Space of modular forms of level 30, weight 8, and trivial character

• Label: 30.8.a
• Level: $30$
• Weight: $8$
• Character orbit: 30.a
• Rep. character: $\chi_{30}(1,\cdot)$
• Character field: $\Q$
• Dimension: $6$
• Newform subspaces: $6$
• Sturm bound: $48$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	30.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(48\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(30))\).

	Total	New	Old
Modular forms	46	6	40
Cusp forms	38	6	32
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.

\(2\)	\(3\)	\(5\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(7\)	\(1\)	\(6\)		\(6\)	\(1\)	\(5\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(+\)	\(-\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)	\(+\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(+\)	\(-\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)	\(+\)		\(5\)	\(1\)	\(4\)		\(4\)	\(1\)	\(3\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)	\(+\)		\(6\)	\(1\)	\(5\)		\(5\)	\(1\)	\(4\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(-\)	\(-\)		\(6\)	\(0\)	\(6\)		\(5\)	\(0\)	\(5\)		\(1\)	\(0\)	\(1\)
Plus space			\(+\)		\(24\)	\(4\)	\(20\)		\(20\)	\(4\)	\(16\)		\(4\)	\(0\)	\(4\)
Minus space			\(-\)		\(22\)	\(2\)	\(20\)		\(18\)	\(2\)	\(16\)		\(4\)	\(0\)	\(4\)

Trace form

6 * q - 54 * q^3 + 384 * q^4 + 228 * q^7 + 4374 * q^9 - 2000 * q^10 - 2244 * q^11 - 3456 * q^12 + 21168 * q^13 + 16224 * q^14 + 24576 * q^16 + 22728 * q^17 + 9072 * q^19 + 67932 * q^21 - 58656 * q^22 - 43368 * q^23 + 93750 * q^25 + 31776 * q^26 - 39366 * q^27 + 14592 * q^28 - 482712 * q^29 - 54000 * q^30 - 85080 * q^31 - 61884 * q^33 + 15264 * q^34 + 145500 * q^35 + 279936 * q^36 - 501480 * q^37 - 1025088 * q^38 + 497232 * q^39 - 128000 * q^40 - 467892 * q^41 + 355104 * q^42 + 1323720 * q^43 - 143616 * q^44 + 1205952 * q^46 - 2645904 * q^47 - 221184 * q^48 + 271782 * q^49 + 984312 * q^51 + 1354752 * q^52 + 3053904 * q^53 + 643500 * q^55 + 1038336 * q^56 + 1585872 * q^57 - 3506592 * q^58 - 1030740 * q^59 + 2256396 * q^61 - 589632 * q^62 + 166212 * q^63 + 1572864 * q^64 + 1414500 * q^65 + 655776 * q^66 + 4646208 * q^67 + 1454592 * q^68 - 1826712 * q^69 - 1372000 * q^70 - 16148664 * q^71 - 7013052 * q^73 + 1456992 * q^74 - 843750 * q^75 + 580608 * q^76 - 1533408 * q^77 - 4450464 * q^78 - 5474664 * q^79 + 3188646 * q^81 - 1659264 * q^82 + 22696608 * q^83 + 4347648 * q^84 - 10339500 * q^85 - 11450496 * q^86 + 681048 * q^87 - 3753984 * q^88 + 3139236 * q^89 - 1458000 * q^90 + 8389512 * q^91 - 2775552 * q^92 - 19927080 * q^93 + 9296256 * q^94 + 9903000 * q^95 + 2123820 * q^97 + 19273344 * q^98 - 1635876 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(30))\) 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	3	5
30.8.a.a	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	✓	✓	✓	30.8.a.a	$1$	$0$	\(-8\)	\(-27\)	\(-125\)	\(-1084\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}-5^{3}q^{5}+6^{3}q^{6}+\cdots\)
30.8.a.b	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	✓	✓	✓	30.8.a.b	$1$	$1$	\(-8\)	\(-27\)	\(125\)	\(416\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+5^{3}q^{5}+6^{3}q^{6}+\cdots\)
30.8.a.c	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	✓	✓	✓	30.8.a.c	$1$	$0$	\(-8\)	\(27\)	\(125\)	\(-232\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+5^{3}q^{5}-6^{3}q^{6}+\cdots\)
30.8.a.d	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	✓	✓	✓	30.8.a.d	$1$	$1$	\(8\)	\(-27\)	\(-125\)	\(-988\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-5^{3}q^{5}-6^{3}q^{6}+\cdots\)
30.8.a.e	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	✓	✓	✓	30.8.a.e	$1$	$0$	\(8\)	\(-27\)	\(125\)	\(512\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+5^{3}q^{5}-6^{3}q^{6}+\cdots\)
30.8.a.f	$30$	$8$	30.a	1.a	$1$	$1$	$1$	$9.372$	\(\Q\)	None	✓	✓	✓	✓	30.8.a.f	$1$	$0$	\(8\)	\(27\)	\(-125\)	\(1604\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}-5^{3}q^{5}+6^{3}q^{6}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(30))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(30)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 2}\)