Space of modular forms of level 3850, weight 2, and trivial character

• Label: 3850.2.a
• Level: $3850$
• Weight: $2$
• Character orbit: 3850.a
• Rep. character: $\chi_{3850}(1,\cdot)$
• Character field: $\Q$
• Dimension: $94$
• Newform subspaces: $53$
• Sturm bound: $1440$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3850.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 53 \)
Sturm bound:			\(1440\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(3\), \(13\), \(17\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(3850))\).

	Total	New	Old
Modular forms	744	94	650
Cusp forms	697	94	603
Eisenstein series	47	0	47

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

\(2\)	\(5\)	\(7\)	\(11\)	Fricke		Total				Cusp				Eisenstein
						All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)	\(+\)		\(42\)	\(8\)	\(34\)		\(40\)	\(8\)	\(32\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(+\)	\(-\)	\(-\)		\(48\)	\(5\)	\(43\)		\(45\)	\(5\)	\(40\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(+\)	\(-\)		\(51\)	\(5\)	\(46\)		\(48\)	\(5\)	\(43\)		\(3\)	\(0\)	\(3\)
\(+\)	\(+\)	\(-\)	\(-\)	\(+\)		\(45\)	\(6\)	\(39\)		\(42\)	\(6\)	\(36\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(+\)	\(-\)		\(50\)	\(5\)	\(45\)		\(47\)	\(5\)	\(42\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)	\(+\)		\(46\)	\(5\)	\(41\)		\(43\)	\(5\)	\(38\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)	\(+\)		\(43\)	\(7\)	\(36\)		\(40\)	\(7\)	\(33\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(-\)	\(-\)		\(47\)	\(7\)	\(40\)		\(44\)	\(7\)	\(37\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(+\)	\(-\)		\(45\)	\(6\)	\(39\)		\(42\)	\(6\)	\(36\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)	\(+\)		\(45\)	\(4\)	\(41\)		\(42\)	\(4\)	\(38\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)	\(+\)		\(48\)	\(3\)	\(45\)		\(45\)	\(3\)	\(42\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(-\)	\(-\)		\(48\)	\(9\)	\(39\)		\(45\)	\(9\)	\(36\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)	\(+\)		\(49\)	\(6\)	\(43\)		\(46\)	\(6\)	\(40\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(-\)	\(-\)		\(47\)	\(8\)	\(39\)		\(44\)	\(8\)	\(36\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(+\)	\(-\)		\(44\)	\(8\)	\(36\)		\(41\)	\(8\)	\(33\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)	\(+\)		\(46\)	\(2\)	\(44\)		\(43\)	\(2\)	\(41\)		\(3\)	\(0\)	\(3\)
Plus space				\(+\)		\(364\)	\(41\)	\(323\)		\(341\)	\(41\)	\(300\)		\(23\)	\(0\)	\(23\)
Minus space				\(-\)		\(380\)	\(53\)	\(327\)		\(356\)	\(53\)	\(303\)		\(24\)	\(0\)	\(24\)

