Space of modular forms of level 408, weight 2, and trivial character

• Label: 408.2.a
• Level: $408$
• Weight: $2$
• Character orbit: 408.a
• Rep. character: $\chi_{408}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $6$
• Sturm bound: $144$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 408 = 2^{3} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	408.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 6 \)
Sturm bound:			\(144\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(408))\).

	Total	New	Old
Modular forms	80	8	72
Cusp forms	65	8	57
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
\(+\)	\(+\)	\(+\)	\(+\)		\(9\)	\(2\)	\(7\)		\(8\)	\(2\)	\(6\)		\(1\)	\(0\)	\(1\)
\(+\)	\(+\)	\(-\)	\(-\)		\(10\)	\(0\)	\(10\)		\(8\)	\(0\)	\(8\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(+\)	\(-\)		\(10\)	\(1\)	\(9\)		\(8\)	\(1\)	\(7\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)	\(+\)		\(9\)	\(1\)	\(8\)		\(7\)	\(1\)	\(6\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(+\)	\(-\)		\(11\)	\(1\)	\(10\)		\(9\)	\(1\)	\(8\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)	\(+\)		\(10\)	\(0\)	\(10\)		\(8\)	\(0\)	\(8\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)	\(+\)		\(10\)	\(0\)	\(10\)		\(8\)	\(0\)	\(8\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(-\)	\(-\)		\(11\)	\(3\)	\(8\)		\(9\)	\(3\)	\(6\)		\(2\)	\(0\)	\(2\)
Plus space			\(+\)		\(38\)	\(3\)	\(35\)		\(31\)	\(3\)	\(28\)		\(7\)	\(0\)	\(7\)
Minus space			\(-\)		\(42\)	\(5\)	\(37\)		\(34\)	\(5\)	\(29\)		\(8\)	\(0\)	\(8\)

Trace form

8 * q + 2 * q^3 + 8 * q^9 - 8 * q^11 + 4 * q^13 - 4 * q^15 - 4 * q^19 + 4 * q^21 - 8 * q^23 + 20 * q^25 + 2 * q^27 - 8 * q^31 + 12 * q^33 - 16 * q^35 + 4 * q^39 + 16 * q^41 - 4 * q^43 - 24 * q^47 + 48 * q^49 + 6 * q^51 - 8 * q^53 - 12 * q^55 - 24 * q^59 - 16 * q^61 + 8 * q^67 - 24 * q^71 - 8 * q^73 + 14 * q^75 - 40 * q^77 + 16 * q^79 + 8 * q^81 - 32 * q^83 - 4 * q^85 + 24 * q^89 + 24 * q^91 - 4 * q^93 + 32 * q^95 + 8 * q^97 - 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(408))\) 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
408.2.a.a	$408$	$2$	408.a	1.a	$1$	$1$	$1$	$3.258$	\(\Q\)	None	✓	✓	✓	✓	408.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(0\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}+q^{9}-q^{11}+3q^{13}+\cdots\)
408.2.a.b	$408$	$2$	408.a	1.a	$1$	$1$	$1$	$3.258$	\(\Q\)	None	✓	✓	✓	✓	408.2.a.b	$1$	$1$	\(0\)	\(1\)	\(-3\)	\(-4\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-3q^{5}-4q^{7}+q^{9}+q^{11}-5q^{13}+\cdots\)
408.2.a.c	$408$	$2$	408.a	1.a	$1$	$1$	$1$	$3.258$	\(\Q\)	None	✓	✓	✓	✓	408.2.a.c	$1$	$0$	\(0\)	\(1\)	\(0\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{7}+q^{9}+2q^{13}-q^{17}+\cdots\)
408.2.a.d	$408$	$2$	408.a	1.a	$1$	$1$	$1$	$3.258$	\(\Q\)	None	✓	✓			408.2.a.d	$1$	$0$	\(0\)	\(1\)	\(2\)	\(-4\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots\)
408.2.a.e	$408$	$2$	408.a	1.a	$1$	$2$	$2$	$3.258$	\(\Q(\sqrt{17}) \)	None	✓	✓	✓	✓	408.2.a.e	$1$	$1$	\(0\)	\(-2\)	\(-1\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta q^{5}+(-2+2\beta )q^{7}+q^{9}+\cdots\)
408.2.a.f	$408$	$2$	408.a	1.a	$1$	$2$	$2$	$3.258$	\(\Q(\sqrt{57}) \)	None	✓	✓	✓		408.2.a.f	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta q^{5}+4q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(408))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(408)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 2}\)