Space of modular forms of level 304, weight 4, and Character 304.i

• Label: 304.4.i
• Level: $304$
• Weight: $4$
• Character orbit: 304.i
• Rep. character: $\chi_{304}(49,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $58$
• Newform subspaces: $8$
• Sturm bound: $160$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 304 = 2^{4} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	304.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 19 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 8 \)
Sturm bound:			\(160\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(304, [\chi])\).

	Total	New	Old
Modular forms	252	62	190
Cusp forms	228	58	170
Eisenstein series	24	4	20

Trace form

58 * q - 5 * q^3 - q^5 + 40 * q^7 - 254 * q^9 + 4 * q^11 - q^13 + 7 * q^15 - 77 * q^17 - 166 * q^19 - 28 * q^21 - 41 * q^23 - 626 * q^25 + 382 * q^27 - q^29 - 872 * q^31 - 10 * q^33 + 528 * q^35 - 4 * q^37 + 370 * q^39 - 139 * q^41 - 167 * q^43 + 300 * q^45 + 103 * q^47 + 1994 * q^49 - 293 * q^51 - 197 * q^53 + 1128 * q^55 + 173 * q^57 - 1061 * q^59 - 1013 * q^61 + 488 * q^63 + 1270 * q^65 - 1667 * q^67 + 1106 * q^69 + 1325 * q^71 - 523 * q^73 - 1900 * q^75 - 1560 * q^77 + 257 * q^79 - 2533 * q^81 - 276 * q^83 - 519 * q^85 - 2142 * q^87 - 853 * q^89 - 2824 * q^91 + 884 * q^93 - 4421 * q^95 + 565 * q^97 + 2004 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(304, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
304.4.i.a	$304$	$4$	304.i	19.c	$3$	$2$	$1$	$17.937$	\(\Q(\sqrt{-3}) \)	None					38.4.c.b	$2$	$0$	\(0\)	\(-5\)	\(12\)	\(-16\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-5+5\zeta_{6})q^{3}+(12-12\zeta_{6})q^{5}+\cdots\)
304.4.i.b	$304$	$4$	304.i	19.c	$3$	$2$	$1$	$17.937$	\(\Q(\sqrt{-3}) \)	None					38.4.c.a	$2$	$0$	\(0\)	\(5\)	\(-3\)	\(64\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5-5\zeta_{6})q^{3}+(-3+3\zeta_{6})q^{5}+2^{5}q^{7}+\cdots\)
304.4.i.c	$304$	$4$	304.i	19.c	$3$	$4$	$2$	$17.937$	\(\Q(\sqrt{-3}, \sqrt{55})\)	None					19.4.c.a	$2$	$0$	\(0\)	\(0\)	\(14\)	\(28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}+(7+\beta _{1}+7\beta _{2})q^{5}+(7+\beta _{3})q^{7}+\cdots\)
304.4.i.d	$304$	$4$	304.i	19.c	$3$	$4$	$2$	$17.937$	\(\Q(\sqrt{-3}, \sqrt{73})\)	None					19.4.c.b	$2$	$0$	\(0\)	\(2\)	\(-19\)	\(-40\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{2})q^{3}+(-9-\beta _{1}+10\beta _{2}-\beta _{3})q^{5}+\cdots\)
304.4.i.e	$304$	$4$	304.i	19.c	$3$	$6$	$3$	$17.937$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None					38.4.c.c	$2$	$0$	\(0\)	\(5\)	\(-1\)	\(-52\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{1}+2\beta _{4})q^{3}+(\beta _{1}-\beta _{3}-\beta _{4}+\cdots)q^{5}+\cdots\)
304.4.i.f	$304$	$4$	304.i	19.c	$3$	$10$	$5$	$17.937$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					76.4.e.a	$2$	$0$	\(0\)	\(-7\)	\(-4\)	\(20\)	$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(\beta _{3}+\beta _{7}+\beta _{8}+\cdots)q^{5}+\cdots\)
304.4.i.g	$304$	$4$	304.i	19.c	$3$	$14$	$7$	$17.937$	\(\mathbb{Q}[x]/(x^{14} - \cdots)\)	None					152.4.i.a	$2$	$0$	\(0\)	\(-5\)	\(5\)	\(28\)	$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{7})q^{3}+(\beta _{7}+\beta _{8})q^{5}+(2+\beta _{2}+\cdots)q^{7}+\cdots\)
304.4.i.h	$304$	$4$	304.i	19.c	$3$	$16$	$8$	$17.937$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None			✓		152.4.i.b	$2$	$0$	\(0\)	\(0\)	\(-5\)	\(8\)	$2^{12}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{1}q^{3}+(-1-\beta _{2}+\beta _{6})q^{5}+(1+\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(304, [\chi])\) into lower level spaces

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