LMFDB - Space of modular forms of level 693, weight 2, and Character 693.i

Defining parameters

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.i (of order $3$ and degree $2$)
Character conductor:	$\operatorname{cond}(\chi)$	$=$	$ 7 $
Character field:			$\Q(\zeta_{3})$
Newform subspaces:			$ 12 $
Sturm bound:			$192$
Trace bound:			$5$
Distinguishing $T_p$:			$2$, $5$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(693, [\chi])$.

	Total	New	Old
Modular forms	208	68	140
Cusp forms	176	68	108
Eisenstein series	32	0	32

Trace form

$ 68 q - 36 q^{4} + 4 q^{5} + 6 q^{7} - 12 q^{8} + O(q^{10}) $

$ 68 q - 36 q^{4} + 4 q^{5} + 6 q^{7} - 12 q^{8} + 6 q^{10} + 8 q^{13} + 4 q^{14} - 36 q^{16} - 6 q^{17} - 10 q^{19} - 8 q^{20} + 4 q^{23} - 42 q^{25} + 14 q^{26} - 64 q^{28} + 24 q^{29} + 6 q^{31} + 32 q^{32} + 48 q^{34} - 42 q^{35} - 4 q^{38} + 24 q^{40} + 24 q^{41} - 52 q^{43} + 4 q^{44} + 10 q^{46} + 22 q^{47} - 4 q^{49} - 88 q^{50} + 54 q^{52} - 10 q^{53} + 16 q^{55} - 30 q^{56} - 2 q^{58} + 40 q^{59} + 28 q^{61} + 68 q^{62} + 44 q^{64} - 10 q^{65} + 28 q^{67} - 52 q^{68} - 6 q^{70} - 12 q^{71} - 46 q^{73} - 8 q^{74} + 32 q^{76} - 2 q^{77} - 20 q^{79} - 2 q^{80} - 30 q^{82} + 84 q^{83} + 80 q^{85} + 6 q^{86} + 12 q^{88} + 6 q^{89} - 26 q^{91} - 124 q^{92} - 22 q^{94} - 24 q^{95} - 84 q^{97} + 18 q^{98} + O(q^{100}) $

Display 100 coefficients 10 coefficients

Decomposition of $S_{2}^{\mathrm{new}}(693, [\chi])$ into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Coefficient ring index	Sato-Tate	$q$-expansion
693.2.i.a	$693$	$2$	693.i	7.c	$3$	$2$	$1$	$5.534$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-2$	$0$	$0$	$5$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+(2+\zeta_{6})q^{7}+\cdots$
693.2.i.b	$693$	$2$	693.i	7.c	$3$	$2$	$1$	$5.534$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-1$	$0$	$4$	$-4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+4\zeta_{6}q^{5}+(-3+\cdots)q^{7}+\cdots$
693.2.i.c	$693$	$2$	693.i	7.c	$3$	$2$	$1$	$5.534$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$0$	$0$	$-4$	$-5$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(2-2\zeta_{6})q^{4}-4\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots$
693.2.i.d	$693$	$2$	693.i	7.c	$3$	$2$	$1$	$5.534$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$0$	$0$	$4$	$-5$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(2-2\zeta_{6})q^{4}+4\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots$
693.2.i.e	$693$	$2$	693.i	7.c	$3$	$2$	$1$	$5.534$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$0$	$0$	$-4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+(-1-2\zeta_{6})q^{7}+\cdots$
693.2.i.f	$693$	$2$	693.i	7.c	$3$	$4$	$2$	$5.534$	$\Q(\sqrt{2}, \sqrt{-3})$	None		$2$	$0$	$-2$	$0$	$-4$	$4$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\beta _{1}-\beta _{2})q^{2}+(-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$
693.2.i.g	$693$	$2$	693.i	7.c	$3$	$6$	$3$	$5.534$	6.0.1783323.2	None		$2$	$0$	$0$	$0$	$-2$	$2$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(\beta _{3}-\beta _{5})q^{2}+(-1-\beta _{1}+\beta _{4}+\beta _{5})q^{4}+\cdots$
693.2.i.h	$693$	$2$	693.i	7.c	$3$	$6$	$3$	$5.534$	$\Q(\zeta_{18})$	None		$2$	$0$	$0$	$0$	$6$	$0$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(\zeta_{18}-\zeta_{18}^{2}-\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots$
693.2.i.i	$693$	$2$	693.i	7.c	$3$	$8$	$4$	$5.534$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$2$	$0$	$4$	$2$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{6})q^{4}+\cdots$
693.2.i.j	$693$	$2$	693.i	7.c	$3$	$10$	$5$	$5.534$	$\mathbb{Q}[x]/(x^{10} + \cdots)$	None		$2$	$0$	$2$	$0$	$-4$	$-1$	$3$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{3}q^{2}+(-2-2\beta _{2}-\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots$
693.2.i.k	$693$	$2$	693.i	7.c	$3$	$12$	$6$	$5.534$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		$2$	$0$	$0$	$0$	$-4$	$6$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{6}q^{2}+(-2+2\beta _{4}-\beta _{8})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots$
693.2.i.l	$693$	$2$	693.i	7.c	$3$	$12$	$6$	$5.534$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		$2$	$0$	$0$	$0$	$4$	$6$	$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{6}q^{2}+(-2+2\beta _{4}-\beta _{8})q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(693, [\chi])$ into lower level spaces

$ S_{2}^{\mathrm{old}}(693, [\chi]) \cong $ $S_{2}^{\mathrm{new}}(21, [\chi])$$^{\oplus 4}$$\oplus$$S_{2}^{\mathrm{new}}(63, [\chi])$$^{\oplus 2}$$\oplus$$S_{2}^{\mathrm{new}}(77, [\chi])$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(231, [\chi])$$^{\oplus 2}$

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Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.i (of order \(3\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 7 \)
Character field:			\(\Q(\zeta_{3})\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(192\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(2\), \(5\)

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Defining parameters

Dimensions

Trace form

Decomposition of \(S_{2}^{\mathrm{new}}(693, [\chi])\) into newform subspaces

Decomposition of \(S_{2}^{\mathrm{old}}(693, [\chi])\) into lower level spaces

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	693.2.i

Level	$693$
Weight	$2$
Character orbit	693.i
Rep. character	$\chi_{693}(100,\cdot)$
Character field	$\Q(\zeta_{3})$
Dimension	$68$
Newform subspaces	$12$
Sturm bound	$192$
Trace bound	$5$