Space of modular forms of level 8424, weight 2, and trivial character

• Label: 8424.2.a
• Level: $8424$
• Weight: $2$
• Character orbit: 8424.a
• Rep. character: $\chi_{8424}(1,\cdot)$
• Character field: $\Q$
• Dimension: $144$
• Newform subspaces: $32$
• Sturm bound: $3024$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 8424 = 2^{3} \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8424.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 32 \)
Sturm bound:			\(3024\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(8424))\).

	Total	New	Old
Modular forms	1560	144	1416
Cusp forms	1465	144	1321
Eisenstein series	95	0	95

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

\(2\)	\(3\)	\(13\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(16\)
\(+\)	\(+\)	\(-\)	\(-\)	\(21\)
\(+\)	\(-\)	\(+\)	\(-\)	\(20\)
\(+\)	\(-\)	\(-\)	\(+\)	\(15\)
\(-\)	\(+\)	\(+\)	\(-\)	\(18\)
\(-\)	\(+\)	\(-\)	\(+\)	\(17\)
\(-\)	\(-\)	\(+\)	\(+\)	\(18\)
\(-\)	\(-\)	\(-\)	\(-\)	\(19\)
Plus space			\(+\)	\(66\)
Minus space			\(-\)	\(78\)

Trace form

\( 144 q - 12 q^{19} + 144 q^{25} - 12 q^{43} + 120 q^{49} - 24 q^{55} - 24 q^{61} - 12 q^{67} + 36 q^{73} - 24 q^{79} + 36 q^{97}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(8424))\) 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	13
8424.2.a.a	$8424$	$2$	8424.a	1.a	$1$	$1$	$1$	$67.266$	\(\Q\)	None	✓	✓			8424.2.a.a	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-3\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}-3q^{7}+3q^{11}+q^{13}+2q^{17}+\cdots\)
8424.2.a.b	$8424$	$2$	8424.a	1.a	$1$	$1$	$1$	$67.266$	\(\Q\)	None	✓				936.2.q.a	$1$	$0$	\(0\)	\(0\)	\(-4\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-4q^{5}+2q^{7}+q^{11}-q^{13}-3q^{17}+\cdots\)
8424.2.a.c	$8424$	$2$	8424.a	1.a	$1$	$1$	$1$	$67.266$	\(\Q\)	None	✓				936.2.q.c	$1$	$2$	\(0\)	\(0\)	\(-2\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-4q^{7}-4q^{11}-q^{13}-3q^{17}+\cdots\)
8424.2.a.d	$8424$	$2$	8424.a	1.a	$1$	$1$	$1$	$67.266$	\(\Q\)	None	✓				936.2.q.b	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{5}-2q^{7}+3q^{11}+q^{13}-q^{17}+\cdots\)
8424.2.a.e	$8424$	$2$	8424.a	1.a	$1$	$1$	$1$	$67.266$	\(\Q\)	None	✓				936.2.q.c	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-4q^{7}+4q^{11}-q^{13}+3q^{17}+\cdots\)
8424.2.a.f	$8424$	$2$	8424.a	1.a	$1$	$1$	$1$	$67.266$	\(\Q\)	None	✓				936.2.q.b	$1$	$1$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-2q^{7}-3q^{11}+q^{13}+q^{17}+\cdots\)
8424.2.a.g	$8424$	$2$	8424.a	1.a	$1$	$1$	$1$	$67.266$	\(\Q\)	None	✓	✓			8424.2.a.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-3\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}-3q^{7}-3q^{11}+q^{13}-2q^{17}+\cdots\)
8424.2.a.h	$8424$	$2$	8424.a	1.a	$1$	$1$	$1$	$67.266$	\(\Q\)	None	✓				936.2.q.a	$1$	$0$	\(0\)	\(0\)	\(4\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+4q^{5}+2q^{7}-q^{11}-q^{13}+3q^{17}+\cdots\)
8424.2.a.i	$8424$	$2$	8424.a	1.a	$1$	$2$	$2$	$67.266$	\(\Q(\sqrt{7}) \)	None	✓	✓			8424.2.a.i	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}-q^{7}+\beta q^{11}-q^{13}+\cdots\)
8424.2.a.j	$8424$	$2$	8424.a	1.a	$1$	$2$	$2$	$67.266$	\(\Q(\sqrt{3}) \)	None	✓	✓			8424.2.a.j	$1$	$0$	\(0\)	\(0\)	\(-2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{5}+\beta q^{7}+(2+\beta )q^{11}+\cdots\)
8424.2.a.k	$8424$	$2$	8424.a	1.a	$1$	$2$	$2$	$67.266$	\(\Q(\sqrt{13}) \)	None	✓	✓			8424.2.a.k	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(5\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{5}+(3-\beta )q^{7}+\beta q^{11}-q^{13}+\cdots\)
