Space of modular forms of level 400, weight 2, and Character 400.y

• Label: 400.2.y
• Level: $400$
• Weight: $2$
• Character orbit: 400.y
• Rep. character: $\chi_{400}(129,\cdot)$
• Character field: $\Q(\zeta_{10})$
• Dimension: $56$
• Newform subspaces: $4$
• Sturm bound: $120$
• Trace bound: $5$

Defining parameters

Level:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	400.y (of order \(10\) and degree \(4\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 25 \)
Character field:			\(\Q(\zeta_{10})\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(120\)
Trace bound:			\(5\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(400, [\chi])\).

	Total	New	Old
Modular forms	264	64	200
Cusp forms	216	56	160
Eisenstein series	48	8	40

Trace form

56 * q + 5 * q^3 - 3 * q^5 + 9 * q^9 - 3 * q^11 - 5 * q^13 + 13 * q^15 - 5 * q^17 + 9 * q^19 + 6 * q^21 + 35 * q^23 + q^25 + 5 * q^27 - 5 * q^29 + 9 * q^31 - 5 * q^33 - 28 * q^35 - 30 * q^37 + q^39 - 5 * q^41 - 7 * q^45 + 5 * q^47 - 48 * q^49 + 62 * q^51 - 10 * q^53 + 13 * q^55 + 21 * q^59 + 7 * q^61 - 30 * q^63 + 4 * q^65 - 25 * q^67 - 9 * q^69 + 17 * q^71 - 5 * q^73 - 11 * q^75 + 30 * q^77 - 7 * q^79 - 39 * q^81 - 25 * q^83 + 20 * q^85 - 55 * q^87 - 18 * q^89 - 30 * q^91 - 113 * q^95 - 25 * q^97 - 76 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(400, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
400.2.y.a	$400$	$2$	400.y	25.e	$10$	$8$	$2$	$3.194$	\(\Q(\zeta_{20})\)	None					50.2.e.a	$2$	$0$	\(0\)	\(0\)	\(-10\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(\zeta_{20}-\zeta_{20}^{2}-\zeta_{20}^{3}+\zeta_{20}^{5}+\zeta_{20}^{6}+\cdots)q^{3}+\cdots\)
400.2.y.b	$400$	$2$	400.y	25.e	$10$	$8$	$2$	$3.194$	8.0.58140625.2	None					100.2.i.a	$2$	$0$	\(0\)	\(0\)	\(5\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(\beta _{1}+\beta _{5}+\beta _{7})q^{3}+(-\beta _{3}+\beta _{4}+\beta _{5}+\cdots)q^{5}+\cdots\)
400.2.y.c	$400$	$2$	400.y	25.e	$10$	$8$	$2$	$3.194$	8.0.58140625.2	None					25.2.e.a	$2$	$0$	\(0\)	\(5\)	\(0\)	\(0\)	$1$	$\mathrm{SU}(2)[C_{10}]$	\(q+(1+\beta _{3}+\beta _{7})q^{3}+(\beta _{3}-\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots\)
400.2.y.d	$400$	$2$	400.y	25.e	$10$	$32$	$8$	$3.194$		None			✓		200.2.q.a	$2$	$0$	\(0\)	\(0\)	\(2\)	\(0\)		$\mathrm{SU}(2)[C_{10}]$

Decomposition of \(S_{2}^{\mathrm{old}}(400, [\chi])\) into lower level spaces

\( S_{2}^{\mathrm{old}}(400, [\chi]) \simeq \) \(S_{2}^{\mathrm{new}}(25, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(50, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(100, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{2}^{\mathrm{new}}(200, [\chi])\)\(^{\oplus 2}\)