Space of modular forms of level 3344, weight 2, and trivial character

• Label: 3344.2.a
• Level: $3344$
• Weight: $2$
• Character orbit: 3344.a
• Rep. character: $\chi_{3344}(1,\cdot)$
• Character field: $\Q$
• Dimension: $90$
• Newform subspaces: $28$
• Sturm bound: $960$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 28 \)
Sturm bound:			\(960\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3344))\).

	Total	New	Old
Modular forms	492	90	402
Cusp forms	469	90	379
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.

\(2\)	\(11\)	\(19\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(55\)	\(10\)	\(45\)		\(53\)	\(10\)	\(43\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(68\)	\(15\)	\(53\)		\(65\)	\(15\)	\(50\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(65\)	\(12\)	\(53\)		\(62\)	\(12\)	\(50\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(58\)	\(7\)	\(51\)		\(55\)	\(7\)	\(48\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(68\)	\(14\)	\(54\)		\(65\)	\(14\)	\(51\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(55\)	\(9\)	\(46\)		\(52\)	\(9\)	\(43\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(58\)	\(9\)	\(49\)		\(55\)	\(9\)	\(46\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(65\)	\(14\)	\(51\)		\(62\)	\(14\)	\(48\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(226\)	\(35\)	\(191\)		\(215\)	\(35\)	\(180\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(266\)	\(55\)	\(211\)		\(254\)	\(55\)	\(199\)		\(12\)	\(0\)	\(12\)

