Space of modular forms of level 72, weight 10, and Character 72.d

• Label: 72.10.d
• Level: $72$
• Weight: $10$
• Character orbit: 72.d
• Rep. character: $\chi_{72}(37,\cdot)$
• Character field: $\Q$
• Dimension: $44$
• Newform subspaces: $4$
• Sturm bound: $120$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 72 = 2^{3} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	72.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(120\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(72, [\chi])\).

	Total	New	Old
Modular forms	112	46	66
Cusp forms	104	44	60
Eisenstein series	8	2	6

Trace form

44 * q - 16 * q^2 + 256 * q^4 + 4800 * q^7 - 12460 * q^8 - 7556 * q^10 + 281540 * q^14 - 892656 * q^16 - 101996 * q^17 - 1652008 * q^20 - 372244 * q^22 + 53944 * q^23 - 16485884 * q^25 + 3235472 * q^26 - 4538056 * q^28 - 11885840 * q^31 - 4976936 * q^32 - 7946440 * q^34 - 17868548 * q^38 + 23255160 * q^40 + 5961740 * q^41 - 57480 * q^44 + 8104376 * q^46 + 105025944 * q^47 + 206155356 * q^49 + 8793944 * q^50 + 21329216 * q^52 + 147165664 * q^55 + 9257864 * q^56 + 292790108 * q^58 + 100360204 * q^62 + 569976880 * q^64 - 154314256 * q^65 + 67716120 * q^68 - 256438728 * q^70 - 151292376 * q^71 - 80287264 * q^73 + 544744664 * q^74 - 102926504 * q^76 - 1042128000 * q^79 - 223446336 * q^80 - 1423587824 * q^82 - 57810148 * q^86 - 1051579984 * q^88 - 286363036 * q^89 - 1302692064 * q^92 - 1700540952 * q^94 + 1133362000 * q^95 - 498399216 * q^97 - 943274520 * q^98

Decomposition of \(S_{10}^{\mathrm{new}}(72, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
72.10.d.a	$72$	$10$	72.d	8.b	$2$	$2$	$2$	$37.083$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-6}) \)		✓			72.10.d.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-8636\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-8\beta q^{2}-2^{9}q^{4}+119\beta q^{5}-4318q^{7}+\cdots\)
72.10.d.b	$72$	$10$	72.d	8.b	$2$	$8$	$8$	$37.083$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					8.10.b.a	$2$	$0$	\(18\)	\(0\)	\(0\)	\(4800\)	$2^{28}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2+\beta _{1})q^{2}+(-54+3\beta _{1}+\beta _{2})q^{4}+\cdots\)
72.10.d.c	$72$	$10$	72.d	8.b	$2$	$16$	$16$	$37.083$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	None		✓			72.10.d.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(18240\)	$2^{56}\cdot 3^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(75+\beta _{2})q^{4}+(2\beta _{1}+\beta _{6}+\cdots)q^{5}+\cdots\)
72.10.d.d	$72$	$10$	72.d	8.b	$2$	$18$	$18$	$37.083$	\(\mathbb{Q}[x]/(x^{18} + \cdots)\)	None			✓		24.10.d.a	$2$	$0$	\(-34\)	\(0\)	\(0\)	\(-9604\)	$2^{72}\cdot 3^{28}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2-\beta _{1})q^{2}+(29+2\beta _{1}+\beta _{5})q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(72, [\chi])\) into lower level spaces

\( S_{10}^{\mathrm{old}}(72, [\chi]) \simeq \) \(S_{10}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(24, [\chi])\)\(^{\oplus 2}\)