Space of modular forms of level 1368, weight 4, and trivial character

• Label: 1368.4.a
• Level: $1368$
• Weight: $4$
• Character orbit: 1368.a
• Rep. character: $\chi_{1368}(1,\cdot)$
• Character field: $\Q$
• Dimension: $67$
• Newform subspaces: $16$
• Sturm bound: $960$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(960\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(1368))\).

	Total	New	Old
Modular forms	736	67	669
Cusp forms	704	67	637
Eisenstein series	32	0	32

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

\(2\)	\(3\)	\(19\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(5\)
\(+\)	\(+\)	\(-\)	\(-\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)	\(10\)
\(+\)	\(-\)	\(-\)	\(+\)	\(10\)
\(-\)	\(+\)	\(+\)	\(-\)	\(5\)
\(-\)	\(+\)	\(-\)	\(+\)	\(8\)
\(-\)	\(-\)	\(+\)	\(+\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)	\(9\)
Plus space			\(+\)	\(35\)
Minus space			\(-\)	\(32\)

Trace form

67 * q + 24 * q^5 + 12 * q^7 - 66 * q^11 - 66 * q^13 + 154 * q^17 + 57 * q^19 + 42 * q^23 + 1379 * q^25 - 194 * q^29 - 60 * q^31 - 606 * q^35 + 206 * q^37 - 474 * q^41 + 570 * q^43 + 1602 * q^47 + 3303 * q^49 - 1694 * q^53 - 1866 * q^55 - 280 * q^59 + 148 * q^61 - 616 * q^65 + 620 * q^67 + 2312 * q^71 + 1514 * q^73 - 574 * q^77 - 1548 * q^79 - 2156 * q^83 - 1462 * q^85 - 618 * q^89 - 1424 * q^91 - 1014 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(1368))\) 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	19
1368.4.a.a	$1368$	$4$	1368.a	1.a	$1$	$2$	$2$	$80.715$	\(\Q(\sqrt{57}) \)	None	✓				152.4.a.a	$1$	$1$	\(0\)	\(0\)	\(5\)	\(-6\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(3-\beta )q^{5}+(-5+4\beta )q^{7}+(11-7\beta )q^{11}+\cdots\)
1368.4.a.b	$1368$	$4$	1368.a	1.a	$1$	$2$	$2$	$80.715$	\(\Q(\sqrt{105}) \)	None	✓				456.4.a.a	$1$	$1$	\(0\)	\(0\)	\(9\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(5-\beta )q^{5}+(1+\beta )q^{7}+(-11+5\beta )q^{11}+\cdots\)
1368.4.a.c	$1368$	$4$	1368.a	1.a	$1$	$2$	$2$	$80.715$	\(\Q(\sqrt{97}) \)	None	✓				456.4.a.b	$1$	$0$	\(0\)	\(0\)	\(13\)	\(3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(7-\beta )q^{5}+(3-3\beta )q^{7}+(19+3\beta )q^{11}+\cdots\)
1368.4.a.d	$1368$	$4$	1368.a	1.a	$1$	$3$	$3$	$80.715$	3.3.7057.1	None	✓				152.4.a.c	$1$	$0$	\(0\)	\(0\)	\(-7\)	\(7\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-2-2\beta _{1}-\beta _{2})q^{5}+(2+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1368.4.a.e	$1368$	$4$	1368.a	1.a	$1$	$3$	$3$	$80.715$	3.3.3221.1	None	✓				152.4.a.b	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-35\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1}-2\beta _{2})q^{5}+(-11+2\beta _{1}+\cdots)q^{7}+\cdots\)
1368.4.a.f	$1368$	$4$	1368.a	1.a	$1$	$3$	$3$	$80.715$	3.3.24665.1	None	✓				456.4.a.c	$1$	$0$	\(0\)	\(0\)	\(1\)	\(7\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(3+2\beta _{1}-3\beta _{2})q^{7}+(-5+\cdots)q^{11}+\cdots\)
1368.4.a.g	$1368$	$4$	1368.a	1.a	$1$	$4$	$4$	$80.715$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		456.4.a.g	$1$	$1$	\(0\)	\(0\)	\(-9\)	\(7\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2})q^{5}+(2-\beta _{1}+\beta _{2})q^{7}+\cdots\)
1368.4.a.h	$1368$	$4$	1368.a	1.a	$1$	$4$	$4$	$80.715$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				456.4.a.d	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-14\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-3+\beta _{1}-\beta _{2})q^{7}+\cdots\)
1368.4.a.i	$1368$	$4$	1368.a	1.a	$1$	$4$	$4$	$80.715$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓				456.4.a.f	$1$	$1$	\(0\)	\(0\)	\(8\)	\(10\)		$+$	$-$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{5}+(3+\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1368.4.a.j	$1368$	$4$	1368.a	1.a	$1$	$4$	$4$	$80.715$	4.4.4914253.1	None	✓				456.4.a.e	$1$	$0$	\(0\)	\(0\)	\(16\)	\(-14\)		$-$	$-$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(4-\beta _{2}-\beta _{3})q^{5}+(-4-\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
1368.4.a.k	$1368$	$4$	1368.a	1.a	$1$	$5$	$5$	$80.715$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	1368.4.a.k	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(10\)		$+$	$+$	$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{2})q^{5}+(2+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1368.4.a.l	$1368$	$4$	1368.a	1.a	$1$	$5$	$5$	$80.715$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		456.4.a.h	$1$	$0$	\(0\)	\(0\)	\(-6\)	\(10\)		$-$	$-$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{5}+(2-\beta _{2})q^{7}+(7-\beta _{1}+\cdots)q^{11}+\cdots\)
1368.4.a.m	$1368$	$4$	1368.a	1.a	$1$	$5$	$5$	$80.715$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓		✓		152.4.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(22\)		$+$	$-$	$-$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(-\beta _{1}-\beta _{2})q^{5}+(4+\beta _{2}-\beta _{3})q^{7}+\cdots\)
1368.4.a.n	$1368$	$4$	1368.a	1.a	$1$	$5$	$5$	$80.715$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	✓	✓	✓	1368.4.a.k	$1$	$1$	\(0\)	\(0\)	\(6\)	\(10\)		$-$	$+$	$+$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+(1-\beta _{2})q^{5}+(2+\beta _{2}-\beta _{3})q^{7}+(5+\cdots)q^{11}+\cdots\)
1368.4.a.o	$1368$	$4$	1368.a	1.a	$1$	$8$	$8$	$80.715$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	1368.4.a.o	$1$	$1$	\(0\)	\(0\)	\(-16\)	\(-4\)		$+$	$+$	$-$	$-$	$2^{9}$	$\mathrm{SU}(2)$	\(q+(-2+\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(-2+\cdots)q^{11}+\cdots\)
1368.4.a.p	$1368$	$4$	1368.a	1.a	$1$	$8$	$8$	$80.715$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓	✓	1368.4.a.o	$1$	$0$	\(0\)	\(0\)	\(16\)	\(-4\)		$-$	$+$	$-$	$+$	$2^{9}$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(2+\beta _{1}+\cdots)q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(1368))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(1368)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(9))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(18))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 12}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 9}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(57))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(76))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(114))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(152))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(171))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(228))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(342))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(456))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(684))\)\(^{\oplus 2}\)