8424.2.a.l	$8424$	$2$	8424.a	1.a	$1$	$2$	$2$	$67.266$	\(\Q(\sqrt{13}) \)	None	✓	✓			8424.2.a.k	$1$	$1$	\(0\)	\(0\)	\(1\)	\(5\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{5}+(3-\beta )q^{7}-\beta q^{11}-q^{13}+\cdots\)
8424.2.a.m	$8424$	$2$	8424.a	1.a	$1$	$2$	$2$	$67.266$	\(\Q(\sqrt{7}) \)	None	✓	✓			8424.2.a.i	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}-q^{7}+\beta q^{11}-q^{13}+\beta q^{19}+\cdots\)
8424.2.a.n	$8424$	$2$	8424.a	1.a	$1$	$2$	$2$	$67.266$	\(\Q(\sqrt{3}) \)	None	✓	✓			8424.2.a.j	$1$	$1$	\(0\)	\(0\)	\(2\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{5}-\beta q^{7}+(-2+\beta )q^{11}+\cdots\)
8424.2.a.o	$8424$	$2$	8424.a	1.a	$1$	$4$	$4$	$67.266$	4.4.9225.1	None	✓	✓			8424.2.a.o	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(1\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+\beta _{1}q^{7}+(1+\beta _{2})q^{11}+\cdots\)
8424.2.a.p	$8424$	$2$	8424.a	1.a	$1$	$4$	$4$	$67.266$	4.4.142776.1	None	✓	✓			8424.2.a.p	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+\beta _{2}q^{11}+\cdots\)
8424.2.a.q	$8424$	$2$	8424.a	1.a	$1$	$4$	$4$	$67.266$	4.4.142776.1	None	✓	✓			8424.2.a.p	$1$	$0$	\(0\)	\(0\)	\(1\)	\(3\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}+(1+\beta _{1}+\beta _{3})q^{7}-\beta _{2}q^{11}+\cdots\)
8424.2.a.r	$8424$	$2$	8424.a	1.a	$1$	$4$	$4$	$67.266$	4.4.9225.1	None	✓	✓			8424.2.a.o	$1$	$1$	\(0\)	\(0\)	\(3\)	\(1\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(1+\beta _{1})q^{5}-\beta _{1}q^{7}+(-1+\beta _{3})q^{11}+\cdots\)
8424.2.a.s	$8424$	$2$	8424.a	1.a	$1$	$5$	$5$	$67.266$	5.5.559701.1	None	✓	✓			8424.2.a.s	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-4\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}+\beta _{4}q^{11}+\cdots\)
8424.2.a.t	$8424$	$2$	8424.a	1.a	$1$	$5$	$5$	$67.266$	5.5.559701.1	None	✓	✓			8424.2.a.s	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-4\)		$+$	$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{5}+(-1+\beta _{1}+\beta _{2})q^{7}-\beta _{4}q^{11}+\cdots\)
8424.2.a.u	$8424$	$2$	8424.a	1.a	$1$	$6$	$6$	$67.266$	6.6.20396961.1	None	✓				936.2.q.d	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{3}q^{5}-\beta _{5}q^{7}+(-1+\beta _{2}-\beta _{4}+\cdots)q^{11}+\cdots\)
8424.2.a.v	$8424$	$2$	8424.a	1.a	$1$	$6$	$6$	$67.266$	6.6.20396961.1	None	✓		✓		936.2.q.d	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{3}q^{5}-\beta _{5}q^{7}+(1-\beta _{2}+\beta _{4})q^{11}+\cdots\)
8424.2.a.w	$8424$	$2$	8424.a	1.a	$1$	$8$	$8$	$67.266$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			8424.2.a.w	$1$	$1$	\(0\)	\(0\)	\(-4\)	\(-6\)		$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(-\beta _{4}+\cdots)q^{11}+\cdots\)
8424.2.a.x	$8424$	$2$	8424.a	1.a	$1$	$8$	$8$	$67.266$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		8424.2.a.x	$1$	$1$	\(0\)	\(0\)	\(-2\)	\(6\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{7}q^{5}+(1+\beta _{6})q^{7}+(-1+\beta _{1}-\beta _{6}+\cdots)q^{11}+\cdots\)
8424.2.a.y	$8424$	$2$	8424.a	1.a	$1$	$8$	$8$	$67.266$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		936.2.q.f	$1$	$1$	\(0\)	\(0\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{6}q^{5}-\beta _{3}q^{7}+(-1+\beta _{1})q^{11}+\cdots\)
8424.2.a.z	$8424$	$2$	8424.a	1.a	$1$	$8$	$8$	$67.266$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				936.2.q.e	$1$	$0$	\(0\)	\(0\)	\(-1\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{5}-\beta _{3}q^{7}+(1+\beta _{2}-\beta _{7})q^{11}+\cdots\)
