Space of modular forms of level 816, weight 4, and trivial character

• Label: 816.4.a
• Level: $816$
• Weight: $4$
• Character orbit: 816.a
• Rep. character: $\chi_{816}(1,\cdot)$
• Character field: $\Q$
• Dimension: $48$
• Newform subspaces: $24$
• Sturm bound: $576$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 816 = 2^{4} \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	816.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 24 \)
Sturm bound:			\(576\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(5\), \(7\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(816))\).

	Total	New	Old
Modular forms	444	48	396
Cusp forms	420	48	372
Eisenstein series	24	0	24

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	Dim
\(+\)	\(+\)	\(+\)	\(+\)	\(7\)
\(+\)	\(+\)	\(-\)	\(-\)	\(4\)
\(+\)	\(-\)	\(+\)	\(-\)	\(5\)
\(+\)	\(-\)	\(-\)	\(+\)	\(8\)
\(-\)	\(+\)	\(+\)	\(-\)	\(6\)
\(-\)	\(+\)	\(-\)	\(+\)	\(6\)
\(-\)	\(-\)	\(+\)	\(+\)	\(6\)
\(-\)	\(-\)	\(-\)	\(-\)	\(6\)
Plus space			\(+\)	\(27\)
Minus space			\(-\)	\(21\)

