Space of modular forms of level 3675, weight 2, and trivial character

• Label: 3675.2.a
• Level: $3675$
• Weight: $2$
• Character orbit: 3675.a
• Rep. character: $\chi_{3675}(1,\cdot)$
• Character field: $\Q$
• Dimension: $129$
• Newform subspaces: $54$
• Sturm bound: $1120$
• Trace bound: $13$

Defining parameters

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 54 \)
Sturm bound:			\(1120\)
Trace bound:			\(13\)
Distinguishing \(T_p\):			\(2\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3675))\).

	Total	New	Old
Modular forms	608	129	479
Cusp forms	513	129	384
Eisenstein series	95	0	95

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

\(3\)	\(5\)	\(7\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(72\)	\(14\)	\(58\)		\(61\)	\(14\)	\(47\)		\(11\)	\(0\)	\(11\)
\(+\)	\(+\)	\(-\)	\(-\)		\(78\)	\(18\)	\(60\)		\(66\)	\(18\)	\(48\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(+\)	\(-\)		\(80\)	\(16\)	\(64\)		\(68\)	\(16\)	\(52\)		\(12\)	\(0\)	\(12\)
\(+\)	\(-\)	\(-\)	\(+\)		\(74\)	\(17\)	\(57\)		\(62\)	\(17\)	\(45\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(+\)	\(-\)		\(80\)	\(18\)	\(62\)		\(68\)	\(18\)	\(50\)		\(12\)	\(0\)	\(12\)
\(-\)	\(+\)	\(-\)	\(+\)		\(74\)	\(12\)	\(62\)		\(62\)	\(12\)	\(50\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(+\)	\(+\)		\(72\)	\(13\)	\(59\)		\(60\)	\(13\)	\(47\)		\(12\)	\(0\)	\(12\)
\(-\)	\(-\)	\(-\)	\(-\)		\(78\)	\(21\)	\(57\)		\(66\)	\(21\)	\(45\)		\(12\)	\(0\)	\(12\)
Plus space			\(+\)		\(292\)	\(56\)	\(236\)		\(245\)	\(56\)	\(189\)		\(47\)	\(0\)	\(47\)
Minus space			\(-\)		\(316\)	\(73\)	\(243\)		\(268\)	\(73\)	\(195\)		\(48\)	\(0\)	\(48\)

