Space of modular forms of level 409, weight 2, and Character 409.m

• Label: 409.2.m
• Level: $409$
• Weight: $2$
• Character orbit: 409.m
• Rep. character: $\chi_{409}(4,\cdot)$
• Character field: $\Q(\zeta_{102})$
• Dimension: $1056$
• Newform subspaces: $1$
• Sturm bound: $68$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 409 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	409.m (of order \(102\) and degree \(32\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 409 \)
Character field:			\(\Q(\zeta_{102})\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(68\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(409, [\chi])\).

	Total	New	Old
Modular forms	1120	1120	0
Cusp forms	1056	1056	0
Eisenstein series	64	64	0

Trace form

1056 * q - 33 * q^2 - 31 * q^3 + 29 * q^4 - 22 * q^5 - 44 * q^6 - 57 * q^7 - 22 * q^8 + 2 * q^9 - 30 * q^10 + 17 * q^11 + 67 * q^12 - 34 * q^13 - 68 * q^14 - 32 * q^15 + 21 * q^16 - 27 * q^17 + 22 * q^18 - 34 * q^19 - 51 * q^20 - 108 * q^21 - 22 * q^22 - 32 * q^23 + 87 * q^24 - 112 * q^25 - 64 * q^26 - 82 * q^27 - 75 * q^28 - 19 * q^29 - 22 * q^30 + 52 * q^32 - 43 * q^33 - 48 * q^34 - 113 * q^35 + 22 * q^36 - 10 * q^37 - 57 * q^38 + 41 * q^39 - 225 * q^40 - 288 * q^41 - 173 * q^42 + 68 * q^43 - 34 * q^44 - 42 * q^45 + 17 * q^46 - 34 * q^47 + 68 * q^48 + 369 * q^49 - 40 * q^50 + 56 * q^51 + 8 * q^52 + 104 * q^53 + 128 * q^54 + 102 * q^55 - 138 * q^56 - 55 * q^57 - 34 * q^58 - 34 * q^59 - 254 * q^60 - 68 * q^61 + 25 * q^62 - 34 * q^63 + 98 * q^64 - 34 * q^65 - 40 * q^67 - 116 * q^68 - 38 * q^69 - 17 * q^70 + 21 * q^71 - 26 * q^72 - 31 * q^73 + 203 * q^74 + 23 * q^75 - 101 * q^76 + 19 * q^77 - 102 * q^78 - 34 * q^79 + 443 * q^80 - 265 * q^81 - 50 * q^82 + 26 * q^83 - 459 * q^84 + 84 * q^85 + 136 * q^86 - 188 * q^87 + 170 * q^88 - 80 * q^89 + 613 * q^90 - 152 * q^91 + 18 * q^92 + 56 * q^93 - 34 * q^94 - 34 * q^95 + 13 * q^96 - 31 * q^97 - 293 * q^98 - 229 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(409, [\chi])\) 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
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
409.2.m.a	$409$	$2$	409.m	409.m	$102$	$1056$	$33$	$3.266$		None		✓	✓	✓	409.2.m.a	$2$	$0$	\(-33\)	\(-31\)	\(-22\)	\(-57\)		$\mathrm{SU}(2)[C_{102}]$