Space of modular forms of level 4312, weight 2, and trivial character

• Label: 4312.2.a
• Level: $4312$
• Weight: $2$
• Character orbit: 4312.a
• Rep. character: $\chi_{4312}(1,\cdot)$
• Character field: $\Q$
• Dimension: $103$
• Newform subspaces: $36$
• Sturm bound: $1344$
• Trace bound: $15$

Defining parameters

Level:	\( N \)	\(=\)	\( 4312 = 2^{3} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4312.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 36 \)
Sturm bound:			\(1344\)
Trace bound:			\(15\)
Distinguishing \(T_p\):			\(3\), \(5\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(4312))\).

	Total	New	Old
Modular forms	704	103	601
Cusp forms	641	103	538
Eisenstein series	63	0	63

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

\(2\)	\(7\)	\(11\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(84\)	\(15\)	\(69\)		\(77\)	\(15\)	\(62\)		\(7\)	\(0\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)		\(88\)	\(11\)	\(77\)		\(80\)	\(11\)	\(69\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)		\(92\)	\(11\)	\(81\)		\(84\)	\(11\)	\(73\)		\(8\)	\(0\)	\(8\)
\(+\)	\(-\)	\(-\)	\(+\)		\(88\)	\(14\)	\(74\)		\(80\)	\(14\)	\(66\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)		\(92\)	\(17\)	\(75\)		\(84\)	\(17\)	\(67\)		\(8\)	\(0\)	\(8\)
\(-\)	\(+\)	\(-\)	\(+\)		\(88\)	\(9\)	\(79\)		\(80\)	\(9\)	\(71\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(+\)	\(+\)		\(84\)	\(10\)	\(74\)		\(76\)	\(10\)	\(66\)		\(8\)	\(0\)	\(8\)
\(-\)	\(-\)	\(-\)	\(-\)		\(88\)	\(16\)	\(72\)		\(80\)	\(16\)	\(64\)		\(8\)	\(0\)	\(8\)
Plus space			\(+\)		\(344\)	\(48\)	\(296\)		\(313\)	\(48\)	\(265\)		\(31\)	\(0\)	\(31\)
Minus space			\(-\)		\(360\)	\(55\)	\(305\)		\(328\)	\(55\)	\(273\)		\(32\)	\(0\)	\(32\)

