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

• Label: 676.2.f
• Level: $676$
• Weight: $2$
• Character orbit: 676.f
• Rep. character: $\chi_{676}(99,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $134$
• Newform subspaces: $10$
• Sturm bound: $182$
• Trace bound: $5$

Defining parameters

Level:	$N$	$=$	$676 = 2^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	676.f (of order $4$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$52$
Character field:			$\Q(i)$
Newform subspaces:			$10$
Sturm bound:			$182$
Trace bound:			$5$
Distinguishing $T_p$:			$3$, $5$, $7$

Dimensions

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

	Total	New	Old
Modular forms	210	174	36
Cusp forms	154	134	20
Eisenstein series	56	40	16

Trace form

134 * q + 2 * q^2 + 6 * q^5 - 8 * q^6 + 8 * q^8 - 86 * q^9 - 4 * q^14 + 20 * q^16 - 6 * q^18 - 4 * q^20 - 16 * q^21 - 32 * q^22 - 32 * q^24 - 12 * q^28 + 8 * q^29 + 12 * q^32 - 8 * q^33 - 8 * q^34 + 22 * q^37 - 44 * q^40 + 2 * q^41 - 24 * q^42 + 28 * q^44 + 2 * q^45 + 20 * q^46 + 136 * q^48 + 42 * q^50 - 60 * q^53 + 32 * q^54 - 48 * q^57 - 20 * q^58 + 12 * q^60 + 4 * q^61 - 4 * q^66 + 56 * q^68 - 8 * q^70 - 4 * q^72 + 54 * q^73 - 76 * q^74 - 32 * q^76 - 40 * q^80 - 42 * q^81 + 48 * q^84 - 4 * q^85 - 32 * q^86 - 22 * q^89 - 56 * q^92 - 32 * q^93 - 52 * q^94 + 24 * q^96 - 2 * q^97 + 2 * q^98

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.f.a	$676$	$2$	676.f	52.f	$4$	$2$	$1$	$5.398$	$\Q(\sqrt{-1})$	$\Q(\sqrt{-13})$		✓			676.2.f.a	$4$	$0$	$-2$	$0$	$0$	$-2$	$1$	$\mathrm{U}(1)[D_{4}]$	$q+(i-1)q^{2}-2 i q^{4}+(-i-1)q^{7}+\cdots$
676.2.f.b	$676$	$2$	676.f	52.f	$4$	$2$	$1$	$5.398$	$\Q(\sqrt{-1})$	$\Q(\sqrt{-1})$					52.2.f.a	$4$	$0$	$-2$	$0$	$6$	$0$	$1$	$\mathrm{U}(1)[D_{4}]$	$q+(-i-1)q^{2}+2 i q^{4}+(3 i+3)q^{5}+\cdots$
676.2.f.c	$676$	$2$	676.f	52.f	$4$	$2$	$1$	$5.398$	$\Q(\sqrt{-1})$	$\Q(\sqrt{-13})$		✓			676.2.f.a	$4$	$0$	$2$	$0$	$0$	$2$	$1$	$\mathrm{U}(1)[D_{4}]$	$q+(-i+1)q^{2}-2 i q^{4}+(i+1)q^{7}+\cdots$
676.2.f.d	$676$	$2$	676.f	52.f	$4$	$4$	$2$	$5.398$	$\Q(\zeta_{12})$	$\Q(\sqrt{-1})$					52.2.l.a	$4$	$0$	$-4$	$0$	$-6$	$0$	$2$	$\mathrm{U}(1)[D_{4}]$	$q+(\beta_{2}-1)q^{2}-2\beta_{2} q^{4}+(\beta_{3}+\beta_{2}-1)q^{5}+\cdots$
676.2.f.e	$676$	$2$	676.f	52.f	$4$	$4$	$2$	$5.398$	$\Q(\zeta_{12})$	$\Q(\sqrt{-1})$					52.2.l.a	$4$	$0$	$4$	$0$	$6$	$0$	$2$	$\mathrm{U}(1)[D_{4}]$	$q+(-\beta_{2}+1)q^{2}-2\beta_{2} q^{4}+(-\beta_{3}-\beta_{2}+1)q^{5}+\cdots$
676.2.f.f	$676$	$2$	676.f	52.f	$4$	$8$	$4$	$5.398$	8.0.157351936.1	None		✓			676.2.f.f	$8$	$0$	$0$	$0$	$0$	$0$	$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(\beta _{6}-\beta _{7})q^{2}+(1-2\beta _{1})q^{3}+(-2\beta _{3}+\cdots)q^{4}+\cdots$
676.2.f.g	$676$	$2$	676.f	52.f	$4$	$8$	$4$	$5.398$	8.0.18939904.2	None					52.2.f.b	$4$	$0$	$4$	$0$	$0$	$0$	$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(-\beta _{3}+\beta _{6}+\beta _{7})q^{2}+(-1+2\beta _{1}+\cdots)q^{3}+\cdots$
676.2.f.h	$676$	$2$	676.f	52.f	$4$	$16$	$8$	$5.398$	16.0.$\cdots$.1	None					52.2.l.b	$4$	$0$	$-2$	$0$	$-12$	$0$	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	$q-\beta _{2}q^{2}+(-\beta _{3}-\beta _{6})q^{3}+(-\beta _{5}-\beta _{8}+\cdots)q^{4}+\cdots$
676.2.f.i	$676$	$2$	676.f	52.f	$4$	$16$	$8$	$5.398$	16.0.$\cdots$.1	None					52.2.l.b	$4$	$0$	$2$	$0$	$12$	$0$	$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	$q+\beta _{2}q^{2}+(-\beta _{3}-\beta _{6})q^{3}+(-\beta _{5}-\beta _{8}+\cdots)q^{4}+\cdots$
676.2.f.j	$676$	$2$	676.f	52.f	$4$	$72$	$36$	$5.398$		None		✓	✓		676.2.f.j	$8$	$0$	$0$	$0$	$0$	$0$		$\mathrm{SU}(2)[C_{4}]$

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

$S_{2}^{\mathrm{old}}(676, [\chi]) \simeq$ $S_{2}^{\mathrm{new}}(52, [\chi])$$^{\oplus 2}$