Space of modular forms of level 338, weight 8, and Character 338.b

• Label: 338.8.b
• Level: $338$
• Weight: $8$
• Character orbit: 338.b
• Rep. character: $\chi_{338}(337,\cdot)$
• Character field: $\Q$
• Dimension: $88$
• Newform subspaces: $11$
• Sturm bound: $364$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 338 = 2 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	338.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(364\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(338, [\chi])\).

	Total	New	Old
Modular forms	332	88	244
Cusp forms	304	88	216
Eisenstein series	28	0	28

Trace form

88 * q - 54 * q^3 - 5632 * q^4 + 57482 * q^9 + 1136 * q^10 + 3456 * q^12 - 5600 * q^14 + 360448 * q^16 + 17252 * q^17 - 13264 * q^22 - 130592 * q^23 - 1165546 * q^25 + 79248 * q^27 + 90702 * q^29 + 585568 * q^30 - 1181932 * q^35 - 3678848 * q^36 - 779920 * q^38 - 72704 * q^40 - 1832224 * q^42 - 2981042 * q^43 - 221184 * q^48 - 9548636 * q^49 - 1932476 * q^51 + 4648578 * q^53 - 5048840 * q^55 + 358400 * q^56 - 14202614 * q^61 + 1860416 * q^62 - 23068672 * q^64 + 7062592 * q^66 - 1104128 * q^68 + 344652 * q^69 - 20903216 * q^74 + 22539702 * q^75 + 5373020 * q^77 + 32332684 * q^79 - 5403568 * q^81 - 17895712 * q^82 - 62233460 * q^87 + 848896 * q^88 + 22099440 * q^90 + 8357888 * q^92 - 13879040 * q^94 - 4853268 * q^95

Decomposition of \(S_{8}^{\mathrm{new}}(338, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
338.8.b.a	$338$	$8$	338.b	13.b	$2$	$2$	$2$	$105.586$	\(\Q(\sqrt{-1}) \)	None					26.8.a.b	$2$	$1$	\(0\)	\(-174\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-8 i q^{2}-87 q^{3}-64 q^{4}-321 i q^{5}+\cdots\)
338.8.b.b	$338$	$8$	338.b	13.b	$2$	$2$	$2$	$105.586$	\(\Q(\sqrt{-1}) \)	None					26.8.a.a	$2$	$0$	\(0\)	\(-78\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-8 i q^{2}-39 q^{3}-64 q^{4}+385 i q^{5}+\cdots\)
338.8.b.c	$338$	$8$	338.b	13.b	$2$	$2$	$2$	$105.586$	\(\Q(\sqrt{-1}) \)	None					26.8.a.c	$2$	$0$	\(0\)	\(-54\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+8 i q^{2}-27 q^{3}-64 q^{4}-245 i q^{5}+\cdots\)
338.8.b.d	$338$	$8$	338.b	13.b	$2$	$2$	$2$	$105.586$	\(\Q(\sqrt{-1}) \)	None					2.8.a.a	$2$	$0$	\(0\)	\(24\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4\beta q^{2}+12 q^{3}-64 q^{4}-105\beta q^{5}+\cdots\)
338.8.b.e	$338$	$8$	338.b	13.b	$2$	$4$	$4$	$105.586$	\(\Q(i, \sqrt{105})\)	None					26.8.a.d	$2$	$0$	\(0\)	\(-24\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-8\beta _{1}q^{2}+(-6+7\beta _{3})q^{3}-2^{6}q^{4}+\cdots\)
338.8.b.f	$338$	$8$	338.b	13.b	$2$	$4$	$4$	$105.586$	\(\Q(i, \sqrt{2305})\)	None					26.8.a.e	$2$	$0$	\(0\)	\(174\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4\beta _{2}q^{2}+(43+\beta _{3})q^{3}-2^{6}q^{4}+(5\beta _{1}+\cdots)q^{5}+\cdots\)
338.8.b.g	$338$	$8$	338.b	13.b	$2$	$6$	$6$	$105.586$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					26.8.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+8\beta _{2}q^{2}+\beta _{3}q^{3}-2^{6}q^{4}+(2\beta _{1}-110\beta _{2}+\cdots)q^{5}+\cdots\)
338.8.b.h	$338$	$8$	338.b	13.b	$2$	$8$	$8$	$105.586$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None					26.8.c.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-8\beta _{2}q^{2}+\beta _{3}q^{3}-2^{6}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
338.8.b.i	$338$	$8$	338.b	13.b	$2$	$16$	$16$	$105.586$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None					26.8.e.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{42}\cdot 13^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{10}q^{2}+\beta _{2}q^{3}-2^{6}q^{4}+(-\beta _{9}+\cdots)q^{5}+\cdots\)
338.8.b.j	$338$	$8$	338.b	13.b	$2$	$18$	$18$	$105.586$	\(\mathbb{Q}[x]/(x^{18} - \cdots)\)	None		✓			338.8.a.o	$2$	$0$	\(0\)	\(-138\)	\(0\)	\(0\)	$2^{6}\cdot 7^{2}\cdot 13^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+8\beta _{3}q^{2}+(-7-\beta _{1}-\beta _{2})q^{3}-2^{6}q^{4}+\cdots\)
338.8.b.k	$338$	$8$	338.b	13.b	$2$	$24$	$24$	$105.586$		None		✓	✓		338.8.a.q	$2$	$0$	\(0\)	\(216\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{8}^{\mathrm{old}}(338, [\chi])\) into lower level spaces

\( S_{8}^{\mathrm{old}}(338, [\chi]) \simeq \) \(S_{8}^{\mathrm{new}}(13, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 2}\)