Space of modular forms of level 150, weight 12, and trivial character

• Label: 150.12.a
• Level: $150$
• Weight: $12$
• Character orbit: 150.a
• Rep. character: $\chi_{150}(1,\cdot)$
• Character field: $\Q$
• Dimension: $35$
• Newform subspaces: $21$
• Sturm bound: $360$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	150.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 21 \)
Sturm bound:			\(360\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(150))\).

	Total	New	Old
Modular forms	342	35	307
Cusp forms	318	35	283
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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(4\)
\(+\)	\(+\)	\(-\)	\(-\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(4\)
\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(4\)
\(-\)	\(+\)	\(-\)	\(+\)	\(5\)
\(-\)	\(-\)	\(+\)	\(+\)	\(5\)
\(-\)	\(-\)	\(-\)	\(-\)	\(4\)
Plus space			\(+\)	\(19\)
Minus space			\(-\)	\(16\)

Trace form

35 * q + 32 * q^2 + 243 * q^3 + 35840 * q^4 - 7776 * q^6 - 66564 * q^7 + 32768 * q^8 + 2066715 * q^9 + 1576964 * q^11 + 248832 * q^12 - 535218 * q^13 - 4739328 * q^14 + 36700160 * q^16 + 12946866 * q^17 + 1889568 * q^18 - 8629828 * q^19 - 53040096 * q^21 + 26517504 * q^22 - 44563248 * q^23 - 7962624 * q^24 + 479552 * q^26 + 14348907 * q^27 - 68161536 * q^28 - 516520182 * q^29 + 223708848 * q^31 + 33554432 * q^32 + 55304856 * q^33 + 626468032 * q^34 + 2116316160 * q^36 + 982129566 * q^37 + 254711680 * q^38 - 1639121994 * q^39 + 687879086 * q^41 - 37666944 * q^42 - 801672828 * q^43 + 1614811136 * q^44 - 2461759232 * q^46 - 6561777744 * q^47 + 254803968 * q^48 + 6291308019 * q^49 + 1952650314 * q^51 - 548063232 * q^52 + 9616568142 * q^53 - 459165024 * q^54 - 4853071872 * q^56 + 6330961620 * q^57 + 2313007680 * q^58 + 10002122500 * q^59 - 22104036206 * q^61 - 4954789376 * q^62 - 3930537636 * q^63 + 37580963840 * q^64 - 704474496 * q^66 + 28415926836 * q^67 + 13257590784 * q^68 - 45832612968 * q^69 + 42551520744 * q^71 + 1934917632 * q^72 - 51675989298 * q^73 - 44505117120 * q^74 - 8836943872 * q^76 + 75411643392 * q^77 + 19438926912 * q^78 - 42762858304 * q^79 + 122037454035 * q^81 - 3821958336 * q^82 - 69372722508 * q^83 - 54313058304 * q^84 - 129838339200 * q^86 - 7974596610 * q^87 + 27153924096 * q^88 + 24562221086 * q^89 + 180733483840 * q^91 - 45632765952 * q^92 - 26276126544 * q^93 + 69121301504 * q^94 - 8153726976 * q^96 - 79953135954 * q^97 + 342959344416 * q^98 + 93118147236 * q^99