Trace form

94 * q - 2 * q^2 - 4 * q^3 + 94 * q^4 - 2 * q^8 + 86 * q^9 - 2 * q^11 - 4 * q^12 - 12 * q^13 - 4 * q^14 + 94 * q^16 + 4 * q^17 - 18 * q^18 + 8 * q^19 + 2 * q^22 - 8 * q^23 + 32 * q^26 - 16 * q^27 + 28 * q^29 - 8 * q^31 - 2 * q^32 - 4 * q^33 + 20 * q^34 + 86 * q^36 - 12 * q^37 + 12 * q^38 - 40 * q^39 + 12 * q^41 + 4 * q^42 + 8 * q^43 - 2 * q^44 + 16 * q^46 - 24 * q^47 - 4 * q^48 + 94 * q^49 - 48 * q^51 - 12 * q^52 + 28 * q^53 + 24 * q^54 - 4 * q^56 - 8 * q^57 + 4 * q^58 - 36 * q^59 + 12 * q^61 + 40 * q^62 + 8 * q^63 + 94 * q^64 + 24 * q^67 + 4 * q^68 + 64 * q^69 + 72 * q^71 - 18 * q^72 - 12 * q^73 + 28 * q^74 + 8 * q^76 + 4 * q^77 + 40 * q^78 - 32 * q^79 + 142 * q^81 - 20 * q^82 + 32 * q^83 - 8 * q^86 - 16 * q^87 + 2 * q^88 + 84 * q^89 + 4 * q^91 - 8 * q^92 + 24 * q^93 + 40 * q^94 - 28 * q^97 - 2 * q^98 + 54 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(3850))\) 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	5	7	11
3850.2.a.a	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.a	$1$	$1$	\(-1\)	\(-1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.b	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.c.c	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.c	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.c	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.d	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.c.a	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.e	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.c.b	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{9}+q^{11}+\cdots\)
3850.2.a.f	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				154.2.a.c	$1$	$0$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.g	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.g	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+q^{7}-q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.h	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.h	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.i	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.i	$1$	$1$	\(-1\)	\(1\)	\(0\)	\(-1\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{3}+q^{4}-q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.j	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.a.g	$1$	$1$	\(-1\)	\(2\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.k	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.a.f	$1$	$1$	\(-1\)	\(2\)	\(0\)	\(-1\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.l	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.l	$1$	$0$	\(-1\)	\(2\)	\(0\)	\(-1\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}+q^{4}-2q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.m	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.a.e	$1$	$0$	\(1\)	\(-2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-2q^{6}-q^{7}+q^{8}+\cdots\)
3850.2.a.n	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.l	$1$	$1$	\(1\)	\(-2\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.o	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				154.2.a.b	$1$	$0$	\(1\)	\(-2\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-2q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.p	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.h	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.q	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.i	$1$	$0$	\(1\)	\(-1\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-q^{3}+q^{4}-q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.r	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.g	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.s	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.a.d	$1$	$0$	\(1\)	\(0\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.t	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.a.c	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.u	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				154.2.a.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.v	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.c.a	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.w	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.c	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$+$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.x	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.c.c	$1$	$0$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-q^{11}+\cdots\)
3850.2.a.y	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.c.b	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}+q^{11}+\cdots\)
3850.2.a.z	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓	✓			3850.2.a.a	$1$	$0$	\(1\)	\(1\)	\(0\)	\(1\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.ba	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.a.a	$1$	$0$	\(1\)	\(2\)	\(0\)	\(-1\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+2q^{6}-q^{7}+q^{8}+\cdots\)
3850.2.a.bb	$3850$	$2$	3850.a	1.a	$1$	$1$	$1$	$30.742$	\(\Q\)	None	✓				770.2.a.b	$1$	$0$	\(1\)	\(2\)	\(0\)	\(1\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.bc	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{33}) \)	None	✓				770.2.a.k	$1$	$1$	\(-2\)	\(-4\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.bd	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{3}) \)	None	✓				770.2.a.j	$1$	$0$	\(-2\)	\(-2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1-\beta )q^{6}+\cdots\)
