Space of modular forms of level 2312, weight 2, and trivial character

• Label: 2312.2.a
• Level: $2312$
• Weight: $2$
• Character orbit: 2312.a
• Rep. character: $\chi_{2312}(1,\cdot)$
• Character field: $\Q$
• Dimension: $68$
• Newform subspaces: $23$
• Sturm bound: $612$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 2312 = 2^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2312.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 23 \)
Sturm bound:			\(612\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(3\), \(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(2312))\).

	Total	New	Old
Modular forms	342	68	274
Cusp forms	271	68	203
Eisenstein series	71	0	71

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

\(2\)	\(17\)	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(81\)	\(14\)	\(67\)		\(64\)	\(14\)	\(50\)		\(17\)	\(0\)	\(17\)
\(+\)	\(-\)	\(-\)		\(89\)	\(20\)	\(69\)		\(71\)	\(20\)	\(51\)		\(18\)	\(0\)	\(18\)
\(-\)	\(+\)	\(-\)		\(90\)	\(18\)	\(72\)		\(72\)	\(18\)	\(54\)		\(18\)	\(0\)	\(18\)
\(-\)	\(-\)	\(+\)		\(82\)	\(16\)	\(66\)		\(64\)	\(16\)	\(48\)		\(18\)	\(0\)	\(18\)
Plus space		\(+\)		\(163\)	\(30\)	\(133\)		\(128\)	\(30\)	\(98\)		\(35\)	\(0\)	\(35\)
Minus space		\(-\)		\(179\)	\(38\)	\(141\)		\(143\)	\(38\)	\(105\)		\(36\)	\(0\)	\(36\)

