Space of modular forms of level 448, weight 4, and Character 448.i

• Label: 448.4.i
• Level: $448$
• Weight: $4$
• Character orbit: 448.i
• Rep. character: $\chi_{448}(65,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $92$
• Newform subspaces: $17$
• Sturm bound: $256$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	448.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 17 \)
Sturm bound:			\(256\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(448, [\chi])\).

	Total	New	Old
Modular forms	408	100	308
Cusp forms	360	92	268
Eisenstein series	48	8	40

Trace form

92 * q + 2 * q^5 - 380 * q^9 - 136 * q^13 - 2 * q^17 - 290 * q^21 - 952 * q^25 - 392 * q^29 + 106 * q^33 - 494 * q^37 - 8 * q^41 - 444 * q^45 - 4 * q^49 - 374 * q^53 + 100 * q^57 + 2162 * q^61 - 252 * q^65 - 100 * q^69 - 2 * q^73 - 2434 * q^77 - 3718 * q^81 - 2844 * q^85 - 850 * q^89 - 1954 * q^93 + 2968 * q^97

Decomposition of \(S_{4}^{\mathrm{new}}(448, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
448.4.i.a	$448$	$4$	448.i	7.c	$3$	$2$	$1$	$26.433$	\(\Q(\sqrt{-3}) \)	None					7.4.c.a	$2$	$0$	\(0\)	\(-7\)	\(7\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-7+7\zeta_{6})q^{3}+7\zeta_{6}q^{5}+(-7-14\zeta_{6})q^{7}+\cdots\)
448.4.i.b	$448$	$4$	448.i	7.c	$3$	$2$	$1$	$26.433$	\(\Q(\sqrt{-3}) \)	None					14.4.c.a	$2$	$1$	\(0\)	\(-5\)	\(-9\)	\(-28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-5+5\zeta_{6})q^{3}-9\zeta_{6}q^{5}+(-21+\cdots)q^{7}+\cdots\)
448.4.i.c	$448$	$4$	448.i	7.c	$3$	$2$	$1$	$26.433$	\(\Q(\sqrt{-3}) \)	None					14.4.c.b	$2$	$0$	\(0\)	\(-1\)	\(7\)	\(-20\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+7\zeta_{6}q^{5}+(-1-18\zeta_{6})q^{7}+\cdots\)
448.4.i.d	$448$	$4$	448.i	7.c	$3$	$2$	$1$	$26.433$	\(\Q(\sqrt{-3}) \)	None					14.4.c.b	$2$	$0$	\(0\)	\(1\)	\(7\)	\(20\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+7\zeta_{6}q^{5}+(1+18\zeta_{6})q^{7}+\cdots\)
448.4.i.e	$448$	$4$	448.i	7.c	$3$	$2$	$1$	$26.433$	\(\Q(\sqrt{-3}) \)	None					14.4.c.a	$2$	$0$	\(0\)	\(5\)	\(-9\)	\(28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(5-5\zeta_{6})q^{3}-9\zeta_{6}q^{5}+(21-14\zeta_{6})q^{7}+\cdots\)
448.4.i.f	$448$	$4$	448.i	7.c	$3$	$2$	$1$	$26.433$	\(\Q(\sqrt{-3}) \)	None					7.4.c.a	$2$	$0$	\(0\)	\(7\)	\(7\)	\(28\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(7-7\zeta_{6})q^{3}+7\zeta_{6}q^{5}+(7+14\zeta_{6})q^{7}+\cdots\)
448.4.i.g	$448$	$4$	448.i	7.c	$3$	$4$	$2$	$26.433$	\(\Q(\sqrt{-3}, \sqrt{37})\)	None					28.4.e.a	$2$	$0$	\(0\)	\(0\)	\(-14\)	\(-24\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{3}+(-7+7\beta _{1}+2\beta _{2}+2\beta _{3})q^{5}+\cdots\)
448.4.i.h	$448$	$4$	448.i	7.c	$3$	$4$	$2$	$26.433$	\(\Q(\sqrt{-3}, \sqrt{37})\)	None					28.4.e.a	$2$	$0$	\(0\)	\(0\)	\(-14\)	\(24\)	$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{2}q^{3}+(-7+7\beta _{1}-2\beta _{2}-2\beta _{3})q^{5}+\cdots\)
448.4.i.i	$448$	$4$	448.i	7.c	$3$	$4$	$2$	$26.433$	\(\Q(\sqrt{-3}, \sqrt{7})\)	None					224.4.i.a	$4$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{1}q^{3}-\beta _{2}q^{5}+(8\beta _{1}+5\beta _{3})q^{7}+\cdots\)
448.4.i.j	$448$	$4$	448.i	7.c	$3$	$6$	$3$	$26.433$	6.0.11163123.4	None					56.4.i.b	$2$	$0$	\(0\)	\(-7\)	\(-3\)	\(-4\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-2\beta _{1}+\beta _{5})q^{3}+(-1+\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
448.4.i.k	$448$	$4$	448.i	7.c	$3$	$6$	$3$	$26.433$	6.0.11163123.4	None					56.4.i.a	$2$	$0$	\(0\)	\(-1\)	\(13\)	\(20\)	$2^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{3}+(5-5\beta _{1}+\beta _{3}+\beta _{4}+2\beta _{5})q^{5}+\cdots\)
448.4.i.l	$448$	$4$	448.i	7.c	$3$	$6$	$3$	$26.433$	6.0.11163123.4	None					56.4.i.a	$2$	$0$	\(0\)	\(1\)	\(13\)	\(-20\)	$2^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta _{3}q^{3}+(5-5\beta _{1}+\beta _{3}+\beta _{4}+2\beta _{5})q^{5}+\cdots\)
448.4.i.m	$448$	$4$	448.i	7.c	$3$	$6$	$3$	$26.433$	6.0.11163123.4	None					56.4.i.b	$2$	$0$	\(0\)	\(7\)	\(-3\)	\(4\)	$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\beta _{1}-\beta _{3}+\beta _{5})q^{3}+(-\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)
448.4.i.n	$448$	$4$	448.i	7.c	$3$	$8$	$4$	$26.433$	8.0.\(\cdots\).1	None					224.4.i.b	$4$	$0$	\(0\)	\(0\)	\(12\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta _{4}-\beta _{5}+\beta _{6}-\beta _{7})q^{3}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
448.4.i.o	$448$	$4$	448.i	7.c	$3$	$12$	$6$	$26.433$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					224.4.i.c	$2$	$0$	\(0\)	\(-6\)	\(-10\)	\(-4\)	$2^{14}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{2}+\beta _{7}-\beta _{8})q^{3}+(\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)
448.4.i.p	$448$	$4$	448.i	7.c	$3$	$12$	$6$	$26.433$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None					224.4.i.d	$4$	$0$	\(0\)	\(0\)	\(6\)	\(0\)	$2^{14}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{3}q^{3}+(-\beta _{1}-\beta _{6})q^{5}+(\beta _{3}-\beta _{9}+\cdots)q^{7}+\cdots\)
448.4.i.q	$448$	$4$	448.i	7.c	$3$	$12$	$6$	$26.433$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None					224.4.i.c	$2$	$0$	\(0\)	\(6\)	\(-10\)	\(4\)	$2^{14}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\beta _{2}-\beta _{7}+\beta _{8})q^{3}+(\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(448, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(448, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(7, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(14, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(28, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(56, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(112, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(224, [\chi])\)\(^{\oplus 2}\)