3850.2.a.be	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{6}) \)	None	✓		✓		770.2.c.d	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-2\)		$+$	$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.bf	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{10}) \)	None	✓	✓			3850.2.a.bf	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}-q^{7}-q^{8}+\cdots\)
3850.2.a.bg	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{2}) \)	None	✓	✓			3850.2.a.bg	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+q^{7}-q^{8}+\cdots\)
3850.2.a.bh	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{7}) \)	None	✓	✓			3850.2.a.bh	$1$	$1$	\(-2\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+q^{7}-q^{8}+\cdots\)
3850.2.a.bi	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{2}) \)	None	✓				770.2.a.i	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(2\)		$+$	$+$	$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+q^{7}-q^{8}+\cdots\)
3850.2.a.bj	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{5}) \)	None	✓				154.2.a.d	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots\)
3850.2.a.bk	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{3}) \)	None	✓	✓			3850.2.a.bk	$1$	$0$	\(-2\)	\(2\)	\(0\)	\(2\)		$+$	$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots\)
3850.2.a.bl	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{3}) \)	None	✓	✓			3850.2.a.bk	$1$	$1$	\(2\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots\)
3850.2.a.bm	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{3}) \)	None	✓				770.2.a.h	$1$	$1$	\(2\)	\(-2\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots\)
3850.2.a.bn	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{2}) \)	None	✓	✓			3850.2.a.bg	$1$	$1$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}-q^{7}+q^{8}+\cdots\)
3850.2.a.bo	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{7}) \)	None	✓	✓	✓		3850.2.a.bh	$1$	$0$	\(2\)	\(0\)	\(0\)	\(-2\)		$-$	$+$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}-q^{7}+q^{8}+\cdots\)
3850.2.a.bp	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{6}) \)	None	✓				770.2.c.d	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.bq	$3850$	$2$	3850.a	1.a	$1$	$2$	$2$	$30.742$	\(\Q(\sqrt{10}) \)	None	✓	✓			3850.2.a.bf	$1$	$0$	\(2\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+q^{7}+q^{8}+\cdots\)
3850.2.a.br	$3850$	$2$	3850.a	1.a	$1$	$3$	$3$	$30.742$	3.3.940.1	None	✓	✓	✓		3850.2.a.br	$1$	$1$	\(-3\)	\(-3\)	\(0\)	\(-3\)		$+$	$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
3850.2.a.bs	$3850$	$2$	3850.a	1.a	$1$	$3$	$3$	$30.742$	3.3.316.1	None	✓	✓			3850.2.a.bs	$1$	$1$	\(-3\)	\(-2\)	\(0\)	\(3\)		$+$	$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta _{1})q^{3}+q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
3850.2.a.bt	$3850$	$2$	3850.a	1.a	$1$	$3$	$3$	$30.742$	3.3.316.1	None	✓				770.2.a.m	$1$	$1$	\(-3\)	\(-2\)	\(0\)	\(3\)		$+$	$+$	$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(1+\beta _{2})q^{6}+\cdots\)
3850.2.a.bu	$3850$	$2$	3850.a	1.a	$1$	$3$	$3$	$30.742$	3.3.892.1	None	✓				770.2.a.l	$1$	$0$	\(3\)	\(0\)	\(0\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+q^{7}+\cdots\)
3850.2.a.bv	$3850$	$2$	3850.a	1.a	$1$	$3$	$3$	$30.742$	3.3.316.1	None	✓	✓			3850.2.a.bs	$1$	$0$	\(3\)	\(2\)	\(0\)	\(-3\)		$-$	$-$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+(1-\beta _{1})q^{6}+\cdots\)
3850.2.a.bw	$3850$	$2$	3850.a	1.a	$1$	$3$	$3$	$30.742$	3.3.940.1	None	✓	✓			3850.2.a.br	$1$	$0$	\(3\)	\(3\)	\(0\)	\(3\)		$-$	$+$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+(1-\beta _{1})q^{6}+\cdots\)
3850.2.a.bx	$3850$	$2$	3850.a	1.a	$1$	$4$	$4$	$30.742$	4.4.10304.1	None	✓		✓		770.2.c.e	$1$	$1$	\(-4\)	\(0\)	\(0\)	\(4\)		$+$	$-$	$-$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+q^{7}+\cdots\)
3850.2.a.by	$3850$	$2$	3850.a	1.a	$1$	$4$	$4$	$30.742$	4.4.10304.1	None	✓		✓		770.2.c.e	$1$	$1$	\(4\)	\(0\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}-q^{7}+\cdots\)
3850.2.a.bz	$3850$	$2$	3850.a	1.a	$1$	$5$	$5$	$30.742$	5.5.117688.1	None	✓		✓		770.2.c.f	$1$	$0$	\(-5\)	\(0\)	\(0\)	\(5\)		$+$	$-$	$-$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}+q^{7}+\cdots\)
3850.2.a.ca	$3850$	$2$	3850.a	1.a	$1$	$5$	$5$	$30.742$	5.5.117688.1	None	✓		✓		770.2.c.f	$1$	$0$	\(5\)	\(0\)	\(0\)	\(-5\)		$-$	$-$	$+$	$-$	$-$	$2^{4}$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{2}q^{3}+q^{4}+\beta _{2}q^{6}-q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(3850))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(3850)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(11))\)\(^{\oplus 12}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(35))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(55))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(70))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(77))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(110))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(154))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(175))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(275))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(350))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(385))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(550))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(770))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(1925))\)\(^{\oplus 2}\)