Space of modular forms of level 50, weight 7, and Character 50.c

• Label: 50.7.c
• Level: $50$
• Weight: $7$
• Character orbit: 50.c
• Rep. character: $\chi_{50}(7,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $18$
• Newform subspaces: $6$
• Sturm bound: $52$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 50 = 2 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	50.c (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(52\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(50, [\chi])\).

	Total	New	Old
Modular forms	102	18	84
Cusp forms	78	18	60
Eisenstein series	24	0	24

Trace form

18 * q - 8 * q^2 + 64 * q^3 - 448 * q^6 + 696 * q^7 + 256 * q^8 - 944 * q^11 + 2048 * q^12 + 4614 * q^13 - 18432 * q^16 - 17554 * q^17 - 12152 * q^18 + 15056 * q^21 + 20544 * q^22 - 39616 * q^23 - 1488 * q^26 - 43160 * q^27 - 22272 * q^28 - 32544 * q^31 + 8192 * q^32 + 137848 * q^33 - 45376 * q^36 - 6414 * q^37 - 57760 * q^38 + 246976 * q^41 + 367424 * q^42 + 215184 * q^43 - 497088 * q^46 - 385824 * q^47 - 65536 * q^48 + 946696 * q^51 + 147648 * q^52 + 49954 * q^53 + 149504 * q^56 - 189680 * q^57 - 473280 * q^58 - 225624 * q^61 - 226976 * q^62 - 289376 * q^63 + 13184 * q^66 + 651936 * q^67 + 561728 * q^68 - 2299744 * q^71 - 388864 * q^72 - 664326 * q^73 + 806400 * q^76 - 347528 * q^77 + 971616 * q^78 + 70798 * q^81 - 724416 * q^82 + 1466904 * q^83 + 1189632 * q^86 + 2642560 * q^87 - 657408 * q^88 - 4548864 * q^91 - 1267712 * q^92 - 4076792 * q^93 + 458752 * q^96 + 871866 * q^97 + 1286072 * q^98

Decomposition of \(S_{7}^{\mathrm{new}}(50, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
50.7.c.a	$50$	$7$	50.c	5.c	$4$	$2$	$1$	$11.503$	\(\Q(\sqrt{-1}) \)	None		✓			50.7.c.a	$2$	$0$	\(-8\)	\(-6\)	\(0\)	\(-234\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4 i-4)q^{2}+(3 i-3)q^{3}+32 i q^{4}+\cdots\)
50.7.c.b	$50$	$7$	50.c	5.c	$4$	$2$	$1$	$11.503$	\(\Q(\sqrt{-1}) \)	None		✓			50.7.c.a	$2$	$0$	\(8\)	\(6\)	\(0\)	\(234\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4 i+4)q^{2}+(-3 i+3)q^{3}+32 i q^{4}+\cdots\)
50.7.c.c	$50$	$7$	50.c	5.c	$4$	$2$	$1$	$11.503$	\(\Q(\sqrt{-1}) \)	None					10.7.c.a	$2$	$0$	\(8\)	\(46\)	\(0\)	\(494\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4 i+4)q^{2}+(-23 i+23)q^{3}+\cdots\)
50.7.c.d	$50$	$7$	50.c	5.c	$4$	$4$	$2$	$11.503$	\(\Q(i, \sqrt{129})\)	None					10.7.c.b	$2$	$0$	\(-16\)	\(18\)	\(0\)	\(202\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4+4\beta _{1})q^{2}+(5+5\beta _{1}-\beta _{2})q^{3}+\cdots\)
50.7.c.e	$50$	$7$	50.c	5.c	$4$	$4$	$2$	$11.503$	\(\Q(i, \sqrt{6})\)	None		✓			50.7.c.e	$2$	$0$	\(-16\)	\(48\)	\(0\)	\(672\)	$5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4-4\beta _{2})q^{2}+(12-12\beta _{2}-\beta _{3})q^{3}+\cdots\)
50.7.c.f	$50$	$7$	50.c	5.c	$4$	$4$	$2$	$11.503$	\(\Q(i, \sqrt{6})\)	None		✓			50.7.c.e	$2$	$0$	\(16\)	\(-48\)	\(0\)	\(-672\)	$5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4+4\beta _{2})q^{2}+(-12+12\beta _{2}-\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(50, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(50, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 2}\)