Space of modular forms of level 8, weight 12, and trivial character

• Label: 8.12.a
• Level: $8$
• Weight: $12$
• Character orbit: 8.a
• Rep. character: $\chi_{8}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $2$
• Sturm bound: $12$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 8 = 2^{3} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	8.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(12\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{12}(\Gamma_0(8))\).

	Total	New	Old
Modular forms	13	3	10
Cusp forms	9	3	6
Eisenstein series	4	0	4

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

\(2\)	Dim
\(+\)	\(2\)
\(-\)	\(1\)

Trace form

3 * q + 20 * q^3 + 4378 * q^5 + 35592 * q^7 + 364351 * q^9 - 437924 * q^11 + 2424354 * q^13 - 10369192 * q^15 + 11571734 * q^17 - 29163996 * q^19 + 42049248 * q^21 + 3749272 * q^23 + 25230069 * q^25 + 67917320 * q^27 - 242433102 * q^29 + 297882912 * q^31 - 592852720 * q^33 + 101747952 * q^35 - 182227398 * q^37 + 826449464 * q^39 + 979775598 * q^41 - 1743278724 * q^43 + 2138827042 * q^45 - 701569872 * q^47 + 2864997627 * q^49 - 9990058520 * q^51 - 531170102 * q^53 + 10134953928 * q^55 - 4445419280 * q^57 - 9014835572 * q^59 - 1840230702 * q^61 + 36447883176 * q^63 - 10820210564 * q^65 - 2485905324 * q^67 - 19932595552 * q^69 + 30489530696 * q^71 + 4016618382 * q^73 - 81254774516 * q^75 + 14365261728 * q^77 + 47757712848 * q^79 + 20946031387 * q^81 - 147967912348 * q^83 + 86218163220 * q^85 + 47261838840 * q^87 + 81753623454 * q^89 + 7178123184 * q^91 - 32147884160 * q^93 - 11404963528 * q^95 - 71551453338 * q^97 + 113298667660 * q^99

Decomposition of \(S_{12}^{\mathrm{new}}(\Gamma_0(8))\) 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
8.12.a.a	$8$	$12$	8.a	1.a	$1$	$1$	$1$	$6.147$	\(\Q\)	None	✓	✓	✓	✓	8.12.a.a	$1$	$1$	\(0\)	\(-36\)	\(-3490\)	\(-55464\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6^{2}q^{3}-3490q^{5}-55464q^{7}-175851q^{9}+\cdots\)
8.12.a.b	$8$	$12$	8.a	1.a	$1$	$2$	$2$	$6.147$	\(\Q(\sqrt{109}) \)	None	✓	✓	✓	✓	8.12.a.b	$1$	$0$	\(0\)	\(56\)	\(7868\)	\(91056\)		$+$	$+$	$2^{7}$	$\mathrm{SU}(2)$	\(q+(28+\beta )q^{3}+(3934-12\beta )q^{5}+(45528+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{12}^{\mathrm{old}}(\Gamma_0(8))\) into lower level spaces

\( S_{12}^{\mathrm{old}}(\Gamma_0(8)) \simeq \) \(S_{12}^{\mathrm{new}}(\Gamma_0(1))\)\(^{\oplus 4}\)\(\oplus\)\(S_{12}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)