Space of modular forms of level 350, weight 10, and Character 350.c

• Label: 350.10.c
• Level: $350$
• Weight: $10$
• Character orbit: 350.c
• Rep. character: $\chi_{350}(99,\cdot)$
• Character field: $\Q$
• Dimension: $80$
• Newform subspaces: $16$
• Sturm bound: $600$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	350.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(600\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(350, [\chi])\).

	Total	New	Old
Modular forms	552	80	472
Cusp forms	528	80	448
Eisenstein series	24	0	24

Trace form

80 * q - 20480 * q^4 - 582540 * q^9 - 120212 * q^11 + 153664 * q^14 + 5242880 * q^16 + 1799136 * q^19 - 701092 * q^21 - 3609792 * q^26 + 6747240 * q^29 + 11538720 * q^31 - 565184 * q^34 + 149130240 * q^36 - 5363488 * q^39 - 82694044 * q^41 + 30774272 * q^44 + 16749568 * q^46 - 461184080 * q^49 + 138291476 * q^51 + 166819392 * q^54 - 39337984 * q^56 - 571685812 * q^59 + 246078492 * q^61 - 1342177280 * q^64 + 457638464 * q^66 - 167120632 * q^69 - 46622224 * q^71 + 1472071552 * q^74 - 460578816 * q^76 - 1856733016 * q^79 + 6273047256 * q^81 + 179479552 * q^84 - 1769892992 * q^86 - 3522551092 * q^89 - 1992993268 * q^91 + 2955249152 * q^94 + 3655379592 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(350, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
350.10.c.a	$350$	$10$	350.c	5.b	$2$	$2$	$2$	$180.263$	\(\Q(\sqrt{-1}) \)	None					70.10.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+16 i q^{2}+120 i q^{3}-256 q^{4}+\cdots\)
350.10.c.b	$350$	$10$	350.c	5.b	$2$	$2$	$2$	$180.263$	\(\Q(\sqrt{-1}) \)	None					14.10.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-16 i q^{2}+6 i q^{3}-256 q^{4}+96 q^{6}+\cdots\)
350.10.c.c	$350$	$10$	350.c	5.b	$2$	$2$	$2$	$180.263$	\(\Q(\sqrt{-1}) \)	None					70.10.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-16 i q^{2}+87 i q^{3}-256 q^{4}+1392 q^{6}+\cdots\)
350.10.c.d	$350$	$10$	350.c	5.b	$2$	$2$	$2$	$180.263$	\(\Q(\sqrt{-1}) \)	None					14.10.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-16 i q^{2}+170 i q^{3}-256 q^{4}+\cdots\)
350.10.c.e	$350$	$10$	350.c	5.b	$2$	$4$	$4$	$180.263$	\(\Q(i, \sqrt{2473})\)	None					70.10.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{4}\beta _{1}q^{2}+(-71\beta _{1}-\beta _{2})q^{3}-2^{8}q^{4}+\cdots\)
350.10.c.f	$350$	$10$	350.c	5.b	$2$	$4$	$4$	$180.263$	\(\Q(i, \sqrt{5881})\)	None					70.10.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{4}\beta _{1}q^{2}+(-67\beta _{1}+\beta _{2})q^{3}-2^{8}q^{4}+\cdots\)
350.10.c.g	$350$	$10$	350.c	5.b	$2$	$4$	$4$	$180.263$	\(\Q(i, \sqrt{3061})\)	None					70.10.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{4}\beta _{1}q^{2}+(55\beta _{1}-\beta _{2})q^{3}-2^{8}q^{4}+\cdots\)
350.10.c.h	$350$	$10$	350.c	5.b	$2$	$4$	$4$	$180.263$	\(\Q(i, \sqrt{541})\)	None					70.10.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{4}\beta _{1}q^{2}+(29\beta _{1}+\beta _{2})q^{3}-2^{8}q^{4}+\cdots\)
350.10.c.i	$350$	$10$	350.c	5.b	$2$	$4$	$4$	$180.263$	\(\Q(i, \sqrt{457})\)	None					70.10.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{4}\beta _{1}q^{2}+(21\beta _{1}-\beta _{2})q^{3}-2^{8}q^{4}+\cdots\)
350.10.c.j	$350$	$10$	350.c	5.b	$2$	$4$	$4$	$180.263$	\(\Q(i, \sqrt{2305})\)	None					14.10.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{4}\beta _{1}q^{2}+(-7\beta _{1}+\beta _{2})q^{3}-2^{8}q^{4}+\cdots\)
350.10.c.k	$350$	$10$	350.c	5.b	$2$	$6$	$6$	$180.263$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					70.10.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{4}\beta _{1}q^{2}+(-22\beta _{1}-\beta _{5})q^{3}-2^{8}q^{4}+\cdots\)
350.10.c.l	$350$	$10$	350.c	5.b	$2$	$6$	$6$	$180.263$	\(\mathbb{Q}[x]/(x^{6} + \cdots)\)	None					70.10.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{4}\beta _{2}q^{2}+(\beta _{1}+69\beta _{2})q^{3}-2^{8}q^{4}+\cdots\)
350.10.c.m	$350$	$10$	350.c	5.b	$2$	$8$	$8$	$180.263$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			350.10.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{4}\beta _{4}q^{2}+(-2\beta _{4}-\beta _{5})q^{3}-2^{8}q^{4}+\cdots\)
350.10.c.n	$350$	$10$	350.c	5.b	$2$	$8$	$8$	$180.263$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			350.10.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{6}\cdot 3^{2}\cdot 5^{4}\cdot 37^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{4}\beta _{4}q^{2}+(40\beta _{4}-\beta _{5})q^{3}-2^{8}q^{4}+\cdots\)
350.10.c.o	$350$	$10$	350.c	5.b	$2$	$10$	$10$	$180.263$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓			350.10.a.r	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{6}\cdot 5^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2^{4}\beta _{6}q^{2}+(\beta _{1}-15\beta _{6})q^{3}-2^{8}q^{4}+\cdots\)
350.10.c.p	$350$	$10$	350.c	5.b	$2$	$10$	$10$	$180.263$	\(\mathbb{Q}[x]/(x^{10} + \cdots)\)	None		✓			350.10.a.q	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{4}\cdot 5^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2^{4}\beta _{5}q^{2}+(\beta _{1}-19\beta _{5})q^{3}-2^{8}q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(350, [\chi])\) into lower level spaces

\( S_{10}^{\mathrm{old}}(350, [\chi]) \simeq \) \(S_{10}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(35, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(70, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(175, [\chi])\)\(^{\oplus 2}\)