Space of modular forms of level 36, weight 7, and Character 36.d

• Label: 36.7.d
• Level: $36$
• Weight: $7$
• Character orbit: 36.d
• Rep. character: $\chi_{36}(19,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $5$
• Sturm bound: $42$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	36.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(42\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(36, [\chi])\).

	Total	New	Old
Modular forms	40	16	24
Cusp forms	32	14	18
Eisenstein series	8	2	6

Trace form

14 * q + 6 * q^2 + 20 * q^4 + 24 * q^5 - 432 * q^8 - 548 * q^10 - 2108 * q^13 + 24 * q^14 - 2032 * q^16 - 2688 * q^17 - 16488 * q^20 - 24744 * q^22 + 30306 * q^25 + 53388 * q^26 + 47856 * q^28 + 33864 * q^29 - 123744 * q^32 - 81884 * q^34 + 31180 * q^37 + 244728 * q^38 + 178720 * q^40 + 6528 * q^41 - 504528 * q^44 - 435552 * q^46 - 41266 * q^49 + 805218 * q^50 + 550792 * q^52 - 84600 * q^53 - 1024128 * q^56 - 567476 * q^58 - 348788 * q^61 + 1162344 * q^62 + 843008 * q^64 - 350832 * q^65 - 1224120 * q^68 - 806352 * q^70 + 122980 * q^73 + 1514940 * q^74 + 1128816 * q^76 + 892128 * q^77 - 1315872 * q^80 - 541244 * q^82 - 368656 * q^85 + 820968 * q^86 + 88704 * q^88 - 1352400 * q^89 + 689856 * q^92 + 719088 * q^94 + 2381188 * q^97 - 1691514 * q^98

Decomposition of \(S_{7}^{\mathrm{new}}(36, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
36.7.d.a	$36$	$7$	36.d	4.b	$2$	$1$	$1$	$8.282$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			36.7.d.a	$2$	$0$	\(-8\)	\(0\)	\(-88\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-8q^{2}+2^{6}q^{4}-88q^{5}-2^{9}q^{8}+704q^{10}+\cdots\)
36.7.d.b	$36$	$7$	36.d	4.b	$2$	$1$	$1$	$8.282$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			36.7.d.a	$2$	$0$	\(8\)	\(0\)	\(88\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+8q^{2}+2^{6}q^{4}+88q^{5}+2^{9}q^{8}+704q^{10}+\cdots\)
36.7.d.c	$36$	$7$	36.d	4.b	$2$	$2$	$2$	$8.282$	\(\Q(\sqrt{-15}) \)	None					4.7.b.a	$2$	$0$	\(-4\)	\(0\)	\(-20\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2-\beta )q^{2}+(-56+4\beta )q^{4}-10q^{5}+\cdots\)
36.7.d.d	$36$	$7$	36.d	4.b	$2$	$4$	$4$	$8.282$	\(\Q(\sqrt{13}, \sqrt{-51})\)	None		✓			36.7.d.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-38+\beta _{3})q^{4}+(-10\beta _{1}+\cdots)q^{5}+\cdots\)
36.7.d.e	$36$	$7$	36.d	4.b	$2$	$6$	$6$	$8.282$	6.0.50898483.1	None			✓		12.7.d.a	$2$	$0$	\(10\)	\(0\)	\(44\)	\(0\)	$2^{15}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2-\beta _{1})q^{2}+(26-\beta _{1}-\beta _{2})q^{4}+(7+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(36, [\chi])\) into lower level spaces

\( S_{7}^{\mathrm{old}}(36, [\chi]) \simeq \) \(S_{7}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{7}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 2}\)