Space of modular forms of level 1850, weight 2, and trivial character

• Label: 1850.2.a
• Level: $1850$
• Weight: $2$
• Character orbit: 1850.a
• Rep. character: $\chi_{1850}(1,\cdot)$
• Character field: $\Q$
• Dimension: $57$
• Newform subspaces: $31$
• Sturm bound: $570$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 1850 = 2 \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1850.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 31 \)
Sturm bound:			\(570\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(1850))\).

	Total	New	Old
Modular forms	296	57	239
Cusp forms	273	57	216
Eisenstein series	23	0	23

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

\(2\)	\(5\)	\(37\)	Fricke	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)	\(8\)
\(+\)	\(-\)	\(+\)	\(-\)	\(9\)
\(+\)	\(-\)	\(-\)	\(+\)	\(5\)
\(-\)	\(+\)	\(+\)	\(-\)	\(8\)
\(-\)	\(+\)	\(-\)	\(+\)	\(4\)
\(-\)	\(-\)	\(+\)	\(+\)	\(5\)
\(-\)	\(-\)	\(-\)	\(-\)	\(11\)
Plus space			\(+\)	\(21\)
Minus space			\(-\)	\(36\)

Trace form

57 * q - q^2 - 2 * q^3 + 57 * q^4 + 4 * q^6 + 8 * q^7 - q^8 + 67 * q^9 + 6 * q^11 - 2 * q^12 + 2 * q^13 + 4 * q^14 + 57 * q^16 + 2 * q^17 - 5 * q^18 + 4 * q^19 + 20 * q^21 - 8 * q^22 + 12 * q^23 + 4 * q^24 - 12 * q^26 + 4 * q^27 + 8 * q^28 + 22 * q^29 - 4 * q^31 - q^32 + 32 * q^33 - 6 * q^34 + 67 * q^36 - q^37 + 44 * q^39 - 8 * q^41 - 8 * q^42 + 16 * q^43 + 6 * q^44 + 2 * q^46 + 4 * q^47 - 2 * q^48 + 81 * q^49 - 12 * q^51 + 2 * q^52 - 18 * q^53 + 4 * q^54 + 4 * q^56 - 4 * q^57 - 4 * q^58 - 44 * q^59 + 14 * q^61 + 6 * q^62 + 68 * q^63 + 57 * q^64 - 36 * q^66 - 2 * q^67 + 2 * q^68 + 16 * q^69 - 5 * q^72 + 20 * q^73 + 5 * q^74 + 4 * q^76 - 28 * q^77 + 38 * q^78 + 60 * q^79 + 81 * q^81 - 2 * q^82 + 24 * q^83 + 20 * q^84 - 24 * q^86 + 20 * q^87 - 8 * q^88 + 38 * q^89 + 56 * q^91 + 12 * q^92 + 24 * q^93 + 24 * q^94 + 4 * q^96 - 14 * q^97 + 7 * q^98 - 40 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(1850))\) 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	37
1850.2.a.a	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓	✓			1850.2.a.a	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}+2q^{6}-4q^{7}-q^{8}+\cdots\)
1850.2.a.b	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓	✓			1850.2.a.b	$1$	$1$	\(-1\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}+2q^{6}-q^{8}+q^{9}+\cdots\)
1850.2.a.c	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓	✓			1850.2.a.c	$1$	$0$	\(-1\)	\(-1\)	\(0\)	\(4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-q^{3}+q^{4}+q^{6}+4q^{7}-q^{8}+\cdots\)
1850.2.a.d	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓				370.2.b.b	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-1\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-q^{7}-q^{8}-3q^{9}-3q^{11}+\cdots\)
1850.2.a.e	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓				370.2.b.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(2\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{7}-q^{8}-3q^{9}-2q^{13}+\cdots\)
1850.2.a.f	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓				370.2.a.d	$1$	$1$	\(-1\)	\(2\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+2q^{3}+q^{4}-2q^{6}-2q^{7}-q^{8}+\cdots\)
1850.2.a.g	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓	✓			1850.2.a.g	$1$	$0$	\(-1\)	\(3\)	\(0\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+3q^{3}+q^{4}-3q^{6}-q^{8}+6q^{9}+\cdots\)
1850.2.a.h	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓	✓			1850.2.a.g	$1$	$1$	\(1\)	\(-3\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-3q^{3}+q^{4}-3q^{6}+q^{8}+6q^{9}+\cdots\)
1850.2.a.i	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓				370.2.a.c	$1$	$1$	\(1\)	\(-2\)	\(0\)	\(-1\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-2q^{3}+q^{4}-2q^{6}-q^{7}+q^{8}+\cdots\)
1850.2.a.j	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓				370.2.b.a	$1$	$1$	\(1\)	\(0\)	\(0\)	\(-2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}-2q^{7}+q^{8}-3q^{9}+2q^{13}+\cdots\)
1850.2.a.k	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓				370.2.a.b	$1$	$1$	\(1\)	\(0\)	\(0\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{8}-3q^{9}-4q^{11}-2q^{13}+\cdots\)
1850.2.a.l	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓				370.2.b.b	$1$	$1$	\(1\)	\(0\)	\(0\)	\(1\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+q^{7}+q^{8}-3q^{9}-3q^{11}+\cdots\)