Trace form

129 * q - 3 * q^2 - q^3 + 123 * q^4 - q^6 - 3 * q^8 + 129 * q^9 + 4 * q^11 - 7 * q^12 - 10 * q^13 + 115 * q^16 - 6 * q^17 - 3 * q^18 - 2 * q^19 - 20 * q^22 + 16 * q^23 - 9 * q^24 + 10 * q^26 - q^27 + 10 * q^29 + 6 * q^31 - 19 * q^32 + 4 * q^33 + 26 * q^34 + 123 * q^36 + 8 * q^37 + 4 * q^38 - 2 * q^39 + 38 * q^41 + 14 * q^43 - 4 * q^44 - 24 * q^46 + 24 * q^47 + q^48 - 6 * q^51 + 10 * q^52 - 54 * q^53 - q^54 + 6 * q^57 - 34 * q^58 + 32 * q^59 - 4 * q^61 + 24 * q^62 + 107 * q^64 + 20 * q^66 + 26 * q^67 - 10 * q^68 - 4 * q^69 + 28 * q^71 - 3 * q^72 - 14 * q^73 - 10 * q^74 - 16 * q^76 + 22 * q^78 - 10 * q^79 + 129 * q^81 - 18 * q^82 - 20 * q^83 + 12 * q^86 + 2 * q^87 + 76 * q^88 - 2 * q^89 + 104 * q^92 + 18 * q^93 - 16 * q^94 + 31 * q^96 + 10 * q^97 + 4 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3675))\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		3	5	7
3675.2.a.a	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				21.2.e.a	$1$	$1$	\(-2\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{9}-2q^{11}+\cdots\)
3675.2.a.b	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				75.2.a.a	$1$	$0$	\(-2\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}-q^{3}+2q^{4}+2q^{6}+q^{9}+2q^{11}+\cdots\)
3675.2.a.c	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				21.2.e.a	$1$	$1$	\(-2\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{2}+q^{3}+2q^{4}-2q^{6}+q^{9}-2q^{11}+\cdots\)
3675.2.a.d	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				105.2.d.a	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.e	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				525.2.i.b	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}-q^{4}+q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.f	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				105.2.a.a	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.g	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				525.2.i.b	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}-q^{4}-q^{6}+3q^{8}+q^{9}+\cdots\)
3675.2.a.h	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				105.2.i.a	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{4}+q^{9}+2q^{12}+q^{13}+\cdots\)
3675.2.a.i	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				105.2.i.a	$1$	$1$	\(0\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-2q^{4}+q^{9}-2q^{12}-q^{13}+\cdots\)
3675.2.a.j	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				15.2.a.a	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.k	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				525.2.i.b	$1$	$1$	\(1\)	\(-1\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}-q^{4}-q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.l	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				105.2.d.a	$1$	$0$	\(1\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.m	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				525.2.i.b	$1$	$1$	\(1\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.n	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				21.2.a.a	$1$	$1$	\(1\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}-q^{4}+q^{6}-3q^{8}+q^{9}+\cdots\)
3675.2.a.o	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				105.2.i.b	$1$	$1$	\(2\)	\(-1\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}-q^{3}+2q^{4}-2q^{6}+q^{9}-6q^{11}+\cdots\)
3675.2.a.p	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				105.2.i.b	$1$	$1$	\(2\)	\(1\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+q^{9}-6q^{11}+\cdots\)
3675.2.a.q	$3675$	$2$	3675.a	1.a	$1$	$1$	$1$	$29.345$	\(\Q\)	None	✓				75.2.a.a	$1$	$0$	\(2\)	\(1\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{2}+q^{3}+2q^{4}+2q^{6}+q^{9}+2q^{11}+\cdots\)
3675.2.a.r	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{5}) \)	None	✓				525.2.a.e	$1$	$0$	\(-3\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{2}+q^{3}+3\beta q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.s	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{2}) \)	None	✓				735.2.a.l	$1$	$0$	\(-2\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}-q^{3}+(1-2\beta )q^{4}+(1+\cdots)q^{6}+\cdots\)
3675.2.a.t	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{2}) \)	None	✓				735.2.a.l	$1$	$1$	\(-2\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.u	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{5}) \)	None	✓	✓			3675.2.a.u	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
3675.2.a.v	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{5}) \)	None	✓	✓			3675.2.a.u	$1$	$0$	\(-1\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
3675.2.a.w	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{13}) \)	None	✓				525.2.a.f	$1$	$1$	\(-1\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-\beta q^{2}+q^{3}+(1+\beta )q^{4}-\beta q^{6}-3q^{8}+\cdots\)
3675.2.a.x	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{2}) \)	None	✓				105.2.i.c	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}-\beta q^{6}-2\beta q^{8}+q^{9}+\cdots\)
3675.2.a.y	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{5}) \)	None	✓				105.2.a.b	$1$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-\beta q^{2}-q^{3}+3q^{4}+\beta q^{6}-\beta q^{8}+\cdots\)
3675.2.a.z	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{2}) \)	None	✓				105.2.i.c	$1$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+\beta q^{6}-2\beta q^{8}+q^{9}+\cdots\)
3675.2.a.ba	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{5}) \)	None	✓	✓			3675.2.a.u	$1$	$0$	\(1\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+(-1+\beta )q^{4}-\beta q^{6}+\cdots\)
3675.2.a.bb	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{13}) \)	None	✓				525.2.a.f	$1$	$1$	\(1\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}-q^{3}+(1+\beta )q^{4}-\beta q^{6}+3q^{8}+\cdots\)
3675.2.a.bc	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{5}) \)	None	✓	✓			3675.2.a.u	$1$	$1$	\(1\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+q^{3}+(-1+\beta )q^{4}+\beta q^{6}+\cdots\)
