Space of modular forms of level 3312, weight 2, and trivial character

• Label: 3312.2.a
• Level: $3312$
• Weight: $2$
• Character orbit: 3312.a
• Rep. character: $\chi_{3312}(1,\cdot)$
• Character field: $\Q$
• Dimension: $55$
• Newform subspaces: $34$
• Sturm bound: $1152$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3312 = 2^{4} \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3312.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 34 \)
Sturm bound:			\(1152\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3312))\).

	Total	New	Old
Modular forms	600	55	545
Cusp forms	553	55	498
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.

\(2\)	\(3\)	\(23\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(66\)	\(6\)	\(60\)		\(61\)	\(6\)	\(55\)		\(5\)	\(0\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)		\(84\)	\(6\)	\(78\)		\(78\)	\(6\)	\(72\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(+\)	\(-\)		\(75\)	\(8\)	\(67\)		\(69\)	\(8\)	\(61\)		\(6\)	\(0\)	\(6\)
\(+\)	\(-\)	\(-\)	\(+\)		\(75\)	\(8\)	\(67\)		\(69\)	\(8\)	\(61\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(+\)	\(-\)		\(72\)	\(5\)	\(67\)		\(66\)	\(5\)	\(61\)		\(6\)	\(0\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)		\(78\)	\(5\)	\(73\)		\(72\)	\(5\)	\(67\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)		\(75\)	\(7\)	\(68\)		\(69\)	\(7\)	\(62\)		\(6\)	\(0\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)		\(75\)	\(10\)	\(65\)		\(69\)	\(10\)	\(59\)		\(6\)	\(0\)	\(6\)
Plus space			\(+\)		\(294\)	\(26\)	\(268\)		\(271\)	\(26\)	\(245\)		\(23\)	\(0\)	\(23\)
Minus space			\(-\)		\(306\)	\(29\)	\(277\)		\(282\)	\(29\)	\(253\)		\(24\)	\(0\)	\(24\)

