Space of modular forms of level 16, weight 10, and trivial character

• Label: 16.10.a
• Level: $16$
• Weight: $10$
• Character orbit: 16.a
• Rep. character: $\chi_{16}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $4$
• Sturm bound: $20$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(20\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(16))\).

	Total	New	Old
Modular forms	21	5	16
Cusp forms	15	4	11
Eisenstein series	6	1	5

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

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(10\)	\(2\)	\(8\)		\(7\)	\(2\)	\(5\)		\(3\)	\(0\)	\(3\)
\(-\)		\(11\)	\(3\)	\(8\)		\(8\)	\(2\)	\(6\)		\(3\)	\(1\)	\(2\)

Trace form

4 * q - 80 * q^3 - 360 * q^5 + 1376 * q^7 + 5812 * q^9 - 10992 * q^11 - 43080 * q^13 + 60448 * q^15 + 172104 * q^17 - 296336 * q^19 - 336768 * q^21 + 1349664 * q^23 - 30468 * q^25 - 5005088 * q^27 + 1723896 * q^29 + 13751680 * q^31 - 3529024 * q^33 - 27870144 * q^35 + 1408792 * q^37 + 46429600 * q^39 + 4797864 * q^41 - 78798192 * q^43 - 6847304 * q^45 + 139372608 * q^47 + 3427300 * q^49 - 202646944 * q^51 - 10028712 * q^53 + 216893536 * q^55 + 25903936 * q^57 - 224197296 * q^59 - 10080648 * q^61 + 293290464 * q^63 - 64788144 * q^65 - 255761744 * q^67 + 121435520 * q^69 - 39174048 * q^71 - 15735768 * q^73 + 276237392 * q^75 - 120707712 * q^77 - 341312064 * q^79 - 113638172 * q^81 + 758656368 * q^83 + 277817008 * q^85 - 1417283424 * q^87 + 573376104 * q^89 + 1721466688 * q^91 - 1004178944 * q^93 - 2153970528 * q^95 - 728181880 * q^97 + 2808900560 * q^99

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(16))\) 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
16.10.a.a	$16$	$10$	16.a	1.a	$1$	$1$	$1$	$8.241$	\(\Q\)	None	✓				4.10.a.a	$1$	$0$	\(0\)	\(-228\)	\(-666\)	\(6328\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-228q^{3}-666q^{5}+6328q^{7}+32301q^{9}+\cdots\)
16.10.a.b	$16$	$10$	16.a	1.a	$1$	$1$	$1$	$8.241$	\(\Q\)	None	✓				8.10.a.b	$1$	$1$	\(0\)	\(-68\)	\(1510\)	\(-10248\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-68q^{3}+1510q^{5}-10248q^{7}-15059q^{9}+\cdots\)
16.10.a.c	$16$	$10$	16.a	1.a	$1$	$1$	$1$	$8.241$	\(\Q\)	None	✓				8.10.a.a	$1$	$1$	\(0\)	\(60\)	\(-2074\)	\(4344\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+60q^{3}-2074q^{5}+4344q^{7}-16083q^{9}+\cdots\)
16.10.a.d	$16$	$10$	16.a	1.a	$1$	$1$	$1$	$8.241$	\(\Q\)	None	✓				2.10.a.a	$1$	$0$	\(0\)	\(156\)	\(870\)	\(952\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+156q^{3}+870q^{5}+952q^{7}+4653q^{9}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(16))\) into lower level spaces

\( S_{10}^{\mathrm{old}}(\Gamma_0(16)) \simeq \) \(S_{10}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 3}\)\(\oplus\)\(S_{10}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)