Space of modular forms of level 816, weight 2, and trivial character

• Label: 816.2.a
• Level: $816$
• Weight: $2$
• Character orbit: 816.a
• Rep. character: $\chi_{816}(1,\cdot)$
• Character field: $\Q$
• Dimension: $16$
• Newform subspaces: $13$
• Sturm bound: $288$
• Trace bound: $19$

Defining parameters

Level:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	816.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 13 \)
Sturm bound:			\(288\)
Trace bound:			\(19\)
Distinguishing \(T_p\):			\(5\), \(7\), \(19\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(816))\).

	Total	New	Old
Modular forms	156	16	140
Cusp forms	133	16	117
Eisenstein series	23	0	23

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
\(+\)	\(+\)	\(+\)	\(+\)		\(16\)	\(1\)	\(15\)		\(14\)	\(1\)	\(13\)		\(2\)	\(0\)	\(2\)
\(+\)	\(+\)	\(-\)	\(-\)		\(23\)	\(4\)	\(19\)		\(20\)	\(4\)	\(16\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(+\)	\(-\)		\(19\)	\(3\)	\(16\)		\(16\)	\(3\)	\(13\)		\(3\)	\(0\)	\(3\)
\(+\)	\(-\)	\(-\)	\(+\)		\(20\)	\(0\)	\(20\)		\(17\)	\(0\)	\(17\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(+\)	\(-\)		\(23\)	\(2\)	\(21\)		\(20\)	\(2\)	\(18\)		\(3\)	\(0\)	\(3\)
\(-\)	\(+\)	\(-\)	\(+\)		\(16\)	\(2\)	\(14\)		\(13\)	\(2\)	\(11\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(+\)	\(+\)		\(20\)	\(2\)	\(18\)		\(17\)	\(2\)	\(15\)		\(3\)	\(0\)	\(3\)
\(-\)	\(-\)	\(-\)	\(-\)		\(19\)	\(2\)	\(17\)		\(16\)	\(2\)	\(14\)		\(3\)	\(0\)	\(3\)
Plus space			\(+\)		\(72\)	\(5\)	\(67\)		\(61\)	\(5\)	\(56\)		\(11\)	\(0\)	\(11\)
Minus space			\(-\)		\(84\)	\(11\)	\(73\)		\(72\)	\(11\)	\(61\)		\(12\)	\(0\)	\(12\)

Trace form

16 * q - 2 * q^3 + 16 * q^9 + 8 * q^11 - 4 * q^19 + 8 * q^23 + 24 * q^25 - 2 * q^27 - 16 * q^29 + 8 * q^31 + 8 * q^33 + 24 * q^35 - 16 * q^37 + 4 * q^39 + 16 * q^41 + 20 * q^43 + 24 * q^47 + 32 * q^49 - 6 * q^51 - 16 * q^53 + 4 * q^55 - 8 * q^57 + 16 * q^65 + 16 * q^67 + 8 * q^71 + 2 * q^75 - 16 * q^77 + 16 * q^81 + 40 * q^83 - 12 * q^87 + 32 * q^89 - 56 * q^91 + 16 * q^95 + 8 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(816))\) 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
816.2.a.a	$816$	$2$	816.a	1.a	$1$	$1$	$1$	$6.516$	\(\Q\)	None	✓				408.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(-3\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-3q^{5}+4q^{7}+q^{9}-q^{11}-5q^{13}+\cdots\)
816.2.a.b	$816$	$2$	816.a	1.a	$1$	$1$	$1$	$6.516$	\(\Q\)	None	✓				102.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(-2\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+q^{9}+4q^{11}-2q^{13}+\cdots\)
816.2.a.c	$816$	$2$	816.a	1.a	$1$	$1$	$1$	$6.516$	\(\Q\)	None	✓		✓	✓	408.2.a.c	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-2\)		$+$	$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}+2q^{13}-q^{17}+\cdots\)
816.2.a.d	$816$	$2$	816.a	1.a	$1$	$1$	$1$	$6.516$	\(\Q\)	None	✓				102.2.a.b	$1$	$0$	\(0\)	\(-1\)	\(0\)	\(-2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{7}+q^{9}+2q^{13}-q^{17}+\cdots\)
816.2.a.e	$816$	$2$	816.a	1.a	$1$	$1$	$1$	$6.516$	\(\Q\)	None	✓				204.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(1\)	\(0\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+q^{5}+q^{9}-5q^{11}-5q^{13}+\cdots\)
816.2.a.f	$816$	$2$	816.a	1.a	$1$	$1$	$1$	$6.516$	\(\Q\)	None	✓				408.2.a.d	$1$	$0$	\(0\)	\(-1\)	\(2\)	\(4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+2q^{5}+4q^{7}+q^{9}-4q^{11}+\cdots\)
816.2.a.g	$816$	$2$	816.a	1.a	$1$	$1$	$1$	$6.516$	\(\Q\)	None	✓				51.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(3\)	\(4\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+3q^{5}+4q^{7}+q^{9}+3q^{11}+\cdots\)
816.2.a.h	$816$	$2$	816.a	1.a	$1$	$1$	$1$	$6.516$	\(\Q\)	None	✓				102.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}+2q^{7}+q^{9}-6q^{13}+\cdots\)
816.2.a.i	$816$	$2$	816.a	1.a	$1$	$1$	$1$	$6.516$	\(\Q\)	None	✓				204.2.a.a	$1$	$1$	\(0\)	\(1\)	\(-1\)	\(-4\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-q^{5}-4q^{7}+q^{9}-3q^{11}+3q^{13}+\cdots\)
816.2.a.j	$816$	$2$	816.a	1.a	$1$	$1$	$1$	$6.516$	\(\Q\)	None	✓				408.2.a.a	$1$	$0$	\(0\)	\(1\)	\(3\)	\(0\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+3q^{5}+q^{9}+q^{11}+3q^{13}+\cdots\)
816.2.a.k	$816$	$2$	816.a	1.a	$1$	$2$	$2$	$6.516$	\(\Q(\sqrt{57}) \)	None	✓		✓		408.2.a.f	$1$	$0$	\(0\)	\(-2\)	\(-1\)	\(-8\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-\beta q^{5}-4q^{7}+q^{9}+(2-\beta )q^{11}+\cdots\)
816.2.a.l	$816$	$2$	816.a	1.a	$1$	$2$	$2$	$6.516$	\(\Q(\sqrt{17}) \)	None	✓		✓		408.2.a.e	$1$	$0$	\(0\)	\(2\)	\(-1\)	\(2\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-\beta q^{5}+(2-2\beta )q^{7}+q^{9}+(4+\cdots)q^{11}+\cdots\)
816.2.a.m	$816$	$2$	816.a	1.a	$1$	$2$	$2$	$6.516$	\(\Q(\sqrt{17}) \)	None	✓		✓	✓	51.2.a.b	$1$	$0$	\(0\)	\(2\)	\(3\)	\(0\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+(1+\beta )q^{5}+q^{9}+(1-\beta )q^{11}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(816))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(816)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 10}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 8}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 6}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(272))\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(\Gamma_0(408))\)\(^{\oplus 2}\)