Trace form

103 * q + 2 * q^3 - 4 * q^5 + 101 * q^9 - 3 * q^11 + 6 * q^13 - 14 * q^15 - 2 * q^17 + 4 * q^19 - 6 * q^23 + 107 * q^25 - 10 * q^27 + 14 * q^29 + 18 * q^31 - 2 * q^33 - 12 * q^37 - 16 * q^39 + 2 * q^41 - 10 * q^45 - 16 * q^47 - 20 * q^51 + 34 * q^53 + 40 * q^57 + 14 * q^59 - 18 * q^61 + 52 * q^65 - 14 * q^67 + 2 * q^69 - 18 * q^71 - 6 * q^73 + 24 * q^75 + 28 * q^79 + 127 * q^81 - 24 * q^83 + 16 * q^85 + 16 * q^87 + 8 * q^89 - 30 * q^93 + 64 * q^95 + 12 * q^97 - 9 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(4312))\) 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	7	11
4312.2.a.a	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓	✓			4312.2.a.a	$1$	$2$	\(0\)	\(-2\)	\(-4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-4q^{5}+q^{9}+q^{11}-2q^{13}+\cdots\)
4312.2.a.b	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓				616.2.a.e	$1$	$0$	\(0\)	\(-2\)	\(-2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2q^{5}+q^{9}-q^{11}+4q^{15}+\cdots\)
4312.2.a.c	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓	✓			4312.2.a.c	$1$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}+q^{9}-q^{11}+2q^{13}-2q^{17}+\cdots\)
4312.2.a.d	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓				616.2.q.a	$1$	$1$	\(0\)	\(-1\)	\(2\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}-2q^{9}-q^{11}+3q^{13}+\cdots\)
4312.2.a.e	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓				616.2.a.d	$1$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{9}-q^{11}+6q^{13}+2q^{19}+4q^{23}+\cdots\)
4312.2.a.f	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓				616.2.a.c	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-3q^{9}-q^{11}-2q^{13}+2q^{17}+\cdots\)
4312.2.a.g	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓				616.2.q.a	$1$	$0$	\(0\)	\(1\)	\(-2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{5}-2q^{9}-q^{11}-3q^{13}+\cdots\)
4312.2.a.h	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓				616.2.a.b	$1$	$1$	\(0\)	\(1\)	\(1\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{5}-2q^{9}+q^{11}+q^{15}+2q^{17}+\cdots\)
4312.2.a.i	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓				616.2.a.a	$1$	$0$	\(0\)	\(2\)	\(-2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-2q^{5}+q^{9}-q^{11}-4q^{13}+\cdots\)
4312.2.a.j	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓	✓			4312.2.a.c	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+q^{9}-q^{11}-2q^{13}+2q^{17}+\cdots\)
4312.2.a.k	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓	✓			4312.2.a.a	$1$	$0$	\(0\)	\(2\)	\(4\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}+4q^{5}+q^{9}+q^{11}+2q^{13}+\cdots\)
4312.2.a.l	$4312$	$2$	4312.a	1.a	$1$	$1$	$1$	$34.431$	\(\Q\)	None	✓				88.2.a.a	$1$	$0$	\(0\)	\(3\)	\(3\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+3q^{5}+6q^{9}-q^{11}+9q^{15}+\cdots\)
4312.2.a.m	$4312$	$2$	4312.a	1.a	$1$	$2$	$2$	$34.431$	\(\Q(\sqrt{2}) \)	None	✓				616.2.q.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{3}-\beta q^{5}-2\beta q^{9}+q^{11}+\cdots\)
4312.2.a.n	$4312$	$2$	4312.a	1.a	$1$	$2$	$2$	$34.431$	\(\Q(\sqrt{17}) \)	None	✓				88.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(-3\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{3}+(-2+\beta )q^{5}+(1+\beta )q^{9}-q^{11}+\cdots\)
4312.2.a.o	$4312$	$2$	4312.a	1.a	$1$	$2$	$2$	$34.431$	\(\Q(\sqrt{2}) \)	None	✓	✓			4312.2.a.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{5}-3q^{9}-q^{11}+\beta q^{17}+\beta q^{19}+\cdots\)
4312.2.a.p	$4312$	$2$	4312.a	1.a	$1$	$2$	$2$	$34.431$	\(\Q(\sqrt{2}) \)	None	✓	✓			4312.2.a.p	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2\beta q^{5}-3q^{9}+q^{11}-\beta q^{13}+2\beta q^{17}+\cdots\)
4312.2.a.q	$4312$	$2$	4312.a	1.a	$1$	$2$	$2$	$34.431$	\(\Q(\sqrt{2}) \)	None	✓	✓			4312.2.a.q	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{5}-q^{9}-q^{11}+2q^{15}+\cdots\)
4312.2.a.r	$4312$	$2$	4312.a	1.a	$1$	$2$	$2$	$34.431$	\(\Q(\sqrt{2}) \)	None	✓	✓			4312.2.a.r	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-\beta q^{5}-q^{9}+q^{11}+2\beta q^{13}+\cdots\)
4312.2.a.s	$4312$	$2$	4312.a	1.a	$1$	$2$	$2$	$34.431$	\(\Q(\sqrt{2}) \)	None	✓	✓			4312.2.a.s	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+\beta q^{5}-q^{9}+q^{11}+2q^{15}+\cdots\)
4312.2.a.t	$4312$	$2$	4312.a	1.a	$1$	$2$	$2$	$34.431$	\(\Q(\sqrt{17}) \)	None	✓				616.2.a.f	$1$	$0$	\(0\)	\(1\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+(2-\beta )q^{5}+(1+\beta )q^{9}+q^{11}+\cdots\)
