-\)		\(80\)	\(0\)	\(80\)		\(66\)	\(0\)	\(66\)		\(14\)	\(0\)	\(14\)
\(+\)	\(-\)	\(+\)	\(-\)		\(77\)	\(0\)	\(77\)		\(63\)	\(0\)	\(63\)		\(14\)	\(0\)	\(14\)
\(+\)	\(-\)	\(-\)	\(+\)		\(73\)	\(0\)	\(73\)		\(59\)	\(0\)	\(59\)		\(14\)	\(0\)	\(14\)
\(-\)	\(+\)	\(+\)	\(-\)		\(77\)	\(16\)	\(61\)		\(70\)	\(16\)	\(54\)		\(7\)	\(0\)	\(7\)
\(-\)	\(+\)	\(-\)	\(+\)		\(67\)	\(10\)	\(57\)		\(60\)	\(10\)	\(50\)		\(7\)	\(0\)	\(7\)
\(-\)	\(-\)	\(+\)	\(+\)		\(70\)	\(9\)	\(61\)		\(63\)	\(9\)	\(54\)		\(7\)	\(0\)	\(7\)
\(-\)	\(-\)	\(-\)	\(-\)		\(74\)	\(16\)	\(58\)		\(67\)	\(16\)	\(51\)		\(7\)	\(0\)	\(7\)
Plus space			\(+\)		\(280\)	\(19\)	\(261\)		\(239\)	\(19\)	\(220\)		\(41\)	\(0\)	\(41\)
Minus space			\(-\)		\(308\)	\(32\)	\(276\)		\(266\)	\(32\)	\(234\)		\(42\)	\(0\)	\(42\)

Trace form

51 * q - 2 * q^3 - q^5 - 2 * q^7 + 43 * q^9 - 4 * q^11 - 2 * q^15 + 6 * q^17 - 4 * q^19 + 8 * q^21 + 10 * q^23 + 51 * q^25 - 8 * q^27 - 2 * q^29 + 16 * q^31 + 4 * q^33 + 6 * q^35 - 10 * q^37 - 2 * q^41 + 10 * q^43 - 9 * q^45 + 22 * q^47 + 39 * q^49 + 16 * q^51 + 26 * q^53 + 12 * q^57 + 12 * q^59 - 18 * q^61 + 46 * q^63 - 10 * q^67 + 16 * q^69 + 20 * q^71 - 26 * q^73 - 2 * q^75 + 36 * q^77 + 4 * q^79 + 63 * q^81 - 6 * q^83 - 6 * q^85 + 40 * q^87 - 2 * q^89 + 12 * q^93 - 12 * q^95 - 30 * q^97 - 40 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3380))\) 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	5	13
3380.2.a.a	$3380$	$2$	3380.a	1.a	$1$	$1$	$1$	$26.989$	\(\Q\)	None	✓				260.2.i.d	$1$	$2$	\(0\)	\(-3\)	\(-1\)	\(-3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-q^{5}-3q^{7}+6q^{9}-3q^{11}+\cdots\)
3380.2.a.b	$3380$	$2$	3380.a	1.a	$1$	$1$	$1$	$26.989$	\(\Q\)	None	✓				260.2.i.d	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+q^{5}+3q^{7}+6q^{9}+3q^{11}+\cdots\)
3380.2.a.c	$3380$	$2$	3380.a	1.a	$1$	$1$	$1$	$26.989$	\(\Q\)	None	✓				20.2.a.a	$1$	$1$	\(0\)	\(-2\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{5}-2q^{7}+q^{9}-2q^{15}+\cdots\)
3380.2.a.d	$3380$	$2$	3380.a	1.a	$1$	$1$	$1$	$26.989$	\(\Q\)	None	✓				260.2.i.c	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-q^{5}+5q^{7}-2q^{9}-5q^{11}+\cdots\)
3380.2.a.e	$3380$	$2$	3380.a	1.a	$1$	$1$	$1$	$26.989$	\(\Q\)	None	✓				260.2.i.c	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(-5\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}-5q^{7}-2q^{9}+5q^{11}+\cdots\)
3380.2.a.f	$3380$	$2$	3380.a	1.a	$1$	$1$	$1$	$26.989$	\(\Q\)	None	✓				260.2.i.a	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-q^{7}-2q^{9}+3q^{11}-q^{15}+\cdots\)
3380.2.a.g	$3380$	$2$	3380.a	1.a	$1$	$1$	$1$	$26.989$	\(\Q\)	None	✓				260.2.i.b	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}+q^{7}-2q^{9}-3q^{11}-q^{15}+\cdots\)
3380.2.a.h	$3380$	$2$	3380.a	1.a	$1$	$1$	$1$	$26.989$	\(\Q\)	None	✓				260.2.i.b	$1$	$1$	\(0\)	\(1\)	\(1\)	\(-1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-q^{7}-2q^{9}+3q^{11}+q^{15}+\cdots\)