Trace form

55 * q - 2 * q^5 - 4 * q^7 - 2 * q^13 + 6 * q^17 - 4 * q^19 + 3 * q^23 + 65 * q^25 - 2 * q^29 + 2 * q^31 - 24 * q^35 - 2 * q^37 - 2 * q^41 + 24 * q^43 - 2 * q^47 + 55 * q^49 - 10 * q^53 + 8 * q^55 - 28 * q^59 - 10 * q^61 + 12 * q^65 - 16 * q^67 - 26 * q^71 - 2 * q^73 - 8 * q^77 - 28 * q^79 + 20 * q^83 - 4 * q^85 + 14 * q^89 - 12 * q^91 - 32 * q^95 - 26 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3312))\) 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	3	23
3312.2.a.a	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				552.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-2q^{7}+2q^{13}+4q^{17}+6q^{19}+\cdots\)
3312.2.a.b	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				46.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+4q^{7}+2q^{11}-2q^{13}+2q^{17}+\cdots\)
3312.2.a.c	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				828.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{7}+4q^{11}-2q^{13}-2q^{17}+\cdots\)
3312.2.a.d	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				138.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{13}-2q^{17}+8q^{19}-q^{23}+\cdots\)
3312.2.a.e	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				552.2.a.c	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+4q^{7}-4q^{11}-2q^{13}+2q^{17}+\cdots\)
3312.2.a.f	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				184.2.a.c	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{7}+6q^{11}-2q^{13}-6q^{17}+6q^{19}+\cdots\)
3312.2.a.g	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				92.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}-q^{13}+6q^{17}-2q^{19}-q^{23}+\cdots\)
3312.2.a.h	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				138.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{7}+2q^{13}-2q^{19}-q^{23}-5q^{25}+\cdots\)
3312.2.a.i	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				184.2.a.d	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}-5q^{13}+6q^{17}-6q^{19}+q^{23}+\cdots\)
3312.2.a.j	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				552.2.a.b	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+2q^{13}-8q^{17}-6q^{19}+q^{23}+\cdots\)
3312.2.a.k	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				69.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{7}+4q^{11}-6q^{13}-4q^{17}-2q^{19}+\cdots\)
3312.2.a.l	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				828.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{7}-4q^{11}-2q^{13}+2q^{17}+\cdots\)
3312.2.a.m	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				552.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{7}-2q^{11}-2q^{13}+4q^{17}+\cdots\)
3312.2.a.n	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				138.2.a.a	$1$	$1$	\(0\)	\(0\)	\(2\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+2q^{7}-6q^{11}-2q^{13}-q^{23}+\cdots\)
3312.2.a.o	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				184.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+4q^{7}-2q^{11}+7q^{13}+4q^{17}+\cdots\)
3312.2.a.p	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				552.2.a.e	$1$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+4q^{7}-2q^{13}+2q^{17}+4q^{19}+\cdots\)
3312.2.a.q	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				92.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}+4q^{7}+2q^{11}-5q^{13}-4q^{17}+\cdots\)
3312.2.a.r	$3312$	$2$	3312.a	1.a	$1$	$1$	$1$	$26.446$	\(\Q\)	None	✓				184.2.a.a	$1$	$1$	\(0\)	\(0\)	\(4\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-2q^{7}-4q^{11}-5q^{13}+2q^{17}+\cdots\)
3312.2.a.s	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{2}) \)	None	✓				276.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}+\beta q^{7}+4\beta q^{11}+4\beta q^{13}+\cdots\)
3312.2.a.t	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{17}) \)	None	✓				184.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+(2-2\beta )q^{11}+(3-\beta )q^{13}+\cdots\)
3312.2.a.u	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{2}) \)	None	✓				207.2.a.b	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2+\beta )q^{5}+(2+\beta )q^{7}-2\beta q^{11}+\cdots\)
3312.2.a.v	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{7}) \)	None	✓				414.2.a.e	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}-2q^{7}+(-1+\beta )q^{11}+\cdots\)
3312.2.a.w	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{5}) \)	None	✓		✓		552.2.a.f	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{5}-2q^{7}+(1+\beta )q^{11}+\cdots\)
3312.2.a.x	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{3}) \)	None	✓				1656.2.a.k	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+(1-\beta )q^{7}+(-2+2\beta )q^{11}+\cdots\)
3312.2.a.y	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{10}) \)	None	✓		✓		276.2.a.a	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(-2-\beta )q^{7}+4q^{13}+(-4+\cdots)q^{17}+\cdots\)
3312.2.a.z	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{7}) \)	None	✓				414.2.a.e	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}-2q^{7}+(1+\beta )q^{11}-2\beta q^{13}+\cdots\)
3312.2.a.ba	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{5}) \)	None	✓				23.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(-1+\beta )q^{7}+(-3+\beta )q^{11}+\cdots\)
3312.2.a.bb	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{5}) \)	None	✓				69.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(-1+\beta )q^{7}+4q^{11}+\cdots\)
3312.2.a.bc	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{5}) \)	None	✓				138.2.a.d	$1$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}-2\beta q^{7}+(-3+\beta )q^{11}+\cdots\)
3312.2.a.bd	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{3}) \)	None	✓				1656.2.a.k	$1$	$0$	\(0\)	\(0\)	\(2\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}+(1+\beta )q^{7}+(2+2\beta )q^{11}+\cdots\)
3312.2.a.be	$3312$	$2$	3312.a	1.a	$1$	$2$	$2$	$26.446$	\(\Q(\sqrt{2}) \)	None	✓				207.2.a.b	$1$	$0$	\(0\)	\(0\)	\(4\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2+\beta )q^{5}+(2-\beta )q^{7}-2\beta q^{11}+(6+\cdots)q^{17}+\cdots\)
3312.2.a.bf	$3312$	$2$	3312.a	1.a	$1$	$3$	$3$	$26.446$	3.3.148.1	None	✓		✓		552.2.a.g	$1$	$0$	\(0\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{5}+(-1-\beta _{1})q^{7}+(1-\beta _{1}-\beta _{2})q^{11}+\cdots\)
3312.2.a.bg	$3312$	$2$	3312.a	1.a	$1$	$4$	$4$	$26.446$	4.4.44688.2	None	✓		✓		1656.2.a.o	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-2\)		$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{5}+\beta _{1}q^{7}+(\beta _{2}-\beta _{3})q^{11}+\cdots\)
3312.2.a.bh	$3312$	$2$	3312.a	1.a	$1$	$4$	$4$	$26.446$	4.4.44688.2	None	✓		✓		1656.2.a.o	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-2\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{5}+(-1-\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3312))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3312)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(23))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(46))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(69))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(92))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(138))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(144))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(184))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(207))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(276))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(368))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(414))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(552))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(828))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1104))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1656))\)\(^{\oplus 2}\)