Space of modular forms of level 162, weight 8, and trivial character

• Label: 162.8.a
• Level: $162$
• Weight: $8$
• Character orbit: 162.a
• Rep. character: $\chi_{162}(1,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $10$
• Sturm bound: $216$
• Trace bound: $5$

Defining parameters

Level:	$N$	$=$	$162 = 2 \cdot 3^{4}$
Weight:	$k$	$=$	$8$
Character orbit:	$[\chi]$	$=$	162.a (trivial)
Character field:			$\Q$
Newform subspaces:			$10$
Sturm bound:			$216$
Trace bound:			$5$
Distinguishing $T_p$:			$5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{8}(\Gamma_0(162))$.

	Total	New	Old
Modular forms	201	28	173
Cusp forms	177	28	149
Eisenstein series	24	0	24

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

$2$	$3$	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$		$52$	$7$	$45$		$46$	$7$	$39$		$6$	$0$	$6$
$+$	$-$	$-$		$49$	$7$	$42$		$43$	$7$	$36$		$6$	$0$	$6$
$-$	$+$	$-$		$50$	$6$	$44$		$44$	$6$	$38$		$6$	$0$	$6$
$-$	$-$	$+$		$50$	$8$	$42$		$44$	$8$	$36$		$6$	$0$	$6$
Plus space		$+$		$102$	$15$	$87$		$90$	$15$	$75$		$12$	$0$	$12$
Minus space		$-$		$99$	$13$	$86$		$87$	$13$	$74$		$12$	$0$	$12$

Trace form

28 * q + 1792 * q^4 + 332 * q^7 + 3984 * q^10 + 16286 * q^13 + 114688 * q^16 + 42590 * q^19 - 70512 * q^22 + 428602 * q^25 + 21248 * q^28 - 504832 * q^31 - 96768 * q^34 - 452590 * q^37 + 254976 * q^40 - 974458 * q^43 + 602016 * q^46 + 4578204 * q^49 + 1042304 * q^52 + 8204976 * q^55 + 206832 * q^58 - 6942430 * q^61 + 7340032 * q^64 + 11626394 * q^67 - 5569728 * q^70 + 11677052 * q^73 + 2725760 * q^76 + 1422440 * q^79 - 2862768 * q^82 - 25452666 * q^85 - 4512768 * q^88 - 18769424 * q^91 + 9219936 * q^94 - 15764302 * q^97

Decomposition of $S_{8}^{\mathrm{new}}(\Gamma_0(162))$ 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
162.8.a.a	$162$	$8$	162.a	1.a	$1$	$1$	$1$	$50.606$	$\Q$	None	✓	✓			162.8.a.a	$1$	$0$	$-8$	$0$	$165$	$-508$		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	$q-8q^{2}+2^{6}q^{4}+165q^{5}-508q^{7}+\cdots$
162.8.a.b	$162$	$8$	162.a	1.a	$1$	$1$	$1$	$50.606$	$\Q$	None	✓	✓			162.8.a.a	$1$	$1$	$8$	$0$	$-165$	$-508$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+8q^{2}+2^{6}q^{4}-165q^{5}-508q^{7}+\cdots$
162.8.a.c	$162$	$8$	162.a	1.a	$1$	$2$	$2$	$50.606$	$\Q(\sqrt{1929})$	None	✓	✓			162.8.a.c	$1$	$0$	$-16$	$0$	$114$	$280$		$+$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	$q-8q^{2}+2^{6}q^{4}+(57-\beta )q^{5}+(140+\cdots)q^{7}+\cdots$
162.8.a.d	$162$	$8$	162.a	1.a	$1$	$2$	$2$	$50.606$	$\Q(\sqrt{1929})$	None	✓	✓			162.8.a.c	$1$	$1$	$16$	$0$	$-114$	$280$		$-$	$+$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	$q+8q^{2}+2^{6}q^{4}+(-57-\beta )q^{5}+(140+\cdots)q^{7}+\cdots$
162.8.a.e	$162$	$8$	162.a	1.a	$1$	$3$	$3$	$50.606$	3.3.69765.1	None	✓				18.8.c.a	$1$	$1$	$-24$	$0$	$54$	$-210$		$+$	$-$	$-$	$2\cdot 3^{4}$	$\mathrm{SU}(2)$	$q-8q^{2}+2^{6}q^{4}+(18+2\beta _{1}-3\beta _{2})q^{5}+\cdots$
162.8.a.f	$162$	$8$	162.a	1.a	$1$	$3$	$3$	$50.606$	3.3.69765.1	None	✓		✓		18.8.c.a	$1$	$1$	$24$	$0$	$-54$	$-210$		$-$	$+$	$-$	$2\cdot 3^{4}$	$\mathrm{SU}(2)$	$q+8q^{2}+2^{6}q^{4}+(-18-2\beta _{1}+3\beta _{2})q^{5}+\cdots$
162.8.a.g	$162$	$8$	162.a	1.a	$1$	$4$	$4$	$50.606$	4.4.43103376.1	None	✓	✓	✓		162.8.a.g	$1$	$1$	$-32$	$0$	$-528$	$560$		$+$	$-$	$-$	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)$	$q-8q^{2}+2^{6}q^{4}+(-132+\beta _{1})q^{5}+\cdots$
162.8.a.h	$162$	$8$	162.a	1.a	$1$	$4$	$4$	$50.606$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓		✓		18.8.c.b	$1$	$0$	$-32$	$0$	$-54$	$44$		$+$	$+$	$+$	$2^{2}\cdot 3^{8}$	$\mathrm{SU}(2)$	$q-8q^{2}+2^{6}q^{4}+(-14-\beta _{1})q^{5}+(12+\cdots)q^{7}+\cdots$
162.8.a.i	$162$	$8$	162.a	1.a	$1$	$4$	$4$	$50.606$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓				18.8.c.b	$1$	$0$	$32$	$0$	$54$	$44$		$-$	$-$	$+$	$2^{2}\cdot 3^{8}$	$\mathrm{SU}(2)$	$q+8q^{2}+2^{6}q^{4}+(14+\beta _{1})q^{5}+(12+\cdots)q^{7}+\cdots$
162.8.a.j	$162$	$8$	162.a	1.a	$1$	$4$	$4$	$50.606$	4.4.43103376.1	None	✓	✓			162.8.a.g	$1$	$0$	$32$	$0$	$528$	$560$		$-$	$-$	$+$	$2^{4}\cdot 3^{8}$	$\mathrm{SU}(2)$	$q+8q^{2}+2^{6}q^{4}+(132+\beta _{1})q^{5}+(140+\cdots)q^{7}+\cdots$

Decomposition of $S_{8}^{\mathrm{old}}(\Gamma_0(162))$ into lower level spaces

$S_{8}^{\mathrm{old}}(\Gamma_0(162)) \simeq$ $S_{8}^{\mathrm{new}}(\Gamma_0(2))$$^{\oplus 5}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(3))$$^{\oplus 8}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(6))$$^{\oplus 4}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(9))$$^{\oplus 6}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(18))$$^{\oplus 3}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(27))$$^{\oplus 4}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(54))$$^{\oplus 2}$$\oplus$$S_{8}^{\mathrm{new}}(\Gamma_0(81))$$^{\oplus 2}$