Trace form

68 * q + 2 * q^3 - 2 * q^5 + 64 * q^9 + 2 * q^11 + 4 * q^13 + 12 * q^21 - 12 * q^23 + 80 * q^25 + 20 * q^27 + 6 * q^29 + 8 * q^31 - 4 * q^35 + 2 * q^37 - 4 * q^39 - 4 * q^41 + 16 * q^43 - 10 * q^45 + 84 * q^49 + 8 * q^53 - 28 * q^55 + 8 * q^57 - 8 * q^59 - 22 * q^61 - 24 * q^63 + 4 * q^65 + 4 * q^69 - 28 * q^71 + 6 * q^75 - 36 * q^77 + 4 * q^79 + 76 * q^81 - 32 * q^83 - 28 * q^87 + 28 * q^89 + 24 * q^91 - 4 * q^93 + 8 * q^95 + 12 * q^97 - 22 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(2312))\) 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	17
2312.2.a.a	$2312$	$2$	2312.a	1.a	$1$	$1$	$1$	$18.461$	\(\Q\)	None	✓				136.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-2q^{11}-6q^{13}+4q^{19}+\cdots\)
2312.2.a.b	$2312$	$2$	2312.a	1.a	$1$	$1$	$1$	$18.461$	\(\Q\)	None	✓				136.2.b.a	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+2q^{7}+q^{9}+2q^{11}-2q^{13}+\cdots\)
2312.2.a.c	$2312$	$2$	2312.a	1.a	$1$	$1$	$1$	$18.461$	\(\Q\)	None	✓				136.2.b.a	$1$	$1$	\(0\)	\(2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{7}+q^{9}-2q^{11}-2q^{13}+\cdots\)
2312.2.a.d	$2312$	$2$	2312.a	1.a	$1$	$1$	$1$	$18.461$	\(\Q\)	None	✓				136.2.a.a	$1$	$0$	\(0\)	\(2\)	\(2\)	\(2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+2q^{5}+2q^{7}+q^{9}+6q^{11}+\cdots\)
2312.2.a.e	$2312$	$2$	2312.a	1.a	$1$	$2$	$2$	$18.461$	\(\Q(\sqrt{13}) \)	None	✓	✓			2312.2.a.e	$1$	$1$	\(0\)	\(-3\)	\(1\)	\(1\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+\beta q^{5}+\beta q^{7}+(1+3\beta )q^{9}+\cdots\)
2312.2.a.f	$2312$	$2$	2312.a	1.a	$1$	$2$	$2$	$18.461$	\(\Q(\sqrt{5}) \)	None	✓	✓			2312.2.a.f	$1$	$1$	\(0\)	\(-1\)	\(-1\)	\(-5\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(1-3\beta )q^{5}+(-3+\beta )q^{7}+\cdots\)
2312.2.a.g	$2312$	$2$	2312.a	1.a	$1$	$2$	$2$	$18.461$	\(\Q(\sqrt{2}) \)	None	✓				136.2.b.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+\beta q^{7}-3q^{9}-2\beta q^{11}+2q^{13}+\cdots\)
2312.2.a.h	$2312$	$2$	2312.a	1.a	$1$	$2$	$2$	$18.461$	\(\Q(\sqrt{2}) \)	None	✓				136.2.k.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-3\beta q^{5}-3\beta q^{7}-q^{9}-\beta q^{11}+\cdots\)
2312.2.a.i	$2312$	$2$	2312.a	1.a	$1$	$2$	$2$	$18.461$	\(\Q(\sqrt{2}) \)	None	✓				136.2.k.c	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{5}+\beta q^{7}-q^{9}-\beta q^{11}+\cdots\)
2312.2.a.j	$2312$	$2$	2312.a	1.a	$1$	$2$	$2$	$18.461$	\(\Q(\sqrt{2}) \)	None	✓				136.2.k.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}+2\beta q^{7}-q^{9}-\beta q^{11}+\cdots\)
2312.2.a.k	$2312$	$2$	2312.a	1.a	$1$	$2$	$2$	$18.461$	\(\Q(\sqrt{2}) \)	None	✓				136.2.k.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2\beta q^{3}+\beta q^{5}-2\beta q^{7}+5q^{9}-2\beta q^{11}+\cdots\)
2312.2.a.l	$2312$	$2$	2312.a	1.a	$1$	$2$	$2$	$18.461$	\(\Q(\sqrt{5}) \)	None	✓	✓			2312.2.a.f	$1$	$0$	\(0\)	\(1\)	\(1\)	\(5\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(-1+3\beta )q^{5}+(3-\beta )q^{7}+\cdots\)
2312.2.a.m	$2312$	$2$	2312.a	1.a	$1$	$2$	$2$	$18.461$	\(\Q(\sqrt{5}) \)	None	✓				136.2.a.c	$1$	$1$	\(0\)	\(2\)	\(-4\)	\(-2\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-2q^{5}+(-1-\beta )q^{7}+(3+\cdots)q^{9}+\cdots\)
2312.2.a.n	$2312$	$2$	2312.a	1.a	$1$	$2$	$2$	$18.461$	\(\Q(\sqrt{13}) \)	None	✓	✓			2312.2.a.e	$1$	$0$	\(0\)	\(3\)	\(-1\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-\beta q^{5}-\beta q^{7}+(1+3\beta )q^{9}+\cdots\)
2312.2.a.o	$2312$	$2$	2312.a	1.a	$1$	$3$	$3$	$18.461$	\(\Q(\zeta_{18})^+\)	None	✓	✓			2312.2.a.o	$1$	$1$	\(0\)	\(-3\)	\(-6\)	\(-6\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-2\beta _{1}+\beta _{2})q^{3}+(-2-\beta _{1}+\cdots)q^{5}+\cdots\)
2312.2.a.p	$2312$	$2$	2312.a	1.a	$1$	$3$	$3$	$18.461$	\(\Q(\zeta_{18})^+\)	None	✓	✓			2312.2.a.p	$1$	$1$	\(0\)	\(-3\)	\(6\)	\(-3\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1})q^{3}+(2+\beta _{1}-\beta _{2})q^{5}+\cdots\)
2312.2.a.q	$2312$	$2$	2312.a	1.a	$1$	$3$	$3$	$18.461$	\(\Q(\zeta_{18})^+\)	None	✓	✓			2312.2.a.p	$1$	$0$	\(0\)	\(3\)	\(-6\)	\(3\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{3}+(-2-\beta _{1}+\beta _{2})q^{5}+\cdots\)
2312.2.a.r	$2312$	$2$	2312.a	1.a	$1$	$3$	$3$	$18.461$	\(\Q(\zeta_{18})^+\)	None	✓	✓			2312.2.a.o	$1$	$0$	\(0\)	\(3\)	\(6\)	\(6\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+2\beta _{1}-\beta _{2})q^{3}+(2+\beta _{1})q^{5}+(2+\cdots)q^{7}+\cdots\)
2312.2.a.s	$2312$	$2$	2312.a	1.a	$1$	$4$	$4$	$18.461$	\(\Q(\zeta_{16})^+\)	None	✓				136.2.n.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+\beta _{3}q^{5}+(-2\beta _{1}-\beta _{3})q^{7}+\cdots\)
2312.2.a.t	$2312$	$2$	2312.a	1.a	$1$	$4$	$4$	$18.461$	\(\Q(\zeta_{16})^+\)	None	✓				136.2.n.b	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(\beta _{1}+\beta _{3})q^{3}+(-2\beta _{1}+\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
2312.2.a.u	$2312$	$2$	2312.a	1.a	$1$	$6$	$6$	$18.461$	6.6.3418281.1	None	✓	✓	✓		2312.2.a.u	$1$	$1$	\(0\)	\(0\)	\(-6\)	\(-3\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(-1-\beta _{3})q^{5}+(-\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
2312.2.a.v	$2312$	$2$	2312.a	1.a	$1$	$6$	$6$	$18.461$	6.6.3418281.1	None	✓	✓			2312.2.a.u	$1$	$0$	\(0\)	\(0\)	\(6\)	\(3\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(1+\beta _{3})q^{5}+(\beta _{1}+2\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
2312.2.a.w	$2312$	$2$	2312.a	1.a	$1$	$12$	$12$	$18.461$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓		✓		136.2.n.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{10}q^{5}+\beta _{6}q^{7}+(2+\beta _{3}+\cdots)q^{9}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(2312))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(2312)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(289))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(578))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1156))\)\(^{\oplus 2}\)