Space of modular forms of level 32, weight 30, and trivial character

• Label: 32.30.a
• Level: $32$
• Weight: $30$
• Character orbit: 32.a
• Rep. character: $\chi_{32}(1,\cdot)$
• Character field: $\Q$
• Dimension: $29$
• Newform subspaces: $5$
• Sturm bound: $120$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 30 \)
Character orbit:	\([\chi]\)	\(=\)	32.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(120\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{30}(\Gamma_0(32))\).

	Total	New	Old
Modular forms	120	29	91
Cusp forms	112	29	83
Eisenstein series	8	0	8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(59\)	\(14\)	\(45\)		\(55\)	\(14\)	\(41\)		\(4\)	\(0\)	\(4\)
\(-\)		\(61\)	\(15\)	\(46\)		\(57\)	\(15\)	\(42\)		\(4\)	\(0\)	\(4\)

Trace form

29 * q - 8701963882 * q^5 + 751629454372313 * q^9 + 4268937488690270 * q^13 + 642780256115780090 * q^17 - 35878598345025424640 * q^21 + 1080104661109300850323 * q^25 + 2010959232001954541646 * q^29 + 14073104545624863499904 * q^33 - 128256412586339161080298 * q^37 + 288115627066209444650882 * q^41 - 3466173195634843698806306 * q^45 + 9935058606032005399037781 * q^49 + 32734272441401204406434470 * q^53 + 79208012608884284779163008 * q^57 + 51838118033536712998250190 * q^61 - 187692989393350514095082204 * q^65 + 3070717424078206437481613568 * q^69 + 3441922520978921236021372722 * q^73 - 3508504276477752070684428032 * q^77 + 18188269625972425643712281189 * q^81 - 25301010545839565913520481364 * q^85 - 75338541145260290599905496638 * q^89 + 49757462283824920689313504256 * q^93 - 102449679283965407040091276918 * q^97

Decomposition of \(S_{30}^{\mathrm{new}}(\Gamma_0(32))\) 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					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2
32.30.a.a	$32$	$30$	32.a	1.a	$1$	$1$	$1$	$170.490$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	✓			32.30.a.a	$2$	$1$	\(0\)	\(0\)	\(-21027633202\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-21027633202q^{5}-3^{29}q^{9}-6381221746069194q^{13}+\cdots\)
32.30.a.b	$32$	$30$	32.a	1.a	$1$	$6$	$6$	$170.490$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	✓			32.30.a.b	$2$	$1$	\(0\)	\(0\)	\(20319712980\)	\(0\)		$+$	$+$	$2^{74}\cdot 3^{10}\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{3}+(3386618830-\beta _{2})q^{5}+\cdots\)
32.30.a.c	$32$	$30$	32.a	1.a	$1$	$7$	$7$	$170.490$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓	✓		32.30.a.c	$1$	$1$	\(0\)	\(-11756888\)	\(-1328883646\)	\(-26\!\cdots\!56\)		$+$	$+$	$2^{101}\cdot 3^{7}\cdot 5^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-1679555+\beta _{1})q^{3}+(-189840610+\cdots)q^{5}+\cdots\)
32.30.a.d	$32$	$30$	32.a	1.a	$1$	$7$	$7$	$170.490$	\(\mathbb{Q}[x]/(x^{7} - \cdots)\)	None	✓	✓			32.30.a.c	$1$	$0$	\(0\)	\(11756888\)	\(-1328883646\)	\(26\!\cdots\!56\)		$-$	$-$	$2^{101}\cdot 3^{7}\cdot 5^{3}\cdot 7$	$\mathrm{SU}(2)$	\(q+(1679555-\beta _{1})q^{3}+(-189840610+\cdots)q^{5}+\cdots\)
32.30.a.e	$32$	$30$	32.a	1.a	$1$	$8$	$8$	$170.490$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None	✓	✓	✓		32.30.a.e	$2$	$0$	\(0\)	\(0\)	\(-5336276368\)	\(0\)		$-$	$-$	$2^{133}\cdot 3^{13}\cdot 5^{2}\cdot 7^{3}$	$\mathrm{SU}(2)$	\(q-\beta _{4}q^{3}+(-667034546+\beta _{1})q^{5}+\cdots\)

Decomposition of \(S_{30}^{\mathrm{old}}(\Gamma_0(32))\) into lower level spaces

\( S_{30}^{\mathrm{old}}(\Gamma_0(32)) \simeq \) \(S_{30}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 6}\)\(\oplus\)\(S_{30}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 5}\)\(\oplus\)\(S_{30}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{30}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 3}\)\(\oplus\)\(S_{30}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 2}\)