4312.2.a.u	$4312$	$2$	4312.a	1.a	$1$	$2$	$2$	$34.431$	\(\Q(\sqrt{2}) \)	None	✓				616.2.q.b	$1$	$1$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{3}-\beta q^{5}+2\beta q^{9}+q^{11}+\cdots\)
4312.2.a.v	$4312$	$2$	4312.a	1.a	$1$	$3$	$3$	$34.431$	\(\Q(\zeta_{14})^+\)	None	✓		✓		616.2.q.c	$1$	$1$	\(0\)	\(-5\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-2-\beta _{2})q^{3}+(1-3\beta _{1}+2\beta _{2})q^{5}+\cdots\)
4312.2.a.w	$4312$	$2$	4312.a	1.a	$1$	$3$	$3$	$34.431$	3.3.229.1	None	✓				616.2.a.g	$1$	$0$	\(0\)	\(-1\)	\(-1\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+\beta _{2}q^{5}+(1-\beta _{1})q^{9}+q^{11}+\cdots\)
4312.2.a.x	$4312$	$2$	4312.a	1.a	$1$	$3$	$3$	$34.431$	\(\Q(\zeta_{14})^+\)	None	✓				616.2.q.c	$1$	$0$	\(0\)	\(5\)	\(2\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(2-\beta _{1})q^{3}+(2-\beta _{1}+3\beta _{2})q^{5}+(3+\cdots)q^{9}+\cdots\)
4312.2.a.y	$4312$	$2$	4312.a	1.a	$1$	$4$	$4$	$34.431$	4.4.89289.1	None	✓				616.2.q.d	$1$	$1$	\(0\)	\(-2\)	\(-4\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-1+\beta _{1})q^{5}+(3+\cdots)q^{9}+\cdots\)
4312.2.a.z	$4312$	$2$	4312.a	1.a	$1$	$4$	$4$	$34.431$	4.4.11348.1	None	✓				616.2.a.h	$1$	$1$	\(0\)	\(-1\)	\(-5\)	\(0\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+(-1+\beta _{1})q^{5}+(2+\beta _{2}-\beta _{3})q^{9}+\cdots\)
4312.2.a.ba	$4312$	$2$	4312.a	1.a	$1$	$4$	$4$	$34.431$	\(\Q(\sqrt{10 +2 \sqrt{17}})\)	None	✓	✓			4312.2.a.ba	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{3}+\beta _{2}q^{5}-\beta _{3}q^{9}+q^{11}+(-\beta _{1}+\cdots)q^{13}+\cdots\)
4312.2.a.bb	$4312$	$2$	4312.a	1.a	$1$	$4$	$4$	$34.431$	\(\Q(\sqrt{7}, \sqrt{15})\)	None	✓	✓	✓		4312.2.a.bb	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}-\beta _{2}q^{5}+(2-\beta _{3})q^{9}+q^{11}+\cdots\)
4312.2.a.bc	$4312$	$2$	4312.a	1.a	$1$	$4$	$4$	$34.431$	\(\Q(\zeta_{24})^+\)	None	✓	✓			4312.2.a.bc	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{2}q^{3}+\beta _{1}q^{5}+3q^{9}+q^{11}+(-\beta _{1}+\cdots)q^{13}+\cdots\)
4312.2.a.bd	$4312$	$2$	4312.a	1.a	$1$	$4$	$4$	$34.431$	4.4.89289.1	None	✓		✓		616.2.q.d	$1$	$0$	\(0\)	\(2\)	\(4\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(1-\beta _{1})q^{5}+(3+\beta _{1}+\cdots)q^{9}+\cdots\)
4312.2.a.be	$4312$	$2$	4312.a	1.a	$1$	$5$	$5$	$34.431$	5.5.352076.1	None	✓		✓		616.2.q.e	$1$	$1$	\(0\)	\(-3\)	\(-4\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{2})q^{3}+(-1+\beta _{1}-\beta _{4})q^{5}+\cdots\)
4312.2.a.bf	$4312$	$2$	4312.a	1.a	$1$	$5$	$5$	$34.431$	5.5.559701.1	None	✓				616.2.q.f	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(0\)		$-$	$+$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}+(1-\beta _{1}+\beta _{2})q^{5}+\beta _{2}q^{9}+\cdots\)
4312.2.a.bg	$4312$	$2$	4312.a	1.a	$1$	$5$	$5$	$34.431$	5.5.559701.1	None	✓		✓		616.2.q.f	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(0\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(-1+\beta _{1}-\beta _{2})q^{5}+\beta _{2}q^{9}+\cdots\)
4312.2.a.bh	$4312$	$2$	4312.a	1.a	$1$	$5$	$5$	$34.431$	5.5.352076.1	None	✓		✓		616.2.q.e	$1$	$0$	\(0\)	\(3\)	\(4\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta _{2})q^{3}+(1-\beta _{1}+\beta _{4})q^{5}+(1+\cdots)q^{9}+\cdots\)
4312.2.a.bi	$4312$	$2$	4312.a	1.a	$1$	$8$	$8$	$34.431$	8.8.\(\cdots\).1	None	✓	✓	✓		4312.2.a.bi	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{3}+(\beta _{3}+\beta _{4}-\beta _{5})q^{5}+(1-\beta _{2}+\cdots)q^{9}+\cdots\)
4312.2.a.bj	$4312$	$2$	4312.a	1.a	$1$	$12$	$12$	$34.431$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None	✓	✓	✓		4312.2.a.bj	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$2^{7}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}-\beta _{9}q^{5}+(3+\beta _{2})q^{9}-q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(4312))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(4312)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(44))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(56))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(88))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(98))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(196))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(308))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(392))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(539))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(616))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1078))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2156))\)\(^{\oplus 2}\)