Trace form

48 * q + 6 * q^3 - 64 * q^7 + 432 * q^9 + 40 * q^11 - 180 * q^19 + 328 * q^23 + 1032 * q^25 + 54 * q^27 + 400 * q^29 - 312 * q^31 + 24 * q^33 - 456 * q^35 + 16 * q^37 - 732 * q^39 - 592 * q^41 + 292 * q^43 + 408 * q^47 + 2912 * q^49 + 306 * q^51 - 752 * q^53 + 612 * q^55 + 168 * q^57 - 576 * q^63 + 560 * q^65 + 720 * q^67 + 1480 * q^71 + 768 * q^73 + 2154 * q^75 - 1904 * q^77 + 64 * q^79 + 3888 * q^81 - 2680 * q^83 - 684 * q^87 + 352 * q^89 + 2280 * q^91 + 6416 * q^95 - 1024 * q^97 + 360 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(816))\) 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
816.4.a.a	$816$	$4$	816.a	1.a	$1$	$1$	$1$	$48.146$	\(\Q\)	None	✓				51.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(-20\)	\(2\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-20q^{5}+2q^{7}+9q^{9}+48q^{11}+\cdots\)
816.4.a.b	$816$	$4$	816.a	1.a	$1$	$1$	$1$	$48.146$	\(\Q\)	None	✓				408.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(-7\)	\(-4\)		$+$	$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-7q^{5}-4q^{7}+9q^{9}+21q^{11}+\cdots\)
816.4.a.c	$816$	$4$	816.a	1.a	$1$	$1$	$1$	$48.146$	\(\Q\)	None	✓				102.4.a.b	$1$	$1$	\(0\)	\(-3\)	\(-5\)	\(32\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-5q^{5}+2^{5}q^{7}+9q^{9}-3^{3}q^{11}+\cdots\)
816.4.a.d	$816$	$4$	816.a	1.a	$1$	$1$	$1$	$48.146$	\(\Q\)	None	✓				204.4.a.a	$1$	$0$	\(0\)	\(-3\)	\(-3\)	\(16\)		$-$	$+$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}-3q^{5}+2^{4}q^{7}+9q^{9}+57q^{11}+\cdots\)
816.4.a.e	$816$	$4$	816.a	1.a	$1$	$1$	$1$	$48.146$	\(\Q\)	None	✓				102.4.a.c	$1$	$0$	\(0\)	\(3\)	\(-12\)	\(22\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-12q^{5}+22q^{7}+9q^{9}+48q^{11}+\cdots\)
816.4.a.f	$816$	$4$	816.a	1.a	$1$	$1$	$1$	$48.146$	\(\Q\)	None	✓				51.4.a.c	$1$	$1$	\(0\)	\(3\)	\(-10\)	\(8\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-10q^{5}+8q^{7}+9q^{9}-12q^{11}+\cdots\)
816.4.a.g	$816$	$4$	816.a	1.a	$1$	$1$	$1$	$48.146$	\(\Q\)	None	✓				102.4.a.a	$1$	$1$	\(0\)	\(3\)	\(-3\)	\(-20\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}-3q^{5}-20q^{7}+9q^{9}+51q^{11}+\cdots\)
816.4.a.h	$816$	$4$	816.a	1.a	$1$	$1$	$1$	$48.146$	\(\Q\)	None	✓				102.4.a.d	$1$	$1$	\(0\)	\(3\)	\(5\)	\(-12\)		$-$	$-$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+5q^{5}-12q^{7}+9q^{9}-37q^{11}+\cdots\)
816.4.a.i	$816$	$4$	816.a	1.a	$1$	$1$	$1$	$48.146$	\(\Q\)	None	✓				408.4.a.a	$1$	$0$	\(0\)	\(3\)	\(6\)	\(24\)		$+$	$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+6q^{5}+24q^{7}+9q^{9}-44q^{11}+\cdots\)
816.4.a.j	$816$	$4$	816.a	1.a	$1$	$1$	$1$	$48.146$	\(\Q\)	None	✓				51.4.a.a	$1$	$0$	\(0\)	\(3\)	\(16\)	\(-34\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+2^{4}q^{5}-34q^{7}+9q^{9}+48q^{11}+\cdots\)
816.4.a.k	$816$	$4$	816.a	1.a	$1$	$2$	$2$	$48.146$	\(\Q(\sqrt{393}) \)	None	✓				102.4.a.f	$1$	$1$	\(0\)	\(-6\)	\(3\)	\(-22\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+(1+\beta )q^{5}+(-12+2\beta )q^{7}+\cdots\)
816.4.a.l	$816$	$4$	816.a	1.a	$1$	$2$	$2$	$48.146$	\(\Q(\sqrt{15}) \)	None	✓				102.4.a.e	$1$	$0$	\(0\)	\(-6\)	\(12\)	\(-16\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(6+\beta )q^{5}+(-8+\beta )q^{7}+9q^{9}+\cdots\)
816.4.a.m	$816$	$4$	816.a	1.a	$1$	$2$	$2$	$48.146$	\(\Q(\sqrt{217}) \)	None	✓				204.4.a.c	$1$	$1$	\(0\)	\(-6\)	\(19\)	\(-18\)		$-$	$+$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-3q^{3}+(10-\beta )q^{5}+(-10+2\beta )q^{7}+\cdots\)
816.4.a.n	$816$	$4$	816.a	1.a	$1$	$2$	$2$	$48.146$	\(\Q(\sqrt{201}) \)	None	✓				204.4.a.b	$1$	$0$	\(0\)	\(6\)	\(3\)	\(2\)		$-$	$-$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+(1+\beta )q^{5}+2\beta q^{7}+9q^{9}+\cdots\)
816.4.a.o	$816$	$4$	816.a	1.a	$1$	$2$	$2$	$48.146$	\(\Q(\sqrt{2}) \)	None	✓				51.4.a.d	$1$	$0$	\(0\)	\(6\)	\(6\)	\(8\)		$-$	$-$	$+$	$+$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	\(q+3q^{3}+(3+\beta )q^{5}+(4+\beta )q^{7}+9q^{9}+\cdots\)
816.4.a.p	$816$	$4$	816.a	1.a	$1$	$2$	$2$	$48.146$	\(\Q(\sqrt{241}) \)	None	✓				408.4.a.c	$1$	$1$	\(0\)	\(6\)	\(7\)	\(-18\)		$+$	$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+3q^{3}+(4-\beta )q^{5}+(-10+2\beta )q^{7}+\cdots\)
816.4.a.q	$816$	$4$	816.a	1.a	$1$	$3$	$3$	$48.146$	3.3.17717.1	None	✓				408.4.a.f	$1$	$0$	\(0\)	\(-9\)	\(-10\)	\(22\)		$+$	$+$	$+$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(-3-\beta _{1})q^{5}+(8-\beta _{1}+\beta _{2})q^{7}+\cdots\)
816.4.a.r	$816$	$4$	816.a	1.a	$1$	$3$	$3$	$48.146$	3.3.23321.1	None	✓		✓		408.4.a.g	$1$	$1$	\(0\)	\(-9\)	\(5\)	\(-20\)		$+$	$+$	$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(2+\beta _{2})q^{5}+(-7+\beta _{1})q^{7}+\cdots\)
816.4.a.s	$816$	$4$	816.a	1.a	$1$	$3$	$3$	$48.146$	3.3.5912.1	None	✓		✓		51.4.a.e	$1$	$0$	\(0\)	\(-9\)	\(8\)	\(8\)		$-$	$+$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q-3q^{3}+(3-\beta _{1})q^{5}+(2+3\beta _{1}+\beta _{2})q^{7}+\cdots\)
816.4.a.t	$816$	$4$	816.a	1.a	$1$	$3$	$3$	$48.146$	3.3.21324.1	None	✓		✓		204.4.a.d	$1$	$1$	\(0\)	\(9\)	\(-19\)	\(8\)		$-$	$-$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-7-\beta _{1}+\beta _{2})q^{5}+(3-\beta _{1}+\cdots)q^{7}+\cdots\)
816.4.a.u	$816$	$4$	816.a	1.a	$1$	$3$	$3$	$48.146$	3.3.4481.1	None	✓		✓		408.4.a.d	$1$	$1$	\(0\)	\(9\)	\(-5\)	\(4\)		$+$	$-$	$+$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1+\beta _{1}+\beta _{2})q^{5}+(2+2\beta _{1}+\cdots)q^{7}+\cdots\)
816.4.a.v	$816$	$4$	816.a	1.a	$1$	$3$	$3$	$48.146$	3.3.12821.1	None	✓				408.4.a.e	$1$	$0$	\(0\)	\(9\)	\(-4\)	\(-28\)		$+$	$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(-1-\beta _{1}+2\beta _{2})q^{5}+(-9+\cdots)q^{7}+\cdots\)
816.4.a.w	$816$	$4$	816.a	1.a	$1$	$4$	$4$	$48.146$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		408.4.a.i	$1$	$0$	\(0\)	\(-12\)	\(-2\)	\(-32\)		$+$	$+$	$+$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q-3q^{3}+\beta _{1}q^{5}+(-9-\beta _{1}+\beta _{2})q^{7}+\cdots\)
816.4.a.x	$816$	$4$	816.a	1.a	$1$	$4$	$4$	$48.146$	\(\mathbb{Q}[x]/(x^{4} - \cdots)\)	None	✓		✓		408.4.a.h	$1$	$0$	\(0\)	\(12\)	\(10\)	\(4\)		$+$	$-$	$-$	$+$	$2^{4}$	$\mathrm{SU}(2)$	\(q+3q^{3}+(2-\beta _{1})q^{5}+(1+\beta _{2}+\beta _{3})q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(816))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(816)) \simeq \) \(S_{4}^{\mathrm{new}}(\Gamma_0(6))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(12))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(16))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 10}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(24))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(34))\)\(^{\oplus 8}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(48))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(51))\)\(^{\oplus 5}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(68))\)\(^{\oplus 6}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(102))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(136))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(204))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(272))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(408))\)\(^{\oplus 2}\)