Space of modular forms of level 409, weight 4, and trivial character

• Label: 409.4.a
• Level: $409$
• Weight: $4$
• Character orbit: 409.a
• Rep. character: $\chi_{409}(1,\cdot)$
• Character field: $\Q$
• Dimension: $102$
• Newform subspaces: $2$
• Sturm bound: $136$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 409 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	409.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(136\)
Trace bound:			\(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(409))\).

	Total	New	Old
Modular forms	104	102	2
Cusp forms	102	102	0
Eisenstein series	2	0	2

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(409\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(56\)	\(55\)	\(1\)		\(55\)	\(55\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)		\(48\)	\(47\)	\(1\)		\(47\)	\(47\)	\(0\)		\(1\)	\(0\)	\(1\)

Trace form

102 * q - 2 * q^2 - 4 * q^3 + 418 * q^4 - 10 * q^7 - 24 * q^8 + 918 * q^9 - 78 * q^10 - 24 * q^11 - 198 * q^12 - 34 * q^13 - 114 * q^14 - 56 * q^15 + 1578 * q^16 - 4 * q^17 - 36 * q^18 - 96 * q^19 + 24 * q^20 + 52 * q^21 - 150 * q^22 + 308 * q^23 - 298 * q^24 + 2446 * q^25 - 294 * q^26 - 112 * q^27 + 12 * q^28 - 106 * q^29 - 748 * q^30 - 102 * q^31 - 124 * q^32 - 208 * q^33 - 78 * q^34 + 184 * q^35 + 4110 * q^36 + 126 * q^37 - 186 * q^38 - 528 * q^39 + 144 * q^40 - 196 * q^41 + 1134 * q^42 - 932 * q^43 + 360 * q^44 - 192 * q^45 - 336 * q^46 + 754 * q^47 - 1808 * q^48 + 4714 * q^49 + 1550 * q^50 + 96 * q^51 + 408 * q^52 - 700 * q^53 + 1410 * q^54 - 196 * q^55 - 3134 * q^56 - 132 * q^57 + 2124 * q^58 - 1324 * q^59 - 352 * q^60 - 1290 * q^61 + 1204 * q^62 - 1862 * q^63 + 8560 * q^64 - 1256 * q^65 - 1566 * q^66 - 392 * q^67 + 464 * q^68 + 188 * q^69 + 408 * q^70 + 556 * q^71 - 1982 * q^72 - 208 * q^73 - 2326 * q^74 - 404 * q^75 - 1402 * q^76 - 960 * q^77 + 1118 * q^78 - 2538 * q^79 + 4870 * q^80 + 8126 * q^81 + 538 * q^82 + 416 * q^83 + 2542 * q^84 - 2900 * q^85 + 1454 * q^86 - 3284 * q^87 - 1100 * q^88 - 1636 * q^89 - 5050 * q^90 - 700 * q^91 + 4716 * q^92 + 1012 * q^93 - 364 * q^94 + 344 * q^95 - 1524 * q^96 + 260 * q^97 - 3864 * q^98 - 1112 * q^99

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(409))\) 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					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		409
409.4.a.a	$409$	$4$	409.a	1.a	$1$	$47$	$47$	$24.132$		None	✓	✓	✓	✓	409.4.a.a	$1$	$1$	\(-15\)	\(-14\)	\(-60\)	\(-82\)		$-$	$-$		$\mathrm{SU}(2)$
409.4.a.b	$409$	$4$	409.a	1.a	$1$	$55$	$55$	$24.132$		None	✓	✓	✓	✓	409.4.a.b	$1$	$0$	\(13\)	\(10\)	\(60\)	\(72\)		$+$	$+$		$\mathrm{SU}(2)$