Space of modular forms of level 1368, weight 2, and trivial character

• Label: 1368.2.a
• Level: $1368$
• Weight: $2$
• Character orbit: 1368.a
• Rep. character: $\chi_{1368}(1,\cdot)$
• Character field: $\Q$
• Dimension: $23$
• Newform subspaces: $15$
• Sturm bound: $480$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 15 \)
Sturm bound:			\(480\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(5\), \(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1368))\).

	Total	New	Old
Modular forms	256	23	233
Cusp forms	225	23	202
Eisenstein series	31	0	31

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
\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)	\(5\)
Plus space			\(+\)	\(10\)
Minus space			\(-\)	\(13\)

Trace form

23 * q - 4 * q^5 - 4 * q^7 + 6 * q^11 - 2 * q^13 - 6 * q^17 - 3 * q^19 - 6 * q^23 + 35 * q^25 + 14 * q^29 - 12 * q^31 - 6 * q^35 - 2 * q^37 - 2 * q^41 - 14 * q^43 - 6 * q^47 + 51 * q^49 + 10 * q^53 - 10 * q^55 - 8 * q^59 + 16 * q^61 + 16 * q^65 - 20 * q^67 + 8 * q^71 + 10 * q^73 + 30 * q^77 - 12 * q^79 + 4 * q^83 + 14 * q^85 + 30 * q^89 - 16 * q^91 + 26 * q^97

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1368))\) 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
1368.2.a.a	$1368$	$2$	1368.a	1.a	$1$	$1$	$1$	$10.924$	\(\Q\)	None	✓	✓			1368.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+6q^{11}+2q^{13}-4q^{17}-q^{19}+\cdots\)
1368.2.a.b	$1368$	$2$	1368.a	1.a	$1$	$1$	$1$	$10.924$	\(\Q\)	None	✓				456.2.a.b	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+4q^{7}+4q^{11}-4q^{13}-6q^{17}+\cdots\)
1368.2.a.c	$1368$	$2$	1368.a	1.a	$1$	$1$	$1$	$10.924$	\(\Q\)	None	✓				456.2.a.d	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}+2q^{13}-2q^{17}-q^{19}-q^{25}+\cdots\)
1368.2.a.d	$1368$	$2$	1368.a	1.a	$1$	$1$	$1$	$10.924$	\(\Q\)	None	✓				456.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-3\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{5}-3q^{7}+5q^{11}-2q^{13}+q^{17}+\cdots\)
1368.2.a.e	$1368$	$2$	1368.a	1.a	$1$	$1$	$1$	$10.924$	\(\Q\)	None	✓	✓	✓	✓	1368.2.a.e	$1$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{11}-2q^{13}-4q^{17}+q^{19}-2q^{23}+\cdots\)
1368.2.a.f	$1368$	$2$	1368.a	1.a	$1$	$1$	$1$	$10.924$	\(\Q\)	None	✓	✓	✓	✓	1368.2.a.e	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{11}-2q^{13}+4q^{17}+q^{19}+2q^{23}+\cdots\)
1368.2.a.g	$1368$	$2$	1368.a	1.a	$1$	$1$	$1$	$10.924$	\(\Q\)	None	✓				152.2.a.b	$1$	$0$	\(0\)	\(0\)	\(0\)	\(3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{7}-2q^{11}+q^{13}+5q^{17}+q^{19}+\cdots\)
1368.2.a.h	$1368$	$2$	1368.a	1.a	$1$	$1$	$1$	$10.924$	\(\Q\)	None	✓				152.2.a.a	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-3q^{7}+3q^{11}-4q^{13}-5q^{17}+\cdots\)
1368.2.a.i	$1368$	$2$	1368.a	1.a	$1$	$1$	$1$	$10.924$	\(\Q\)	None	✓				456.2.a.c	$1$	$0$	\(0\)	\(0\)	\(3\)	\(-3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{5}-3q^{7}+q^{11}-2q^{13}+5q^{17}+\cdots\)
1368.2.a.j	$1368$	$2$	1368.a	1.a	$1$	$1$	$1$	$10.924$	\(\Q\)	None	✓	✓			1368.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-6q^{11}+2q^{13}+4q^{17}-q^{19}+\cdots\)
1368.2.a.k	$1368$	$2$	1368.a	1.a	$1$	$2$	$2$	$10.924$	\(\Q(\sqrt{17}) \)	None	✓		✓		456.2.a.f	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+\beta q^{7}+(-4+\beta )q^{11}-2\beta q^{13}+\cdots\)
1368.2.a.l	$1368$	$2$	1368.a	1.a	$1$	$2$	$2$	$10.924$	\(\Q(\sqrt{41}) \)	None	✓		✓		456.2.a.e	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-3\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(-2+\beta )q^{7}+(-2+\beta )q^{11}+\cdots\)
1368.2.a.m	$1368$	$2$	1368.a	1.a	$1$	$3$	$3$	$10.924$	3.3.892.1	None	✓	✓	✓		1368.2.a.m	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}+\cdots\)
1368.2.a.n	$1368$	$2$	1368.a	1.a	$1$	$3$	$3$	$10.924$	3.3.961.1	None	✓		✓	✓	152.2.a.c	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(1-\beta _{2})q^{7}+(2-\beta _{1})q^{11}+\cdots\)
1368.2.a.o	$1368$	$2$	1368.a	1.a	$1$	$3$	$3$	$10.924$	3.3.892.1	None	✓	✓	✓		1368.2.a.m	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1368))\) into lower level spaces

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