Space of modular forms of level 75, weight 13, and Character 75.d

• Label: 75.13.d
• Level: $75$
• Weight: $13$
• Character orbit: 75.d
• Rep. character: $\chi_{75}(74,\cdot)$
• Character field: $\Q$
• Dimension: $70$
• Newform subspaces: $4$
• Sturm bound: $130$
• Trace bound: $6$

Defining parameters

Level:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 13 \)
Character orbit:	\([\chi]\)	\(=\)	75.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 15 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(130\)
Trace bound:			\(6\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{13}(75, [\chi])\).

	Total	New	Old
Modular forms	126	74	52
Cusp forms	114	70	44
Eisenstein series	12	4	8

Trace form

70 * q + 138344 * q^4 - 70658 * q^6 + 804962 * q^9 + 313864504 * q^16 + 27575044 * q^19 + 152746668 * q^21 + 229445274 * q^24 - 3538940020 * q^31 + 1157935524 * q^34 + 5922698642 * q^36 - 3143631068 * q^39 - 51250395216 * q^46 - 76156067586 * q^49 + 31111451752 * q^51 - 906630868 * q^54 - 380411343940 * q^61 + 768188938688 * q^64 + 1088440356110 * q^66 - 378296632128 * q^69 - 1331005463716 * q^76 - 1451620556 * q^79 + 1285051675150 * q^81 + 103452724152 * q^84 - 3823471734344 * q^91 + 2316170835744 * q^94 + 1500981923366 * q^96 + 1305185996680 * q^99

Decomposition of \(S_{13}^{\mathrm{new}}(75, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
75.13.d.a	$75$	$13$	75.d	15.d	$2$	$2$	$2$	$68.550$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-3}) \)					3.13.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-729 i q^{3}-4096 q^{4}-153502 i q^{7}+\cdots\)
75.13.d.b	$75$	$13$	75.d	15.d	$2$	$4$	$4$	$68.550$	\(\Q(i, \sqrt{26})\)	None					3.13.b.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}\cdot 3^{4}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+(-3^{3}\beta _{1}-3\beta _{2})q^{3}+4328q^{4}+\cdots\)
75.13.d.c	$75$	$13$	75.d	15.d	$2$	$32$	$32$	$68.550$		None					15.13.c.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$
75.13.d.d	$75$	$13$	75.d	15.d	$2$	$32$	$32$	$68.550$		None		✓			75.13.c.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$\mathrm{SU}(2)[C_{2}]$

Decomposition of \(S_{13}^{\mathrm{old}}(75, [\chi])\) into lower level spaces

\( S_{13}^{\mathrm{old}}(75, [\chi]) \simeq \) \(S_{13}^{\mathrm{new}}(15, [\chi])\)\(^{\oplus 2}\)