Space of modular forms of level 36, weight 9, and Character 36.d

• Label: 36.9.d
• Level: $36$
• Weight: $9$
• Character orbit: 36.d
• Rep. character: $\chi_{36}(19,\cdot)$
• Character field: $\Q$
• Dimension: $19$
• Newform subspaces: $4$
• Sturm bound: $54$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 36 = 2^{2} \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	36.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(54\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(36, [\chi])\).

	Total	New	Old
Modular forms	52	21	31
Cusp forms	44	19	25
Eisenstein series	8	2	6

Trace form

19 * q - 2 * q^2 - 332 * q^4 + 170 * q^5 + 1504 * q^8 + 12908 * q^10 + 39974 * q^13 - 17784 * q^14 + 127120 * q^16 + 110378 * q^17 + 2552 * q^20 + 421656 * q^22 + 1295865 * q^25 - 123316 * q^26 + 248016 * q^28 - 399190 * q^29 + 2067808 * q^32 + 787988 * q^34 - 518458 * q^37 - 4146120 * q^38 - 6733120 * q^40 + 993770 * q^41 + 6839280 * q^44 + 5290176 * q^46 - 18765101 * q^49 - 26476998 * q^50 - 19633432 * q^52 - 734614 * q^53 + 50597568 * q^56 + 19674332 * q^58 + 17764742 * q^61 - 82481256 * q^62 - 60826112 * q^64 - 1123820 * q^65 + 141693992 * q^68 + 158548368 * q^70 + 73820102 * q^73 - 203803780 * q^74 - 95408208 * q^76 + 5254272 * q^77 + 305234912 * q^80 + 129868244 * q^82 - 8406740 * q^85 - 388412568 * q^86 - 255189312 * q^88 + 7922090 * q^89 + 506168064 * q^92 + 454893648 * q^94 - 182185210 * q^97 - 707159906 * q^98

Decomposition of \(S_{9}^{\mathrm{new}}(36, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
36.9.d.a	$36$	$9$	36.d	4.b	$2$	$1$	$1$	$14.666$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				4.9.b.a	$2$	$0$	\(-16\)	\(0\)	\(1054\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2^{4}q^{2}+2^{8}q^{4}+1054q^{5}-2^{12}q^{8}+\cdots\)
36.9.d.b	$36$	$9$	36.d	4.b	$2$	$2$	$2$	$14.666$	\(\Q(\sqrt{-39}) \)	None					4.9.b.b	$2$	$0$	\(20\)	\(0\)	\(-1220\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(10-\beta )q^{2}+(-56-20\beta )q^{4}-610q^{5}+\cdots\)
36.9.d.c	$36$	$9$	36.d	4.b	$2$	$8$	$8$	$14.666$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None					12.9.d.a	$2$	$0$	\(-6\)	\(0\)	\(336\)	\(0\)	$2^{26}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{1})q^{2}+(-6+\beta _{1}+\beta _{3})q^{4}+\cdots\)
36.9.d.d	$36$	$9$	36.d	4.b	$2$	$8$	$8$	$14.666$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓			36.9.d.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{28}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+(-53-\beta _{5})q^{4}+(3\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{9}^{\mathrm{old}}(36, [\chi])\) into lower level spaces

\( S_{9}^{\mathrm{old}}(36, [\chi]) \simeq \) \(S_{9}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{9}^{\mathrm{new}}(12, [\chi])\)\(^{\oplus 2}\)