Space of modular forms of level 3332, weight 2, and trivial character

• Label: 3332.2.a
• Level: $3332$
• Weight: $2$
• Character orbit: 3332.a
• Rep. character: $\chi_{3332}(1,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $21$
• Sturm bound: $1008$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(1008\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3332))\).

	Total	New	Old
Modular forms	528	54	474
Cusp forms	481	54	427
Eisenstein series	47	0	47

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

\(2\)	\(7\)	\(17\)	Fricke	Dim
\(-\)	\(+\)	\(+\)	\(-\)	\(15\)
\(-\)	\(+\)	\(-\)	\(+\)	\(11\)
\(-\)	\(-\)	\(+\)	\(+\)	\(11\)
\(-\)	\(-\)	\(-\)	\(-\)	\(17\)
Plus space			\(+\)	\(22\)
Minus space			\(-\)	\(32\)

Trace form

54 * q + 2 * q^3 + 4 * q^5 + 58 * q^9 + 2 * q^11 + 12 * q^15 + 2 * q^17 + 2 * q^23 + 58 * q^25 + 8 * q^27 + 6 * q^31 - 4 * q^33 + 20 * q^37 + 24 * q^39 + 8 * q^41 + 32 * q^43 + 8 * q^45 - 16 * q^47 - 2 * q^51 - 12 * q^53 + 16 * q^55 + 40 * q^57 + 4 * q^59 + 4 * q^61 + 4 * q^65 + 16 * q^67 + 4 * q^69 + 14 * q^71 - 8 * q^73 - 38 * q^75 + 42 * q^79 + 62 * q^81 + 20 * q^89 - 4 * q^93 - 16 * q^95 + 24 * q^97 - 6 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3332))\) 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	17
3332.2.a.a	$3332$	$2$	3332.a	1.a	$1$	$1$	$1$	$26.606$	\(\Q\)	None	✓				476.2.i.c	$1$	$0$	\(0\)	\(-3\)	\(4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+4q^{5}+6q^{9}+q^{11}+3q^{13}+\cdots\)
3332.2.a.b	$3332$	$2$	3332.a	1.a	$1$	$1$	$1$	$26.606$	\(\Q\)	None	✓				476.2.i.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{9}-5q^{11}-5q^{13}-q^{17}+\cdots\)
3332.2.a.c	$3332$	$2$	3332.a	1.a	$1$	$1$	$1$	$26.606$	\(\Q\)	None	✓				476.2.i.a	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}-2q^{9}-3q^{11}+5q^{13}+\cdots\)
3332.2.a.d	$3332$	$2$	3332.a	1.a	$1$	$1$	$1$	$26.606$	\(\Q\)	None	✓				476.2.i.a	$1$	$0$	\(0\)	\(1\)	\(-4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-2q^{9}-3q^{11}-5q^{13}+\cdots\)
3332.2.a.e	$3332$	$2$	3332.a	1.a	$1$	$1$	$1$	$26.606$	\(\Q\)	None	✓				476.2.i.b	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{9}-5q^{11}+5q^{13}+q^{17}+\cdots\)
3332.2.a.f	$3332$	$2$	3332.a	1.a	$1$	$1$	$1$	$26.606$	\(\Q\)	None	✓				476.2.i.c	$1$	$0$	\(0\)	\(3\)	\(-4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-4q^{5}+6q^{9}+q^{11}-3q^{13}+\cdots\)
3332.2.a.g	$3332$	$2$	3332.a	1.a	$1$	$2$	$2$	$26.606$	\(\Q(\sqrt{5}) \)	None	✓	✓			3332.2.a.g	$1$	$1$	\(0\)	\(-3\)	\(-5\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(-2-\beta )q^{5}+(-1+\cdots)q^{9}+\cdots\)
3332.2.a.h	$3332$	$2$	3332.a	1.a	$1$	$2$	$2$	$26.606$	\(\Q(\sqrt{3}) \)	None	✓				68.2.a.a	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-2\beta q^{5}+(1-2\beta )q^{9}+\cdots\)
3332.2.a.i	$3332$	$2$	3332.a	1.a	$1$	$2$	$2$	$26.606$	\(\Q(\sqrt{21}) \)	None	✓	✓			3332.2.a.i	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-1+\beta )q^{5}+(2+\beta )q^{9}+(2+\cdots)q^{11}+\cdots\)
