Space of modular forms of level 3450, weight 2, and Character 3450.d

• Label: 3450.2.d
• Level: $3450$
• Weight: $2$
• Character orbit: 3450.d
• Rep. character: $\chi_{3450}(2899,\cdot)$
• Character field: $\Q$
• Dimension: $68$
• Newform subspaces: $27$
• Sturm bound: $1440$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 27 \)
Sturm bound:			\(1440\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(7\), \(11\), \(13\), \(17\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3450, [\chi])\).

	Total	New	Old
Modular forms	744	68	676
Cusp forms	696	68	628
Eisenstein series	48	0	48

Trace form

68 * q - 68 * q^4 + 4 * q^6 - 68 * q^9 + 68 * q^16 - 8 * q^19 + 8 * q^21 - 4 * q^24 + 24 * q^26 - 8 * q^29 - 24 * q^34 + 68 * q^36 - 8 * q^39 + 56 * q^41 + 8 * q^46 - 84 * q^49 - 4 * q^54 - 32 * q^59 + 64 * q^61 - 68 * q^64 - 8 * q^69 + 32 * q^71 - 24 * q^74 + 8 * q^76 + 16 * q^79 + 68 * q^81 - 8 * q^84 - 64 * q^86 - 40 * q^89 + 64 * q^91 - 16 * q^94 + 4 * q^96

Decomposition of \(S_{2}^{\mathrm{new}}(3450, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3450.2.d.a	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					690.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+i q^{8}+\cdots\)
3450.2.d.b	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None		✓			3450.2.a.h	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+i q^{3}-q^{4}-q^{6}+3 i q^{7}+\cdots\)
3450.2.d.c	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None		✓			3450.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+i q^{3}-q^{4}-q^{6}+i q^{7}+\cdots\)
3450.2.d.d	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					690.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+2 i q^{7}+\cdots\)
3450.2.d.e	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					690.2.a.h	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+i q^{8}+\cdots\)
3450.2.d.f	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					138.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}+i q^{3}-q^{4}-q^{6}-i q^{8}+\cdots\)
3450.2.d.g	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					138.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+2 i q^{7}+\cdots\)
3450.2.d.h	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None		✓			3450.2.a.m	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+5 i q^{7}+\cdots\)
3450.2.d.i	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					690.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}-i q^{3}-q^{4}-q^{6}+i q^{8}+\cdots\)
3450.2.d.j	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					138.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-i q^{3}-q^{4}+q^{6}+2 i q^{7}+\cdots\)
3450.2.d.k	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None		✓			3450.2.a.e	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+i q^{7}+\cdots\)
3450.2.d.l	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None		✓			3450.2.a.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-i q^{3}-q^{4}+q^{6}+5 i q^{7}+\cdots\)
3450.2.d.m	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					690.2.a.j	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-i q^{3}-q^{4}+q^{6}-i q^{8}+\cdots\)
3450.2.d.n	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					690.2.a.b	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+4 i q^{7}+\cdots\)
3450.2.d.o	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None		✓			3450.2.a.g	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+3 i q^{7}+\cdots\)
3450.2.d.p	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None		✓			3450.2.a.f	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+i q^{7}+\cdots\)
3450.2.d.q	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					690.2.a.c	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-i q^{3}-q^{4}+q^{6}+2 i q^{7}+\cdots\)
3450.2.d.r	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					690.2.a.i	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+i q^{8}+\cdots\)
3450.2.d.s	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					690.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+4 i q^{7}+\cdots\)
3450.2.d.t	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					690.2.a.k	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-i q^{2}+i q^{3}-q^{4}+q^{6}+i q^{8}+\cdots\)
3450.2.d.u	$3450$	$2$	3450.d	5.b	$2$	$2$	$2$	$27.548$	\(\Q(\sqrt{-1}) \)	None					690.2.a.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+i q^{2}-i q^{3}-q^{4}+q^{6}+2 i q^{7}+\cdots\)
3450.2.d.v	$3450$	$2$	3450.d	5.b	$2$	$4$	$4$	$27.548$	\(\Q(i, \sqrt{17})\)	None					690.2.a.l	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}-q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
3450.2.d.w	$3450$	$2$	3450.d	5.b	$2$	$4$	$4$	$27.548$	\(\Q(i, \sqrt{73})\)	None		✓			3450.2.a.bh	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}-\beta _{2}q^{3}-q^{4}-q^{6}-3\beta _{2}q^{7}+\cdots\)
3450.2.d.x	$3450$	$2$	3450.d	5.b	$2$	$4$	$4$	$27.548$	\(\Q(i, \sqrt{5})\)	None					138.2.a.d	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{2}-\beta _{1}q^{3}-q^{4}+q^{6}+2\beta _{2}q^{7}+\cdots\)
3450.2.d.y	$3450$	$2$	3450.d	5.b	$2$	$4$	$4$	$27.548$	\(\Q(i, \sqrt{6})\)	None		✓			3450.2.a.bf	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}+q^{6}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
3450.2.d.z	$3450$	$2$	3450.d	5.b	$2$	$4$	$4$	$27.548$	\(\Q(i, \sqrt{57})\)	None		✓			3450.2.a.bd	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{2}+\beta _{2}q^{3}-q^{4}+q^{6}-3\beta _{2}q^{7}+\cdots\)
3450.2.d.ba	$3450$	$2$	3450.d	5.b	$2$	$6$	$6$	$27.548$	6.0.181494784.1	None		✓	✓		3450.2.a.bp	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{2}+\beta _{4}q^{3}-q^{4}-q^{6}-\beta _{3}q^{7}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3450, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3450, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(75, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(115, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(230, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(345, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(575, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(690, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1150, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1725, [\chi])\)\(^{\oplus 2}\)