1850.2.a.m	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓	✓			1850.2.a.c	$1$	$1$	\(1\)	\(1\)	\(0\)	\(-4\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{3}+q^{4}+q^{6}-4q^{7}+q^{8}+\cdots\)
1850.2.a.n	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓	✓			1850.2.a.b	$1$	$0$	\(1\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{8}+q^{9}+\cdots\)
1850.2.a.o	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓				370.2.a.a	$1$	$0$	\(1\)	\(2\)	\(0\)	\(1\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+2q^{6}+q^{7}+q^{8}+\cdots\)
1850.2.a.p	$1850$	$2$	1850.a	1.a	$1$	$1$	$1$	$14.772$	\(\Q\)	None	✓	✓			1850.2.a.a	$1$	$0$	\(1\)	\(2\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}+q^{4}+2q^{6}+4q^{7}+q^{8}+\cdots\)
1850.2.a.q	$1850$	$2$	1850.a	1.a	$1$	$2$	$2$	$14.772$	\(\Q(\sqrt{33}) \)	None	✓				370.2.a.f	$1$	$1$	\(-2\)	\(-4\)	\(0\)	\(3\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-2q^{3}+q^{4}+2q^{6}+(1+\beta )q^{7}+\cdots\)
1850.2.a.r	$1850$	$2$	1850.a	1.a	$1$	$2$	$2$	$14.772$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓		1850.2.a.r	$1$	$1$	\(-2\)	\(-2\)	\(0\)	\(0\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(-1+\beta )q^{3}+q^{4}+(1-\beta )q^{6}+\cdots\)
1850.2.a.s	$1850$	$2$	1850.a	1.a	$1$	$2$	$2$	$14.772$	\(\Q(\sqrt{6}) \)	None	✓				370.2.b.c	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+(-2-\beta )q^{7}+\cdots\)
1850.2.a.t	$1850$	$2$	1850.a	1.a	$1$	$2$	$2$	$14.772$	\(\Q(\sqrt{5}) \)	None	✓				74.2.a.b	$1$	$0$	\(-2\)	\(1\)	\(0\)	\(2\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta q^{3}+q^{4}-\beta q^{6}+2\beta q^{7}+\cdots\)
1850.2.a.u	$1850$	$2$	1850.a	1.a	$1$	$2$	$2$	$14.772$	\(\Q(\sqrt{13}) \)	None	✓				74.2.a.a	$1$	$0$	\(2\)	\(-3\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1-\beta )q^{3}+q^{4}+(-1-\beta )q^{6}+\cdots\)
1850.2.a.v	$1850$	$2$	1850.a	1.a	$1$	$2$	$2$	$14.772$	\(\Q(\sqrt{6}) \)	None	✓				370.2.b.c	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+\beta q^{6}+(2-\beta )q^{7}+\cdots\)
1850.2.a.w	$1850$	$2$	1850.a	1.a	$1$	$2$	$2$	$14.772$	\(\Q(\sqrt{6}) \)	None	✓	✓			1850.2.a.r	$1$	$0$	\(2\)	\(2\)	\(0\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
1850.2.a.x	$1850$	$2$	1850.a	1.a	$1$	$2$	$2$	$14.772$	\(\Q(\sqrt{3}) \)	None	✓				370.2.a.e	$1$	$0$	\(2\)	\(2\)	\(0\)	\(6\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(1+\beta )q^{3}+q^{4}+(1+\beta )q^{6}+\cdots\)
1850.2.a.y	$1850$	$2$	1850.a	1.a	$1$	$3$	$3$	$14.772$	3.3.1524.1	None	✓	✓	✓		1850.2.a.y	$1$	$1$	\(-3\)	\(-1\)	\(0\)	\(-4\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(-1+\cdots)q^{7}+\cdots\)
1850.2.a.z	$1850$	$2$	1850.a	1.a	$1$	$3$	$3$	$14.772$	3.3.892.1	None	✓				370.2.a.g	$1$	$0$	\(-3\)	\(0\)	\(0\)	\(1\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta _{2}q^{3}+q^{4}-\beta _{2}q^{6}+\beta _{1}q^{7}+\cdots\)
1850.2.a.ba	$1850$	$2$	1850.a	1.a	$1$	$3$	$3$	$14.772$	3.3.148.1	None	✓	✓			1850.2.a.ba	$1$	$0$	\(-3\)	\(3\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+(1+\beta _{2})q^{3}+q^{4}+(-1-\beta _{2})q^{6}+\cdots\)
1850.2.a.bb	$1850$	$2$	1850.a	1.a	$1$	$3$	$3$	$14.772$	3.3.148.1	None	✓	✓	✓		1850.2.a.ba	$1$	$1$	\(3\)	\(-3\)	\(0\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+(-1-\beta _{2})q^{3}+q^{4}+(-1-\beta _{2})q^{6}+\cdots\)
1850.2.a.bc	$1850$	$2$	1850.a	1.a	$1$	$3$	$3$	$14.772$	3.3.1524.1	None	✓	✓			1850.2.a.y	$1$	$0$	\(3\)	\(1\)	\(0\)	\(4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta _{1}q^{3}+q^{4}+\beta _{1}q^{6}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
1850.2.a.bd	$1850$	$2$	1850.a	1.a	$1$	$5$	$5$	$14.772$	5.5.1791440.1	None	✓		✓		370.2.b.d	$1$	$0$	\(-5\)	\(0\)	\(0\)	\(1\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
1850.2.a.be	$1850$	$2$	1850.a	1.a	$1$	$5$	$5$	$14.772$	5.5.1791440.1	None	✓		✓		370.2.b.d	$1$	$0$	\(5\)	\(0\)	\(0\)	\(-1\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(1850))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(1850)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(37))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(50))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(74))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(185))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(370))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(925))\)\(^{\oplus 2}\)