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}$