Space of modular forms of level 350, weight 4, and Character 350.c

• Label: 350.4.c
• Level: $350$
• Weight: $4$
• Character orbit: 350.c
• Rep. character: $\chi_{350}(99,\cdot)$
• Character field: $\Q$
• Dimension: $28$
• Newform subspaces: $14$
• Sturm bound: $240$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 14 \)
Sturm bound:			\(240\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(350, [\chi])\).

	Total	New	Old
Modular forms	192	28	164
Cusp forms	168	28	140
Eisenstein series	24	0	24

Trace form

28 * q - 112 * q^4 - 96 * q^9 + 28 * q^11 - 56 * q^14 + 448 * q^16 + 96 * q^19 + 308 * q^21 + 72 * q^26 - 784 * q^29 - 288 * q^31 + 968 * q^34 + 384 * q^36 + 1280 * q^39 - 1372 * q^41 - 112 * q^44 + 224 * q^46 - 1372 * q^49 + 2996 * q^51 + 648 * q^54 + 224 * q^56 + 460 * q^59 - 900 * q^61 - 1792 * q^64 - 712 * q^66 + 56 * q^69 + 2448 * q^71 + 1472 * q^74 - 384 * q^76 - 248 * q^79 + 3444 * q^81 - 1232 * q^84 - 464 * q^86 + 492 * q^89 - 644 * q^91 - 1664 * q^94 - 4392 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(350, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
350.4.c.a	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None		✓			350.4.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+10 i q^{3}-4 q^{4}-20 q^{6}+\cdots\)
350.4.c.b	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None					14.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+8 i q^{3}-4 q^{4}-16 q^{6}+\cdots\)
350.4.c.c	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None		✓			350.4.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+4 i q^{3}-4 q^{4}-8 q^{6}+\cdots\)
350.4.c.d	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None					70.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+4 i q^{3}-4 q^{4}-8 q^{6}+\cdots\)
350.4.c.e	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None		✓			350.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+3 i q^{3}-4 q^{4}-6 q^{6}+\cdots\)
350.4.c.f	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None		✓			350.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+2 i q^{3}-4 q^{4}-4 q^{6}+\cdots\)
350.4.c.g	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None					14.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+2 i q^{3}-4 q^{4}-4 q^{6}+\cdots\)
350.4.c.h	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None					70.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}+i q^{3}-4 q^{4}+2 q^{6}+7 i q^{7}+\cdots\)
350.4.c.i	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None		✓			350.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}+i q^{3}-4 q^{4}+2 q^{6}+7 i q^{7}+\cdots\)
350.4.c.j	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None					70.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}+3 i q^{3}-4 q^{4}+6 q^{6}+\cdots\)
350.4.c.k	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None					70.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}+5 i q^{3}-4 q^{4}+10 q^{6}+\cdots\)
350.4.c.l	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None					70.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}+7 i q^{3}-4 q^{4}+14 q^{6}+\cdots\)
350.4.c.m	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None		✓			350.4.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}+8 i q^{3}-4 q^{4}+16 q^{6}+\cdots\)
350.4.c.n	$350$	$4$	350.c	5.b	$2$	$2$	$2$	$20.651$	\(\Q(\sqrt{-1}) \)	None					70.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}+8 i q^{3}-4 q^{4}+16 q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(350, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)