Space of modular forms of level 150, weight 8, and trivial character

• Label: 150.8.a
• Level: $150$
• Weight: $8$
• Character orbit: 150.a
• Rep. character: $\chi_{150}(1,\cdot)$
• Character field: $\Q$
• Dimension: $21$
• Newform subspaces: $19$
• Sturm bound: $240$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(240\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(150))\).

	Total	New	Old
Modular forms	222	21	201
Cusp forms	198	21	177
Eisenstein series	24	0	24

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
\(+\)	\(+\)	\(+\)	\(+\)		\(30\)	\(3\)	\(27\)		\(27\)	\(3\)	\(24\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)		\(26\)	\(2\)	\(24\)		\(23\)	\(2\)	\(21\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(27\)	\(3\)	\(24\)		\(24\)	\(3\)	\(21\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(28\)	\(3\)	\(25\)		\(25\)	\(3\)	\(22\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(27\)	\(2\)	\(25\)		\(24\)	\(2\)	\(22\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(28\)	\(3\)	\(25\)		\(25\)	\(3\)	\(22\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(27\)	\(3\)	\(24\)		\(24\)	\(3\)	\(21\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(29\)	\(2\)	\(27\)		\(26\)	\(2\)	\(24\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(113\)	\(12\)	\(101\)		\(101\)	\(12\)	\(89\)		\(12\)	\(0\)	\(12\)
Minus space			\(-\)		\(109\)	\(9\)	\(100\)		\(97\)	\(9\)	\(88\)		\(12\)	\(0\)	\(12\)

Trace form

21 * q - 8 * q^2 + 27 * q^3 + 1344 * q^4 - 216 * q^6 + 1348 * q^7 - 512 * q^8 + 15309 * q^9 - 2060 * q^11 + 1728 * q^12 - 17366 * q^13 + 5376 * q^14 + 86016 * q^16 - 16122 * q^17 - 5832 * q^18 + 2124 * q^19 - 106704 * q^21 + 1920 * q^23 - 13824 * q^24 - 47312 * q^26 + 19683 * q^27 + 86272 * q^28 - 98034 * q^29 + 118728 * q^31 - 32768 * q^32 - 136080 * q^33 - 567984 * q^34 + 979776 * q^36 + 537946 * q^37 + 826208 * q^38 + 674838 * q^39 + 1404754 * q^41 - 14688 * q^42 - 1167308 * q^43 - 131840 * q^44 - 1606080 * q^46 + 3079680 * q^47 + 110592 * q^48 + 3120621 * q^49 - 902934 * q^51 - 1111424 * q^52 - 3839982 * q^53 - 157464 * q^54 + 344064 * q^56 - 2257092 * q^57 + 3839472 * q^58 - 2657980 * q^59 + 12091590 * q^61 + 324416 * q^62 + 982692 * q^63 + 5505024 * q^64 - 3762720 * q^66 - 1355372 * q^67 - 1031808 * q^68 + 2733480 * q^69 + 7138200 * q^71 - 373248 * q^72 + 4353154 * q^73 + 14209392 * q^74 + 135936 * q^76 + 13088640 * q^77 + 5271696 * q^78 + 5532552 * q^79 + 11160261 * q^81 + 6771888 * q^82 - 24926076 * q^83 - 6829056 * q^84 + 14617824 * q^86 + 442422 * q^87 - 3005198 * q^89 - 26731248 * q^91 + 122880 * q^92 + 19031976 * q^93 - 7228800 * q^94 - 884736 * q^96 + 1936306 * q^97 - 32555208 * q^98 - 1501740 * q^99

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(150))\) 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
150.8.a.a	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓				30.8.a.f	$1$	$0$	\(-8\)	\(-27\)	\(0\)	\(-1604\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}-1604q^{7}+\cdots\)
150.8.a.b	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓	✓			150.8.a.b	$1$	$1$	\(-8\)	\(-27\)	\(0\)	\(-349\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}-349q^{7}+\cdots\)
150.8.a.c	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓	✓			150.8.a.c	$1$	$0$	\(-8\)	\(-27\)	\(0\)	\(391\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}+391q^{7}+\cdots\)
150.8.a.d	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓				30.8.c.a	$1$	$1$	\(-8\)	\(-27\)	\(0\)	\(1126\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}+1126q^{7}+\cdots\)
150.8.a.e	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓				6.8.a.a	$1$	$0$	\(-8\)	\(-27\)	\(0\)	\(1576\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}+1576q^{7}+\cdots\)
150.8.a.f	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓	✓			150.8.a.f	$1$	$1$	\(-8\)	\(27\)	\(0\)	\(-1427\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}-1427q^{7}+\cdots\)
150.8.a.g	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓				30.8.a.e	$1$	$1$	\(-8\)	\(27\)	\(0\)	\(-512\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}-2^{9}q^{7}+\cdots\)
150.8.a.h	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓	✓			150.8.a.h	$1$	$0$	\(-8\)	\(27\)	\(0\)	\(713\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+713q^{7}+\cdots\)
150.8.a.i	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓				30.8.a.d	$1$	$1$	\(-8\)	\(27\)	\(0\)	\(988\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+988q^{7}+\cdots\)
150.8.a.j	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓	✓			150.8.a.h	$1$	$1$	\(8\)	\(-27\)	\(0\)	\(-713\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}-713q^{7}+\cdots\)
150.8.a.k	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓				30.8.a.c	$1$	$1$	\(8\)	\(-27\)	\(0\)	\(232\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+232q^{7}+\cdots\)
150.8.a.l	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓	✓			150.8.a.f	$1$	$0$	\(8\)	\(-27\)	\(0\)	\(1427\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+1427q^{7}+\cdots\)
150.8.a.m	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓				30.8.c.a	$1$	$1$	\(8\)	\(27\)	\(0\)	\(-1126\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}-1126q^{7}+\cdots\)
150.8.a.n	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓				30.8.a.b	$1$	$0$	\(8\)	\(27\)	\(0\)	\(-416\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}-416q^{7}+\cdots\)
150.8.a.o	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓	✓			150.8.a.c	$1$	$1$	\(8\)	\(27\)	\(0\)	\(-391\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}-391q^{7}+\cdots\)
150.8.a.p	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓	✓			150.8.a.b	$1$	$0$	\(8\)	\(27\)	\(0\)	\(349\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}+349q^{7}+\cdots\)
150.8.a.q	$150$	$8$	150.a	1.a	$1$	$1$	$1$	$46.858$	\(\Q\)	None	✓				30.8.a.a	$1$	$0$	\(8\)	\(27\)	\(0\)	\(1084\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+6^{3}q^{6}+1084q^{7}+\cdots\)
150.8.a.r	$150$	$8$	150.a	1.a	$1$	$2$	$2$	$46.858$	\(\Q(\sqrt{2641}) \)	None	✓		✓		30.8.c.b	$1$	$0$	\(-16\)	\(54\)	\(0\)	\(-564\)		$+$	$-$	$-$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+(-282+\cdots)q^{7}+\cdots\)
150.8.a.s	$150$	$8$	150.a	1.a	$1$	$2$	$2$	$46.858$	\(\Q(\sqrt{2641}) \)	None	✓		✓		30.8.c.b	$1$	$0$	\(16\)	\(-54\)	\(0\)	\(564\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 5$	$\mathrm{SU}(2)$	\(q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}-6^{3}q^{6}+(282+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(150))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(150)) \simeq \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)