Space of modular forms of level 650, weight 4, and Character 650.b

• Label: 650.4.b
• Level: $650$
• Weight: $4$
• Character orbit: 650.b
• Rep. character: $\chi_{650}(599,\cdot)$
• Character field: $\Q$
• Dimension: $54$
• Newform subspaces: $16$
• Sturm bound: $420$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	650.b (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 16 \)
Sturm bound:			\(420\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(650, [\chi])\).

	Total	New	Old
Modular forms	326	54	272
Cusp forms	302	54	248
Eisenstein series	24	0	24

Trace form

54 * q - 216 * q^4 + 8 * q^6 - 414 * q^9 - 68 * q^11 - 176 * q^14 + 864 * q^16 - 76 * q^19 + 280 * q^21 - 32 * q^24 + 156 * q^26 + 360 * q^29 - 760 * q^31 - 272 * q^34 + 1656 * q^36 + 1608 * q^41 + 272 * q^44 - 1504 * q^46 - 2150 * q^49 - 696 * q^51 - 2024 * q^54 + 704 * q^56 + 8 * q^59 - 280 * q^61 - 3456 * q^64 - 2056 * q^66 - 4328 * q^69 - 1184 * q^71 + 1536 * q^74 + 304 * q^76 + 128 * q^79 + 4614 * q^81 - 1120 * q^84 + 1264 * q^86 - 2400 * q^89 + 1612 * q^91 - 1520 * q^94 + 128 * q^96 + 7432 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(650, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
650.4.b.a	$650$	$4$	650.b	5.b	$2$	$2$	$2$	$38.351$	\(\Q(\sqrt{-1}) \)	None					130.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+4 i q^{3}-4 q^{4}-8 q^{6}+\cdots\)
650.4.b.b	$650$	$4$	650.b	5.b	$2$	$2$	$2$	$38.351$	\(\Q(\sqrt{-1}) \)	None					26.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+3 i q^{3}-4 q^{4}-6 q^{6}+\cdots\)
650.4.b.c	$650$	$4$	650.b	5.b	$2$	$2$	$2$	$38.351$	\(\Q(\sqrt{-1}) \)	None		✓			650.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+3 i q^{3}-4 q^{4}-6 q^{6}+\cdots\)
650.4.b.d	$650$	$4$	650.b	5.b	$2$	$2$	$2$	$38.351$	\(\Q(\sqrt{-1}) \)	None					26.4.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{2}+i q^{3}-4 q^{4}-2 q^{6}-35 i q^{7}+\cdots\)
650.4.b.e	$650$	$4$	650.b	5.b	$2$	$2$	$2$	$38.351$	\(\Q(\sqrt{-1}) \)	None					130.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}+2 i q^{3}-4 q^{4}+4 q^{6}+\cdots\)
650.4.b.f	$650$	$4$	650.b	5.b	$2$	$2$	$2$	$38.351$	\(\Q(\sqrt{-1}) \)	None					26.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}+4 i q^{3}-4 q^{4}+8 q^{6}+\cdots\)
650.4.b.g	$650$	$4$	650.b	5.b	$2$	$2$	$2$	$38.351$	\(\Q(\sqrt{-1}) \)	None		✓			650.4.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2 i q^{2}+7 i q^{3}-4 q^{4}+14 q^{6}+\cdots\)
650.4.b.h	$650$	$4$	650.b	5.b	$2$	$4$	$4$	$38.351$	\(\Q(i, \sqrt{454})\)	None		✓			650.4.a.n	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}+4\beta _{1}q^{3}-4q^{4}-8q^{6}+\cdots\)
650.4.b.i	$650$	$4$	650.b	5.b	$2$	$4$	$4$	$38.351$	\(\Q(i, \sqrt{105})\)	None		✓			650.4.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{2}q^{2}+(\beta _{1}-3\beta _{2})q^{3}-4q^{4}+(-8+\cdots)q^{6}+\cdots\)
650.4.b.j	$650$	$4$	650.b	5.b	$2$	$4$	$4$	$38.351$	\(\Q(i, \sqrt{145})\)	None		✓			650.4.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{2}q^{2}+(\beta _{1}-\beta _{2})q^{3}-4q^{4}+(-4+\cdots)q^{6}+\cdots\)
650.4.b.k	$650$	$4$	650.b	5.b	$2$	$4$	$4$	$38.351$	\(\Q(i, \sqrt{5})\)	None					130.4.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}+(\beta _{1}-3\beta _{2})q^{3}-4q^{4}+(-2+\cdots)q^{6}+\cdots\)
650.4.b.l	$650$	$4$	650.b	5.b	$2$	$4$	$4$	$38.351$	\(\Q(i, \sqrt{19})\)	None					130.4.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{2}+(-\beta _{1}+\beta _{3})q^{3}-4q^{4}+\cdots\)
650.4.b.m	$650$	$4$	650.b	5.b	$2$	$4$	$4$	$38.351$	\(\Q(i, \sqrt{51})\)	None					130.4.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+2\beta _{1}q^{2}+(-\beta _{1}-\beta _{3})q^{3}-4q^{4}+\cdots\)
650.4.b.n	$650$	$4$	650.b	5.b	$2$	$4$	$4$	$38.351$	\(\Q(i, \sqrt{30})\)	None					130.4.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{2}+(4\beta _{1}-\beta _{2})q^{3}-4q^{4}+(8+\cdots)q^{6}+\cdots\)
650.4.b.o	$650$	$4$	650.b	5.b	$2$	$4$	$4$	$38.351$	\(\Q(i, \sqrt{10})\)	None					130.4.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{1}q^{2}+(6\beta _{1}-\beta _{2})q^{3}-4q^{4}+(12+\cdots)q^{6}+\cdots\)
650.4.b.p	$650$	$4$	650.b	5.b	$2$	$8$	$8$	$38.351$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		✓	✓		650.4.a.w	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-2\beta _{2}q^{2}+\beta _{1}q^{3}-4q^{4}-2\beta _{3}q^{6}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(650, [\chi])\) into lower level spaces

\( S_{4}^{\mathrm{old}}(650, [\chi]) \simeq \) \(S_{4}^{\mathrm{new}}(10, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 2}\)