Space of modular forms of level 650, weight 2, and Character 650.e

• Label: 650.2.e
• Level: $650$
• Weight: $2$
• Character orbit: 650.e
• Rep. character: $\chi_{650}(451,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $42$
• Newform subspaces: $11$
• Sturm bound: $210$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	650.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(210\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(650, [\chi])\).

	Total	New	Old
Modular forms	232	42	190
Cusp forms	184	42	142
Eisenstein series	48	0	48

Trace form

42 * q + q^2 - 21 * q^4 - 4 * q^7 - 2 * q^8 - 21 * q^9 + 8 * q^11 - 3 * q^13 - 21 * q^16 - 5 * q^17 - 10 * q^18 + 16 * q^19 + 8 * q^21 - 4 * q^22 + 4 * q^23 - 17 * q^26 - 24 * q^27 - 4 * q^28 + 17 * q^29 + 40 * q^31 + q^32 + 20 * q^33 - 26 * q^34 - 21 * q^36 + 15 * q^37 + 24 * q^38 - 44 * q^39 - 7 * q^41 + 12 * q^42 + 8 * q^43 - 16 * q^44 + 4 * q^46 + 48 * q^47 - 5 * q^49 - 16 * q^51 - 6 * q^52 + 50 * q^53 - 12 * q^54 - 64 * q^57 - 13 * q^58 + 5 * q^61 - 4 * q^62 - 44 * q^63 + 42 * q^64 - 48 * q^66 + 4 * q^67 - 5 * q^68 - 4 * q^69 + 24 * q^71 + 5 * q^72 - 78 * q^73 + 25 * q^74 + 16 * q^76 - 24 * q^77 + 20 * q^78 + 16 * q^79 + 27 * q^81 - 25 * q^82 - 4 * q^84 + 8 * q^86 + 40 * q^87 - 4 * q^88 - 50 * q^89 + 28 * q^91 - 8 * q^92 + 44 * q^93 - 20 * q^94 - 2 * q^97 - 7 * q^98 - 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(650, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
650.2.e.a	$650$	$2$	650.e	13.c	$3$	$2$	$1$	$5.190$	\(\Q(\sqrt{-3}) \)	None					130.2.e.b	$2$	$1$	\(-1\)	\(-2\)	\(0\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
650.2.e.b	$650$	$2$	650.e	13.c	$3$	$2$	$1$	$5.190$	\(\Q(\sqrt{-3}) \)	None					130.2.e.a	$2$	$0$	\(1\)	\(0\)	\(0\)	\(-3\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}-3\zeta_{6}q^{7}-q^{8}+\cdots\)
650.2.e.c	$650$	$2$	650.e	13.c	$3$	$2$	$1$	$5.190$	\(\Q(\sqrt{-3}) \)	None					26.2.c.a	$2$	$0$	\(1\)	\(0\)	\(0\)	\(4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{4}+4\zeta_{6}q^{7}-q^{8}+\cdots\)
650.2.e.d	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		✓			650.2.e.d	$2$	$0$	\(-2\)	\(-1\)	\(0\)	\(1\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{1})q^{2}-\beta _{3}q^{3}+\beta _{1}q^{4}+(-\beta _{2}+\cdots)q^{6}+\cdots\)
650.2.e.e	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		✓			650.2.e.e	$2$	$0$	\(-2\)	\(1\)	\(0\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1-\beta _{2})q^{2}+(1-\beta _{1}+\beta _{2})q^{3}+\cdots\)
650.2.e.f	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\zeta_{12})\)	None					130.2.e.d	$2$	$0$	\(-2\)	\(2\)	\(0\)	\(-2\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q-\beta_1 q^{2}+(-\beta_{2}+\beta_1)q^{3}+(\beta_1-1)q^{4}+\cdots\)
650.2.e.g	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\sqrt{-3}, \sqrt{13})\)	None		✓			650.2.e.e	$2$	$0$	\(2\)	\(-1\)	\(0\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+\beta _{2}q^{4}+(1-\beta _{1}+\cdots)q^{6}+\cdots\)
650.2.e.h	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\sqrt{-3}, \sqrt{10})\)	None					130.2.e.c	$2$	$0$	\(2\)	\(0\)	\(0\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{2})q^{2}-\beta _{1}q^{3}+\beta _{2}q^{4}+(-\beta _{1}+\cdots)q^{6}+\cdots\)
650.2.e.i	$650$	$2$	650.e	13.c	$3$	$4$	$2$	$5.190$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		✓			650.2.e.d	$2$	$0$	\(2\)	\(1\)	\(0\)	\(-1\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{1})q^{2}+\beta _{3}q^{3}+\beta _{1}q^{4}+(-\beta _{2}+\cdots)q^{6}+\cdots\)
650.2.e.j	$650$	$2$	650.e	13.c	$3$	$6$	$3$	$5.190$	6.0.591408.1	None					130.2.n.a	$2$	$0$	\(-3\)	\(0\)	\(0\)	\(5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{4})q^{2}-\beta _{5}q^{3}-\beta _{4}q^{4}+(-\beta _{3}+\cdots)q^{6}+\cdots\)
650.2.e.k	$650$	$2$	650.e	13.c	$3$	$6$	$3$	$5.190$	6.0.591408.1	None					130.2.n.a	$2$	$0$	\(3\)	\(0\)	\(0\)	\(-5\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+\beta _{4}q^{2}+(\beta _{3}-\beta _{5})q^{3}+(-1+\beta _{4}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(650, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(650, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 2}\)