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(150))\) 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
150.12.a.a	$150$	$12$	150.a	1.a	$1$	$1$	$1$	$115.251$	\(\Q\)	None	✓				6.12.a.c	$1$	$0$	\(-32\)	\(-243\)	\(0\)	\(-32936\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.b	$150$	$12$	150.a	1.a	$1$	$1$	$1$	$115.251$	\(\Q\)	None	✓				30.12.a.f	$1$	$0$	\(-32\)	\(-243\)	\(0\)	\(-10556\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.c	$150$	$12$	150.a	1.a	$1$	$1$	$1$	$115.251$	\(\Q\)	None	✓				30.12.a.d	$1$	$1$	\(-32\)	\(243\)	\(0\)	\(-29348\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.d	$150$	$12$	150.a	1.a	$1$	$1$	$1$	$115.251$	\(\Q\)	None	✓				30.12.a.e	$1$	$1$	\(-32\)	\(243\)	\(0\)	\(5152\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.e	$150$	$12$	150.a	1.a	$1$	$1$	$1$	$115.251$	\(\Q\)	None	✓				30.12.a.c	$1$	$1$	\(32\)	\(-243\)	\(0\)	\(-56672\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.f	$150$	$12$	150.a	1.a	$1$	$1$	$1$	$115.251$	\(\Q\)	None	✓				6.12.a.b	$1$	$1$	\(32\)	\(-243\)	\(0\)	\(50008\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.g	$150$	$12$	150.a	1.a	$1$	$1$	$1$	$115.251$	\(\Q\)	None	✓				6.12.a.a	$1$	$0$	\(32\)	\(243\)	\(0\)	\(-72464\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.h	$150$	$12$	150.a	1.a	$1$	$1$	$1$	$115.251$	\(\Q\)	None	✓				30.12.a.a	$1$	$0$	\(32\)	\(243\)	\(0\)	\(22876\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.i	$150$	$12$	150.a	1.a	$1$	$1$	$1$	$115.251$	\(\Q\)	None	✓				30.12.a.b	$1$	$0$	\(32\)	\(243\)	\(0\)	\(57376\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.j	$150$	$12$	150.a	1.a	$1$	$2$	$2$	$115.251$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓			150.12.a.j	$1$	$1$	\(-64\)	\(-486\)	\(0\)	\(24058\)		$+$	$+$	$-$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.k	$150$	$12$	150.a	1.a	$1$	$2$	$2$	$115.251$	\(\Q(\sqrt{1129}) \)	None	✓				30.12.c.a	$1$	$1$	\(-64\)	\(-486\)	\(0\)	\(27738\)		$+$	$+$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.l	$150$	$12$	150.a	1.a	$1$	$2$	$2$	$115.251$	\(\Q(\sqrt{499}) \)	None	✓	✓	✓		150.12.a.l	$1$	$0$	\(-64\)	\(-486\)	\(0\)	\(65438\)		$+$	$+$	$+$	$+$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.m	$150$	$12$	150.a	1.a	$1$	$2$	$2$	$115.251$	\(\Q(\sqrt{94291}) \)	None	✓	✓			150.12.a.m	$1$	$0$	\(-64\)	\(486\)	\(0\)	\(21394\)		$+$	$-$	$-$	$+$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.n	$150$	$12$	150.a	1.a	$1$	$2$	$2$	$115.251$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓		150.12.a.n	$1$	$1$	\(-64\)	\(486\)	\(0\)	\(37214\)		$+$	$-$	$+$	$-$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.o	$150$	$12$	150.a	1.a	$1$	$2$	$2$	$115.251$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓			150.12.a.n	$1$	$0$	\(64\)	\(-486\)	\(0\)	\(-37214\)		$-$	$+$	$-$	$+$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.p	$150$	$12$	150.a	1.a	$1$	$2$	$2$	$115.251$	\(\Q(\sqrt{94291}) \)	None	✓	✓	✓		150.12.a.m	$1$	$1$	\(64\)	\(-486\)	\(0\)	\(-21394\)		$-$	$+$	$+$	$-$	$2^{3}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.q	$150$	$12$	150.a	1.a	$1$	$2$	$2$	$115.251$	\(\Q(\sqrt{499}) \)	None	✓	✓			150.12.a.l	$1$	$1$	\(64\)	\(486\)	\(0\)	\(-65438\)		$-$	$-$	$-$	$-$	$2^{4}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.r	$150$	$12$	150.a	1.a	$1$	$2$	$2$	$115.251$	\(\Q(\sqrt{1129}) \)	None	✓				30.12.c.a	$1$	$1$	\(64\)	\(486\)	\(0\)	\(-27738\)		$-$	$-$	$-$	$-$	$2\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.s	$150$	$12$	150.a	1.a	$1$	$2$	$2$	$115.251$	\(\mathbb{Q}[x]/(x^{2} - \cdots)\)	None	✓	✓	✓		150.12.a.j	$1$	$0$	\(64\)	\(486\)	\(0\)	\(-24058\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 3\cdot 5$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}+6^{5}q^{6}+\cdots\)
150.12.a.t	$150$	$12$	150.a	1.a	$1$	$3$	$3$	$115.251$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		30.12.c.b	$1$	$0$	\(-96\)	\(729\)	\(0\)	\(-67384\)		$+$	$-$	$-$	$+$	$2^{3}\cdot 3\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q-2^{5}q^{2}+3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)
150.12.a.u	$150$	$12$	150.a	1.a	$1$	$3$	$3$	$115.251$	\(\mathbb{Q}[x]/(x^{3} - \cdots)\)	None	✓		✓		30.12.c.b	$1$	$0$	\(96\)	\(-729\)	\(0\)	\(67384\)		$-$	$+$	$-$	$+$	$2^{3}\cdot 3\cdot 5^{3}$	$\mathrm{SU}(2)$	\(q+2^{5}q^{2}-3^{5}q^{3}+2^{10}q^{4}-6^{5}q^{6}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(150))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(150)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 12}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(3))\)\(^{\oplus 6}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 8}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 3}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(25))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(30))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 2}\)