3380.2.a.i	$3380$	$2$	3380.a	1.a	$1$	$1$	$1$	$26.989$	\(\Q\)	None	✓				260.2.i.a	$1$	$1$	\(0\)	\(1\)	\(1\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}+q^{7}-2q^{9}-3q^{11}+q^{15}+\cdots\)
3380.2.a.j	$3380$	$2$	3380.a	1.a	$1$	$1$	$1$	$26.989$	\(\Q\)	None	✓				260.2.a.a	$1$	$1$	\(0\)	\(2\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
3380.2.a.k	$3380$	$2$	3380.a	1.a	$1$	$3$	$3$	$26.989$	\(\Q(\zeta_{14})^+\)	None	✓	✓			3380.2.a.k	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}-\beta _{1}q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
3380.2.a.l	$3380$	$2$	3380.a	1.a	$1$	$3$	$3$	$26.989$	\(\Q(\zeta_{14})^+\)	None	✓	✓	✓		3380.2.a.k	$1$	$1$	\(0\)	\(-1\)	\(3\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+\beta _{1}q^{7}+(-1+\beta _{2})q^{9}+\cdots\)
3380.2.a.m	$3380$	$2$	3380.a	1.a	$1$	$3$	$3$	$26.989$	3.3.756.1	None	✓				260.2.f.a	$1$	$1$	\(0\)	\(0\)	\(-3\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-q^{5}-\beta _{2}q^{7}+(1+\beta _{2})q^{9}+\cdots\)
3380.2.a.n	$3380$	$2$	3380.a	1.a	$1$	$3$	$3$	$26.989$	3.3.756.1	None	✓				260.2.f.a	$1$	$0$	\(0\)	\(0\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+q^{5}+\beta _{2}q^{7}+(1+\beta _{2})q^{9}+\cdots\)
3380.2.a.o	$3380$	$2$	3380.a	1.a	$1$	$3$	$3$	$26.989$	3.3.564.1	None	✓				260.2.a.b	$1$	$0$	\(0\)	\(2\)	\(-3\)	\(2\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}-q^{5}+(1+\beta _{1})q^{7}+(4+\cdots)q^{9}+\cdots\)
3380.2.a.p	$3380$	$2$	3380.a	1.a	$1$	$4$	$4$	$26.989$	4.4.4752.1	None	✓		✓		260.2.x.a	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(-6\)		$-$	$+$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}-q^{5}+(-2-\beta _{2}+\beta _{3})q^{7}+\cdots\)
3380.2.a.q	$3380$	$2$	3380.a	1.a	$1$	$4$	$4$	$26.989$	4.4.4752.1	None	✓				260.2.x.a	$1$	$0$	\(0\)	\(2\)	\(4\)	\(6\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+q^{5}+(2+\beta _{2}-\beta _{3})q^{7}+\cdots\)
3380.2.a.r	$3380$	$2$	3380.a	1.a	$1$	$9$	$9$	$26.989$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		3380.2.a.r	$1$	$0$	\(0\)	\(-1\)	\(-9\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-q^{5}+(-1+\beta _{1}+\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
3380.2.a.s	$3380$	$2$	3380.a	1.a	$1$	$9$	$9$	$26.989$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓	✓	✓		3380.2.a.r	$1$	$0$	\(0\)	\(-1\)	\(9\)	\(1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+q^{5}+(1-\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3380))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3380)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(65))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(130))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(169))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(260))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(338))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(676))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(845))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1690))\)\(^{\oplus 2}\)