Space of modular forms of level 304, weight 3, and Character 304.z

• Label: 304.3.z
• Level: $304$
• Weight: $3$
• Character orbit: 304.z
• Rep. character: $\chi_{304}(33,\cdot)$
• Character field: $\Q(\zeta_{18})$
• Dimension: $114$
• Newform subspaces: $4$
• Sturm bound: $120$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	304.z (of order \(18\) and degree \(6\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q(\zeta_{18})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(120\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(304, [\chi])\).

	Total	New	Old
Modular forms	516	126	390
Cusp forms	444	114	330
Eisenstein series	72	12	60

Trace form

114 * q + 6 * q^3 - 6 * q^5 + 3 * q^7 + 6 * q^9 + 3 * q^11 - 6 * q^13 + 6 * q^15 - 6 * q^17 - 42 * q^19 + 21 * q^21 + 6 * q^23 - 6 * q^25 + 9 * q^27 - 6 * q^29 + 9 * q^31 + 81 * q^33 - 69 * q^35 + 12 * q^39 - 66 * q^41 + 198 * q^43 - 3 * q^45 + 150 * q^47 - 276 * q^49 + 150 * q^51 - 30 * q^53 + 75 * q^55 - 6 * q^57 + 6 * q^59 + 66 * q^61 + 219 * q^63 - 369 * q^65 + 486 * q^67 + 423 * q^69 + 150 * q^71 - 198 * q^73 + 282 * q^77 - 330 * q^79 + 99 * q^81 - 429 * q^83 - 438 * q^85 - 429 * q^87 + 354 * q^89 - 909 * q^91 - 33 * q^93 - 402 * q^95 - 258 * q^97 - 237 * q^99

Decomposition of \(S_{3}^{\mathrm{new}}(304, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
304.3.z.a	$304$	$3$	304.z	19.f	$18$	$12$	$2$	$8.283$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					19.3.f.a	$2$	$0$	\(0\)	\(0\)	\(-6\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{18}]$	\(q+(-\beta _{1}+\beta _{4}+\beta _{5}-\beta _{6}+\beta _{7}+\beta _{8}+\cdots)q^{3}+\cdots\)
304.3.z.b	$304$	$3$	304.z	19.f	$18$	$18$	$3$	$8.283$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None					76.3.j.a	$2$	$0$	\(0\)	\(6\)	\(0\)	\(-9\)	$1$	$\mathrm{SU}(2)[C_{18}]$	\(q+(1-\beta _{4}-\beta _{5}+\beta _{7}+\beta _{10}-\beta _{12}+\cdots)q^{3}+\cdots\)
304.3.z.c	$304$	$3$	304.z	19.f	$18$	$24$	$4$	$8.283$		None					38.3.f.a	$2$	$0$	\(0\)	\(6\)	\(0\)	\(18\)		$\mathrm{SU}(2)[C_{18}]$
304.3.z.d	$304$	$3$	304.z	19.f	$18$	$60$	$10$	$8.283$		None			✓		152.3.r.a	$2$	$0$	\(0\)	\(-6\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{18}]$

Decomposition of \(S_{3}^{\mathrm{old}}(304, [\chi])\) into lower level spaces

\( S_{3}^{\mathrm{old}}(304, [\chi]) \simeq \) \(S_{3}^{\mathrm{new}}(19, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(38, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(76, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{3}^{\mathrm{new}}(152, [\chi])\)\(^{\oplus 2}\)