Space of modular forms of level 1300, weight 2, and Character 1300.c

• Label: 1300.2.c
• Level: $1300$
• Weight: $2$
• Character orbit: 1300.c
• Rep. character: $\chi_{1300}(1249,\cdot)$
• Character field: $\Q$
• Dimension: $18$
• Newform subspaces: $6$
• Sturm bound: $420$
• Trace bound: $11$

Defining parameters

Level:	\( N \)	\(=\)	\( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1300.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(420\)
Trace bound:			\(11\)
Distinguishing \(T_p\):			\(3\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(1300, [\chi])\).

	Total	New	Old
Modular forms	228	18	210
Cusp forms	192	18	174
Eisenstein series	36	0	36

Trace form

\( 18 q - 30 q^{9} + 4 q^{19} + 20 q^{21} - 4 q^{29} - 4 q^{31} - 16 q^{41} - 38 q^{49} - 32 q^{51} + 44 q^{59} + 36 q^{61} - 36 q^{69} + 24 q^{71} + 4 q^{79} + 114 q^{81} - 32 q^{89} - 12 q^{91} - 16 q^{99}+O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(1300, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1300.2.c.a	$1300$	$2$	1300.c	5.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-1}) \)	None					260.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{3}-2 i q^{7}-q^{9}+4 q^{11}+\cdots\)
1300.2.c.b	$1300$	$2$	1300.c	5.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-1}) \)	None		✓			1300.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{3}+2 i q^{7}+2 q^{9}-2 q^{11}+\cdots\)
1300.2.c.c	$1300$	$2$	1300.c	5.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-1}) \)	None					52.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+2 i q^{7}+3 q^{9}-2 q^{11}-i q^{13}+\cdots\)
1300.2.c.d	$1300$	$2$	1300.c	5.b	$2$	$2$	$2$	$10.381$	\(\Q(\sqrt{-1}) \)	None		✓			1300.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+3 i q^{7}+3 q^{9}+3 q^{11}+i q^{13}+\cdots\)
1300.2.c.e	$1300$	$2$	1300.c	5.b	$2$	$4$	$4$	$10.381$	\(\Q(i, \sqrt{33})\)	None		✓			1300.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{2})q^{7}+(-6+\beta _{3})q^{9}+\cdots\)
1300.2.c.f	$1300$	$2$	1300.c	5.b	$2$	$6$	$6$	$10.381$	6.0.5089536.1	None			✓		260.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-\beta _{1}-\beta _{4})q^{3}+(-\beta _{4}-\beta _{5})q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(1300, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(1300, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(65, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(130, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(260, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(325, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(650, [\chi])\)\(^{\oplus 2}\)