Space of modular forms of level 2850, weight 2, and Character 2850.d

• Label: 2850.2.d
• Level: $2850$
• Weight: $2$
• Character orbit: 2850.d
• Rep. character: $\chi_{2850}(799,\cdot)$
• Character field: $\Q$
• Dimension: $56$
• Newform subspaces: $24$
• Sturm bound: $1200$
• Trace bound: $26$

Defining parameters

Level:	\( N \)	\(=\)	\( 2850 = 2 \cdot 3 \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2850.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(1200\)
Trace bound:			\(26\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(2850, [\chi])\).

	Total	New	Old
Modular forms	624	56	568
Cusp forms	576	56	520
Eisenstein series	48	0	48

Trace form

56 * q - 56 * q^4 + 4 * q^6 - 56 * q^9 + 8 * q^11 + 56 * q^16 - 4 * q^19 - 4 * q^24 + 24 * q^26 + 8 * q^29 - 40 * q^31 - 8 * q^34 + 56 * q^36 + 16 * q^39 + 40 * q^41 - 8 * q^44 - 16 * q^46 - 24 * q^49 - 32 * q^51 - 4 * q^54 + 64 * q^59 - 8 * q^61 - 56 * q^64 + 32 * q^69 - 64 * q^71 - 56 * q^74 + 4 * q^76 + 40 * q^79 + 56 * q^81 + 16 * q^86 + 40 * q^89 + 96 * q^91 + 48 * q^94 + 4 * q^96 - 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(2850, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
2850.2.d.a	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.e	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+i q^{3}-q^{4}-q^{6}+4 i q^{7}+\cdots\)
2850.2.d.b	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					114.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+i q^{8}+\cdots\)
2850.2.d.c	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+i q^{3}-q^{4}-q^{6}+4 i q^{7}+\cdots\)
2850.2.d.d	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+2 i q^{7}+\cdots\)
2850.2.d.e	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.h	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+2 i q^{7}+\cdots\)
2850.2.d.f	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+4 i q^{7}+\cdots\)
2850.2.d.g	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None		✓			2850.2.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+i q^{3}-q^{4}-q^{6}-i q^{8}+\cdots\)
2850.2.d.h	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None		✓			2850.2.a.o	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+2 i q^{7}+\cdots\)
2850.2.d.i	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+i q^{8}+\cdots\)
2850.2.d.j	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+2 i q^{7}+\cdots\)
2850.2.d.k	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.m	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-i q^{3}-q^{4}+q^{6}+4 i q^{7}+\cdots\)
2850.2.d.l	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-i q^{3}-q^{4}+q^{6}+2 i q^{7}+\cdots\)
2850.2.d.m	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None		✓			2850.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+2 i q^{7}+\cdots\)
2850.2.d.n	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-i q^{3}-q^{4}+q^{6}+2 i q^{7}+\cdots\)
2850.2.d.o	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None		✓			2850.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+4 i q^{7}+\cdots\)
2850.2.d.p	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					114.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+4 i q^{7}+\cdots\)
2850.2.d.q	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+2 i q^{7}+\cdots\)
2850.2.d.r	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-i q^{3}-q^{4}+q^{6}+2 i q^{7}+\cdots\)
2850.2.d.s	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					114.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+4 i q^{7}+\cdots\)
2850.2.d.t	$2850$	$2$	2850.d	5.b	$2$	$2$	$2$	$22.757$	\(\Q(\sqrt{-1}) \)	None					570.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-i q^{3}-q^{4}+q^{6}+2 i q^{7}+\cdots\)
2850.2.d.u	$2850$	$2$	2850.d	5.b	$2$	$4$	$4$	$22.757$	\(\Q(\zeta_{12})\)	None		✓			2850.2.a.be	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{2}+\beta_1 q^{3}-q^{4}-q^{6}+(-\beta_{2}+\beta_1)q^{7}+\cdots\)
2850.2.d.v	$2850$	$2$	2850.d	5.b	$2$	$4$	$4$	$22.757$	\(\Q(i, \sqrt{10})\)	None		✓			2850.2.a.bf	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}-q^{6}+\beta _{2}q^{7}+\cdots\)
2850.2.d.w	$2850$	$2$	2850.d	5.b	$2$	$4$	$4$	$22.757$	\(\Q(i, \sqrt{6})\)	None		✓			2850.2.a.bc	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-\beta _{1}q^{3}-q^{4}+q^{6}+(2\beta _{1}+\cdots)q^{7}+\cdots\)
2850.2.d.x	$2850$	$2$	2850.d	5.b	$2$	$4$	$4$	$22.757$	\(\Q(\zeta_{12})\)	None		✓			2850.2.a.bd	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_1 q^{2}-\beta_1 q^{3}-q^{4}+q^{6}+(-\beta_{2}+\beta_1)q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(2850, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(2850, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(95, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(190, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(285, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(475, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(570, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(950, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1425, [\chi])\)\(^{\oplus 2}\)