3675.2.a.bd	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{2}) \)	None	✓				147.2.a.d	$1$	$1$	\(2\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-q^{3}+(1+2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.be	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{3}) \)	None	✓				105.2.i.d	$1$	$0$	\(2\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-q^{3}+(2+2\beta )q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.bf	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{2}) \)	None	✓				147.2.a.d	$1$	$0$	\(2\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+q^{3}+(1+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
3675.2.a.bg	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{3}) \)	None	✓				105.2.i.d	$1$	$0$	\(2\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}+q^{3}+(2+2\beta )q^{4}+(1+\beta )q^{6}+\cdots\)
3675.2.a.bh	$3675$	$2$	3675.a	1.a	$1$	$2$	$2$	$29.345$	\(\Q(\sqrt{5}) \)	None	✓				525.2.a.e	$1$	$0$	\(3\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1+\beta )q^{2}-q^{3}+3\beta q^{4}+(-1-\beta )q^{6}+\cdots\)
3675.2.a.bi	$3675$	$2$	3675.a	1.a	$1$	$3$	$3$	$29.345$	3.3.148.1	None	✓				105.2.d.b	$1$	$1$	\(-1\)	\(-3\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.bj	$3675$	$2$	3675.a	1.a	$1$	$3$	$3$	$29.345$	3.3.148.1	None	✓				105.2.d.b	$1$	$0$	\(1\)	\(3\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(1+\beta _{1}+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.bk	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	\(\Q(\sqrt{10 +4 \sqrt{2}})\)	None	✓				735.2.a.n	$1$	$1$	\(-4\)	\(-4\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}+\beta _{3})q^{2}-q^{3}+(2+\beta _{1}+\cdots)q^{4}+\cdots\)
3675.2.a.bl	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	\(\Q(\sqrt{10 +4 \sqrt{2}})\)	None	✓				735.2.a.n	$1$	$0$	\(-4\)	\(4\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1-\beta _{1}+\beta _{3})q^{2}+q^{3}+(2+\beta _{1}+\cdots)q^{4}+\cdots\)
3675.2.a.bm	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	\(\Q(\sqrt{7 +2 \sqrt{2}})\)	None	✓	✓			3675.2.a.bm	$1$	$0$	\(-2\)	\(-4\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-q^{3}+\beta _{2}q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
3675.2.a.bn	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	4.4.11344.1	None	✓				105.2.q.a	$1$	$1$	\(-2\)	\(-4\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
3675.2.a.bo	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	\(\Q(\sqrt{7 +2 \sqrt{2}})\)	None	✓	✓			3675.2.a.bm	$1$	$1$	\(-2\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+q^{3}+\beta _{2}q^{4}+(-1+\cdots)q^{6}+\cdots\)
3675.2.a.bp	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	4.4.11344.1	None	✓				105.2.q.a	$1$	$1$	\(-2\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
3675.2.a.bq	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	4.4.88404.1	None	✓		✓		525.2.i.i	$1$	$0$	\(-1\)	\(-4\)	\(0\)	\(0\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.br	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	4.4.88404.1	None	✓				525.2.i.i	$1$	$0$	\(-1\)	\(4\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
3675.2.a.bs	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	\(\Q(\zeta_{24})^+\)	None	✓				735.2.d.f	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+\beta _{2}q^{4}-\beta _{1}q^{6}+\beta _{3}q^{8}+\cdots\)
3675.2.a.bt	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	\(\Q(\sqrt{14 +2 \sqrt{5}})\)	None	✓				735.2.d.c	$2$	$1$	\(0\)	\(-4\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}-q^{3}+(2-\beta _{3})q^{4}+\beta _{2}q^{6}+\cdots\)
3675.2.a.bu	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	\(\Q(\zeta_{24})^+\)	None	✓				735.2.d.f	$2$	$1$	\(0\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+\beta _{2}q^{4}+\beta _{1}q^{6}+\beta _{3}q^{8}+\cdots\)
3675.2.a.bv	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	\(\Q(\sqrt{14 +2 \sqrt{5}})\)	None	✓				735.2.d.c	$2$	$0$	\(0\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q-\beta _{2}q^{2}+q^{3}+(2-\beta _{3})q^{4}-\beta _{2}q^{6}+\cdots\)
3675.2.a.bw	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	4.4.88404.1	None	✓				525.2.i.i	$1$	$0$	\(1\)	\(-4\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}-q^{3}+(2+\beta _{2})q^{4}-\beta _{1}q^{6}+\cdots\)
3675.2.a.bx	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	4.4.88404.1	None	✓				525.2.i.i	$1$	$0$	\(1\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta _{1}q^{2}+q^{3}+(2+\beta _{2})q^{4}+\beta _{1}q^{6}+\cdots\)
3675.2.a.by	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	\(\Q(\sqrt{7 +2 \sqrt{2}})\)	None	✓	✓			3675.2.a.bm	$1$	$1$	\(2\)	\(-4\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-q^{3}+\beta _{2}q^{4}+(-1+\beta _{1}+\cdots)q^{6}+\cdots\)
3675.2.a.bz	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	4.4.11344.1	None	✓				105.2.q.a	$1$	$0$	\(2\)	\(-4\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
3675.2.a.ca	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	\(\Q(\sqrt{7 +2 \sqrt{2}})\)	None	✓	✓			3675.2.a.bm	$1$	$0$	\(2\)	\(4\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+q^{3}+\beta _{2}q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
3675.2.a.cb	$3675$	$2$	3675.a	1.a	$1$	$4$	$4$	$29.345$	4.4.11344.1	None	✓				105.2.q.a	$1$	$0$	\(2\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+(1-\beta _{1})q^{2}+q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3675))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3675)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(15))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(21))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(49))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(75))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(105))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(147))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(245))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(525))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(735))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\)\(^{\oplus 2}\)