Space of modular forms of level 10, weight 7, and Character 10.c

• Label: 10.7.c
• Level: $10$
• Weight: $7$
• Character orbit: 10.c
• Rep. character: $\chi_{10}(3,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $6$
• Newform subspaces: $2$
• Sturm bound: $10$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 10 = 2 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	10.c (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(10\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(10, [\chi])\).

	Total	New	Old
Modular forms	22	6	16
Cusp forms	14	6	8
Eisenstein series	8	0	8

Trace form

6 * q + 8 * q^2 - 64 * q^3 + 180 * q^5 + 224 * q^6 - 696 * q^7 - 256 * q^8 + 2280 * q^10 + 472 * q^11 - 2048 * q^12 - 4614 * q^13 + 3320 * q^15 - 6144 * q^16 + 17554 * q^17 + 12152 * q^18 - 9920 * q^20 - 46408 * q^21 - 20544 * q^22 + 39616 * q^23 - 10350 * q^25 + 49584 * q^26 + 43160 * q^27 + 22272 * q^28 - 65280 * q^30 - 94608 * q^31 - 8192 * q^32 - 137848 * q^33 + 217240 * q^35 + 139328 * q^36 + 6414 * q^37 + 57760 * q^38 - 49920 * q^40 + 39352 * q^41 - 367424 * q^42 - 215184 * q^43 - 6710 * q^45 + 248544 * q^46 + 385824 * q^47 + 65536 * q^48 - 86600 * q^50 - 53168 * q^51 - 147648 * q^52 - 49954 * q^53 - 318840 * q^55 - 74752 * q^56 + 189680 * q^57 + 473280 * q^58 - 186880 * q^60 - 76728 * q^61 + 226976 * q^62 + 289376 * q^63 - 35130 * q^65 - 70912 * q^66 - 651936 * q^67 - 561728 * q^68 + 500640 * q^70 - 1126768 * q^71 + 388864 * q^72 + 664326 * q^73 + 1258600 * q^75 + 241920 * q^76 + 347528 * q^77 - 971616 * q^78 - 184320 * q^80 + 1686226 * q^81 + 724416 * q^82 - 1466904 * q^83 - 820710 * q^85 - 2975136 * q^86 - 2642560 * q^87 + 657408 * q^88 + 1096040 * q^90 + 2448672 * q^91 + 1267712 * q^92 + 4076792 * q^93 - 699400 * q^95 - 229376 * q^96 - 871866 * q^97 - 1286072 * q^98

Decomposition of \(S_{7}^{\mathrm{new}}(10, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
10.7.c.a	$10$	$7$	10.c	5.c	$4$	$2$	$1$	$2.301$	\(\Q(\sqrt{-1}) \)	None		✓			10.7.c.a	$2$	$0$	\(-8\)	\(-46\)	\(-150\)	\(-494\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-4 i-4)q^{2}+(23 i-23)q^{3}+\cdots\)
10.7.c.b	$10$	$7$	10.c	5.c	$4$	$4$	$2$	$2.301$	\(\Q(i, \sqrt{129})\)	None		✓	✓		10.7.c.b	$2$	$0$	\(16\)	\(-18\)	\(330\)	\(-202\)	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(4+4\beta _{1})q^{2}+(-5+4\beta _{1}-\beta _{3})q^{3}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(10, [\chi])\) into lower level spaces

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