8424.2.a.ba	$8424$	$2$	8424.a	1.a	$1$	$8$	$8$	$67.266$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓				936.2.q.f	$1$	$1$	\(0\)	\(0\)	\(1\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{6}q^{5}-\beta _{3}q^{7}+(1-\beta _{1})q^{11}-q^{13}+\cdots\)
8424.2.a.bb	$8424$	$2$	8424.a	1.a	$1$	$8$	$8$	$67.266$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓		✓		936.2.q.e	$1$	$0$	\(0\)	\(0\)	\(1\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{5}-\beta _{3}q^{7}+(-1-\beta _{2}+\beta _{7})q^{11}+\cdots\)
8424.2.a.bc	$8424$	$2$	8424.a	1.a	$1$	$8$	$8$	$67.266$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			8424.2.a.x	$1$	$0$	\(0\)	\(0\)	\(2\)	\(6\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{7}q^{5}+(1+\beta _{6})q^{7}+(1-\beta _{1}+\beta _{6}+\cdots)q^{11}+\cdots\)
8424.2.a.bd	$8424$	$2$	8424.a	1.a	$1$	$8$	$8$	$67.266$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓			8424.2.a.w	$1$	$0$	\(0\)	\(0\)	\(4\)	\(-6\)		$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{5}+(-1+\beta _{2})q^{7}+(\beta _{4}-\beta _{5}+\cdots)q^{11}+\cdots\)
8424.2.a.be	$8424$	$2$	8424.a	1.a	$1$	$11$	$11$	$67.266$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		936.2.q.g	$1$	$0$	\(0\)	\(0\)	\(-3\)	\(4\)		$-$	$-$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q+\beta _{8}q^{5}-\beta _{6}q^{7}-\beta _{4}q^{11}+q^{13}+\cdots\)
8424.2.a.bf	$8424$	$2$	8424.a	1.a	$1$	$11$	$11$	$67.266$	\(\mathbb{Q}[x]/(x^{11} - \cdots)\)	None	✓		✓		936.2.q.g	$1$	$0$	\(0\)	\(0\)	\(3\)	\(4\)		$+$	$+$	$-$	$-$	$3^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{8}q^{5}-\beta _{6}q^{7}+\beta _{4}q^{11}+q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(8424))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(8424)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(26))\)\(^{\oplus 15}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(27))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(36))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(39))\)\(^{\oplus 16}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(52))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(54))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(72))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(78))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(81))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(104))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(108))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(117))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(156))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(162))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(216))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(234))\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(312))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(324))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(351))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(468))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(648))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(702))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(936))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1053))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1404))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2106))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(2808))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(4212))\)\(^{\oplus 2}\)