Space of modular forms of level 306, weight 2, and trivial character

• Label: 306.2.a
• Level: $306$
• Weight: $2$
• Character orbit: 306.a
• Rep. character: $\chi_{306}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $6$
• Sturm bound: $108$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 306 = 2 \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	306.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(108\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\), \(23\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(306))\).

	Total	New	Old
Modular forms	62	8	54
Cusp forms	47	8	39
Eisenstein series	15	0	15

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(3\)	\(17\)	Fricke		Total				Cusp				Eisenstein
					All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)	\(+\)		\(5\)	\(0\)	\(5\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(9\)	\(2\)	\(7\)		\(7\)	\(2\)	\(5\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(7\)	\(1\)	\(6\)		\(5\)	\(1\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(8\)	\(2\)	\(6\)		\(6\)	\(2\)	\(4\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(8\)	\(0\)	\(8\)		\(6\)	\(0\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(7\)	\(0\)	\(7\)		\(5\)	\(0\)	\(5\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(9\)	\(2\)	\(7\)		\(7\)	\(2\)	\(5\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(27\)	\(1\)	\(26\)		\(20\)	\(1\)	\(19\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(35\)	\(7\)	\(28\)		\(27\)	\(7\)	\(20\)		\(8\)	\(0\)	\(8\)

Trace form

8 * q + 8 * q^4 + 6 * q^5 + 4 * q^7 + 2 * q^10 - 2 * q^11 + 4 * q^13 + 4 * q^14 + 8 * q^16 + 2 * q^17 + 8 * q^19 + 6 * q^20 + 2 * q^22 + 4 * q^25 - 4 * q^26 + 4 * q^28 + 14 * q^29 - 4 * q^31 - 2 * q^34 - 8 * q^35 - 18 * q^37 + 2 * q^40 - 12 * q^41 - 4 * q^43 - 2 * q^44 - 24 * q^46 - 16 * q^47 + 8 * q^49 + 12 * q^50 + 4 * q^52 - 4 * q^53 - 40 * q^55 + 4 * q^56 - 6 * q^58 - 12 * q^59 - 26 * q^61 - 20 * q^62 + 8 * q^64 - 28 * q^65 - 12 * q^67 + 2 * q^68 - 32 * q^70 - 24 * q^73 + 10 * q^74 + 8 * q^76 + 24 * q^77 + 8 * q^79 + 6 * q^80 - 4 * q^82 - 4 * q^83 + 2 * q^85 - 28 * q^86 + 2 * q^88 + 32 * q^89 - 24 * q^91 - 16 * q^94 + 24 * q^95 - 20 * q^97 - 8 * q^98

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(306))\) 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					A-L signs			Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	3	17
306.2.a.a	$306$	$2$	306.a	1.a	$1$	$1$	$1$	$2.443$	\(\Q\)	None	✓		✓	✓	34.2.a.a	$1$	$1$	\(-1\)	\(0\)	\(0\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}-4q^{7}-q^{8}-6q^{11}+2q^{13}+\cdots\)
306.2.a.b	$306$	$2$	306.a	1.a	$1$	$1$	$1$	$2.443$	\(\Q\)	None	✓		✓	✓	102.2.a.c	$1$	$0$	\(-1\)	\(0\)	\(2\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+2q^{5}-q^{8}-2q^{10}+4q^{11}+\cdots\)
306.2.a.c	$306$	$2$	306.a	1.a	$1$	$1$	$1$	$2.443$	\(\Q\)	None	✓				102.2.a.b	$1$	$0$	\(1\)	\(0\)	\(0\)	\(2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+2q^{7}+q^{8}+2q^{13}+2q^{14}+\cdots\)
306.2.a.d	$306$	$2$	306.a	1.a	$1$	$1$	$1$	$2.443$	\(\Q\)	None	✓				102.2.a.a	$1$	$0$	\(1\)	\(0\)	\(4\)	\(-2\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+4q^{5}-2q^{7}+q^{8}+4q^{10}+\cdots\)
306.2.a.e	$306$	$2$	306.a	1.a	$1$	$2$	$2$	$2.443$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓	✓	306.2.a.e	$1$	$0$	\(-2\)	\(0\)	\(0\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}+q^{4}+\beta q^{5}+(2+\beta )q^{7}-q^{8}+\cdots\)
306.2.a.f	$306$	$2$	306.a	1.a	$1$	$2$	$2$	$2.443$	\(\Q(\sqrt{6}) \)	None	✓	✓	✓	✓	306.2.a.e	$1$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+q^{4}+\beta q^{5}+(2-\beta )q^{7}+q^{8}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(306))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(306)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(153))\)\(^{\oplus 2}\)