Space of modular forms of level 6900, weight 2, and Character 6900.f

• Label: 6900.2.f
• Level: $6900$
• Weight: $2$
• Character orbit: 6900.f
• Rep. character: $\chi_{6900}(6349,\cdot)$
• Character field: $\Q$
• Dimension: $68$
• Newform subspaces: $19$
• Sturm bound: $2880$
• Trace bound: $21$

Defining parameters

Level:	\( N \)	\(=\)	\( 6900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6900.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 19 \)
Sturm bound:			\(2880\)
Trace bound:			\(21\)
Distinguishing \(T_p\):			\(7\), \(11\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(6900, [\chi])\).

	Total	New	Old
Modular forms	1476	68	1408
Cusp forms	1404	68	1336
Eisenstein series	72	0	72

Trace form

\( 68 q - 68 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(6900, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
6900.2.f.a	$6900$	$2$	6900.f	5.b	$2$	$2$	$2$	$55.097$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+3iq^{7}-q^{9}-5q^{11}-iq^{13}+\cdots\)
6900.2.f.b	$6900$	$2$	6900.f	5.b	$2$	$2$	$2$	$55.097$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+3iq^{7}-q^{9}-2q^{11}-2iq^{13}+\cdots\)
6900.2.f.c	$6900$	$2$	6900.f	5.b	$2$	$2$	$2$	$55.097$	\(\Q(\sqrt{-1}) \)	None		$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-q^{9}+6iq^{13}+2iq^{17}-6q^{19}+\cdots\)
6900.2.f.d	$6900$	$2$	6900.f	5.b	$2$	$2$	$2$	$55.097$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+4iq^{7}-q^{9}+2iq^{13}-6iq^{17}+\cdots\)
6900.2.f.e	$6900$	$2$	6900.f	5.b	$2$	$2$	$2$	$55.097$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+iq^{7}-q^{9}-4iq^{13}+3iq^{17}+\cdots\)
6900.2.f.f	$6900$	$2$	6900.f	5.b	$2$	$2$	$2$	$55.097$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-iq^{3}+5iq^{7}-q^{9}+4iq^{13}+3iq^{17}+\cdots\)
6900.2.f.g	$6900$	$2$	6900.f	5.b	$2$	$2$	$2$	$55.097$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+3iq^{7}-q^{9}+q^{11}+iq^{13}+\cdots\)
6900.2.f.h	$6900$	$2$	6900.f	5.b	$2$	$4$	$4$	$55.097$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}-\beta _{1}q^{7}-q^{9}+(-2+\beta _{3})q^{11}+\cdots\)
6900.2.f.i	$6900$	$2$	6900.f	5.b	$2$	$4$	$4$	$55.097$	\(\Q(\zeta_{12})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}q^{3}+(\zeta_{12}-2\zeta_{12}^{2})q^{7}-q^{9}+\cdots\)
6900.2.f.j	$6900$	$2$	6900.f	5.b	$2$	$4$	$4$	$55.097$	\(\Q(i, \sqrt{73})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-\beta _{2}q^{7}-q^{9}+(-1+\beta _{3})q^{11}+\cdots\)
6900.2.f.k	$6900$	$2$	6900.f	5.b	$2$	$4$	$4$	$55.097$	\(\Q(i, \sqrt{10})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}+(2\beta _{1}-\beta _{2})q^{7}-q^{9}-4\beta _{1}q^{13}+\cdots\)
6900.2.f.l	$6900$	$2$	6900.f	5.b	$2$	$4$	$4$	$55.097$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{8}q^{3}-\zeta_{8}^{2}q^{7}-q^{9}-4\zeta_{8}^{3}q^{11}+\cdots\)
6900.2.f.m	$6900$	$2$	6900.f	5.b	$2$	$4$	$4$	$55.097$	\(\Q(i, \sqrt{13})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(\beta _{1}+3\beta _{2})q^{7}-q^{9}+\beta _{3}q^{11}+\cdots\)
6900.2.f.n	$6900$	$2$	6900.f	5.b	$2$	$4$	$4$	$55.097$	\(\Q(i, \sqrt{21})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(\beta _{1}-\beta _{2})q^{7}-q^{9}+\beta _{3}q^{11}+\cdots\)
6900.2.f.o	$6900$	$2$	6900.f	5.b	$2$	$4$	$4$	$55.097$	\(\Q(i, \sqrt{15})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+3\beta _{1}q^{7}-q^{9}+(1+\beta _{2})q^{11}+\cdots\)
6900.2.f.p	$6900$	$2$	6900.f	5.b	$2$	$4$	$4$	$55.097$	\(\Q(i, \sqrt{17})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+\beta _{1}q^{7}-q^{9}+(2-2\beta _{3})q^{11}+\cdots\)
6900.2.f.q	$6900$	$2$	6900.f	5.b	$2$	$4$	$4$	$55.097$	\(\Q(i, \sqrt{13})\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}+(\beta _{1}+3\beta _{2})q^{7}-q^{9}+(2+\beta _{3})q^{11}+\cdots\)
6900.2.f.r	$6900$	$2$	6900.f	5.b	$2$	$6$	$6$	$55.097$	6.0.158155776.1	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(-\beta _{2}-\beta _{5})q^{7}-q^{9}+(1+\cdots)q^{11}+\cdots\)
6900.2.f.s	$6900$	$2$	6900.f	5.b	$2$	$8$	$8$	$55.097$	\(\mathbb{Q}[x]/(x^{8} + \cdots)\)	None		$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{4}q^{3}+(-\beta _{4}+\beta _{5})q^{7}-q^{9}+\beta _{2}q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(6900, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(6900, [\chi]) \cong \) \(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1380, [\chi])\)\(^{\oplus 2}\)