Trace form

90 * q + 4 * q^5 + 90 * q^9 - 6 * q^11 + 4 * q^13 - 12 * q^15 - 4 * q^17 + 16 * q^21 + 12 * q^23 + 86 * q^25 + 12 * q^27 + 20 * q^29 + 20 * q^37 + 24 * q^39 - 4 * q^41 + 36 * q^45 + 8 * q^47 + 90 * q^49 + 40 * q^51 + 4 * q^53 + 8 * q^55 + 40 * q^59 + 20 * q^61 + 40 * q^63 - 24 * q^65 + 16 * q^69 + 16 * q^71 - 20 * q^73 - 12 * q^75 + 74 * q^81 + 40 * q^83 + 24 * q^85 + 24 * q^87 - 36 * q^89 + 16 * q^91 - 48 * q^93 - 4 * q^97 - 18 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\) 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	11	19
3344.2.a.a	$3344$	$2$	3344.a	1.a	$1$	$1$	$1$	$26.702$	\(\Q\)	None	✓				418.2.a.c	$1$	$2$	\(0\)	\(-3\)	\(-2\)	\(-1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-2q^{5}-q^{7}+6q^{9}-q^{11}+\cdots\)
3344.2.a.b	$3344$	$2$	3344.a	1.a	$1$	$1$	$1$	$26.702$	\(\Q\)	None	✓				836.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(2\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{5}+q^{9}-q^{11}+4q^{13}+\cdots\)
3344.2.a.c	$3344$	$2$	3344.a	1.a	$1$	$1$	$1$	$26.702$	\(\Q\)	None	✓				1672.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}-2q^{7}-2q^{9}+q^{11}+\cdots\)
3344.2.a.d	$3344$	$2$	3344.a	1.a	$1$	$1$	$1$	$26.702$	\(\Q\)	None	✓				209.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}+4q^{7}-2q^{9}-q^{11}+\cdots\)
3344.2.a.e	$3344$	$2$	3344.a	1.a	$1$	$1$	$1$	$26.702$	\(\Q\)	None	✓				1672.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+3q^{7}-2q^{9}+q^{11}+\cdots\)
3344.2.a.f	$3344$	$2$	3344.a	1.a	$1$	$1$	$1$	$26.702$	\(\Q\)	None	✓				1672.2.a.a	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+4q^{7}-3q^{9}+q^{11}+2q^{13}+\cdots\)
3344.2.a.g	$3344$	$2$	3344.a	1.a	$1$	$1$	$1$	$26.702$	\(\Q\)	None	✓				418.2.a.b	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{7}-3q^{9}-q^{11}-2q^{13}+\cdots\)
3344.2.a.h	$3344$	$2$	3344.a	1.a	$1$	$1$	$1$	$26.702$	\(\Q\)	None	✓				418.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-2\)	\(3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}+3q^{7}-2q^{9}+q^{11}+\cdots\)
3344.2.a.i	$3344$	$2$	3344.a	1.a	$1$	$1$	$1$	$26.702$	\(\Q\)	None	✓				836.2.a.a	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{9}+q^{11}-2q^{13}+\cdots\)
3344.2.a.j	$3344$	$2$	3344.a	1.a	$1$	$2$	$2$	$26.702$	\(\Q(\sqrt{13}) \)	None	✓				1672.2.a.d	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}-\beta q^{5}+(-1+\beta )q^{7}+\beta q^{9}+\cdots\)
3344.2.a.k	$3344$	$2$	3344.a	1.a	$1$	$2$	$2$	$26.702$	\(\Q(\sqrt{17}) \)	None	✓				418.2.a.e	$1$	$1$	\(0\)	\(-1\)	\(4\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+2q^{5}+(-2+\beta )q^{7}+(1+\beta )q^{9}+\cdots\)
3344.2.a.l	$3344$	$2$	3344.a	1.a	$1$	$2$	$2$	$26.702$	\(\Q(\sqrt{21}) \)	None	✓				418.2.a.f	$1$	$0$	\(0\)	\(1\)	\(3\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2-\beta )q^{5}+(3-\beta )q^{7}+(2+\beta )q^{9}+\cdots\)
3344.2.a.m	$3344$	$2$	3344.a	1.a	$1$	$2$	$2$	$26.702$	\(\Q(\sqrt{2}) \)	None	✓				836.2.a.c	$1$	$1$	\(0\)	\(2\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}-\beta q^{7}+2\beta q^{9}-q^{11}+\cdots\)
3344.2.a.n	$3344$	$2$	3344.a	1.a	$1$	$2$	$2$	$26.702$	\(\Q(\sqrt{2}) \)	None	✓				209.2.a.b	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-q^{5}+(2+\beta )q^{7}+2\beta q^{9}+\cdots\)
3344.2.a.o	$3344$	$2$	3344.a	1.a	$1$	$2$	$2$	$26.702$	\(\Q(\sqrt{13}) \)	None	✓				418.2.a.d	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}+(-1+\beta )q^{5}+\beta q^{7}+(1+\cdots)q^{9}+\cdots\)
3344.2.a.p	$3344$	$2$	3344.a	1.a	$1$	$3$	$3$	$26.702$	3.3.469.1	None	✓				418.2.a.h	$1$	$0$	\(0\)	\(-1\)	\(5\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(2-\beta _{1})q^{5}+\beta _{2}q^{7}+(1+\beta _{2})q^{9}+\cdots\)
3344.2.a.q	$3344$	$2$	3344.a	1.a	$1$	$3$	$3$	$26.702$	3.3.621.1	None	✓				418.2.a.g	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(6\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{1}-\beta _{2})q^{5}+(2+\beta _{1}+\cdots)q^{7}+\cdots\)
3344.2.a.r	$3344$	$2$	3344.a	1.a	$1$	$3$	$3$	$26.702$	3.3.229.1	None	✓		✓		1672.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+\beta _{1}q^{5}+(-1-\beta _{1})q^{7}+\beta _{2}q^{9}+\cdots\)
3344.2.a.s	$3344$	$2$	3344.a	1.a	$1$	$4$	$4$	$26.702$	4.4.13676.1	None	✓				1672.2.a.f	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{3})q^{5}+(1-\beta _{1})q^{7}+\cdots\)
3344.2.a.t	$3344$	$2$	3344.a	1.a	$1$	$5$	$5$	$26.702$	5.5.246832.1	None	✓		✓		209.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(-5\)	\(-6\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(-1-\beta _{1}+\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots\)
3344.2.a.u	$3344$	$2$	3344.a	1.a	$1$	$5$	$5$	$26.702$	5.5.3979184.1	None	✓				1672.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(7\)	\(-8\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{4}q^{3}+(1+\beta _{1})q^{5}+(-2+\beta _{1})q^{7}+\cdots\)
3344.2.a.v	$3344$	$2$	3344.a	1.a	$1$	$6$	$6$	$26.702$	6.6.576096652.1	None	✓		✓		1672.2.a.j	$1$	$1$	\(0\)	\(-4\)	\(1\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{3}+\beta _{4}q^{5}-\beta _{3}q^{7}+(1+\cdots)q^{9}+\cdots\)
3344.2.a.w	$3344$	$2$	3344.a	1.a	$1$	$6$	$6$	$26.702$	6.6.744786576.1	None	✓		✓		836.2.a.e	$1$	$0$	\(0\)	\(-1\)	\(5\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{5})q^{5}+(1-\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
3344.2.a.x	$3344$	$2$	3344.a	1.a	$1$	$6$	$6$	$26.702$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓		✓		836.2.a.d	$1$	$0$	\(0\)	\(2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{5}q^{5}-\beta _{4}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
3344.2.a.y	$3344$	$2$	3344.a	1.a	$1$	$6$	$6$	$26.702$	6.6.57500224.1	None	✓				1672.2.a.h	$1$	$0$	\(0\)	\(4\)	\(-4\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1-\beta _{4})q^{5}+(1-\beta _{1}+\cdots)q^{7}+\cdots\)
3344.2.a.z	$3344$	$2$	3344.a	1.a	$1$	$6$	$6$	$26.702$	6.6.106392688.1	None	✓		✓		1672.2.a.i	$1$	$0$	\(0\)	\(4\)	\(-3\)	\(3\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{3}+(-1+\beta _{5})q^{5}+\beta _{5}q^{7}+\cdots\)
3344.2.a.ba	$3344$	$2$	3344.a	1.a	$1$	$7$	$7$	$26.702$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓		✓		209.2.a.d	$1$	$1$	\(0\)	\(-2\)	\(2\)	\(-10\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-\beta _{4}q^{5}+(-2-\beta _{2}+\beta _{5}+\cdots)q^{7}+\cdots\)
3344.2.a.bb	$3344$	$2$	3344.a	1.a	$1$	$9$	$9$	$26.702$	\(\mathbb{Q}[x]/(x^{9} - \cdots)\)	None	✓		✓		1672.2.a.k	$1$	$0$	\(0\)	\(-1\)	\(6\)	\(-3\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1+\beta _{7})q^{5}-\beta _{8}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3344))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3344)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(176))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(209))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(304))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(418))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(836))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1672))\)\(^{\oplus 2}\)