Space of modular forms of level 306, weight 2, and Character 306.g

• Label: 306.2.g
• Level: $306$
• Weight: $2$
• Character orbit: 306.g
• Rep. character: $\chi_{306}(55,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $16$
• Newform subspaces: $7$
• Sturm bound: $108$
• Trace bound: $7$

Defining parameters

Level:	\( N \)	\(=\)	\( 306 = 2 \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	306.g (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 17 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(108\)
Trace bound:			\(7\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(306, [\chi])\).

	Total	New	Old
Modular forms	124	16	108
Cusp forms	92	16	76
Eisenstein series	32	0	32

Trace form

16 * q - 16 * q^4 + 2 * q^5 - 4 * q^7 + 2 * q^10 + 6 * q^11 + 12 * q^13 + 4 * q^14 + 16 * q^16 + 6 * q^17 - 2 * q^20 + 6 * q^22 - 16 * q^23 + 4 * q^28 + 6 * q^29 - 28 * q^31 - 6 * q^34 + 16 * q^35 - 6 * q^37 - 2 * q^40 + 16 * q^41 - 6 * q^44 - 8 * q^46 - 48 * q^47 - 16 * q^50 - 12 * q^52 - 8 * q^55 - 4 * q^56 - 26 * q^58 + 26 * q^61 - 28 * q^62 - 16 * q^64 + 48 * q^65 + 52 * q^67 - 6 * q^68 - 8 * q^71 + 4 * q^73 + 18 * q^74 - 48 * q^79 + 2 * q^80 + 44 * q^82 - 14 * q^85 + 20 * q^86 - 6 * q^88 + 56 * q^91 + 16 * q^92 - 24 * q^95 - 64 * q^97 + 36 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(306, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
306.2.g.a	$306$	$2$	306.g	17.c	$4$	$2$	$1$	$2.443$	\(\Q(\sqrt{-1}) \)	None		✓			306.2.g.a	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{2}-q^{4}+(-2 i-2)q^{5}+(3 i-3)q^{7}+\cdots\)
306.2.g.b	$306$	$2$	306.g	17.c	$4$	$2$	$1$	$2.443$	\(\Q(\sqrt{-1}) \)	None					34.2.c.a	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{2}-q^{4}+(-2 i-2)q^{5}+(2 i-2)q^{7}+\cdots\)
306.2.g.c	$306$	$2$	306.g	17.c	$4$	$2$	$1$	$2.443$	\(\Q(\sqrt{-1}) \)	None		✓			306.2.g.c	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{2}-q^{4}+(-2 i-2)q^{5}+(-i+1)q^{7}+\cdots\)
306.2.g.d	$306$	$2$	306.g	17.c	$4$	$2$	$1$	$2.443$	\(\Q(\sqrt{-1}) \)	None					34.2.c.b	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{2}-q^{4}+(i+1)q^{5}+i q^{8}+\cdots\)
306.2.g.e	$306$	$2$	306.g	17.c	$4$	$2$	$1$	$2.443$	\(\Q(\sqrt{-1}) \)	None		✓			306.2.g.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+i q^{2}-q^{4}+(2 i+2)q^{5}+(3 i-3)q^{7}+\cdots\)
306.2.g.f	$306$	$2$	306.g	17.c	$4$	$2$	$1$	$2.443$	\(\Q(\sqrt{-1}) \)	None		✓			306.2.g.c	$2$	$0$	\(0\)	\(0\)	\(4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-i q^{2}-q^{4}+(2 i+2)q^{5}+(-i+1)q^{7}+\cdots\)
306.2.g.g	$306$	$2$	306.g	17.c	$4$	$4$	$2$	$2.443$	\(\Q(\zeta_{8})\)	None			✓		102.2.f.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(8\)	$2^{2}$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta_{2} q^{2}-q^{4}+(\beta_{2}+\beta_1+1)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(306, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(306, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(34, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(51, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(102, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(153, [\chi])\)\(^{\oplus 2}\)