Space of modular forms of level 3420, weight 2, and Character 3420.f

• Label: 3420.2.f
• Level: $3420$
• Weight: $2$
• Character orbit: 3420.f
• Rep. character: $\chi_{3420}(1369,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $6$
• Sturm bound: $1440$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.f (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 5 \)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(1440\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(3420, [\chi])\).

	Total	New	Old
Modular forms	744	46	698
Cusp forms	696	46	650
Eisenstein series	48	0	48

Trace form

46 * q + 3 * q^5 - 10 * q^11 - 2 * q^19 + 3 * q^25 + 4 * q^29 - 8 * q^31 + 31 * q^35 - 12 * q^41 - 28 * q^49 + 25 * q^55 - 12 * q^59 - 14 * q^61 + 16 * q^65 - 4 * q^71 - 48 * q^79 - 11 * q^85 + 8 * q^89 - 16 * q^91 + q^95

Decomposition of \(S_{2}^{\mathrm{new}}(3420, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
3420.2.f.a	$3420$	$2$	3420.f	5.b	$2$	$4$	$4$	$27.309$	\(\Q(\sqrt{-2}, \sqrt{5})\)	None					380.2.c.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{5}+(\beta _{1}-\beta _{3})q^{7}+(3-\beta _{2})q^{11}+\cdots\)
3420.2.f.b	$3420$	$2$	3420.f	5.b	$2$	$6$	$6$	$27.309$	6.0.350464.1	None					1140.2.f.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+(\beta _{1}-\beta _{4})q^{5}-\beta _{5}q^{7}+(1-\beta _{2})q^{11}+\cdots\)
3420.2.f.c	$3420$	$2$	3420.f	5.b	$2$	$6$	$6$	$27.309$	6.0.14077504.2	None					380.2.c.b	$2$	$0$	\(0\)	\(0\)	\(1\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{5}+(-\beta _{3}-\beta _{4})q^{7}+(-3-\beta _{1}+\cdots)q^{11}+\cdots\)
3420.2.f.d	$3420$	$2$	3420.f	5.b	$2$	$8$	$8$	$27.309$	\(\Q(\sqrt{2}, \sqrt{-3}, \sqrt{-7})\)	None		✓			3420.2.f.d	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{5}-\beta _{5}q^{7}-\beta _{1}q^{11}-\beta _{5}q^{13}+\cdots\)
3420.2.f.e	$3420$	$2$	3420.f	5.b	$2$	$10$	$10$	$27.309$	\(\mathbb{Q}[x]/(x^{10} - \cdots)\)	None					1140.2.f.b	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{7}q^{5}+\beta _{4}q^{7}+(-1+\beta _{5})q^{11}+\cdots\)
3420.2.f.f	$3420$	$2$	3420.f	5.b	$2$	$12$	$12$	$27.309$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	None		✓	✓		3420.2.f.f	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{5}+\beta _{5}q^{7}+\beta _{3}q^{11}+(-\beta _{2}+\cdots)q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(3420, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(3420, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(30, [\chi])\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(45, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(60, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(90, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(95, [\chi])\)\(^{\oplus 9}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(180, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(190, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(285, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(380, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(570, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(855, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1140, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(1710, [\chi])\)\(^{\oplus 2}\)