Space of modular forms of level 676, weight 2, and Character 676.e

• Label: 676.2.e
• Level: $676$
• Weight: $2$
• Character orbit: 676.e
• Rep. character: $\chi_{676}(529,\cdot)$
• Character field: $\Q(\zeta_{3})$
• Dimension: $24$
• Newform subspaces: $7$
• Sturm bound: $182$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 676 = 2^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	676.e (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 13 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 7 \)
Sturm bound:			\(182\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(676, [\chi])\).

	Total	New	Old
Modular forms	224	24	200
Cusp forms	140	24	116
Eisenstein series	84	0	84

Trace form

24 * q + q^3 + 2 * q^5 - 3 * q^7 - 7 * q^9 - 5 * q^11 + 12 * q^15 + 8 * q^17 - q^19 - 22 * q^21 - 9 * q^23 + 26 * q^25 - 38 * q^27 + 8 * q^29 + 12 * q^31 - 15 * q^33 + 18 * q^35 - 4 * q^37 + 6 * q^41 - 5 * q^43 + 9 * q^45 + 4 * q^47 - 13 * q^49 - 10 * q^51 - 6 * q^53 - 14 * q^55 + 26 * q^57 + 5 * q^59 - 2 * q^61 + 2 * q^63 + 9 * q^67 + 13 * q^69 - 17 * q^71 - 26 * q^73 - 13 * q^75 + 18 * q^77 + 20 * q^79 - 16 * q^81 - 48 * q^83 - 3 * q^85 - 25 * q^87 - 3 * q^89 - 28 * q^93 - 18 * q^95 + 13 * q^97 + 60 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(676, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
676.2.e.a	$676$	$2$	676.e	13.c	$3$	$2$	$1$	$5.398$	\(\Q(\sqrt{-3}) \)	None					52.2.e.a	$2$	$1$	\(0\)	\(-3\)	\(-4\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-3+3\zeta_{6})q^{3}-2q^{5}+\zeta_{6}q^{7}-6\zeta_{6}q^{9}+\cdots\)
676.2.e.b	$676$	$2$	676.e	13.c	$3$	$2$	$1$	$5.398$	\(\Q(\sqrt{-3}) \)	None					52.2.a.a	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(-2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q-2q^{5}-2\zeta_{6}q^{7}+3\zeta_{6}q^{9}+(-2+2\zeta_{6})q^{11}+\cdots\)
676.2.e.c	$676$	$2$	676.e	13.c	$3$	$2$	$1$	$5.398$	\(\Q(\sqrt{-3}) \)	None					52.2.a.a	$2$	$0$	\(0\)	\(0\)	\(4\)	\(2\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+2q^{5}+2\zeta_{6}q^{7}+3\zeta_{6}q^{9}+(2-2\zeta_{6})q^{11}+\cdots\)
676.2.e.d	$676$	$2$	676.e	13.c	$3$	$2$	$1$	$5.398$	\(\Q(\sqrt{-3}) \)	None					52.2.e.b	$2$	$0$	\(0\)	\(2\)	\(6\)	\(-4\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(2-2\zeta_{6})q^{3}+3q^{5}-4\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
676.2.e.e	$676$	$2$	676.e	13.c	$3$	$4$	$2$	$5.398$	\(\Q(\zeta_{12})\)	None					52.2.h.a	$4$	$0$	\(0\)	\(2\)	\(0\)	\(0\)	$3$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-\beta_1+1)q^{3}-\beta_{2} q^{7}+2\beta_1 q^{9}+\cdots\)
676.2.e.f	$676$	$2$	676.e	13.c	$3$	$6$	$3$	$5.398$	6.0.64827.1	None		✓			676.2.a.g	$2$	$0$	\(0\)	\(0\)	\(-16\)	\(-1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{2}-\beta _{3}-\beta _{4})q^{3}+(-2-\beta _{2}+\cdots)q^{5}+\cdots\)
676.2.e.g	$676$	$2$	676.e	13.c	$3$	$6$	$3$	$5.398$	6.0.64827.1	None		✓			676.2.a.g	$2$	$0$	\(0\)	\(0\)	\(16\)	\(1\)	$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(\beta _{1}-\beta _{2}-\beta _{3}-\beta _{4})q^{3}+(2+\beta _{2}+\cdots)q^{5}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(676, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(676, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(26, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(52, [\chi])\)\(^{\oplus 2}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(169, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(338, [\chi])\)\(^{\oplus 2}\)