3332.2.a.j	$3332$	$2$	3332.a	1.a	$1$	$2$	$2$	$26.606$	\(\Q(\sqrt{13}) \)	None	✓				476.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1-\beta )q^{5}+\beta q^{9}+(-4+2\beta )q^{13}+\cdots\)
3332.2.a.k	$3332$	$2$	3332.a	1.a	$1$	$2$	$2$	$26.606$	\(\Q(\sqrt{13}) \)	None	✓				476.2.a.c	$1$	$0$	\(0\)	\(1\)	\(-1\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+\beta )q^{5}+\beta q^{9}+4q^{11}+\cdots\)
3332.2.a.l	$3332$	$2$	3332.a	1.a	$1$	$2$	$2$	$26.606$	\(\Q(\sqrt{5}) \)	None	✓				476.2.a.b	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1-\beta )q^{5}+(-2+\beta )q^{9}+(-4+\cdots)q^{11}+\cdots\)
3332.2.a.m	$3332$	$2$	3332.a	1.a	$1$	$2$	$2$	$26.606$	\(\Q(\sqrt{21}) \)	None	✓	✓			3332.2.a.i	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(1-\beta )q^{5}+(2+\beta )q^{9}+(2-2\beta )q^{11}+\cdots\)
3332.2.a.n	$3332$	$2$	3332.a	1.a	$1$	$2$	$2$	$26.606$	\(\Q(\sqrt{13}) \)	None	✓				476.2.a.a	$1$	$0$	\(0\)	\(3\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(2-\beta )q^{5}+(1+3\beta )q^{9}+\cdots\)
3332.2.a.o	$3332$	$2$	3332.a	1.a	$1$	$2$	$2$	$26.606$	\(\Q(\sqrt{5}) \)	None	✓	✓			3332.2.a.g	$1$	$0$	\(0\)	\(3\)	\(5\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(2+\beta )q^{5}+(-1+3\beta )q^{9}+\cdots\)
3332.2.a.p	$3332$	$2$	3332.a	1.a	$1$	$3$	$3$	$26.606$	3.3.257.1	None	✓				476.2.i.d	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1+\beta _{1})q^{5}+\beta _{2}q^{9}-\beta _{2}q^{11}+\cdots\)
3332.2.a.q	$3332$	$2$	3332.a	1.a	$1$	$3$	$3$	$26.606$	3.3.257.1	None	✓		✓		476.2.i.d	$1$	$1$	\(0\)	\(1\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1-\beta _{1})q^{5}+\beta _{2}q^{9}-\beta _{2}q^{11}+\cdots\)
3332.2.a.r	$3332$	$2$	3332.a	1.a	$1$	$4$	$4$	$26.606$	4.4.113481.1	None	✓		✓		476.2.i.e	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+\beta _{1}q^{5}+(3+\beta _{2})q^{9}+\cdots\)
3332.2.a.s	$3332$	$2$	3332.a	1.a	$1$	$4$	$4$	$26.606$	4.4.113481.1	None	✓				476.2.i.e	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}-\beta _{1}q^{5}+(3+\beta _{2})q^{9}+\cdots\)
3332.2.a.t	$3332$	$2$	3332.a	1.a	$1$	$8$	$8$	$26.606$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		3332.2.a.t	$1$	$1$	\(0\)	\(-4\)	\(-4\)	\(0\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{7}q^{5}+(1+2\beta _{1}+\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)
3332.2.a.u	$3332$	$2$	3332.a	1.a	$1$	$8$	$8$	$26.606$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		3332.2.a.t	$1$	$0$	\(0\)	\(4\)	\(4\)	\(0\)		$-$	$+$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{7}q^{5}+(1+2\beta _{1}+\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3332))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3332)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(119))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(238))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(476))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(833))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1666))\)\(^{\oplus 2}\)